1 Introduction
The last two decades have seen an explosion of physics research on the behavior of isolated quantum systems in which both disorder and interactions are present. The appearance of these two features has been linked to the existence of materials that fail to thermalize and consequently cannot be described using equilibrium statistical mechanics. These materials are presumed to remain insulators at nonzero temperature, a phenomenon called many-body localization (MBL). We refer the reader to the physics reviews [Reference Abanin, Altman, Bloch and Serbyn1,Reference Alet and Laflorencie9,Reference Nandkishore and Huse38] for the general description of this phenomenon. MBL-type behavior has been observed in cold atoms experiments [Reference Lukin, Rispoli, Schittko, Tai, Kaufman, Choi, Khemani, Léonard and Greiner31, Reference Schreiber, Hodgman, Bordia, Lüschen, Fischer, Vosk, Altman, Schneider and Bloch41]. The stability of the MBL phase for infinite systems and all times remains a topic of intense debate [Reference Kiefer-Emmanouilidis, Unanyan, Fleischhauer and Sirker25, Reference Morningstar, Colmenarez, Khemani, Luitz and Huse35, Reference Sels and Polkovnikov43, Reference Sierant, Delande and Zakrzewski44, Reference Sierant and Zakrzewski45].
In this paper, we consider the random spin-
$\tfrac 12$
Heisenberg XXZ chain in the Ising phase, a one-dimensional random quantum spin system. This is the most studied model in the context of MBL both in the physics and mathematics literature (going back to [Reference Pal and Huse39, Reference Žnidarič, Prosen and Prelovšek47]). It can be mapped by the Jordan-Wigner transformation into an interacting spinless fermionic model closely related to the disordered Fermi-Hubbard Hamiltonian, a paradigmatic model in condensed matter physics that provides crucial insights into the electronic and magnetic properties of materials. One interesting feature of the random one-dimensional XXZ quantum spin system is the emergence of a many-body localization-delocalization transition. (In contrast, prototypical non-interacting one-dimensional random Schrödinger operators do not exhibit a phase transition and are completely localized.) Numerical evidence for this transition in the disordered XXZ model has been provided in a number of simulations (e.g., [Reference Agarwal, Gopalakrishnan, Knap, Müller and Demler3, Reference Bardarson, Pollmann and Moore10, Reference Baygan, Lim and Sheng11, Reference Luitz, Laflorencie and Alet30, Reference Pal and Huse39]), but remains contested on theoretical grounds (e.g., [Reference De Roeck, Huveneers, Müller and Schiulaz14]).
Until quite recently, mathematical results related to the proposed MBL characteristics, including zero-velocity Lieb-Robinson bounds, exponential clustering, quasi-locality, slow spreading of information and area laws, have been confined to quasi-free systems. The latter are models whose study can effectively be reduced to one of a (disordered) one-particle Hamiltonian. Examples of such systems include the XY spin chain in a random transversal field (going back to [Reference Klein and Perez29]; see [Reference Abdul-Rahman, Nachtergaele, Sims and Stolz2] for a review on this topic), the disordered Tonks-Girardeau gas [Reference Seiringer and Warzel42] and systems of quantum harmonic oscillators [Reference Nachtergaele, Sims and Stolz36]. Another direction of research considers the effect of many-body interaction on a single-particle localization (rather than MBL) within the framework of the effective field theories. This allows to consider a realistic Hilbert space for a single particle, such as
$\ell ^2(\mathbb Z^d)$
, rather than finite dimensional ones that are typically used in the MBL context. In particular, the persistence of the dynamical localization in the Hartree-Fock approximation for the disordered Hubbard model has been established in [Reference Ducatez16, Reference Matos and Schenker34].
In the last few years, there has been some (modest) progress in understanding genuine many-body systems, all of which is concerned with the XXZ model, either in the quasi-periodic setting (where the exponential clustering property for the ground state of the André-Aubry model has been established [Reference Mastropietro32, Reference Mastropietro33]) or in the droplet spectrum regime in the random case [Reference Beaud and Warzel12, Reference Elgart, Klein and Stolz19]. In the latter case, several MBL manifestations have been established, including some that have never been previously discussed in the physics literature [Reference Elgart, Klein and Stolz18].
While not exactly solvable, the XXZ spin chain does have a symmetry; namely, it preserves the particle number. This enables a reduction to an infinite system of discrete N-body Schrödinger operators on the fermionic subspaces of
$\mathbb Z^N$
[Reference Fischbacher and Stolz21, Reference Nachtergaele and Starr37]. For the XXZ spin chain in the Ising phase, in the absence of a magnetic field, the low energy eigenstates above the ground state are characterized by a droplet regime. In this regime, spins form a droplet (i.e., a single cluster of down spins (particles) in a sea of up spins). This reduction has been effectively exploited inside the droplet spectrum (the interval
$ I_1$
in (2.14) below) using methods that resemble the fractional moment method for random Schrödinger operators, yielding the small number of rigorous results [Reference Beaud and Warzel12, Reference Elgart, Klein and Stolz19]. However, these methods seem to be inadequate above this energy interval (i.e, inside the multi-cluster spectrum), and a new set of ideas that do not rely on a reduction to Schrödinger operators are required to tackle this case.
In this paper, we extend the energy interval for which MBL holds well beyond the droplet spectrum, deep inside the multi-cluster spectrum. We develop a suitable method, formulated and proved in terms of spin systems concepts. In particular, our method does not rely on the reduction of the XXZ Hamiltonian to a direct sum of Schrodinger operators (and the subsequent analysis that uses single-particle tools).
Localization phenomenon in condensed matter physics is usually associated with non-spreading of wave packets in a disordered medium. Experimentally, it is observed in semiconductors whose properties are predominantly caused by crystal defects or impurities, as well as in the variety of other systems. This phenomenon is by now well understood for quantum single particle models. A prototypical system studied in this context is the Anderson Hamiltonian
$H_A$
, which is a self-adjoint operator acting on the Hilbert space
$\mathcal H=\ell ^2(\mathbb Z^d)$
of the form
$H_A=-\Delta +\lambda V_\omega $
. Here,
$\Delta $
is the (discrete) Laplacian describing the kinetic hopping,
$V_\omega $
is a randomly generated multiplication operator (
$\omega $
is the random parameter) describing the electric potential, and
$\lambda $
is a parameter measuring the strength of the disorder.
Let us denote by
$\delta _x\in \mathcal H$
the indicator of
$x\in \mathbb Z^d$
, and fix the random parameter
$\omega $
. An important feature of
$H_A$
as a map on
$\ell ^2(\mathbb Z^d)$
is its locality, meaning
$\langle \delta _x,H_A\delta _y\rangle =0$
if
$\left \lvert x-y \right \rvert>1$
. As a consequence, the resolvent
$(H_A-z)^{-1} $
retain a measure of locality, which we will call quasi-locality, given by the Combes-Thomas estimate
$$ \begin{align}\begin{aligned} \left\lvert \langle \delta_x,(H_A-z)^{-1} \delta_y\rangle \right\rvert\le C_z \mathrm{e}^{-m_z\left\lvert x-y \right\rvert}, \end{aligned}\end{align} $$
where
$C_z$
and
$m_z$
are constants independent of
$\omega $
such that
$C_z <\infty $
and
$m_z>0$
if
$z\in \mathbb C$
is outside the spectrum of
$H_A$
. Maps given by smooth functions of
$H_A$
also express a measure of quasi-locality – namely,
$$ \begin{align}\begin{aligned} \left\lvert \langle \delta_x,f(H_A)\delta_y\rangle \right\rvert \le C_{f,n} \left( 1 + \left\lvert x-y \right\rvert \right)^{-n}, \end{aligned}\end{align} $$
where
$C_{f,n} <\infty $
for all
$n\in \mathbb N$
and infinitely differentiable functions f. Moreover, these quasi-locality estimate hold with the same constants for the restriction
$H_A^\Lambda $
of
$H_A$
to a finite volume
$\Lambda \subset \mathbb Z^d$
. (See, for example, [Reference Kirsch26, Reference Germinet and Klein24, Reference Remling40].)
The two mainstream approaches for proving localization in the single particle setting, the multi-scale analysis (MSA) and the fractional moment method (FMM), going back to [Reference von Dreifus and Klein15, Reference Fröhlich and Spencer23, Reference Fröhlich, Martinelli, Scoppola and Spencer22] and [Reference Aizenman4, Reference Aizenman and Molchanov6], respectively – establish localization for the (random) Anderson model
$H_A$
by proving quasi-locality estimates for the finite volume resolvent inside the spectrum of
$H_A$
. In particular, the fractional moment method shows that, fixing
$s\in (0,1)$
, for large disorder
$\lambda $
, we have
$$ \begin{align}\begin{aligned} \mathbb E\left\{ \left\lvert \left\langle {\delta_x,(H^\Lambda_A-E)^{-1} \delta_y } \right\rangle \right\rvert^s \right\}\le C\mathrm{e}^{-m \left\lvert x-y \right\rvert}, \end{aligned}\end{align} $$
for all finite
$\Lambda \subset \mathbb Z^d$
,
$ x,y\in \Lambda $
, and energies
$E\in \mathbb R$
, where the constants
$C<\infty $
and
$m>0$
are in dependent of
$\Lambda $
. Moreover, one also gets a quasi-locality estimate for Borel functions of
$H^A$
(dynamical localization),
$$ \begin{align}\begin{aligned} \mathbb E\left\{ \sup_{f} \left\lvert \left\langle {\delta_x,f(H^\Lambda_A)\delta_y } \right\rangle \right\rvert \right\}\le C\mathrm{e}^{-m \left\lvert x-y \right\rvert}, \end{aligned}\end{align} $$
where the supremum is taken over all Borel functions on
$\mathbb R$
bounded by one. Various manifestations of one-particle localization, such as non-spreading of wave packets, vanishing of conductivity in response to electric field, and statistics of the spacing between nearby energy levels, can be derived from these quasi-locality estimates. (See, for example, [Reference Aizenman and Warzel8].) On the mathematical level, the quasi-locality estimates provides an effective description of single particle localization.
The MSA and the FMM prove localization for random Schrödinger operators, both in the discrete and continuum settings. We refer the reader to the lecture notes [Reference Kirsch26, Reference Klein27] and the monograph [Reference Aizenman and Warzel8] for an introduction to the multi-scale analysis and the fractional moment method, respectively.
Both methods have been extended to quantum system consisting of an arbitrary, but fixed, number of interacting particles, showing that many characteristics of single-particle localization remain valid in this case (e.g., [Reference Aizenman and Warzel7, Reference Chulaevsky and Suhov13, Reference Klein and Nguyen28]). But truly many-body systems (where the number of particles is proportional to the system’s size) present new challenges. A major difficulty lies in the fact that the concepts of MBL proposed in the physics literature are not easily tractable on the mathematical level, and it is not clear what could be chosen as the fundamental description of the theory from which other properties can be derived, as in a single particle case. For example, the available concept of quasi-locality in the many-body systems looks very different from the one for single particle quantum systems.
To introduce a simple many-body system Hamiltonian, we consider a finite graph
$\mathbf {\Gamma }=\left ( \mathcal V,\mathcal E \right )$
(where
$\mathcal V$
is the set of vertices and
$\mathcal E$
is the set of edges) and a family
$\left \{ \mathcal H_i \right \}_{i\in \mathcal V}$
of Hilbert spaces. The Hilbert space of the subsystem associated with a set
$X\subset \mathcal V$
is given by
$\mathcal H_X=\bigotimes _{i\in X}\mathcal H_i$
, and the full Hilbert space (we ignore particles’ statistics) is
$\mathcal H_{\mathcal V}$
. For each
$X\subset \mathcal V$
, one introduces the algebra of observables
$\mathcal A_X$
measurable in this subsystem, which is the collection
$\mathcal B(\mathcal H_X)$
of bounded linear operators on the Hilbert space
$\mathcal {H}_X$
. An observable
$\mathcal O\in \mathcal A_{\mathcal V}$
is said to be supported by
$X\subset \mathcal V$
if 
, where
$\mathcal O_{X}\in \mathcal A_{X}$
(i.e., if
$\mathcal O$
acts trivially on
$\mathcal H_{\mathcal V\setminus X}$
). Slightly abusing the notation, we will usually identify
$\mathcal O$
with
$\mathcal O_{X}$
and call X a support for
$\mathcal O$
. Since we are primarily interested here in understanding the way particles interact, the structure of a single particle Hilbert space
$\mathcal H_i$
will be only of marginal importance for us. So we will be considering the simplest possible realization of such system, where each
$\mathcal H_i$
is the two dimensional vector space
$\mathbb C^2$
describing a spin-
$\frac 12$
particle.
We next describe the interactions between our spins. We again are going to consider the simplest possible arrangement, where only nearest neighboring spins are allowed to interact. Explicitly, for each pair of vertices
$(i,j)\in \mathcal V$
that share an edge (i.e.,
$\left \{ i,j \right \} \in \mathcal {E}$
), we pick an observable (called an interaction)
$h_{i,j}\in \mathcal A_{\left \{ i,j \right \}}$
such that
$h_{i,j}=h^*_{i,j}$
, an observable (called a local transverse field)
$v_i=v_i^*\in \mathcal {A}_{\left \{ i \right \}}$
, and associate a Hamiltonian
$H^{\mathcal V}=\sum _{\left \{ i,j \right \} \in \mathcal E}h_{i,j}+\sum _{i\in {\mathcal V}} v_i$
with our spin system. In particular,
$H^{\mathcal V}$
is the sum of local observables and is consequently referred to as a local Hamiltonian. Locality is manifested by
$[[H^{\mathcal V},\mathcal O],\mathcal O^\prime ]=0$
for any pair of observables
$\mathcal O\in \mathcal A_X$
,
$\mathcal O^\prime \in \mathcal A_Y$
, with
$\operatorname {\mathrm {dist}}(X,Y)>1$
. (To compare it with the concept of (single particle) locality for the map
$H_A$
, we need to define a local observable for the space
$\ell ^2(\mathbb Z^d)$
. We will say that an observable
$\mathcal O\in \mathcal L(\ell ^2(\mathbb Z^d))$
has support
$X\subset \mathbb Z^d$
if
$\mathcal O=\mathcal O_{X} \oplus 0_{\ell ^2(\mathbb Z^d\setminus X)}$
with
$\mathcal O_{X}\in \mathcal L(\ell ^2(X))$
. With this definition, locality of the map
$H_A$
(i.e., the property
$\langle \delta _x,H_A\delta _y\rangle =0$
whenever
$\left \lvert x-y \right \rvert>1$
) is equivalent to the statement that
$[[H_{A},\mathcal O],\mathcal O']=0$
for any pair of observables
$\mathcal O,\mathcal O'$
with
$\operatorname {\mathrm {dist}}(\operatorname {\mathrm {supp}}(\mathcal O),\operatorname {\mathrm {supp}}(\mathcal O'))>1$
.)
The XXZ spin chain is defined as above on finite subgraphs
$\Lambda $
of the graph
$\mathbb Z$
(see Section 2.1). Consider
$\Lambda \subset \mathbb Z$
connected, and let
$\left \lvert \Lambda \right \rvert $
be its cardinality. We say we have a particle at the site
$i\in \Lambda $
if we have spin down in the copy
$\mathcal {H}_i$
of
$\mathbb C^2$
. Let
$\mathcal {N}_i$
be the orthogonal projection onto configurations with a particle at the site i, and set
$[i]^\Lambda _p= \left \{ j\in \Lambda , \left \lvert j-i \right \rvert \le p \right \}$
for
$p=0,1,\ldots $
. Given
$B\subset \Lambda $
, let
$P_+^B$
be the orthogonal projection onto configurations with no particles in B. In the Ising phase,
$H^\Lambda $
is a
$2$
-local, gapped, frustration-free system, and
$P_+^\Lambda $
describes the projection onto the ground state of
$H^\Lambda $
(see Remark 2.3).
We can now informally state our main results. We first prove that the resolvent
$R^\Lambda _z=(H^\Lambda -z)^{-1}$
exhibits quasi-locality in the form (see Lemma 3.1 and Remark 3.2)
$$ \begin{align}\begin{aligned} \left\lVert \mathcal{N}_i R_z^ \Lambda P_+^{[i]^\Lambda_p} \right\rVert\le C_z \mathrm{e}^{-m_z p}, \end{aligned}\end{align} $$
where
$C_z$
and
$m_z$
are constants, independent of
$\Lambda $
and of the transverse field, such that
$C_z <\infty $
and
$m_z>0$
if
$z\in \mathbb C$
is outside the spectrum of
$H^\Lambda $
. We also establish the many-body analogue of (1.2):
$$ \begin{align}\begin{aligned} \left\lVert \mathcal{N}_i f(H^ \Lambda) P_+^{[i]^\Lambda_p} \right\rVert\le C_{f,n} \left( 1 +p \right)^{-n}, \end{aligned}\end{align} $$
where
$C_{f,n} <\infty $
for all
$n\in \mathbb N$
and infinitely differentiable functions f on
$\mathbb R$
with compact support. (See Appendix B.)
We next consider the random XXZ spin chain (see Definition 2.2). The relations (1.5)–(1.6) suggest, by analogy with random Schrödinger operators, that localization should be manifested as quasi-locality inside the spectrum of
$H^\Lambda $
. This is indeed what we prove in Theorem 2.4. We introduce increasing energy intervals
$I_{\le k} $
,
$k=0,1,2,\ldots $
, in (2.14) and prove that quasi-locality of the form given in (1.5) holds for the resolvent for energies in
$I_{\le k} $
for any fixed
$ k$
. In particular, given
$s\in (0,\frac 1 3)$
, we prove, in the appropriate (k dependent) parameter region, that
$$ \begin{align}\begin{aligned} \mathbb E\left\{ \left\lVert \mathcal{N}_i R_E^ \Lambda P_+^{[i]^\Lambda_p} \right\rVert^s \right\}\le C_k\left\lvert \Lambda \right\rvert^{\xi_k} \mathrm{e}^{-m_k p} \quad\text{for all}\quad E\in I_{\le k} , \end{aligned}\end{align} $$
where the constants
$C_k<\infty $
,
$ \xi _k>0$
,
$m_k>0$
do not depend on
$\Lambda $
. As a consequence, we derive a quasi-locality estimate for Borel functions of
$H^\Lambda $
(Corollary 2.6):
$$ \begin{align}\begin{aligned} \mathbb E\left( \sup_f {\left\lVert \mathcal{N}_i f(H^\Lambda)P_+^{[i]^\Lambda_p} \right\rVert} \right)\le C_k\left\lvert \Lambda \right\rvert^{\xi_k} \mathrm{e}^{-m_k p}, \end{aligned}\end{align} $$
where the supremum is taken over all Borel functions on
$\mathbb R$
that are equal to zero outside the interval
$I_{\le k}$
and bounded by one.
While the estimates (1.7) and (1.8) are very natural from the mathematical perspective, it is far from obvious whether they yield any of the MBL-type features proposed by physicists. Nevertheless, in a sequel to this paper [Reference Elgart and Klein17], we derive slow propagation of information, a putative MBL manifestation, from Theorem 2.4 and Corollary 2.6, for any
$k\in \mathbb N$
.
In the droplet spectrum, [Reference Elgart, Klein and Stolz19, Theorem 2.1] imply Corollary 2.6 (with
$k=1$
), and a converse can established using [Reference Elgart and Klein17, Remark 3.3]. While [Reference Elgart, Klein and Stolz19] and the follow-up paper [Reference Elgart, Klein and Stolz18] contain several MBL-type properties such as the (dynamical) exponential clustering property, (properly defined) zero-velocity Lieb-Robinson bounds, and slow propagation (non-spreading) of information, they are all derived using [Reference Elgart, Klein and Stolz19, Theorem 2.1] as the starting point. We stress that [Reference Elgart, Klein and Stolz19, Theorem 2.1], by its very nature, can only hold in the droplet regime, so while it provides us with very strong consequences in the
$k=1$
case, we do not expect the methods of [Reference Elgart, Klein and Stolz19, Reference Elgart, Klein and Stolz18] to be of any use in the multi-cluster case – that is, for
$k\ge 2$
.
Although the methods derived in this work are not universal (which is typical for many-body results), they are sufficiently powerful for investigation of MBL phenomena in this context, as shown in [Reference Elgart and Klein17]. We have to admit, however, that in the physics literature, MBL is usually associated with energies that are not fixed (as we assumed in this work) but are comparable with the system size
$\left \lvert \Lambda \right \rvert $
. We do not expect that our techniques will be sufficient to probe such energies. To be able to do so would require non-perturbative techniques similar to the ones used in the investigations of one dimensional random Schrödinger operators.
The model description and main results (Theorem 2.4 and Corollary 2.6) are presented in Section 2. In Section 3, we outline the main ideas used in the proof of Theorem 2.4, which is completed in Section 4. Corollary 2.6 is proven in Section 5. Appendix A contains some useful identities. Appendix B contains the proof of of the many-body quasi-locality estimate (1.6).
Throughout the paper, we will use generic constants
$C, c,m$
, etc., whose values will be allowed to change from line to line, even in a displayed equation. These constants will not depend on subsets of
$\mathbb Z$
, but they will, in general, depend on the parameters of the model introduced in Section 2.1 (such as
$\mu $
, k,
$\Delta _0$
,
$ \lambda _0$
and s). When necessary, we will indicate the dependence of a constant on k explicitly by writing it as
$C_k, m_k$
, etc. These constants can always be estimated from the arguments, but we will not track the changes to avoid complicating the arguments.
2 Model description and main results
2.1 Model description
The random XXZ quantum spin-
$\frac 12$
chain on an finite subset
$\Lambda $
of
$\mathbb Z$
is given by a self-adjoint Hamiltonian
$H^\Lambda $
acting on the finite dimensional Hilbert space
$\mathcal H_\Lambda =\otimes _{i\in \Lambda } \mathcal {H}_i$
, where
$\mathcal {H}_i=\mathbb C^2$
for each
$i\in \Lambda $
. For a vector
$\phi \in \mathbb C^2$
, we let
$\phi _i$
denote the vector as an element of
$\mathcal {H}_i$
; for an operator (
$2\times 2$
matrix) A on
$\mathbb C^2$
, we let
$A_i$
denote the operator acting on
$\mathcal {H}_i$
.
We consider only finite subsets of
$\mathbb Z$
, so by a subset of
$\mathbb Z$
we will always mean a finite subset. If
$S\subset T\subset \mathbb Z$
, and
$A_S$
is an operator on
$\mathcal {H}_S$
, we consider
$A_S$
as operator on
$\mathcal {H}_T$
by identifying it with 
, where 
denotes the identity operator on
$\mathcal {H}_R$
. We thus identify
$\mathcal {A}_S$
with a subset of
$\mathcal {A}_T$
, where
$\mathcal {A}_R$
denotes the algebra of bounded operators on
$\mathcal {H}_R$
.
We now fix
$\Lambda \subset \mathbb Z$
, and consider
$\Lambda $
as a subgraph of
$\mathbb Z$
. We denote by
$\operatorname {\mathrm {dist}}_\Lambda $
the graph distance in
$\Lambda $
, which can be infinite if
$\Lambda $
is not a connected subset of
$\mathbb Z$
. We write
$K^c=\Lambda \setminus K$
for
$K\subset \Lambda $
. To define
$H^\Lambda $
, we introduce some notation and definitons.
-
1. By
$\sigma ^{x,y,z}$
and
$\sigma ^\pm =\frac 12( \sigma ^x \pm i \sigma ^y)$
we will denote the standard Pauli matrices and ladder operators, respectively. -
2. By
$\uparrow \rangle : = \begin {pmatrix} 1 \\ 0 \end {pmatrix} $
and
$\downarrow \rangle = \begin {pmatrix} 0 \\ 1 \end {pmatrix} $
we will denote the elements of the canonical basis of
$\mathbb C^2$
, called spin-up and spin-down, respectively. Letting , we note that
$\mathcal {N} \uparrow \rangle =0$
and
$\mathcal {N} \downarrow \rangle =\downarrow \rangle $
, and interpret
$\downarrow \rangle $
as a particle. -
3.
$\mathcal {N}_i$
, the matrix
$ \mathcal {N}$
acting on
$\mathcal {H}_i$
, is the projection onto the spin-down state (also called the local number operator) at site i. Given
$S\subset \Lambda $
,
$\mathcal {N}_{S} = \sum _{i\in S} \mathcal {N}_i$
is the total (spin-down) number operator in S. -
4. The total number operator
$\mathcal {N}_\Lambda $
has eigenvalues
$0,1,2,\ldots , \left \lvert \Lambda \right \rvert $
. (
$\left \lvert S \right \rvert $
denotes the cardinality of
$S\subset \mathbb Z$
.) We set
$\mathcal {H}_\Lambda ^{\left (N\right )}=\operatorname {\mathrm {Ran}}\left ( \chi _N(\mathcal N_\Lambda ) \right )$
, obtaining the Hilbert space decomposition
$ \mathcal {H}_\Lambda = \bigoplus _{N=0}^{\left \lvert \Lambda \right \rvert } \mathcal {H}_\Lambda ^{\left (N\right )}$
. We will use the notation
$\chi ^\Lambda _{N}=\chi _{\left \{ N \right \}}(\mathcal N_\Lambda )$
. -
5. The canonical (orthonormal) basis
$\Phi _\Lambda $
for
$\mathcal {H}_\Lambda $
is constructed as follows: Let
$\Omega _\Lambda =\phi _\emptyset =\otimes _{i \in \Lambda }\uparrow \rangle _i$
be the vacuum state. Then (2.1)where
$$ \begin{align}\begin{aligned} & \Phi_\Lambda =\left\{ \phi_A=\left( \prod_{i\in A}\sigma_i^- \right)\Omega_\Lambda:\ A\subset \Lambda \right\} =\bigcup_{N=0}^{\left\lvert \Lambda \right\rvert} \Phi_\Lambda^{\left(N\right)}, \end{aligned}\end{align} $$
$\Phi _\Lambda ^{\left (N\right )}= \left \{ \phi _A: A\subset \Lambda , \ \left \lvert A \right \rvert =N \right \}$
. Note that
$ \Phi _\Lambda ^{\left (0\right )}=\left \{ \Omega _\Lambda \right \}$
.
, we note that 
We now define the free XXZ quantum spin-
$\frac 12$
Hamiltonian on
$\Lambda \subset \mathbb Z$
by
$$ \begin{align} H_0^\Lambda=H_0^\Lambda(\Delta)=-\tfrac{1}{2\Delta} \boldsymbol{\Delta}^\Lambda + \mathcal{W}^\Lambda \quad\text{on}\quad \mathcal{H}_\Lambda, \end{align} $$
where
$$ \begin{align} \boldsymbol{\Delta}^\Lambda &= \sum_{\left\{ i,i+1 \right\}\subset \Lambda} \left( \sigma_i^+\sigma_{i+1}^-+\sigma_i^-\sigma_{i+1}^+ \right), \end{align} $$
$$ \begin{align} \mathcal{W}^\Lambda & = \mathcal{N}_\Lambda -\sum_{\left\{ i,i+1 \right\}\subset \Lambda} \mathcal{N}_i\mathcal{N}_{i+1} , \end{align} $$
and
$\Delta>1$
is the anisotropy parameter, specifying the Ising phase (
$\Delta =1$
selects the Heisenberg chain and
$\Delta =\infty $
corresponds to the the Ising chain).
We will consider the XXZ model in the presence of a transversal field
$\lambda V^\Lambda _\omega $
, given by
$V^\Lambda _\omega =\sum _{i\in \Lambda } \omega _i \mathcal {N}_i$
, where
$\omega _i\ge 0$
, and the parameter
$\lambda>0$
is used to modulate the strength of the field. The full Hamiltonian is then
$$ \begin{align} H^\Lambda=H^\Lambda_\omega =H^\Lambda_\omega(\Delta,\lambda)= H_0^\Lambda(\Delta)+\lambda V_\omega^\Lambda= -\tfrac{1}{2\Delta} \boldsymbol{\Delta}^\Lambda + \mathcal{W}^\Lambda+\lambda V_\omega^\Lambda. \end{align} $$
Remark 2.1.
-
1. The operator
$\boldsymbol {\Delta }^\Lambda $
can be viewed as the analog of the Laplacian operator on
$\mathcal {H}_\Lambda $
. -
2.
$\mathcal {N}_i$
is diagonalized by the canonical basis for all
$i\in \Lambda $
:
$\mathcal {N}_i \phi _A= \phi _A$
if
$i\in A$
and
$0$
otherwise. It follows that the total number operator
$\mathcal {N}_\Lambda $
is also diagonalized by the canonical basis:
$\mathcal {N}_\Lambda \phi _A= \left \lvert A \right \rvert \phi _A$
. -
3.
$\mathcal {W}^\Lambda $
, the number of clusters operator, is diagonalized by the canonical basis:
$\mathcal {W}^\Lambda \phi _A= W^\Lambda _A \phi _A$
, where
$W^\Lambda _A\in [0,\left \lvert A \right \rvert ]\cap \mathbb Z$
is the number of clusters of A in
$\Lambda $
(i.e., the number of connected components of A in
$\Lambda $
(considered as a subgraph of
$\mathbb Z$
)). -
4.
$V^\Lambda _\omega $
is diagonalized by the canonical basis:
$V^\Lambda _\omega \phi _A= \omega ^{\left (A\right )} \phi _A$
, where
$\omega ^{\left (A\right )}=\sum _{i\in A} \omega _i$
. -
5. The operators
$\mathcal {N}_\Lambda $
,
$\mathcal {W}^\Lambda $
, and
$V^\Lambda _\omega $
commute. -
6. The XXZ Hamiltonian
$H^\Lambda $
preserves the total particle number, (2.6)
$$ \begin{align} [H^\Lambda,\mathcal{N}_\Lambda]= -\tfrac{1}{2\Delta}[ \boldsymbol{\Delta}^\Lambda,\mathcal{N}_\Lambda] =0. \end{align} $$
We will consider the XXZ model in the presence of a random transversal field; that is,
$\omega = \left \{ \omega _i \right \}_{i\in \mathbb Z}$
is a family of random variables. More precisely, we make the following definition.
Definition 2.2. The random XXZ spin Hamiltonian on
$\Lambda \subset \mathbb Z$
is the operator
$H^\Lambda =H^\Lambda _\omega (\Delta ,\lambda )$
given in (2.5), where
$\Delta>1$
,
$\lambda>0$
, and
$\omega = \left \{ \omega _i \right \}_{i\in \mathbb Z}$
is a family of independent identically distributed random variables, whose common probability distribution
$\mu $
satisfies
$$ \begin{align} \left\{ 0,1 \right\}\subset \operatorname{\mathrm{supp}} \mu\subset[0,1] \end{align} $$
and is assumed to be absolutely continuous with a bounded density.
From now on,
$H^\Lambda $
always denotes the random XXZ spin Hamiltonian on
$\Lambda $
. The corresponding resolvent is given by
$R_E^\Lambda =\left ( H^\Lambda -E \right )^{-1}$
, which is well-defined for almost every energy
$E\in \mathbb R$
. We set
$\omega _S = \left \{ \omega _i \right \}_{i\in S}$
for
$S\subset \mathbb Z$
and denote the corresponding expectation and probability by
$\mathbb E_S$
and
$\mathbb P_S$
.
It is convenient to introduce the local interaction terms
$$ \begin{align} {h}_{i,i+1}= - \mathcal{N}_{i}\mathcal{N}_{i+1} -\tfrac{1}{2\Delta}\left( \sigma_i^+\sigma_{i+1}^-+\sigma_i^-\sigma_{i+1}^+ \right), \end{align} $$
which allows us to rewrite
$$ \begin{align} H_0^\Lambda=\sum_{\left\{ i,i+1 \right\}\subset \Lambda} {h}_{i,i+1} + \mathcal{N}_\Lambda. \end{align} $$
It can be verified that on
$\mathcal {H}_{\left \{ i,i+1 \right \}}=\mathcal {H}_i^2\otimes \mathcal {H}_{i+1}^2$
, we have
$$ \begin{align} {\tfrac 12 \left( \mathcal{N}_i+\mathcal{N}_{i+1} \right)-\mathcal{N}_i\mathcal{N}_{i+1}} \mp \tfrac 12 \left( \sigma_i^+\sigma_{i+1}^-+\sigma_i^-\sigma_{i+1}^+ \right) \ge 0, \end{align} $$
which implies that
$\mathcal {W}_\Lambda \pm \tfrac 1 2\boldsymbol {\Delta }_\Lambda \ge 0$
; that is,
$$ \begin{align} - 2\mathcal{W}_\Lambda \le - \boldsymbol{\Delta}_\Lambda \le 2\mathcal{W}_\Lambda. \end{align} $$
It follows that
$$ \begin{align} \left( 1- \tfrac{1}{\Delta} \right) \mathcal{W}^\Lambda\le H_{0}^\Lambda \le\left( 1+\tfrac 1 \Delta \right) \mathcal{W}^\Lambda, \quad\text{so}\quad \left( 1- \tfrac{1}{\Delta} \right) \mathcal{W}^\Lambda\le H^\Lambda. \end{align} $$
We conclude that the spectrum of
$H^\Lambda $
is of the form
$$ \begin{align} \sigma(H^\Lambda)=\left\{ 0 \right\} \cup \left( \left[1 -\tfrac 1 \Delta, \infty \right ) \cap \sigma(H^\Lambda) \right). \end{align} $$
The lower bound in (2.12) suggests the introduction of the energy thresholds
$k\left ( 1- \tfrac {1}{\Delta } \right )$
,
$ k=0, 1,2\ldots $
. We define the energy intervals
$$ \begin{align}\begin{aligned} \widehat I_{\le k}&=\left(-\infty, (k+1)\left( 1- \tfrac{1}{\Delta} \right)\right), \quad \widehat I_{ k}= \left[1- \tfrac{1}{\Delta}, (k+1)\left( 1- \tfrac{1}{\Delta} \right)\right),\\ I_{\le k}&= \left(-\infty, (k+\tfrac 3 4)\left( 1- \tfrac{1}{\Delta} \right)\right) ,\quad I_k= \left[1- \tfrac{1}{\Delta}, (k+\tfrac 3 4)\left( 1- \tfrac{1}{\Delta} \right)\right). \end{aligned}\end{align} $$
We call
$\widehat I_{ k}$
the k-cluster spectrum.
Given
$\emptyset \ne S\subset \Lambda$
, we define the orthogonal projections
$P_\pm ^{S}$
on
$\mathcal {H}_\Lambda $
by
$P_+^{S}$
is the orthogonal projection onto states with no particles in the set S;
$P_-^{S}$
is the orthogonal projection onto states with at least one particle in S. We also set
Remark 2.3. In the Ising phase (i.e.,
$\Delta>1$
), we have (2.12) and (2.13) for all
$\Lambda \subset \mathbb Z$
. It follows that the XXZ chain Hamiltonian
$H^\Lambda $
has ground state
$\Omega _\Lambda $
and the ground state energy is
$0$
(
$H_\Lambda \Omega _\Lambda $
=0), and, moreover, the ground state energy is gapped. This makes
$H^\Lambda $
a
$2$
-local, gapped, frustration-free system. These features, plus the preservation of the total particle number, make the XXZ model especially amenable to analysis. In particular, the number of eigenstates of
$H^\Lambda $
in the intervals
$I_{\le k}$
grows only polynomially in the volume of
$\Lambda $
(not exponentially as the dimension of
$\mathcal {H}_\Lambda $
) as shown in Lemma 3.5 below.
2.2 Main results
Our main result establishes quasi-locality for the resolvent of the random XXZ chain inside the spectrum of
$H^\Lambda $
.
Theorem 2.4 (Quasi-locality for resolvents)
Fix
$\Delta _0>1$
,
$ \lambda _0>0$
, and let
$s\in (0,\frac 1 3)$
. Then for all
$k\in {\mathbb {N}}^0$
, there exist constants
$D_k,F_k,\xi _k, \theta _k>0$
(depending on k,
$\Delta _0$
,
$ \lambda _0$
and s) such that, for all
$\Delta \ge \Delta _0$
and
$\lambda \ge \lambda _0$
with
$\lambda \Delta ^2\ge D_k$
,
$\Lambda \subset \mathbb Z$
finite, and energy
$E\in I_{\le k}$
, we have
$$ \begin{align}\begin{aligned} \mathbb E\left\{ \left\lVert P_-^{A}R_E^\Lambda P_+^{B} \right\rVert^s \right\}\le F_k\left\lvert \Lambda \right\rvert^{\xi_k} \mathrm{e}^{-\theta_k\operatorname{\mathrm{dist}}_\Lambda(A,B^c)} , \end{aligned}\end{align} $$
for
$A\subset B\subset \Lambda $
with A connected in
$\Lambda $
.
The theorem is proven in Section 4.
Remark 2.5. If A is not connected in
$\Lambda $
, the theorem still holds with (2.17) replaced by
$$ \begin{align}\begin{aligned} \mathbb E\left\{ \left\lVert P_-^{A}R_E^\Lambda P_+^{B} \right\rVert^s \right\}\le F_k {\Upsilon}^\Lambda_A \left\lvert \Lambda \right\rvert^{\xi_k} \mathrm{e}^{-\theta_k\operatorname{\mathrm{dist}}_\Lambda(A,B^c)} , \end{aligned}\end{align} $$
where
${\Upsilon }^\Lambda _A$
denotes the number of connected components of A in
$\Lambda $
. This follows from (2.17) and
$$ \begin{align} P_-^{A}=\sum_{j=1}^{{\Upsilon}^\Lambda_A} P_+^{\bigcup_{i=i}^{j-1} A_i}P_-^{A_j}, \end{align} $$
where
$A_j$
,
$j=1,2,\ldots , {\Upsilon }^\Lambda _A$
, are the connected components of A in
$\Lambda $
.
As a consequence of Theorem 2.4, we prove the following quasi-locality estimate for Borel functions of
$H^\Lambda $
. By
$B(I_{\le k})$
we denote the collection of Borel functions on
$\mathbb R$
that are equal to zero outside the interval
$I_{\le k}$
.
Corollary 2.6 (Quasi-locality for Borel functions)
Assume the hypotheses and conclusions of Theorem 2.4, Then for all
$k\in {\mathbb {N}}^0$
, there exist constants
$\widetilde F_k,\widetilde \xi _k, \widetilde \theta _k>0$
(depending on k,
$\Delta _0$
,
$ \lambda _0$
and s) such that, for all
$\Delta \ge \Delta _0$
and
$\lambda \ge \lambda _0$
with
$\lambda \Delta ^2\ge D_k$
, and
$\Lambda \subset \mathbb Z$
finite, we have
$$ \begin{align}\begin{aligned} \mathbb E_{\Lambda}\left( \sup_{\substack{f\in B(I_{\le k}):\\\|f\|_\infty\le1}}\left\lVert P_-^{A}f(H^\Lambda) P_+^{B} \right\rVert \right)\le \widetilde F_k\left\lvert \Lambda \right\rvert^{\widetilde \xi_k} \mathrm{e}^{-\widetilde \theta_k\operatorname{\mathrm{dist}}_\Lambda(A,B^c)} , \end{aligned}\end{align} $$
for all
$A\subset B\subset \Lambda $
, A connected in
$\Lambda $
.
The proof of the Corollary is given in Section 5.
3 Key ingredients for the proofs
In this section, we collect a number of definitions, statements and lemmas that will facilitate the proof of Theorem 2.4.
$\Lambda $
will always denote a finite subset of
$\mathbb Z$
, and
$A\subset \Lambda $
will always denote a nonempty subset connected in
$\Lambda $
. (
$B\subset \Lambda $
,
$S\subset \Lambda $
, etc., may not be connected in
$\Lambda $
.)
3.1 Some definitions
-
○ Given
$ M\subset \Lambda $
and
$q\in \mathbb Z$
, we define enlarged (for
$q\ge 0$
) and trimmed (for
$q<0$
) set
$[M]^\Lambda _q $
by (3.1)Note that
$$ \begin{align}\begin{aligned} {[M]^\Lambda_q } &:=\begin{cases}\left\{ x\in\Lambda: \operatorname{\mathrm{dist}}_\Lambda\left( x,M \right)\le q \right\} &\;\text{if}\; q\in {\mathbb{N}}^0=\left\{ 0 \right\} \cup \mathbb N \\ \left\{ x\in\Lambda:\operatorname{\mathrm{dist}}_\Lambda\left( x, M^c \right)\ge 1-q \right\}= M\setminus [M^c]^\Lambda_{-q}&\;\text{if}\; q\in-\mathbb N\\ \left\{ x\in \Lambda: \operatorname{\mathrm{dist}}_\Lambda\left( x,M \right)<\infty \right\}= \bigcup_{p\in {\mathbb{N}}^0} [M]^\Lambda_p & \;\text{if}\;q=\infty \end{cases}. \end{aligned}\end{align} $$
$ [M]^\Lambda _{-\left \lvert M \right \rvert }=\emptyset $
. Moreover,
$[M]^\Lambda _\infty = [M]^\Lambda _{\left \lvert \Lambda \right \rvert -1} $
is the connected component of
$\Lambda $
containing M, and we have (3.2)We define
$$ \begin{align}[H^\Lambda, P_\pm^{[M]^\Lambda_\infty}]=0.\end{align} $$
$\partial _{ex}^\Lambda M$
(the external boundary of M in
$\Lambda $
),
$\partial _{in}^\Lambda M$
(the inner boundary of M in
$\Lambda $
), and
$ \partial ^\Lambda M$
(the boundary of
$M $
in
$\Lambda $
), by (3.3)It follows that
$$ \begin{align}\begin{aligned} \partial_{ex}^\Lambda M&:=\left\{ x\in\Lambda:\ \operatorname{\mathrm{dist}}_\Lambda\left( x,M \right)=1 \right\}= [M]^\Lambda_1\setminus M,\\ \partial_{in}^\Lambda M&:=\left\{ x\in\Lambda:\ \operatorname{\mathrm{dist}}_\Lambda \left( x, M^c \right)=1 \right\}=M\setminus [M]^\Lambda_{-1},\\ \partial^\Lambda M &:= \partial_{in}^\Lambda M \cup \partial_{ex}^\Lambda M. \end{aligned}\end{align} $$
(3.4)and we have
$$ \begin{align}\begin{aligned} ]M[^\Lambda_q:= [M]^\Lambda_{q+1}\setminus [M]^\Lambda_{q}&=\begin{cases} {\partial^\Lambda_{ex}{[M]^\Lambda_{q}}}, & q\in {\mathbb{N}}^0\\ {\partial^\Lambda_{in }{[M]^\Lambda_{q+1}}} & q\in-\mathbb N \end{cases}, \end{aligned}\end{align} $$
(3.5)If
$$ \begin{align}\begin{aligned} ]M[^\Lambda_p= ] M^c[^\Lambda_{-p-1} \quad\text{for}\quad p\in\mathbb Z. \end{aligned}\end{align} $$
$M=\left \{ j \right \}$
, we write
${[j]^\Lambda _q }={[\left \{ j \right \}]^\Lambda _q }$
.
-
○ Given
$A\subset B \subset \Lambda $
, we let
$\rho ^\Lambda (A,B)$
be the largest
$q\in {\mathbb {N}}^0 \cup \left \{ \infty \right \}$
such that
$ [A]^\Lambda _q \subset B$
; that is, (3.6)It will be more convenient to use
$$ \begin{align}\begin{aligned} \rho^\Lambda (A,B)= \sup \left\{ q\in {\mathbb{N}}^0: [A]^\Lambda_q \subset B \right\} =\operatorname{\mathrm{dist}}_\Lambda (A, B^c)-1. \end{aligned}\end{align} $$
$ \rho ^\Lambda (A,B)$
instead of
$\operatorname {\mathrm {dist}}_\Lambda (A, B^c)$
in the proofs.Note that (3.7)
$$ \begin{align}\begin{aligned} \rho^\Lambda (A,B)=\infty \iff \operatorname{\mathrm{dist}}_\Lambda (A, B^c)=\infty \iff [A]^\Lambda_\infty \subset B. \end{aligned}\end{align} $$
-
○ It follows from (3.2) and (3.7) that
(3.8)so it suffices to prove Theorem 2.4 for
$$ \begin{align}\begin{aligned} P_-^A R_{E}^{\Lambda} P_+^{B}=0 \quad\text{if}\quad A\subset B\subset \Lambda \quad\text{and}\quad\rho^\Lambda(A,B)=\infty, \end{aligned}\end{align} $$
$\rho ^\Lambda (A,B)<\infty $
. Moreover, since
$A \subset B$
, we have
$[A]^\Lambda _{\rho ^\Lambda (A,B)}\subset B$
, and hence, (3.9)so without loss of generality, it suffices to prove (2.17) for
$$ \begin{align}\begin{aligned} \left\lVert P_-^{A}R_E^\Lambda P_+^{B} \right\rVert\le \left\lVert P_-^{A}R_E^\Lambda P_+^{[A]^\Lambda_{\rho^\Lambda(A,B)}} \right\rVert, \end{aligned}\end{align} $$
$B=[A]^\Lambda _{\rho }$
with
$\rho \in {\mathbb {N}}^0$
.
-
○ Given
$K\subset \Lambda $
, we consider the operator acting on
$\mathcal {H}_\Lambda $
. We also consider the operators on
$\mathcal {H}_\Lambda $
given by (3.10)
$$ \begin{align}\begin{aligned} H^{K,K^c}= H^{K}+H^{K^c}, \quad R^{K,K^c}_E=\left( H^{K,K^c}-E \right)^{-1}, \quad \Gamma^K=H^\Lambda-H^{K,K^c}. \end{aligned}\end{align} $$
acting on 
3.2 Quasi-locality for resolvents
The following lemma and remark yields (deterministic) quasi-locality for the resolvent of the XXZ chain outside the spectrum of
$H^\Lambda $
.
Lemma 3.1. Let
${\Theta }\subset {\Lambda }$
, and consider the Hilbert space
$\mathcal {H}_{\Lambda }$
. Let the operator
$T\in \mathcal A_{\Lambda }$
be of the form
$$ \begin{align}\begin{aligned}T=T^{\Theta}+T^{{ \Theta^c}};\quad\text{where}\quad T^{\Theta}\in\mathcal A_{\Theta}\quad\text{and}\quad T^{{ \Theta^c}}\in\mathcal A_{{ \Theta^c}}, \end{aligned}\end{align} $$
and let
$\mathcal {X}\in \mathcal {A}_{\Lambda }$
be a projection such that
$[ \mathcal {X},T]=0$
and
$[\mathcal {X}, P_\pm ^K]=0 $
for all
$K\subset {\Theta }$
.
Suppose
-
1. For all
$K \subset {\Theta } $
, we have
$[P_-^{K},T]P_+^{[K]_1^{\Theta }}=0$
. -
2. For all
$K \subset {\Theta } $
, with K connected in
$\Theta $
, we have
$\left \lVert [P_-^{K},T] \right \rVert \le \gamma $
. -
3.
$T_{\mathcal {X}}$
, the restriction of the operator T to
$\operatorname {\mathrm {Ran}} \mathcal {X}$
, is invertible with
$ \left \lVert T_{\mathcal {X}}^{-1} \right \rVert _{\operatorname {\mathrm {Ran}} \mathcal {X}} \le \eta ^{-1}$
, where
$\eta>0$
.
Then for all
$A\subset B\subset {\Theta }$
, with A connected in
${\Theta }$
, we have
$$ \begin{align}\begin{aligned} \left\lVert P_-^{A}\,T_{\mathcal{X}}^{-1}\,P_+^{B} \right\rVert_{\operatorname{\mathrm{Ran}} \mathcal{X}}\le \eta^{-1} \mathrm{e}^{-m \rho^{\Theta} (A,B)} , \quad\text{with}\quad m=\ln\left( {\gamma^{-1}}\eta \right). \end{aligned}\end{align} $$
Proof. We consider first the case 
. Let
$A\subset B\subset {\Theta }$
, with A connected in
${\Theta }$
. Let
$1\le t \le \rho ^{\Theta } (A,B)$
, so
$[A]_{t}^\Theta \subset B$
. We have
$$ \begin{align}\begin{aligned} P_-^{ A}\,T^{-1}\,P_+^{ B}=T^{-1}[T,P_-^{ A}]T^{-1}\,P_+^{B}={T^{-1}[T,P_-^{ A}]} P_-^{ [A]^\Theta_1}T^{-1}\,P_+^{ B}, \end{aligned}\end{align} $$
using condition (i) of the Lemma. Proceeding recursively, we get
$$ \begin{align}\begin{aligned} P_-^{ A}\,T^{-1}\,P_+^{B}= \left( \prod_{p=0}^{t-1}{T^{-1}[T,P_-^{ [A]_p^\Theta}]} \right) P_-^{ [A]^\Theta_{t}}T^{-1}\,P_+^{B}. \end{aligned}\end{align} $$
Since A is connected in
$\Theta $
,
$[ A]_{r}^{\Theta }$
,
$r=1,2,\ldots ,t$
, are also connected in
$\Theta $
. Using assumptions (ii) and (iii), we get
$$ \begin{align}\begin{aligned} \left\lVert P_-^{A}\,T^{-1}\,P_+^{ B} \right\rVert\le \left( \gamma \eta^{-1} \right)^t \eta^{-1}. \end{aligned}\end{align} $$
Since (3.15) holds for all
$1\le t \le \rho ^{\Theta } (A,B)$
, we get
$$ \begin{align}\begin{aligned} \left\lVert P_-^{A}\,T^{-1}\,P_+^{B} \right\rVert \le \eta^{-1} \mathrm{e}^{-m \rho^{\Theta} (A,B)} , \quad\text{with}\quad m=\ln\left( {\gamma^{-1}}\eta \right). \end{aligned}\end{align} $$
If condition (iii) holds with a projection
$\mathcal {X}\in \mathcal {A}_{\Lambda }$
such that
$[ \mathcal {X},T]=0$
and
$[\mathcal {X}, P_\pm ^K]=0 $
for all
$K\subset {\Theta }$
, then
$\widetilde T= T\mathcal {X} + \eta (1-\mathcal {X})$
satisfies conditions (i), (ii), and condition (iii) with 
, and the estimate (3.16) for
$\widetilde T$
implies (3.12).
Remark 3.2. Lemma 3.1 yields quasi-locality for the resolvent of the operator
$H^\Lambda $
. The operator
$H^\Lambda -z$
satisfies the hypotheses of Lemma 3.1 for
$z\notin \sigma (H^\Lambda )$
, with
$\Theta =\Lambda $
,
$\gamma = \frac 1 {\Delta }$
(use (A.6)), 
, and
$\eta = \operatorname {\mathrm {dist}} (z,\sigma (H^\Lambda ))$
. It follows that, with
$R_z^\Lambda = (H^\Lambda -z)^{-1}$
, for all
$A\subset B\subset \Lambda $
, we have
$$ \begin{align}\begin{aligned} \left\lVert P_-^{A}R_z^ \Lambda P_+^{B} \right\rVert\le \left( \operatorname{\mathrm{dist}} (z,\sigma(H^\Lambda)) \right)^{-1} \mathrm{e}^{-m \rho^{\Theta} (A,B)} , \;\text{with}\; m=\ln\left( \Delta \operatorname{\mathrm{dist}} (z,\sigma(H^\Lambda)) \right). \end{aligned}\end{align} $$
From now on, we fix
$\Delta _0>5$
,
$ \lambda _0>0$
, and assume
$\Delta \ge \Delta _0$
and
$\lambda \ge \lambda _0$
. The constants will depend on
$\Delta _0$
and
$\lambda _0$
.
Given
$m \in {\mathbb {N}}^0$
, we set
$Q_m^\Lambda =\chi _{\left \{ m \right \}}\left ( \mathcal {W}^\Lambda \right )$
, the orthogonal projection onto configurations with exactly m clusters, and let
$Q_B^\Lambda =\chi _{B}\left ( \mathcal {W}^\Lambda \right )=\sum _{m\in B} Q_m^\Lambda $
for
$ B\subset {\mathbb {N}}^0$
. Note that
$Q_0^\Lambda = P_+^\Lambda $
and
$Q_{\mathbb {N}}^\Lambda = \chi _{\mathbb N}({\mathcal {N}}^\Lambda )$
. For
$k\in \mathbb N$
, we set
$$ \begin{align}\begin{aligned} Q_{\le k}^\Lambda =Q_{\left\{ 1,2,\ldots,k \right\}}^\Lambda =\sum_{ m=1}^k Q_m^\Lambda \quad\text{and}\quad \widehat Q_{\le k}^\Lambda =Q_{\le k}^\Lambda + \tfrac {k+1} k Q_0^\Lambda. \end{aligned}\end{align} $$
We also set
$$ \begin{align}\begin{aligned} \widehat H_0^{ \Lambda}&=H^{\Lambda}+\left( 1- \tfrac{1}{\Delta} \right) Q_0^\Lambda,\\ \widehat H_k^{ \Lambda}&=H^{\Lambda}+{k}\left( 1- \tfrac{1}{\Delta} \right) \widehat Q_{\le k}^{\Lambda} \quad\text{for}\quad k\in \mathbb N. \end{aligned}\end{align} $$
We use the notation
$$ \begin{align}\begin{aligned} \widehat R^{\Lambda}_{k,E}&= \left( \widehat H_k^\Lambda -E \right)^{-1} \; \ \text{for}\; \ E \notin \sigma(\widehat H_k^\Lambda), \ k\in {\mathbb{N}}^0. \end{aligned}\end{align} $$
It follows from (2.12) and (2.14) that for
$k\in {\mathbb {N}}^0$
, we have
$$ \begin{align}\begin{aligned} \widehat H_k^{ \Lambda}\ge \left( k+1 \right) \left( 1- \tfrac{1}{\Delta} \right) \quad\text{and}\quad \left( \widehat H_k^{ \Lambda}-E \right) \ge \tfrac 1{4}\left( 1- \tfrac{1}{\Delta} \right) \; \ \text{for}\; \ E \in I_{\le k}. \end{aligned}\end{align} $$
For
$k\in {\mathbb {N}}^0$
and
$E\in I_{\le k}$
, the operator
$T= \widehat H_k^{ \Lambda }-E$
satisfies the assumptions of Lemma 3.1 with
$\Theta =\Lambda $
,
$\gamma = \frac 1 {\Delta }$
, 
and
$\eta = \frac 1{4}\left ( 1- \frac {1}{\Delta } \right ) $
(see (3.21)). In this case,
$ m =\ln \frac {\Delta -1}{4}$
, and hence, for
$A\subset B\subset {\Lambda }$
, (3.12) yields
$$ \begin{align}\begin{aligned} \left\lVert P_-^{A}\widehat R^{\Lambda}_{k,E}P_+^{B} \right\rVert\le \tfrac 4 {1- \frac{1}{\Delta}}\mathrm{e}^{-\left( \ln \frac {\Delta-1} {4} \right)\rho^{\Lambda} (A,B)}. \end{aligned}\end{align} $$
To have decay in (3.22), we need
$ \frac {\Delta -1} {4}>1$
; that is,
$\Delta>5 $
. In the proof of Theorem 2.4, we will fix
$\Delta _0> 5 $
and
$\lambda _0>0$
and require
$\Delta \ge \Delta _0$
and
$\lambda \ge \lambda _0$
. In this case, we have
$\tfrac 4 {1- \frac {1}{\Delta }} \le \tfrac 4 {1-\frac 1 {\Delta _0}} $
and
$ \ln \tfrac {\Delta -1} 4\ge \ln \tfrac {\Delta _0-1} 4$
, so we have
$$ \begin{align}\begin{aligned} \left\lVert P_-^{A}\widehat R^{\Lambda}_{k,E}P_+^{B} \right\rVert\le C_0 \mathrm{e}^{-m_0 \rho^\Lambda (A,B)}, \;\text{with}\; C_0= \tfrac 4 {1-\frac 1 {\Delta_0}}, \; m_0= \ln \tfrac {\Delta_0-1} 4>0. \end{aligned}\end{align} $$
It follows from (3.2), which also holds for the operator
$\widehat H_k^{ \Lambda }$
, that
$$ \begin{align}\begin{aligned} P_-^M R_{E}^{\Lambda} P_+^{[M]^\Lambda_\infty}=0 \quad\text{and}\quad P_-^M\widehat R_{{k,E}}^{\Lambda} P_+^{[M]^\Lambda_\infty}=0 \quad\text{for}\quad M\subset \Lambda. \end{aligned}\end{align} $$
Remark 3.3. We will prove Theorem 2.4 with
$\Delta _0> 5$
to simplify our analysis. The proof can be extended to arbitrary
$\Delta _0> 1$
with minor modifications. Specifically, for
$1<\Delta _0 \le 5$
, we need to improve the decay rate in (3.22), which is derived from the lower bound in (3.21). To do so, we would replace
$\widehat H_k^{ \Lambda }$
in the proof by
$\widehat H_{k+r}^{ \Lambda }$
, where
$r\in \mathbb N$
, so (3.21) yields
$\widehat H_{k+r}^{ \Lambda }-E \ge (r+ \frac 1 4)\left ( 1- \tfrac {1}{\Delta } \right )$
for
$E\in I_{\le k}$
, leading to
$m_0= \ln \left ( (r + \frac 1 4) \left ( \Delta _0-1 \right ) \right )>0 $
for an appropriate choice of r.
3.3 An a priori estimate
The first step toward the proof of Theorem 2.4 is to understand why the expression on the left-hand side of (2.17) is actually finite. A useful technical device for this purpose is the following bound, where
$\left \lVert T \right \rVert _{HS}$
denotes the Hilbert-Schmidt norm of the operator T.
Lemma 3.4 (A priori estimate)
Let
$i,j\in \Lambda $
(
$i=j$
is allowed), and let
$T_{1},T_2$
be a pair of Hilbert-Schmidt operators on
$\mathcal {H}_\Lambda $
that are
$\omega _{\left \{ i,j \right \}}$
-independent. Then we have
$$ \begin{align} \mathbb E_{\left\{ i,j \right\}}\left( \left\lVert T_1\mathcal{N}_iR_E^\Lambda\mathcal{N}_j T_2 \right\rVert_{HS}^s \right) \le C\lambda^{-s}\left\lVert T_1 \right\rVert^s_{HS}\left\lVert T_2 \right\rVert^s_{HS} \;\text{for all}\; E\in \mathbb R \;\text{and}\; s\in (0,1). \end{align} $$
The lemma follows from [Reference Aizenman, Elgart, Naboko, Schenker and Stolz5, Proposition 3.2], used with
$U_1=\mathcal {N}_j$
,
$U_2=\mathcal {N}_k$
there, and the layer-cake representation for a non-negative random variable
$X_\omega $
:
$\mathbb E(X_\omega ^s)=\int _0^\infty \mathbb P(X_\omega>t^{1/s})\, dt$
for
$s\in (0,1)$
.
The Hilbert-Schmidt operators for Lemma 3.4 are provided by the following result.
Lemma 3.5. Let
$k\in \mathbb N$
. Then
$$ \begin{align} \left\lVert Q_{\le k}^{\Lambda} \right\rVert_{HS}&\le \sqrt{k} \left\lvert \Lambda \right\rvert^{k}, \end{align} $$
$$ \begin{align} \operatorname{\mathrm{tr}} \chi_{\widehat I_{\le k}}(H^\Lambda)&\le k\left\lvert \Lambda \right\rvert^{2k}+1. \end{align} $$
Proof. For
$m\ge 1$
and
$N \ge 1$
, we have the rough estimate
$$ \begin{align}\begin{aligned} \operatorname{\mathrm{tr}} \chi^\Lambda_N Q_{m}^{\Lambda}\le \left\lvert \Lambda \right\rvert^m N^{m-1}. \end{aligned}\end{align} $$
Thus,
$$ \begin{align} \operatorname{\mathrm{tr}} \chi^\Lambda_N Q_{\le k}^{\Lambda}\le \sum_{m=1}^k \left\lvert \Lambda \right\rvert^m N^{m-1}= \tfrac 1 N \tfrac {(\left\lvert \Lambda \right\rvert N)^{k+1}-(\left\lvert \Lambda \right\rvert N)}{(\left\lvert \Lambda \right\rvert N)-1}\le k \left\lvert \Lambda \right\rvert^k N^{k-1}. \end{align} $$
It follows that
$$ \begin{align} \operatorname{\mathrm{tr}} Q_{\le k}^{\Lambda}\le k \left\lvert \Lambda \right\rvert^k \sum_{N=1}^{\left\lvert \Lambda \right\rvert} N^{k-1}\le k\left\lvert \Lambda \right\rvert^{2k}. \end{align} $$
To prove (3.27), let
$\widehat {H}^\Lambda _k$
be as in (3.19), and note that (3.21) implies
$\operatorname {\mathrm {tr}} \chi _{\widehat I_{\le k}}(\widehat {H}^\Lambda _k)=0$
. Since the spectral shift is bounded by the rank of the perturbation, it follows from (3.19) that
$$ \begin{align} \operatorname{\mathrm{tr}} \chi_{\widehat I_{\le k}}(H^\Lambda)\le \operatorname{\mathrm{tr}} \chi_{\widehat I_{\le k}}(\widehat{H}^\Lambda_k)+\operatorname{\mathrm{Rank}} \left( k \left( 1- \tfrac{1}{\Delta} \right) \widehat Q_{\le k}^{\Lambda} \right)= \operatorname{\mathrm{tr}} \widehat Q_{\le k}^{\Lambda}= \operatorname{\mathrm{tr}} Q_{\le k}^{\Lambda}+1. \end{align} $$
Lemmas 3.4 and 3.5 yield the a priori estimate
$$ \begin{align}\begin{aligned} \mathbb E_{\left\{ i,j \right\}}{\left\lVert Q^\Lambda_{\le k}\mathcal{N}_i R^{\Lambda}_{E}\mathcal{N}_j Q^\Lambda_{\le k} \right\rVert_{HS}^s}\le C\lambda^{-s}k ^s\left\lvert \Lambda \right\rvert^{2sk} \;\text{for all}\; i,j\in \Lambda \;\text{and}\; s\in (0,1). \end{aligned}\end{align} $$
More generally, we have
$$ \begin{align}\begin{aligned} \mathbb E_{\left\{ A\cup B \right\}}{\left\lVert Q^\Lambda_{\le k}P_-^A R^{\Lambda}_{E}P_-^B Q^\Lambda_{\le k} \right\rVert_{HS}^s}\le C\lambda^{-s}k ^s\left\lvert \Lambda \right\rvert^{2sk}\left\lvert A \right\rvert\left\lvert B \right\rvert\quad\text{for}\quad \emptyset \ne A,B\subset\Lambda. \end{aligned}\end{align} $$
Those a priori estimates are only useful if we can ‘dress’ the resolvent with factors of
$Q^\Lambda _{\le k}$
on both sides. To be able to do so, we will decorate
$R^{\Lambda }_{E}$
with resolvents of positive operators that satisfy the quasi-locality property.
3.4 Dressing resolvents with Hilbert-Schmidt operators
For
$k=1,2,\ldots $
, and
$E\in I_{\le k}$
, we use the resolvent identity
$$ \begin{align}\begin{aligned} R_{E}^{\Lambda}=\widehat R^{\Lambda}_{k,E}+ k\left( 1- \tfrac{1}{\Delta} \right) R^{\Lambda}_{E} \widehat Q^\Lambda_{\le k}\widehat R^{\Lambda}_{k,E}=\widehat R^{\Lambda}_{k,E}+ k\left( 1- \tfrac{1}{\Delta} \right)\widehat R^{\Lambda}_{k,E}\widehat Q^\Lambda_{\le k} R^{\Lambda}_{E}. \end{aligned}\end{align} $$
Using it twice, we get
$$ \begin{align}\begin{aligned} R_{E}^{\Lambda}=\widehat R^{\Lambda}_{k,E}+ k\left( 1- \tfrac{1}{\Delta} \right)\widehat R^{\Lambda}_{k,E} \widehat Q^\Lambda_{\le k} \widehat R^{\Lambda}_{k,E}+k^2\left( 1- \tfrac{1}{\Delta} \right)^2\widehat R^{\Lambda}_{k,E}\widehat Q^\Lambda_{\le k} R^{\Lambda}_{E} \widehat Q^\Lambda_{\le k}\widehat R^{\Lambda}_{k,E}. \end{aligned}\end{align} $$
We use the notation
$(p)_+=\max \left ( p,0 \right )$
for
$p\in \mathbb R$
.
Lemma 3.6. Let
$\mathcal {X}$
denote a spectral projection of
$\mathcal {N}_\Lambda $
(say, 
or
$\mathcal {X}= \chi _N^\Lambda $
). Let
$A\subset B\subset \Lambda $
, and
$1\le t=\rho ^\Lambda \left ( A, B \right )<\infty $
. Let
$E\in I_{\le k}$
, and let
$m_0$
be as in (3.23).
-
1. We have the following estimate on operator norms:
(3.36)
$$ \begin{align}\begin{aligned} &{\left\lVert \mathcal{X} P_-^AR_E^\Lambda P_+^B \right\rVert}\le C_k\Big(\left\lvert \Lambda \right\rvert\mathrm{e}^{- m_0 t} + \sum_{p=-\left\lvert A \right\rvert}^{\left\lvert \Lambda \right\rvert} \sum_{q=-\left\lvert A \right\rvert}^{\left\lvert \Lambda \right\rvert}\mathrm{e}^{-m_0(p)_+}\mathrm{e}^{-m_0\left( t-q-1 \right)_+}\left\lVert \mathcal{X} F^\Lambda_{p,q}(E,A) \right\rVert\Big),\\ &\text{where}\quad F^\Lambda_{p,q}(E,A)=Q^\Lambda_{\le k}P_+^{[A]^\Lambda_{p}}P_-^{]A[^\Lambda_{p}} R^{\Lambda}_{E} P_+^{[A]^\Lambda_q}P_-^{]A[^\Lambda_{q}}Q^\Lambda_{\le k} \quad\text{for}\quad p,q\in \mathbb Z. \end{aligned}\end{align} $$
-
2. We have the following estimates on Hilbert-Schmidt norms:
(3.37)Moreover, for
$$ \begin{align}\begin{aligned} \left\lVert \mathcal{X} P_-^AR_E^\Lambda P_+^B Q^\Lambda_{\le k} \right\rVert_{HS} \le C_k\Big(\left\lvert \Lambda \right\rvert^{k}\mathrm{e}^{-m_0 t}+ \sum_{q=-\left\lvert A \right\rvert}^{\left\lvert \Lambda \right\rvert} \mathrm{e}^{-m_0\left( q \right)_+} {\left\lVert \mathcal{X} Q^\Lambda_{\le k}P_-^{]A[_{q}}R^{\Lambda}_{E}P_+^BQ^\Lambda_{\le k} \right\rVert}_{HS}\Big). \end{aligned}\end{align} $$
$s\in (0,1)$
, we have (3.38)
$$ \begin{align}\begin{aligned} \mathbb E\left( \left\lVert \mathcal{X} P_-^AR_E^\Lambda P_+^B Q^\Lambda_{\le k} \right\rVert_{HS} ^s \right) \le C_{k,s} \left\lvert \Lambda \right\rvert^{2sk+3}. \end{aligned}\end{align} $$
Proof. Let
$A\subset B\subset \Lambda $
, A connected in
$\Lambda $
. Since
$\mathcal {X}$
commutes with all the relevants operators, we will just do the proof for
$\mathcal {X}=I$
.
Using (3.35), (3.18) and (3.23), we get
$$ \begin{align}\begin{aligned} {\left\lVert P_-^AR_E^\Lambda P_+^B \right\rVert}\le C_0\mathrm{e}^{-m_0t}&+k{\left\lVert P_-^A\widehat R^{\Lambda}_{k,E}Q^\Lambda_{\le k}\widehat R^{\Lambda}_{k,E} P_+^B \right\rVert}+ k^2 {\left\lVert P_-^A\widehat R^{\Lambda}_{k,E} Q^\Lambda_{\le k} R^{\Lambda}_{E}Q^\Lambda_{\le k}\widehat R^{\Lambda}_{k,E}P_+^B \right\rVert}. \end{aligned}\end{align} $$
Using (3.24), (A.7) and the fact that
$Q^\Lambda _{\le k}$
commutes with
$P_\pm $
operators, we get
$$ \begin{align}\begin{aligned} &{\left\lVert P_-^A\widehat R^{\Lambda}_{k,E} Q^\Lambda_{\le k+1}\widehat R^{\Lambda}_{k,E} P_+^B \right\rVert}\le \sum_{q=-\left\lvert A \right\rvert}^{\left\lvert \Lambda \right\rvert} {\left\lVert D_q \right\rVert}{\left\lVert E_q \right\rVert}, \end{aligned}\end{align} $$
where
$$ \begin{align}\begin{aligned} D_q=P_-^A\widehat R^{\Lambda}_{k,E} P_+^{[A]_{q}} \mbox{ and } E_q=P_-^{]A[_{q}}\widehat R^{\Lambda}_{k,E}P_+^B. \end{aligned}\end{align} $$
Using (3.21), (3.23) and
$ ]A[_{q} \subset B$
for
$q+1\le t$
, we get
$$ \begin{align}\begin{aligned} {\left\lVert D_q \right\rVert}\le C_0 \mathrm{e}^{-m_0(q)_+} \;\text{and}\; {\left\lVert E_q \right\rVert}\le C_0 \mathrm{e}^{-m_0\left( t-q-1 \right)_+} \;\text{for all}\; q\in \mathbb Z. \end{aligned}\end{align} $$
It follows that
$$ \begin{align}\begin{aligned} {\left\lVert P_-^A\widehat R^{\Lambda}_{k,E}Q^\Lambda_{\le k+1}\widehat R^{\Lambda}_{k,E} P_+^B \right\rVert}\le C_0^2 \sum_{q=-\left\lvert A \right\rvert}^{\left\lvert \Lambda \right\rvert}\mathrm{e}^{-m_0(q)_+}\mathrm{e}^{-m_0\left( t-q-1 \right)_+}\le C_0^\prime \left\lvert \Lambda \right\rvert\mathrm{e}^{- m_0 t}. \end{aligned}\end{align} $$
This leaves us with the estimation of the last term in (3.39). To this end, we use (3.24), (A.7) and (3.42) to obtain
$$ \begin{align}\begin{aligned} &{\left\lVert P_-^A\widehat R^{\Lambda}_{k,E}Q^\Lambda_{\le k+1} R^{\Lambda}_{E}Q^\Lambda_{\le k+1}\widehat R^{\Lambda}_{k,E} P_+^B \right\rVert}\le \sum_{p=-\left\lvert A \right\rvert}^{\left\lvert \Lambda \right\rvert} \sum_{q=-\left\lvert A \right\rvert}^{\left\lvert \Lambda \right\rvert} {\left\lVert D_p \right\rVert}\left\lVert F_{p,q} \right\rVert{\left\lVert E_q \right\rVert}\\ & \qquad \qquad\le C _0^2 \sum_{p=-\left\lvert A \right\rvert}^{\left\lvert \Lambda \right\rvert} \sum_{q=-\left\lvert A \right\rvert}^{\left\lvert \Lambda \right\rvert}\mathrm{e}^{-m_0(p)_+}\mathrm{e}^{-m_0\left( t-q-1 \right)_+}\left\lVert F_{p,q} \right\rVert, \end{aligned}\end{align} $$
where
$F_{p,q}=F_{p,q}^\Lambda (E,A)$
is as in (3.36) for
$p,q \in \mathbb Z$
.
Combining (3.39), (3.43) and (3.44), we get (3.36).
To prove (3.37), we proceed as in (3.39) using (3.34), exploit
$\left \lVert T_1T_2 \right \rVert _{HS}\le \left \lVert T_1 \right \rVert \left \lVert T_2 \right \rVert _{HS}$
, and use (3.26), obtaining
$$ \begin{align}\begin{aligned} &\left\lVert P_-^AR_E^\Lambda P_+^B Q^\Lambda_{\le k} \right\rVert_{HS} \le C_k \mathrm{e}^{-m_0 t}\left\lvert \Lambda \right\rvert^k + k{\left\lVert P_-^A\widehat R^{\Lambda}_{k,E}Q^\Lambda_{\le k} R^{\Lambda}_{E} P_+^B Q^\Lambda_{\le k} \right\rVert}_{HS}. \end{aligned}\end{align} $$
We then use (3.24), (A.7) and (3.42) to get
$$ \begin{align}\begin{aligned} {\left\lVert P_-^A\widehat R^{\Lambda}_{k,E}Q^\Lambda_{\le k} R^{\Lambda}_{E} P_+^B Q^\Lambda_{\le k} \right\rVert}_{HS} &\le \sum_{q=-\left\lvert A \right\rvert}^{\left\lvert \Lambda \right\rvert} {\left\lVert D_q \right\rVert} {\left\lVert Q^\Lambda_{\le k}P_-^{]A[_{q}}R^{\Lambda}_{E}P_+^BQ^\Lambda_{\le k} \right\rVert}_{HS} \\ &\le \sum_{q=-\left\lvert A \right\rvert}^{\left\lvert \Lambda \right\rvert} C_0 \mathrm{e}^{-m_0\left( q \right)_+} {\left\lVert Q^\Lambda_{\le k}P_-^{]A[_{q}}R^{\Lambda}_{E}P_+^BQ^\Lambda_{\le k} \right\rVert}_{HS}. \end{aligned}\end{align} $$
Given
$s\in (0,1)$
, it follows from (3.37) and (3.33) that
$$ \begin{align}\begin{aligned} &\mathbb E\left( \left\lVert \mathcal{X} P_-^AR_E^\Lambda P_+^B Q^\Lambda_{\le k} \right\rVert_{HS}^s \right) \le C_{k,s} \left\lvert \Lambda \right\rvert^{2sk+3}. \end{aligned}\end{align} $$
3.5 Large deviation estimate
Using a large deviation argument, we get the following refinement of (3.33). Recall we may assume
$\rho ^\Lambda (A,B)<\infty $
in view of (3.8).
Lemma 3.7. Let
$k\in \mathbb N$
. Let
$A\subset B\subset \Lambda $
, with
$\rho ^\Lambda (A,B)<\infty $
. Given
$s\in (0,\frac 12)$
, there exist constants
$C_{k,s} ,c_\mu>0$
such that for all
$E \in I_{\le k}$
, we have
$$ \begin{align}\begin{aligned} \mathbb E\left( \left\lVert \chi^\Lambda_NQ^\Lambda_{\le k}P_-^A R^{\Lambda}_{E}P_+^B Q^\Lambda_{\le k} \right\rVert_{HS}^s \right) \le C_{k,s} \left\lvert \Lambda \right\rvert^{2(sk+1)} \left( \mathrm{e}^{- c_\mu N}+ \mathrm{e}^{-m_0 \rho^\Lambda (A,B)} \right). \end{aligned}\end{align} $$
In particular,
$$ \begin{align}\begin{aligned} \mathbb E{\left\lVert \chi^\Lambda_NQ^\Lambda_{\le k} P_-^A R^{\Lambda}_{E}P_+^B Q^\Lambda_{\le k} \right\rVert^s} \le C_{k,s}\left\lvert \Lambda \right\rvert^{2(sk+1)} \mathrm{e}^{-m_{0,\mu}\rho^\Lambda (A,B)} \; \ \text{if}\; \ 8kN \ge\rho^\Lambda (A,B), \end{aligned}\end{align} $$
where
$m_ {0,\mu }>0$
.
Proof. Recall
$\mathcal {H}_\Lambda ^{\left (N\right )}= \operatorname {\mathrm {Ran}} \chi ^\Lambda _N$
, and let
$\mathcal {H}_\Lambda ^{\left (N,k\right )}= \operatorname {\mathrm {Ran}} \chi ^\Lambda _N Q^\Lambda _{\le k}$
. Recall also that the restriction of
$V_\omega ^\Lambda $
to
$\mathcal {H}_\Lambda ^{\left (N\right )}$
is diagonalized by the canonical basis
$ \Phi _\Lambda ^{\left (N\right )}$
as in Remark 2.1(iii).
Let us first assume that N is such that
$N \lambda \bar {\mu }\ge 2k\left ( 1- \tfrac {1}{\Delta } \right )$
, where
$ \bar {\mu }$
denotes the mean of the probability distribution
$\mu $
(see Definition 2.2). The standard large deviation estimate (Cramer’s Theorem) gives
$$ \begin{align}\begin{aligned} \mathbb P\left\{ \lambda \omega^{\left(M\right)} < k\left( 1- \tfrac{1}{\Delta} \right) \right\}\le \mathbb P\left\{ \omega^{\left(M\right)} < N \tfrac {\bar{\mu}} 2 \right\}\le \mathrm{e}^{- c_\mu N}\;\text{for all}\; M\subset \Lambda \;\text{with}\; \left\lvert M \right\rvert=N, \end{aligned}\end{align} $$
where
$c_\mu $
is a constant depending only on the probability distribution
$\mu $
. This implies that there exists
$C_k>0$
such that
$$ \begin{align}\begin{aligned} \mathbb P\left\{ \lambda \omega^{\left(M\right)}< k\left( 1- \tfrac{1}{\Delta} \right) \right\}\le C_k\mathrm{e}^{- c_\mu N}\;\text{for all}\; N\in \mathbb N \;\text{and }\; M\subset \Lambda \;\text{with}\; \left\lvert M \right\rvert=N. \end{aligned}\end{align} $$
It follows that for the event
$$ \begin{align}\begin{aligned} {\mathcal B_k^N}=\left\{ \exists M\subset \Lambda \;\text{with}\; \left\lvert M \right\rvert=N, \; W^\Lambda_M =k\;\text{and}\; \lambda \omega^{\left(M\right)}< k\left( 1- \tfrac{1}{\Delta} \right) \right\}, \end{aligned}\end{align} $$
we have
$$ \begin{align}\begin{aligned} &\mathbb P_{\Lambda}\left( \mathcal B_k^N \right) \le C_k\mathrm{e}^{- c_\mu N}\operatorname{\mathrm{tr}} Q_{\le k}^{\Lambda,N} \le C_k \left\lvert \Lambda \right\rvert^{2k} \mathrm{e}^{- c_\mu N} \quad\text{for}\quad N=1,2\ldots, \left\lvert \Lambda \right\rvert, \end{aligned}\end{align} $$
where we also used Lemma 3.5. On the complementary event
$\left ( \mathcal B_k^N \right )^c$
, we have
$$ \begin{align} \lambda V_\omega \chi^\Lambda_N Q^\Lambda_{\le k} \ge k\left( 1- \tfrac{1}{\Delta} \right) \chi^\Lambda_N Q^\Lambda_{\le k}. \end{align} $$
If (3.54) holds, we conclude that
$$ \begin{align}\begin{aligned} H^{\Lambda,N}&\ge \left( 1- \tfrac{1}{\Delta} \right) \mathcal{W}^\Lambda + \lambda V_\omega= \left( Q_{\le k}^{\Lambda,N} + Q_{\ge k+1 }^{\Lambda,N} \right)\left( \left( 1- \tfrac{1}{\Delta} \right) \mathcal{W}^\Lambda + \lambda V_\omega \right)\\ & \ge \left( 1- \tfrac{1}{\Delta} \right) Q_{\ge k+1 }^{\Lambda,N} \mathcal{W}^\Lambda +Q_{\le k}^{\Lambda,N} \left( \left( 1- \tfrac{1}{\Delta} \right) \mathcal{W}^\Lambda + \lambda V_\omega \right)\ge (k+1) \left( 1- \tfrac{1}{\Delta} \right). \end{aligned}\end{align} $$
We deduce that for
$\omega \in \left ( \mathcal B_k^N \right )^c$
and
$E \in I_{\le k}$
, we have
$$ \begin{align}\begin{aligned} H^{\Lambda,N}-E\ge (k+1) \left( 1- \tfrac{1}{\Delta} \right)- (k+\tfrac 34 ) \left( 1- \tfrac{1}{\Delta} \right) = \tfrac 14 \left( 1- \tfrac{1}{\Delta} \right). \end{aligned}\end{align} $$
Proceeding as in the derivation of (3.23), it follows from Lemma 3.1 and Remark 3.2 that for
$\omega \in \left ( \mathcal B_k^N \right )^c$
, we have, for
$A\subset B\subset {\Lambda }$
with A connected in
${\Lambda }$
, that
$$ \begin{align}\begin{aligned} {\left\lVert \chi^\Lambda_NP_-^{A} R_E^\Lambda P_+^{B} \right\rVert}\le C_0 \mathrm{e}^{-m_0 \rho^\Lambda (A,B)}. \end{aligned}\end{align} $$
Given
$E \in I_{\le k}$
, and letting
$T =\chi _N Q_{\le k}^\Lambda P_-^{A} R_E^\Lambda P_+^{B}Q_{\le k}^\Lambda $
, we obtain
$$ \begin{align}\begin{aligned} \mathbb E\left( \left\lVert T \right\rVert_{HS}^s \right) & \le \mathbb E\left( \chi_{\mathcal B_k^N}\left\lVert T \right\rVert_{HS}^s \right)+\mathbb E\left( \chi_{\left( \mathcal B_k^N \right)^c}\left\lVert T \right\rVert_{HS}^s \right)\\ &\le \left( \mathbb P \left( \mathcal B_k^N \right) \right)^{\frac 1 2} \left( \mathbb E\left( \left\lVert T \right\rVert_{HS}^{2s} \right) \right)^{\frac 12}+ C_0 \mathrm{e}^{-m_0 \rho^\Lambda(A,B)}\left\lVert \chi_N Q_{\le k}^{\Lambda} \right\rVert_{HS}^s \\ &\le C_{k,s} \left\lvert \Lambda \right\rvert^{2(sk+1)} \left( \mathrm{e}^{- {\frac 1 2} c_\mu N}+ \mathrm{e}^{-m_0 \rho^\Lambda (A,B)} \right), \end{aligned}\end{align} $$
where we used (3.53), Lemma 3.5 and (3.33) with
$2s$
instead of s. This estimate is (3.48), up to a redefinition of the constant
$c_\mu $
.
3.6 Decoupling of resolvents
We now illustrate the basic idea that allows us to obtain the exponential decay of the left-hand side in (2.17), analogous to the decoupling argument in the single particle localization literature. For this purpose, we will consider a more convenient object than the one in (2.17). To do so, let
$A\subset M\subset B\subset \Lambda $
, and consider
$P_+^{M^c} P_-^AR_E^\Lambda P_+^{B}.$
Let
$K\subset \mathbb Z$
be such that
$M\subset [K]_{-1}\subset K\subset [K]_1\subset B$
. The resolvent identity yields (recall (3.10))
$$ \begin{align}\begin{aligned} P_+^{M^c} P_-^{A}R_E^\Lambda P_+^{B}& = - P_+^{M^c} P_-^{A}R^{K,K^c}_E \Gamma^K R_E^\Lambda P_+^{B} =- P_+^{M^c} P_-^{A}R^{K,K^c}_EP_+^{K^c} \Gamma^K R_E^\Lambda P_+^{B}\\ & =- P_+^{M^c} P_-^{A}R_E^{K}P_+^{K^c}\Gamma^K R^\Lambda_E P_+^{B}, \end{aligned}\end{align} $$
where we used that
$P_-^{A} R^{K,K^c}_E P_+^{K}=0$
by (3.2) since
$[A]_\infty ^K \subset K$
,
$P_+^{M^c}R^{K,K^c}_E=P_+^{M^c} R^{K,K^c}_EP_+^{K^c}$
by (3.2) since
$K^c\subset M^c$
, and
$R^{K,K^c}_EP_+^{K^c}=R^{K}_EP_+^{K^c}$
. Using the specific structure of the XXZ Hamiltonian – that is, (A.3)–(A.5) – we have
$P_+^{K^c}\Gamma ^K= P_+^{K^c}P_-^{ \partial ^\Lambda K} \Gamma ^K P_-^{ \partial ^\Lambda K}= P_+^{K^c}P_-^{\partial _{in}^\Lambda K}\Gamma ^K P_-^{\partial _{ex}^\Lambda K} $
, so it follows from (3.59) that
$$ \begin{align}\begin{aligned} P_+^{M^c} P_-^{A}R_E^\Lambda P_+^{B}=-P_+^{M^c} P_-^{A}R_E^{K}P_-^{\partial_{in}^\Lambda K}P_+^{K^c}\Gamma^K P_-^{\partial_{ex}^\Lambda K}R^\Lambda_E P_+^{B}. \end{aligned}\end{align} $$
We now use the resolvent identity for the operator
$H^{ [K]_1^\Lambda ,([K]^\Lambda _1)^c}$
and (A.3), obtaining
$$ \begin{align}\begin{aligned} P_-^{\partial_{ex}^\Lambda K}R_E^\Lambda P_+^{B}=- P_-^{\partial_{ex}^\Lambda K}R_E^\Lambda P_-^{\partial_{ex}^\Lambda K} \Gamma^{[K]_1} P_-^{\partial_{ex}^\Lambda [K]_1} P_+^{[K]_1}R_E^{[K]_1^c} P_+^{B}. \end{aligned}\end{align} $$
Combining (3.60)–(3.61), we obtain
$$ \begin{align}\begin{aligned} &P_+^{M^c} P_-^{A}R_E^\Lambda P_+^{B}= \\& (P_-^{A}{P_+^{M^c\cap K}R_E^{K}P_-^{\partial_{in}^\Lambda K}})P_+^{K^c}\Gamma^K \left( P_-^{\partial_{ex}^\Lambda K}R_E^\Lambda P_-^{\partial_{ex}^\Lambda K} \right) \Gamma^{[K]_1} P_+^{[K]_1}{\left( P_-^{\partial_{ex}^\Lambda [K]_1} R_E^{[K]_1^c} P_+^{B\cap [K]_1^c} \right)}. \end{aligned}\end{align} $$
This is the basic decoupling formula, in a sense that the expressions in the first and last parentheses on the last line are statistically independent and of the same form as the left-hand side of (2.17). So, if we can perform the averaging over the random variables at sites
$r\in \partial _{ex}^\Lambda K$
to get rid of the middle resolvent, we will effectively decouple the system into pieces supported by the disjoint subsets K and
$[K]_1^c$
. (Note that these pieces do not depend on the random variables at sites
$r\in \partial _{ex}^\Lambda K$
.) This decoupling will be performed using the a priori estimate (3.33), after we dress the corresponding resolvents with Hilbert-Schmidt operators on both sides as in Lemma 3.6. In broad strokes, we then will extract the (initial) exponential decay from the expression in the first parenthesis in (3.62) using reduction to lower energies and obtain the full exponential decay using a sub-harmonicity argument. We flesh out details of this process as we proceed with the proof.
3.7 Clusters classification
In preparation to initiate the FMM, we first inspect the structure of states in
$\operatorname {\mathrm {Ran}} Q^\Lambda _{\le k}$
. Since
$Q^\Lambda _{\le k}$
is a multiplication operator in the canonical basis
$\left \{ \Phi _\Lambda ^{\left (N\right )} \right \}_{N=0}^{\left \lvert \Lambda \right \rvert }$
introduced in (2.1), we just need to consider the elements
$\varphi _M$
of this basis with M that belong to a set
$\mathcal S_{N,k}^\Lambda :=\left \{ M\subset \Lambda :\ \left \lvert M \right \rvert =N,\ 1\le W^\Lambda _M\le k \right \}$
,
$N\ge 1$
. (Recall that
$W^\Lambda _M$
is the number of clusters of the configuration M – that is, the number of connected components of M in the graph
$\Lambda $
.) Denoting by
$ \pi _{\varphi }$
the orthogonal projection onto
$\mathbb C\varphi $
, given
$M\in \mathcal S^{\Lambda }_{N,k}$
, we abuse the notation and write
$\pi _M$
for
$\pi _{\varphi _M}$
, so
$\pi _{M} = \left ( \prod _{j\in M}\mathcal {N}_j \right ) P_+^{M^c}$
, and note that
$ \chi _N^\Lambda Q^\Lambda _{\le k}= \sum _{M\in \mathcal S_{N,k}^\Lambda } \pi _M. $
Given
$ A\subset \Lambda $
, we set
$\mathcal S^{\Lambda ,A}_{N,k}=\left \{ M\in \mathcal S^{\Lambda }_{N,k}: \ M\cap A\ne \emptyset \right \}$
, and note that
$ \chi _N^\Lambda Q^\Lambda _{\le k} P_-^A= \sum _{M\in \mathcal S_{N,k}^{\Lambda ,A}} \pi _M. $
We set
$$ \begin{align} \gamma_A(M) = \max_{x\in M} \operatorname{\mathrm{dist}}_\Lambda(x,A) \le \operatorname{\mathrm{diam}}_\Lambda (M)= \max_{x,y\in M} \operatorname{\mathrm{dist}}_\Lambda(x,y) \quad\text{for}\quad M\in \mathcal S^{\Lambda,A}_{N,k}. \end{align} $$
Note that
$\operatorname {\mathrm {diam}}_\Lambda (M)=N-1$
for
$k=1$
and
$\operatorname {\mathrm {diam}}_\Lambda (M)\ge N\ge 2$
for
$k\ge 2$
.
If
$ 8kN < \rho ^\Lambda (A,B)$
, we will use the following lemma.
Lemma 3.8. Fix
$k\ge 2$
. Let
$A\subset B\subset \Lambda $
be such that
$8kN < \rho ^\Lambda (A,B)<\infty $
, and let
$M\in \mathcal S^{\Lambda ,A}_{N,k}$
.
-
1. Suppose
$\gamma _A(M) < 4kN$
. Then setting
$Z=[A]^\Lambda _{6kN}$
, we have (3.64)
$$ \begin{align}\begin{aligned} A \cup M\subset [Z]_{-1}\subset Z \subset [Z]_1\subset B; \;\rho^\Lambda(A\cup M,Z)\ge 2kN; \quad \rho^\Lambda(Z,B)\ge 2kN. \end{aligned}\end{align} $$
-
2. Suppose
$ \rho ^\Lambda (A,B)\le 2 \gamma _A(M)$
. Let
$ d_\rho := \left \lfloor \tfrac { \rho ^\Lambda (A,B)}{6k}\right \rfloor $
. Then there exists
$a \in \left \{ 1,2,\ldots , 3k-1 \right \}$
, such that, letting
$K= [A]^\Lambda _{a d_\rho }$
, we have (3.65)Moreover, letting
$$ \begin{align}\begin{aligned} \rho^\Lambda \left( \partial^\Lambda K, \Lambda\setminus M \right) \ge d_\rho -1. \end{aligned}\end{align} $$
$M_1= M\cap K$
and
$M_2= M\cap K^c $
, we have
$K\subset B$
and
$M_{i}\neq \emptyset $
for
$i=1,2$
.
-
3. Suppose
$ 8kN< 2\gamma _A(M)< \rho ^\Lambda (A,B )$
. Let
$ d_\gamma := \left \lfloor \frac { \gamma _A(M)}{3k}\right \rfloor $
. Then there exists
$a \in \left \{ 1,2,\ldots , 3k-1 \right \}$
, such that, letting (3.66)we have
$$ \begin{align}\begin{aligned} K= [A]^\Lambda_{a d_\gamma}\cup\left( [A]^\Lambda_{\gamma_A(M)+d_\gamma}\setminus [A]^\Lambda_{a d_\gamma+1} \right), \end{aligned}\end{align} $$
(3.67)Moreover, letting
$$ \begin{align}\begin{aligned} \rho^\Lambda \left( \partial^\Lambda K, \Lambda\setminus M \right) \ge d_\gamma -1. \end{aligned}\end{align} $$
$M_1= M\cap [A]^\Lambda _{j d_\gamma }$
and
$M_2= M\cap [A]^\Lambda _{\gamma _A(M)}\setminus [A]^\Lambda _{j d_\gamma +1}$
, we have
$M_1\cup M_2=M\subset K\subset B$
and
$M_{i}\neq \emptyset $
for
$i=1,2$
.
Proof. Part (i) is obvious. To prove Parts (ii) and (iii), let
$d=d_\rho $
in Part (ii), and
$d=d_\gamma $
in Part (ii); note that
$ d \ge N$
in both cases. We set
$Y_{a}=[A]^\Lambda _{{a} {d}}\setminus [A]^\Lambda _{({a}-1) {d}}\subset B$
for
${a}=1,2,\ldots , 3k$
; note
$3k d\le \frac {\rho ^\Lambda (A,B)}2$
in both cases.
The set M consists of s clusters where
$2\le s \le k$
, so
$N\ge 2$
. Each cluster has length
$\le N-1$
, so it can intersect at most two of the
$Y_{a}$
’s (as
$ d \ge N$
); hence, M can intersect at most
$2k$
of the distinct
$Y_{a}$
’s. Thus, there exists
${a}_* \in \left \{ 1,2,\ldots , 3k-1 \right \}$
such that
$$ \begin{align}\begin{aligned} M\cap \left( Y_{{a}_*}\cup Y_{{a}_*+1} \right)=\emptyset, \end{aligned}\end{align} $$
and
$M_1= M\cap [A]^\Lambda _{{({a}_*-1)}d}\ne \emptyset $
since
$A\cap M\neq \emptyset $
.
To prove Part (ii) with
$d=d_\rho $
, set
$K= [A]^\Lambda _{{a}_* d_\rho }\subset B$
. Then
$M_1= M\cap K\ne \emptyset $
since
$A\cap M\neq \emptyset $
, and
$M_2= M\cap (\Lambda \setminus K)\ne \emptyset $
as
$\rho ^\Lambda (A,B) \le 2\gamma _A(M)$
by hypothesis. Moreover, (3.65) holds due to (3.68).
To prove Part (iii) with
$d=d_\gamma $
, let K be given in (3.66). Then
$M_1= M\cap K\ne \emptyset $
since
$A\cap M\neq \emptyset $
, and
$M_2= M\cap (\Lambda \setminus K)\ne \emptyset $
as
$\rho ^\Lambda (A,B) \le 2\gamma _A(M)$
by hypothesis. Moreover, (3.65) holds due to (3.68).
Motivated by Lemma 3.8, given
$A\subset B\subset \Lambda $
with
$ 8kN< \rho ^\Lambda (A,B)<\infty $
, we decompose
$\mathcal S^{\Lambda ,A}_{N,k}$
into three distinct groups:
-
1. Small
$\gamma _A(M)$
:
$M\in \mathcal {G}_1^{\Lambda ,N}(A,B)$
if
$2\gamma _A(M)\le 8kN < \rho ^\Lambda (A,B) $
. -
2. Large
$\gamma _A(M)$
:
$M\in \mathcal {G}_2^{\Lambda ,N}(A,B)$
if
$ 8kN< \rho ^\Lambda (A,B)\le 2 \gamma _A(M)$
. -
3. Intermediate
$\gamma _A(M)$
:
$M\in \mathcal {G}_3^{\Lambda ,N}(A,B)$
if
$ 8kN< 2\gamma _A(M)< \rho ^\Lambda (A,B)$
.
Note that for
$ 8kN< \rho ^\Lambda (A,B)<\infty $
, we have
$$ \begin{align} \chi_N^\Lambda Q^\Lambda_{\le k} P_-^A= \sum_{1=1}^3 \pi_{ \mathcal{G}_i^{\Lambda,N}(A,B)}, \quad\text{where}\quad \pi_{ \mathcal{G}_i^{\Lambda,N}(A,B)},= \sum_{M\in \mathcal{G}_i^{\Lambda,N}(A,B)} \pi_M. \end{align} $$
3.8 Decoupling revisited
We will need to estimate
$\chi ^\Lambda _N Q^\Lambda _{\le k}P_-^AR_E^\Lambda P_+^{B}Q^\Lambda _{\le k}$
. If
$ 8kN\ge \rho ^\Lambda (A,B)$
, we use (3.49). If
$ 8kN< \rho ^\Lambda (A,B)$
, we note that
$$ \begin{align} \pi_{M}\chi^\Lambda_{N}Q^\Lambda_{\le k}P_-^A R^{\Lambda}_{E}P_+^B Q^\Lambda_{\le k}=\pi_{M}P_-^A R^{\Lambda}_{E}P_+^B Q^\Lambda_{\le k} \quad\text{for}\quad M \in S^{\Lambda,A}_{N,k}. \end{align} $$
We will use different strategies for
$M\in \mathcal {G}_i=\mathcal {G}_i^{\Lambda ,N}(A,B)$
,
$i=1,2,3$
.
If
$M\in \mathcal {G}_1$
, we use the decoupling argument of Section 3.6, getting (3.62) with
$K=[A]^\Lambda _{8kN}$
. The estimation for the expression in the first parenthesis in (3.62) will be performed using directly the a priori estimate (3.48) and (3.64). (No energy reduction.) This yields exponential decay in
$\gamma _A(M)$
for this type of contributions, and the sub-harmonicity argument concludes the analysis.
To handle
$M\in \mathcal {G}_2$
, we consider
$K, M_1,M_2$
as in Lemma 3.8(ii), set
$S=[\partial K]^\Lambda _{d_\gamma -1} $
, and note that
$$ \begin{align}\begin{aligned} {\pi_M}P_-^A R^{\Lambda}_{E}P_+^B Q^\Lambda_{\le k}={\pi_M}P_+^{S}P_-^{K}P_-^{K^c}P_-^A R^{\Lambda}_{E}P_+^B Q^\Lambda_{\le k}. \end{aligned}\end{align} $$
Using
$M_1 \subset B$
, we get
$$ \begin{align}\begin{aligned} P_+^{S} P_-^{ K}P_-^{K^c} R_E^\Lambda P_+^{B}=-\left( P_+^{S}P_-^{K}P_-^{K^c} R^{K,K^c}_E P_-^{\partial^\Lambda K} \right)\Gamma^{\mathcal{K}} P_-^{\partial^\Lambda K} R^\Lambda_E P_+^{B}. \end{aligned}\end{align} $$
The expression in parenthesis is estimated by reduction to lower energies
$E'\in I_{\le k-1}$
, allowing the use of the induction hypothesis (in k) together with the estimate (3.65) to obtain exponential decay in
$\rho ^\Lambda (A,B)$
.
If
$M\in \mathcal {G}_3$
, we use a decoupling based on Lemma 3.8(iii), we get exponential decay in
$\gamma _A(M)$
from the induction hypothesis (in k), and the sub-harmonicity argument concludes the analysis.
3.9 Reduction to lower energies
We first observe that
$P_-^A R^{\Lambda }_{E}P_+^B=P_-^A \widehat R^{\Lambda }_{0,E}P_+^B$
decays exponentially in
$\rho ^\Lambda (A,B)$
for
$E\le \tfrac 34\left ( 1- \tfrac {1}{\Delta } \right )$
due to (3.23) with
$k=0$
; that is, Theorem 2.4 holds for k=0. Suppose now that we already established (2.17) for all energies
$E\in I_{\le k-1}$
and we want to push the allowable energies to the interval
$I_{\le k}$
. The principal idea here is to observe that if
$\emptyset \ne K \subsetneq \Lambda $
, then we have the nontrivial decoupling
$H^{K,K^c}= H^{K} + H^{K^c}$
, and
$R_E^{K,K^c}$
can be decomposed as
$$ \begin{align}\begin{aligned} R_E^{K,K^c} =\sum_{\nu\in \sigma (H^{K^c})} {R_{E-\nu}^{K}\otimes \pi_{\kappa_\nu}}, \end{aligned}\end{align} $$
where
$\left \{ \kappa _\nu \right \}_{\nu \in \sigma (H^{K^c})}$
is an orthonormal basis for
$\mathcal {H}_{K^c}$
that diagonalizes
$H^{K^c}$
:
$H^{K^c}\kappa _\nu =\nu \kappa _\nu $
. In particular, if
$K_1\subset K $
and
$K_2\subset K^c$
, we deduce that
$$ \begin{align}\begin{aligned} P_-^{K_1}P_-^{K_2}R_E^{K,K^c} =\sum_{\nu\in \sigma (H^{K^c})\cap[1- \frac{1}{\Delta},\infty)} \left( P_-^{K_1}R_{E-\nu}^{K} \right)\otimes \left( P_-^{K_2}\pi_{\kappa_\nu} \right), \end{aligned}\end{align} $$
since
$P_-^{K_2}\pi _{\kappa _0}=0$
, and we have
$\min _{\nu \in \sigma (H^{K^c})\setminus \left \{ 0 \right \}}\,\nu \ge 1- \frac {1}{\Delta }$
. This is exactly the type of setup we have in (3.71)–(3.72). It means that the factor
$P_-^{K_1}P_-^{K_2}$
allows us effectively to lower the energy
$E\in I_{\le k}$
to
$E-\nu \in I_{\le k-1}$
and therefore use the induction hypothesis to obtain exponential decay (we of course still need to control the summation over
$\nu $
on the right-hand side of (3.74)).
4 Proof of the main theorem
In this section, we prove Theorem 2.4. We fix
$\Delta _0> 5$
and
$\lambda _0>0$
, and assume
$\Delta \ge \Delta _0$
and
$\lambda \ge \lambda _0$
. As discussed in Remark 3.3, the argument can be modified for
$\Delta _0>1$
.
The proof proceeds by induction on k. Theorem 2.4 holds for
$k=0$
, since in this case, (2.17) follows from (3.23) with
$F_0=C_0$
,
$\xi _0=0$
and
$\theta _0=m_0$
as
$P_-^A R_E^\Lambda = P_-^A R_{0,E}^\Lambda $
. Given
$k\in \mathbb N$
, we assume the theorem holds for
$k-1$
, and we will prove the theorem holds for k.
We now fix
$k\in \mathbb N$
and
$\Lambda \subset \mathbb Z$
, finite and nonempty. We also fix
$A\subset B \subset \Lambda $
, where A is a nonempty subset connected in
$\Lambda $
; it follows that
$[A]^\Lambda _p$
is also connected in
$\Lambda $
and
$\left \lvert ]A[^\Lambda _{p} \right \rvert \le 2$
for all
$p\in \mathbb Z$
.
To derive the bound (2.17) from Lemma 3.6(i), we will estimate
$\mathbb E \left ( \left \lVert F^\Lambda _{p,q}(E,A) \right \rVert _{HS}^s \right )$
for
$p,q= -\left \lvert A \right \rvert , -\left \lvert A \right \rvert +1,\ldots , \left \lvert \Lambda \right \rvert $
for
$E\in I_{\le k}$
, where
$F^\Lambda _{p,q}(E,A)$
is given in (3.36). The estimate (3.33) gives the a priori bound (
$F_{p,q}=F^\Lambda _{p,q}(E,A)$
)
$$ \begin{align}\begin{aligned} \mathbb E \left\lVert F_{p,q} \right\rVert_{HS}^s\le C\lambda_0^{-s}k ^s\left\lvert \Lambda \right\rvert^{2sk+2}. \end{aligned}\end{align} $$
Since
$F_{p,q}=F_{q,p}^*$
, we may assume
$ p\le q$
. If
$p=q$
, we use (4.1); if
$p<q$
, we note that
$$ \begin{align}\begin{aligned} \left\lVert F_{p,q} \right\rVert_{HS}&\le \left\lVert Q^\Lambda_{\le k}P_-^{ ]A[^{\Lambda}_p} R^{\Lambda}_{E} P_+^{[A]^{\Lambda}_q}Q^\Lambda_{\le k} \right\rVert_{HS} \le \sum_{j\in ]A[^{\Lambda}_p}\left\lVert Q^\Lambda_{\le k}\mathcal{N}_j R^{\Lambda}_{E} P_+^{[j]^{\Lambda}_{q-p-1}}Q^\Lambda_{\le k} \right\rVert_{HS}, \end{aligned}\end{align} $$
where we used
$[ ]A[^{\Lambda }_{p}]^{\Lambda }_{q-p-1}\subset [A]^{\Lambda }_q $
for
$p<q$
.
For
$r\in {\mathbb {N}}^0$
and
$E\in I_{\le k}$
, we set
$$ \begin{align}\begin{aligned} f^{\Lambda}(k,E,r)=\max_{\Theta \subset\Lambda }\max_{j\in \Theta} \mathbb E\left( \left\lVert Q_{\le k}^{\Theta} \mathcal{N}_j R^{\Theta}_E P_+^{[j]^{\Theta}_r}Q_{\le k}^{\Theta} \right\rVert_{HS}^s \right) \end{aligned}\end{align} $$
and prove the following lemma.
Lemma 4.1. Let
$k\in \mathbb N$
,
$s\in (0,\frac 13)$
, and assume Theorem 2.4 holds for
$k-1$
. Then there exist constants
$D_k,C_k, \zeta _k , m_k>0$
(depending on k,
$\Delta _0$
,
$ \lambda _0$
and s), such that such that, for all
$\Delta \ge \Delta _0$
and
$\lambda \ge \lambda _0$
with
$\lambda \Delta ^2\ge D_k$
,
$\Lambda \subset \mathbb Z$
finite, energy
$E\in I_{\le k}$
, and
$r\in {\mathbb {N}}^0$
, we have
$$ \begin{align}\begin{aligned} f^{\Lambda}(k,E,r)\le C_k\left\lvert \Lambda \right\rvert^{\zeta_k} \mathrm{e}^{-m_k r}. \end{aligned}\end{align} $$
To finish the proof of the theorem, we assume that
$\Delta \ge \Delta _0$
and
$\lambda \ge \lambda _0$
with
$\lambda \Delta ^2\ge D_k$
as in the lemma. Then, since
$\mathbb E \left ( \left \lVert F_{p,q} \right \rVert \right )_{HS}^s\le 2 f^\Lambda (k,E, \left \lvert q-p \right \rvert -1)$
for
$\left \lvert q-p \right \rvert \ge 1$
, and we have (4.1) for
$q=p$
, we obtain
$$ \begin{align}\begin{aligned} &\mathbb E\, \left( \sum_{p=-\left\lvert A \right\rvert}^{\left\lvert \Lambda \right\rvert} \sum_{q=-\left\lvert A \right\rvert}^{\left\lvert \Lambda \right\rvert}\mathrm{e}^{-m_0(p)_+}\mathrm{e}^{-m_0\left( t-q-1 \right)_+}\left\lVert F_{p,q} \right\rVert \right)^s \le C_k \left\lvert \Lambda \right\rvert^{\zeta_k} \mathrm{e}^{-s m_k t}. \end{aligned}\end{align} $$
The estimate (2.17) now follows from (3.36) and (4.5) (recall (3.6)), so Theorem 2.4 holds for k.
To complete the proof of Theorem 2.4, we need to prove Lemma 4.1. To do so, we need the following lemma.
Lemma 4.2. Let
$k\in \mathbb N$
,
$s\in (0,\frac 13)$
, and assume Theorem 2.4 holds for
$k-1$
. Then there exist constants
$C_k, \zeta _k , m_k>0$
(depending on k,
$\Delta _0$
,
$ \lambda _0$
and s), such that, for all
$\Delta \ge \Delta _0$
and
$\lambda \ge \lambda _0$
,
$j \in \Lambda \subset \mathbb Z$
finite, energy
$E\in I_{\le k}$
,
$N\in \mathbb N$
, and
$r\in {\mathbb {N}}^0$
such that
$8kN < r$
, we have
$$ \begin{align}\begin{aligned} G^\Lambda_N(r)&= \mathbb E\,\left( \left\lVert \chi_N^{\Lambda}Q_{\le k}^{\Lambda} \mathcal{N}_j R^{\Lambda}_E P_+^{[j]^\Lambda_r}Q_{\le k}^{\Lambda} \right\rVert_{HS}^s \right)\\& \le C_k \left( \left\lvert \Lambda \right\rvert^{\zeta_k} \mathrm{e}^{-m_k r} + \mathrm{e}^{- m_k N} \left( \lambda \Delta^2 \right)^{-s} \sum_{p=0}^{r} \mathrm{e}^{-m_k \left( r-p \right)} f_N^\Lambda(p) \right). \end{aligned}\end{align} $$
Proof. Let
$k\in \mathbb N$
,
$s\in (0,\frac 13)$
, and assume Theorem 2.4 holds for
$k-1$
. Let
$j \in \Lambda \subset \mathbb Z$
finite and
$E\in I_{\le k}$
,
$N\in \mathbb N$
, and
$r\in {\mathbb {N}}^0$
such that
$8kN < r$
. Let
$G^\Lambda _N(r)$
be as in (4.6). It follows from (3.69), setting
$\mathcal {G}^N_i =\mathcal {G}_i^{\Lambda ,N}(\left \{ j \right \}, [j]^\Lambda _r)$
,
$i=1,2,3$
(see Section 3.7), that
$$ \begin{align}\begin{aligned} G^\Lambda_N(r)&\le \sum_{i=1}^3 G_i(r), \;\text{where}\; G_i(r)=G^{\Lambda,N}_i(r)= \mathbb E\left( \left\lVert \pi_{ \mathcal{G}^N_i }\mathcal{N}_j R^{\Lambda}_E P_+^{[j]^\Lambda_r}Q_{\le k}^{\Lambda} \right\rVert_{HS}^s \right). \end{aligned}\end{align} $$
To estimate
$G_1(r)$
, we use (3.62) with
$M= [j]_{4kN}^\Lambda $
and
$K= [j]_{6kN}^\Lambda $
, (3.25) and (A.6), obtaining
$$ \begin{align}\begin{aligned} &G_1(r) \le C \left( \lambda \Delta^2 \right)^{-s} \,\mathbb E_K\left( \left\lVert Y \right\rVert_{HS}^s \right)\,\mathbb E_{\left( [K]_1^\Lambda \right)^c} \left( \left\lVert Z \right\rVert_{HS}^s \right);\\ & \quad Y:= \chi_N^{K} Q^K_{\le k}{P_+^{K\setminus M}R_E^{K}P_-^{\partial_{in}^{\Lambda} K}},\quad Z:=P_-^{\partial_{ex}^{\Lambda} [K]_1} R_E^{\left( [K]_1^\Lambda \right)^c}P_+^{[j]^\Lambda_r\cap\left( [K]_1^\Lambda \right)^c}Q^{\left( [K]_1^\Lambda \right)^c}_{\le k}\chi_N^{\left( [K]_1^\Lambda \right)^c}. \end{aligned}\end{align} $$
To estimate
$\mathbb E_K\left ( \left \lVert Y \right \rVert _{HS}^s \right )$
, note that
$$ \begin{align}\begin{aligned} \left\lVert Y \right\rVert_{HS} \le \sum_{u\in \partial_{in}^{\Lambda} K} {\left\lVert Y_u \right\rVert_{HS}}, \;\text{where}\; Y_u= \chi_N^{K} Q^K_{\le k}P_+^{K\setminus M}R_E^{K} \mathcal{N}_u, \;\text{and}\; \left\lvert \partial_{in}^{\Lambda} K \right\rvert\le 2. \end{aligned}\end{align} $$
Using (3.37) and
$\rho ^K(\partial _{in}^{\Lambda } K,K\setminus M) \ge 2kN$
, for
$u\in \partial ^{\Lambda } _{in} K$
, we get
$$ \begin{align}\begin{aligned} \mathbb E_K\left( \left\lVert Y_u \right\rVert_{HS}^s \right)&\le C_k^s\left( \left\lvert K \right\rvert^{sk}\mathrm{e}^{-sm_0 2kN}+ \sum_{q=-1}^{\left\lvert K \right\rvert} \mathrm{e}^{-sm_0\left( q \right)_+} \mathbb E\left( \Big\|{\chi_N^{K} Q^K_{\le k}P_-^{]u[^K_{q}}R^K_{E}P_+^{K\setminus M} Q^K_{\le k}}^s_{HS}\Big\| \right) \right)\\ & \le C_{k,s}\left( \left\lvert K \right\rvert^{2sk+1} \mathrm{e}^{-sm_02kN}+ 2 \sum_{q=-1}^{2kN-1} \mathrm{e}^{-sm_0\left( q \right)_+} f_N^K(2kN-q-1) \right) \\ & \le C_{k,s} \mathrm{e}^{- m_{0,k}^\prime k N}, \end{aligned}\end{align} $$
where we used the a priori bounds (3.33) and (3.48).
Similarly,
$$ \begin{align}\begin{aligned} \left\lVert Z \right\rVert_{HS}\le\sum_{u\in \partial_{ex}^{\Lambda} [K]_1} {\left\lVert Z_u \right\rVert_{HS}}, \;\text{where}\; Z_u= \mathcal{N}_u R_E^{\left( [K]_1^\Lambda \right)^c}P_+^{[j]^\Lambda_r\cap \left( [K]_1^\Lambda \right)^c}Q^{\left( [K]_1^\Lambda \right)^c}_{\le k}\chi_N^{\left( [K]_1^\Lambda \right)^c}, \end{aligned}\end{align} $$
and
$ \left \lvert \partial _{ex}^{\Lambda } [K]_1 \right \rvert \le 2$
. Using (3.37), for
$u\in \partial _{ex} [K]_1$
, we get
$$ \begin{align}\begin{aligned} &\mathbb E_{\left( [K]_1^\Lambda \right)^c}\left( \left\lVert Z_u \right\rVert_{HS}^s \right) \le C_k^s\left(\left\lvert \Lambda \right\rvert^{sk}\mathrm{e}^{-sm_0 (r-6kN-2)}+ \right. \\ & \quad \left. \sum_{q=-1}^{\left\lvert \Lambda \right\rvert} \mathrm{e}^{-sm_0\left( q \right)_+} \mathbb E\Big({\Big\|{\chi_N^{{\left( [K]_1^\Lambda \right)^c}} Q^{\left( [K]_1^\Lambda \right)^c}_{\le k} P_-^{]u[^{\left( [K]_1^\Lambda \right)^c}_{q}} R_E^{\left( [K]_1^\Lambda \right)^c} P_+^{[j]^\Lambda_r\cap \left( [K]_1^\Lambda \right)^c} Q^{\left( [K]_1^\Lambda \right)^c}_{\le k}}\Big\|^s_{HS}}\Big)\right)\\ & \le C_k^s\left( \left\lvert \Lambda \right\rvert^{2sk+2} \mathrm{e}^{-sm_0 (r-6kN-2)}+ \sum_{q=-1}^{r-6kN-3} \mathrm{e}^{-sm_0\left( q \right)_+} f_N^\Lambda(r-6kN-q-3) \right)\\ & = C_k^s \left( \left\lvert \Lambda \right\rvert^{2sk+2} \mathrm{e}^{-sm_0 (r-6kN-2)}+ \sum_{p=0}^{r-6kN-2} \mathrm{e}^{-sm_0\left( r-p-6kN -3 \right)_+} f_N^\Lambda(p) \right). \end{aligned}\end{align} $$
Combining (4.8)–(4.12), we get
$$ \begin{align}\begin{aligned} &G_1(r) \le C \left( \lambda \Delta^2 \right)^{-s} \mathrm{e}^{- m_k^\prime k N}\Big({ \left\lvert \Lambda \right\rvert^{2sk+2} \mathrm{e}^{-m_k^\prime r}+ \sum_{p=0}^{r} \mathrm{e}^{-m_k^\prime \left( r-p \right)} f_N^\Lambda(p)}\Big), \end{aligned}\end{align} $$
for an appropriate
$m_k^\prime>0$
.
To estimate
$G_2(r)$
, we note that it follows from Lemma 3.8(ii), letting
$$ \begin{align}\begin{aligned} K(a)=[ j]^{\Lambda}_{ad_\rho}\quad\text{and}\quad S(a)=[\partial^{\Lambda} K(a)]^{\Lambda}_{d_\rho-1} \quad\text{for}\quad a\in\mathbb N, \end{aligned}\end{align} $$
that
$$ \begin{align}\begin{aligned} G_2(r) \le \sum_{a=1}^{3k-1} G^{\left(a\right)}_2(r), \quad G^{\left(a\right)}_2(r)= \mathbb E\left( \left\lVert \chi_N^{\Lambda} Q^\Lambda_{\le k}P_+^{S(a)}P_-^{K(a)}P_-^{(K(a))^c}\mathcal{N}_j R^{\Lambda}_E P_+^{[j]^\Lambda_r}Q_{\le k}^{\Lambda} \right\rVert_{HS}^s \right). \end{aligned}\end{align} $$
To estimate
$ G^{\left (a\right )}_2(r)$
, we use (3.71) and (3.72), the Cauchy-Schwarz inequality and Hölder’s inequality (recall
$3s <1$
) to get (we mostly omit a from the notation)
$$ \begin{align}\begin{aligned} G^{\left(a\right)}_2(r) &\le C \Delta^{-s} \left( \mathbb E \left\lVert Y \right\rVert^{2s} \right)^{1/2}\left( \mathbb E\left\lVert Z \right\rVert^{2s}_{HS} \right)^{1/2}\\ & \le C \Delta^{-s} \left( \mathbb E\left\lVert Y \right\rVert^{s} \right)^{1/4} \left( \mathbb E\left\lVert Y \right\rVert^{3s} \right)^{1/4}\left( \mathbb E\left\lVert Z \right\rVert^{2s}_{HS} \right)^{1/2} , \end{aligned}\end{align} $$
where
$$ \begin{align}\begin{aligned} Y=\chi_N^{\Lambda} Q^\Lambda_{\le k}P_+^{S(a)}P_-^{K(a)}P_-^{{(K(a))^c}}\mathcal{N}_jR_E^{K(a), (K(a))^c}P_-^{\partial ^{\Lambda} K(a)} \;\text{and}\; Z=P_-^{\partial^{\Lambda} K(a)} R^\Lambda_E P_+^{[j]^\Lambda_r}Q_{\le k}^{\Lambda}\chi_N^\Lambda. \end{aligned}\end{align} $$
It follows immediately from (3.38) that
$$ \begin{align}\begin{aligned} \mathbb E\left\lVert Z \right\rVert^{2s}_{HS}\le C \left\lvert \Lambda \right\rvert^{4sk+3} \quad\text{and}\quad \mathbb E\left\lVert Y \right\rVert^{3s}\le C \left\lvert \Lambda \right\rvert^{6sk+3}, \end{aligned}\end{align} $$
where we used
$\left \lvert \partial ^{\Lambda } K(a) \right \rvert \le 4$
since
$K(a)$
is connected, and hence, we have
$$ \begin{align}\begin{aligned} G^{\left(a\right)}_2(r) \le C \Delta^{-s} \left\lvert \Lambda \right\rvert^{\frac 72 sk+\frac 94} \left( \mathbb E\left\lVert Y \right\rVert^{s} \right)^{1/4}. \end{aligned}\end{align} $$
To estimate
$E\left \lVert Y \right \rVert ^{s}$
, we use ( the dependence on a is being ommitted)
$$ \begin{align}\begin{aligned} \left\lVert Y \right\rVert\le \sum_{x\in\partial^{\Lambda} K}\left\lVert Y_x \right\rVert, \quad\text{with}\quad Y_x=\chi_N^{\Lambda} Q^\Lambda_{\le k}P_+^{S}P_-^{K}P_-^{K^c}\mathcal{N}_jR_E^{K,K^c}\mathcal{N}_x. \end{aligned}\end{align} $$
We consider first the case
$x\in \partial ^{\Lambda }_{in}K$
. Using (3.74), we can further decompose
$Y_x$
as
$$ \begin{align}\begin{aligned} Y_x =\sum_{ \nu\in \sigma (H^{K^c})\cap [1- \frac{1}{\Delta},\infty)} Y_{x,\nu}, \quad Y_{x,\nu}= \chi_N^{\Lambda} Q^\Lambda_{\le k}P_+^{S}P_-^{K}P_-^{K^c}\mathcal{N}_j\left( R_{E-\nu}^{K}\otimes \pi_{\kappa_\nu} \right)\mathcal{N}_x. \end{aligned}\end{align} $$
Note that
$$ \begin{align}\begin{aligned} \left\lVert Y_x \right\rVert=\max_\nu\left\lVert Y_{x,\nu} \right\rVert\le \sum_{ \nu\in \sigma (H^{K^c})\cap [1- \frac{1}{\Delta},k(1- \frac{1}{\Delta}))} \left\lVert Y_{x,\nu} \right\rVert+\max_{ \nu\in \sigma (H^{K^c})\cap [k(1- \frac{1}{\Delta}),\infty)} \left\lVert Y_{x,\nu} \right\rVert. \end{aligned}\end{align} $$
Clearly, we can bound
$$ \begin{align}\begin{aligned} \left\lVert Y_{x,\nu} \right\rVert\le \left\lVert P_+^{S} \left( R_{E-\nu}^{K}\otimes \pi_{\kappa_\nu} \right)\mathcal{N}_x \right\rVert\le \left\lVert P_+^{S\cap K}R_{E-\nu}^{K} \mathcal{N}_x \right\rVert. \end{aligned}\end{align} $$
For
$\nu \ge 1- \frac {1}{\Delta }$
, we have
$E-\nu \in I_{\le k-1} $
for
$E\in I_{\le k}$
(recall (2.14)). For
$\nu \in \sigma (H^{K^c})\cap [1- \frac {1}{\Delta },k(1- \frac {1}{\Delta })) $
, we use the induction hypothesis for Theorem 2.4 and the statistical independence of
$H^{K^c}$
and
$\left \{ \omega _i \right \}_{i\in K}$
to conclude that
$$ \begin{align}\begin{aligned} \mathbb E{\left\lVert Y_{x,\nu} \right\rVert^{s}}\le \mathbb E_{K}{{\left\lVert P_+^{S\cap K}R_{E-\nu}^{K} \mathcal{N}_x \right\rVert}^{s}} \le C_{k-1}\left\lvert \Lambda \right\rvert^{\xi_{k-1}}\mathrm{e}^{-\theta_{k-1} \frac r {6k}}, \end{aligned}\end{align} $$
where we used (3.65).
For
$\nu \in \sigma (H^{K^c})\cap [k(1- \frac {1}{\Delta }),\infty )$
,
$E-\nu \le \frac 34 \left ( 1- \tfrac {1}{\Delta } \right )$
, and in this case,
$$ \begin{align}\begin{aligned} P_+^{S\cap K}R_{E-\nu}^{K} \mathcal{N}_x= P_+^{S\cap K}\widehat R_{E-\nu}^{K} \mathcal{N}_x, \end{aligned}\end{align} $$
so it follows from (3.23) with
$k=0$
, using (3.65), that
$$ \begin{align}\begin{aligned} \left\lVert Y_{x,\nu} \right\rVert \le C_0 \mathrm{e}^{-m_{0} \frac r {6k}}. \end{aligned}\end{align} $$
Using (4.22), (4.24), (4.26) and (3.27), we get
$$ \begin{align}\begin{aligned} \mathbb E\left\lVert Y_{x} \right\rVert^{s}& \le C\left\lvert \Lambda \right\rvert^{\xi_{k-1}}\mathrm{e}^{-\theta_{k-1} \frac r {6k}}\operatorname{\mathrm{tr}} \chi_{ [1- \frac{1}{\Delta},k(1- \frac{1}{\Delta}))}(H^{K^c}) \le C_k\left\lvert \Lambda \right\rvert^{\xi_{k-1}+2k}\,\mathrm{e}^{-\frac {\theta_{k-1}}{6k} r }. \end{aligned}\end{align} $$
Similar considerations show that the estimate (4.27) holds also for
$x\in \partial _{ex}^{\Lambda }K $
.
Combining (4.20) and (4.27) and recalling
$\left \lvert \partial ^{\Lambda } K \right \rvert \le 4$
, we get
$$ \begin{align}\begin{aligned} \mathbb E\left\lVert Y \right\rVert^{s}\le C_k \left\lvert \Lambda \right\rvert^{\xi_{k-1}+2k}\,\mathrm{e}^{-\frac {\theta_{k-1}}{6k} r }. \end{aligned}\end{align} $$
Combining (4.19) and (4.28), we see that
$$ \begin{align}\begin{aligned} G^{\left(a\right)}_2(r) \le C_k \Delta^{-s}\left\lvert \Lambda \right\rvert^{\zeta_k}\,\mathrm{e}^{-\frac {\theta_{k-1}}{24 k} r }. \end{aligned}\end{align} $$
It now follows from (4.15) and (4.29) that
$$ \begin{align}\begin{aligned} G_2(r) \le C_k \Delta^{-s}\left\lvert \Lambda \right\rvert^{\zeta_k}\,\mathrm{e}^{-\frac {\theta_{k-1}}{24 k} r }\le C_k \Delta^{-s}\left\lvert \Lambda \right\rvert^{\zeta_k} \mathrm{e}^{- \theta_{k-1}^{{\prime\prime}} r }. \end{aligned}\end{align} $$
To estimate
$ G_3(r)$
, given
$4kN< \gamma < \frac r 2$
, we let
$ d_\gamma := \left \lfloor \frac { \gamma }{3k}\right \rfloor $
. Given
$a\in \left \{ 1,2,\ldots ,3k-1 \right \} $
, we let
$K(a,\gamma )$
be as in (3.66) with
$A=\left \{ j \right \}$
, and let
$K_1(a,\gamma )=[j ]^\Lambda _{a d_\gamma }$
, the connected component of
$K(a,\gamma )$
that contains j. We also set
$K_2(a,\gamma )=K(a,\gamma )\setminus K_1(a,\gamma )$
,
$S(a,\gamma )=[\partial ^\Lambda K(a,\gamma )]^\Lambda _{d_\gamma -1}$
, and
$T(a,\gamma )=[j]^\Lambda _{\gamma _{\left \{ j \right \}}(M)}$
. It follows from Lemma 3.8(iii) that
$$ \begin{align}\begin{aligned} \pi_M=\pi_M P_+^{T(a,\gamma_{\left\{ j \right\}}(M))}P_+^{S(a,\gamma_{\left\{ j \right\}}(M))}P_-^{K_1(a,\gamma_{\left\{ j \right\}}(M))}P_-^{K_2(a,\gamma_{\left\{ j \right\}}(M))}, \end{aligned}\end{align} $$
for some
$a\in \left \{ 1,2,\ldots ,3k-1 \right \}$
, and hence,
$$ \begin{align}\begin{aligned} G_3(r) &\le \sum_{\gamma=4kN+1}^{\left\lfloor \frac r 2\right \rfloor} \sum_{a=1}^{3k-1} G^{\left(a,\gamma\right)}_3(r), \quad\text{where}\quad \\ G^{\left(a,\gamma\right)}_3(r)& = \mathbb E\left( \left\lVert \chi_N^{\Lambda} Q^\Lambda_{\le k}P_+^{T(a,\gamma)}P_+^{S(a,\gamma)}P_-^{K_1(a,\gamma)}P_-^{K_2(a,\gamma)}\mathcal{N}_j R^{\Lambda}_E P_+^{[j]^\Lambda_r}Q_{\le k}^{\Lambda} \right\rVert_{HS}^s \right). \end{aligned}\end{align} $$
To estimate
$G^{\left (a,\gamma \right )}_3(r)$
, we start with the following analogue of (4.8) (we mostly omit
$(a,\gamma )$
from the notation):
$$ \begin{align}\begin{aligned} &G^{\left(a,\gamma\right)}_3(r)\le C \left( \lambda \Delta^2 \right)^{-s} \,\mathbb E_K\left( \left\lVert Y \right\rVert^s \right)\,\mathbb E_{[K]_1^c} \left( \left\lVert Z \right\rVert_{HS}^s \right);\\ & \quad Y:= \chi_N^{K} Q^K_{\le k}P_+^{T\cap K}P_+^{S}P_-^{K_1}P_-^{K_2}R_E^{K}P_-^{\partial^\Lambda_{in} K},\\ & \quad Z:=P_-^{\partial_{ex}^\Lambda [K]_1} R_E^{\left( [K]_1^\Lambda \right)^c}P_+^{[j]^\Lambda_r\cap{\left( [K]_1^\Lambda \right)^c}}Q^{{\left( [K]_1^\Lambda \right)^c}}_{\le k}\chi_N^{{\left( [K]_1^\Lambda \right)^c}}. \end{aligned}\end{align} $$
Proceeding exactly as in (4.11)–(4.12), we get
$$ \begin{align}\begin{aligned} \mathbb E_{\left( [K]_1^\Lambda \right)^c}\left( \left\lVert Z \right\rVert_{HS}^s \right) \le C_k^s\Big({ \left\lvert \Lambda \right\rvert^{\xi_k} \mathrm{e}^{-sm_0 (r-\gamma -d_\gamma)}+ \sum_{p=0}^{r-(\gamma +d_\gamma)-2} \mathrm{e}^{-sm_0\left( r-p-(\gamma +d_\gamma) -3 \right)_+} f_N^\Lambda(p)}\Big). \end{aligned}\end{align} $$
We estimate
$E\left \lVert Y \right \rVert ^{s}$
similarly to (4.20)–(4.28). We have
$$ \begin{align}\begin{aligned} \left\lVert Y \right\rVert\le \sum_{x\in \partial_{in} K}\left\lVert Y_x \right\rVert, \quad\text{where}\quad Y_x=\chi_N^{K} Q^K_{\le k}P_+^{T\cap K}P_+^{S}P_-^{K_1}P_-^{K_2}R_E^{K}\mathcal{N}_x. \end{aligned}\end{align} $$
We consider first the case
$x=x_i\in \partial _{in}{\left ( [K]_1^\Lambda \right )^c}K_i$
,
$i\in \left \{ 1,2 \right \}$
, and
$i'=\left \{ 1,2 \right \}\setminus \left \{ i \right \}$
. Using (3.74), we can further decompose
$Y_x$
as
$$ \begin{align}\begin{aligned} Y_{x_i} =\sum_{ \nu\in \sigma (H^{K_{i'}})\cap [1- \frac{1}{\Delta},\infty)} Y_{x_i,\nu}, \quad Y_{x_i,\nu}= P_+^{S} P_-^{K_{i'}} \left( R_{E-\nu}^{K_i}\otimes \pi_{\kappa_\nu} \right)\mathcal{N}_x. \end{aligned}\end{align} $$
Note that
$$ \begin{align}\begin{aligned} \left\lVert Y_{x_i} \right\rVert=\max_\nu\left\lVert Y_{{x_i} ,\nu} \right\rVert\le \sum_{ \nu\in \sigma (H^{K_{i'}})\cap [1- \frac{1}{\Delta},k(1- \frac{1}{\Delta}))} \left\lVert Y_{{x_i} ,\nu} \right\rVert+\max_{ \nu\in \sigma (H^{K_{i'}})\cap [k(1- \frac{1}{\Delta}),\infty)} \left\lVert Y_{{x_i} ,\nu} \right\rVert. \end{aligned}\end{align} $$
Clearly, we can bound
$$ \begin{align}\begin{aligned} \left\lVert Y_{{x_i} ,\nu} \right\rVert\le \left\lVert P_+^{S} \left( R_{E-\nu}^{K_i}\otimes \pi_{\kappa_\nu} \right)\mathcal{N}_{x_i} \right\rVert\le \left\lVert P_+^{S\cap K_i}R_{E-\nu}^{K_i} \mathcal{N}_{x_i} \right\rVert. \end{aligned}\end{align} $$
For
$\nu \ge 1- \frac {1}{\Delta }$
, we have
$E-\nu \in I_{\le k-1} $
for
$E\in I_{\le k}$
(recall (2.14)). For
$\nu \in \sigma (H^{K_{i'}})\cap [1- \frac {1}{\Delta },k(1- \frac {1}{\Delta })) $
, we use the induction hypothesis for Theorem 2.4 and the statistical independence of
$H^{K_{i'}}$
and
$\left \{ \omega _i \right \}_{i\in K_i}$
to conclude that
$$ \begin{align}\begin{aligned} \mathbb E{\left\lVert Y_{x,\nu} \right\rVert^{s}}\le \mathbb E_{K_i}\left\lVert P_+^{S\cap K_i}R_{E-\nu}^{K_i} \mathcal{N}_{x_i} \right\rVert^s \le C_{k-1}\left\lvert K_i \right\rvert^{\xi_{k-1}}\mathrm{e}^{-\theta_{k-1} d_\gamma}\le C_{k-1}\left\lvert \gamma \right\rvert^{\xi_{k-1}}\mathrm{e}^{-\theta_{k-1}d_\gamma}. \end{aligned}\end{align} $$
For
$\nu \in \sigma (H^{K_{i'}})\cap [k(1- \frac {1}{\Delta }),\infty )$
,
$E-\nu \le \frac 34 \left ( 1- \tfrac {1}{\Delta } \right )$
, and in this case,
$$\begin{align*}P_+^{S\cap K_i}R_{E-\nu}^{K_i} \mathcal{N}_x= P_+^{S\cap K_i}\widehat R_{E-\nu}^{K_i} \mathcal{N}_{x_i},\end{align*}$$
so it follows from (3.23) with
$k=0$
that
$$ \begin{align}\begin{aligned} \left\lVert Y_{x_i,\nu} \right\rVert \le C_0 \mathrm{e}^{-m_{0} d_\gamma}. \end{aligned}\end{align} $$
Using (4.37), (4.39), (4.40) and (3.27), we get
$$ \begin{align}\begin{aligned} \mathbb E\left\lVert Y_{x_i} \right\rVert^{s}& \le C\gamma^{\xi_{k-1}}\mathrm{e}^{-\frac {\theta_{k-1}}{3k} \gamma}\operatorname{\mathrm{tr}} \chi_{ [1- \frac{1}{\Delta},k(1- \frac{1}{\Delta}))}(H^{K_{i'}}) \le C_k\gamma^{\xi_{k-1}+2k}\,\mathrm{e}^{-\frac {\theta_{k-1}}{3k} \gamma}. \end{aligned}\end{align} $$
Combining (4.35) and (4.41) and recalling
$\left \lvert \partial _{in}^\Lambda K_i \right \rvert \le 4$
, we get
$$ \begin{align}\begin{aligned} \mathbb E\left\lVert Y \right\rVert^{s}\le C_k \left\lvert \gamma \right\rvert^{\xi_{k-1}+2k}\,\mathrm{e}^{-\frac {\theta_{k-1}}{3k} \gamma} \le C_k\mathrm{e}^{- \theta^{\prime}_{k-1} \gamma}. \end{aligned}\end{align} $$
Combining (4.33), (4.34) and (4.42), we get
$$ \begin{align}\begin{aligned} G^{\left(a,\gamma\right)}_3(r)\le C_k \left( \lambda \Delta^2 \right)^{-s}\mathrm{e}^{- \theta^\prime_{k-1} \gamma} \Big({ \left\lvert \Lambda \right\rvert^{2sk+2} \mathrm{e}^{-s \theta^\prime_{k-1} r}+ \sum_{p=0}^{r } \mathrm{e}^{-s \theta^\prime_{k-1} \left( r-p \right)}f_N^\Lambda(p) }\Big). \end{aligned}\end{align} $$
It follows from (4.32) and (4.43) that
$$ \begin{align}\begin{aligned} G_3(r) \le C_k \left( \lambda \Delta^2 \right)^{-s} \mathrm{e}^{-\widehat m N}\Big({ \left\lvert \Lambda \right\rvert^{2sk+2} \mathrm{e}^{-s\theta^\prime_{k-1}r}+ \sum_{p=0}^{r} \mathrm{e}^{-s \theta^\prime_{k-1} \left( r-p \right)}f_N^\Lambda(p) }\Big). \end{aligned}\end{align} $$
Putting together (4.7), (4.13), (4.30) and (4.44), we obtain (4.6).
We can now prove Lemma 4.1.
Proof of Lemma 4.1
For
$\Lambda \subset \mathbb Z$
finite,
$E\in I_{\le k}$
,
$N\in \mathbb N$
, and
$r\in {\mathbb {N}}^0$
, we set
$$ \begin{align}\begin{aligned} f^\Lambda_{N}(r)= f_N^{\Lambda}(k,E,r)=\max_{\Theta \subset\Lambda }\max_{j\in \Theta} \mathbb E\left( \left\lVert \chi_N^\Theta Q_{\le k}^{\Theta} \mathcal{N}_j R^{\Theta}_E P_+^{[j]^{\Theta}_r}Q_{\le k}^{\Theta} \right\rVert_{HS}^s \right). \end{aligned}\end{align} $$
Note that
$f^\Lambda _N(r)$
is monotone increasing in
$\Lambda $
, and it follows from (3.33) that
$$ \begin{align}\begin{aligned} \max_{r\in {\mathbb{N}}^0} f_N^\Lambda(r) \le C\lambda^{-s}k ^s\left\lvert \Lambda \right\rvert^{2sk+1}. \end{aligned}\end{align} $$
Moreover, if
$8kN \ge r$
, it follows from (3.49) that
$$ \begin{align}\begin{aligned} f^\Lambda_{N}(r)\le C_{k,s}\left\lvert \Lambda \right\rvert^{2(sk+1)} e^{-m_{0,\mu}r}. \end{aligned}\end{align} $$
If
$8kN < r$
, we use Lemma 4.2. Since this lemma holds for arbitrary finite subsets of
$\mathbb Z$
, it follows from (4.6) that for
$8kN < r$
, we have
$$ \begin{align} f_N^\Lambda(r) \le C_k \left( \left\lvert \Lambda \right\rvert^{\zeta_k} \mathrm{e}^{-m_k r} + \mathrm{e}^{- m_k N} \left( \lambda \Delta^2 \right)^{-s} \sum_{p=0}^{r} \mathrm{e}^{-m_k \left( r-p \right)} f_N^\Lambda(p) \right), \end{align} $$
for all
$\Lambda \subset \mathbb Z$
finite. Combining with (4.47), we get (with possibly slightly different constants C,
$m_k>0$
,
$\zeta _k>0$
)
$$ \begin{align}\begin{aligned} f^\Lambda(r) \le \sum_{N=1} ^{\left\lvert \Lambda \right\rvert} f_N^\Lambda(r) \le C\left( \left\lvert \Lambda \right\rvert^{\zeta_k} \mathrm{e}^{-m_k r} + \left( \lambda \Delta^2 \right)^{-s} \sum_{p=0}^{r} \mathrm{e}^{-m_k \left( r-p \right)} \right). \end{aligned}\end{align} $$
The proof can now be completed by a standard subharmonicity argument. Let
$h^\Lambda (r)= f^\Lambda (r) - 2C\left \lvert \Lambda \right \rvert ^{{\zeta _k}} \mathrm {e}^{-m_k \frac r2}$
, and take
$\Delta \ge \Delta _0$
and
$\lambda \ge \lambda _0$
such that
$$ \begin{align}\begin{aligned} 2C \left( \lambda \Delta^2 \right)^{-s} \sum_{q=-\infty}^\infty \mathrm{e}^{-m_k \frac {\left\lvert q \right\rvert} 2}\le1. \end{aligned}\end{align} $$
Then (4.49) implies that
$$ \begin{align}\begin{aligned} h^\Lambda(r) &\le C\left\lvert \Lambda \right\rvert^{{\zeta_k}} \mathrm{e}^{-m_k r}- 2C\left\lvert \Lambda \right\rvert^{{\zeta_k}} \mathrm{e}^{-m_k \frac r2}\\ & \qquad + C \left( \lambda \Delta^2 \right)^{-s} \sum_{p=0}^{r} \mathrm{e}^{-m_k \left( r-p \right)} \left( h^\Lambda(p)+ 2C\left\lvert \Lambda \right\rvert^{{\zeta_k}} \mathrm{e}^{-m_k \frac p2} \right)\\ & \le C\left\lvert \Lambda \right\rvert^{{\zeta_k}} \left( \mathrm{e}^{-m_k r}-\mathrm{e}^{-m_k \frac r2} \right)+C \left( \lambda \Delta^2 \right)^{-s} \sum_{p=0}^{r} \mathrm{e}^{-m_k \left( r-p \right)} h^\Lambda(p), \end{aligned}\end{align} $$
for all
$r\in {\mathbb {N}}^0$
. In addition, it follows from (4.46) that
$$ \begin{align}\begin{aligned} R=\sup_{r\in{\mathbb{N}}^0}h^\Lambda (r)\le \sup_{r\in{\mathbb{N}}^0}f^\Lambda (r)\le C\left\lvert \Lambda \right\rvert^{2sk+3}<\infty. \end{aligned}\end{align} $$
We claim that
$R\le 0$
, which implies that (4.4) holds (with different constants), finishing the proof of Lemma 4.1. Indeed, suppose that
$R>0$
. Then it follows from (4.51) and (4.50) that
$$ \begin{align}\begin{aligned} R\le C \left( \lambda \Delta^2 \right)^{-s} \sup_{r\in{\mathbb{N}}^0}\left( \sum_{p=0}^{\left\lvert \Lambda \right\rvert}\mathrm{e}^{-m_k\left\lvert r-p \right\rvert} \right) R \le C \left( \lambda \Delta^2 \right)^{-s} \left( \sum_{q=-\infty}^\infty \mathrm{e}^{-m_k\frac {\left\lvert q \right\rvert} 2} \right)R \le \tfrac 12 R, \end{aligned}\end{align} $$
a contradiction.
The proof of Theorem 2.4 is complete.
5 Quasi-locality in expectation
In this section, we prove Corollary 2.6. To do so, we first extract from Theorem 2.4 a probabilistic statement (cf. [Reference Elgart, Tautenhahn and Veselić20, Proposition 5.1] and [Reference Elgart, Klein and Stolz19, Lemma 7.2]).
We fix
$k\in \mathbb N$
and let
$s, \theta _k, \xi _k$
be as in (2.17), slightly modified so (2.17) holds with
$\rho ^\Lambda (A,B)$
substituted for
$\operatorname {\mathrm {dist}}_\Lambda (A,B^c)$
(recall (3.6)).
We fix a finite subset
$\Lambda $
of
$\mathbb Z$
. Given
$\emptyset \ne K\subset \Lambda $
, we let
$H^{K^\prime }$
be the restriction of
$H^K$
to
$\operatorname {\mathrm {Ran}} P_-^K=\operatorname {\mathrm {Ran}} \chi _{\mathbb N}(\mathcal {N}_K)$
,
$K^c=\Lambda \setminus K$
(we allow
$K^c=\emptyset $
), and consider
$H^{{K^\prime ,K^c}} =H^{K^\prime }+ H^{K^c} $
,
$ \Gamma ^{{K^\prime ,K^c}} =H^\Lambda -H^{{K^\prime ,K^c}} $
,
$R_E^{{K^\prime ,K^c}} =(H^{{K^\prime ,K^c}} -E)^{-1}$
, operators on
$ \operatorname {\mathrm {Ran}} P_-^K \oplus \mathcal {H}_{K^c}$
. Given an interval I and an operator H, we set
$\sigma _I(H)=\sigma (H)\cap I$
.
We start by proving Wegner-like estimates for the XXZ model.
Lemma 5.1. Let
$\emptyset \ne K\subset \Lambda $
.
-
1. Consider the open interval
$I\subset I_k$
. Then (5.1)
$$ \begin{align}\begin{aligned} \mathbb P_K \left\{ \sigma_I(H^{{K^\prime,K^c}} ) \ne \emptyset \right\} \le C_k \lambda^{-1} \left\lvert I \right\rvert \left\lvert \Lambda \right\rvert^{2k+1}. \end{aligned}\end{align} $$
-
2. Let
$0< \delta < \frac 1 4{\left ( 1- \frac {1}{\Delta } \right )}$
. Then (recall (2.14)) (5.2)
$$ \begin{align}\begin{aligned} \mathbb P\left\{ \operatorname{\mathrm{dist}} \left\{ \sigma_{\widehat I_k}(H^{{K^\prime,K^c} }), \sigma_{\widehat I_k} (H^{K^c}) \right\}<\delta \right\}\le C_k \lambda^{-1} \delta \left\lvert \Lambda \right\rvert^{4k+1}. \end{aligned}\end{align} $$
Proof. To prove Part (i), recall (3.27) (it applies to
$H^{\left ( K^\prime ,K^c \right )} $
), and let
$E_1 \le E_2 \le \ldots $
be the at most
$Ck\left \lvert \Lambda \right \rvert ^{2k}$
eigenvalues of
$H^{{K^\prime ,K^c} }$
in
$\widehat I_{\le k}$
, counted with multiplicity, which we consider as functions of
$ \omega _{K}$
for fixed
$\omega _{K^c}$
. Since
$\mathcal {N}_K \ge 1$
, each
$E_n( \omega _{K})$
is a monotone function on
$\mathbb R^{\left \lvert K \right \rvert }$
. Let
$e=(1,1,\ldots ,1)\in \mathbb R^{\left \lvert K \right \rvert }$
. We have
$E_n(\omega _{K}+te) - E_n(\omega _{K}) \ge \lambda t$
for all
$t>0$
and all n by the min-max principle, so we can apply Stollmann’s Lemma [Reference Stollmann46] to get
$$ \begin{align} \mathbb P_{K} \{E_n({\omega_K}) \in I \} \le C \left\lvert I \right\rvert \lambda^{-1} \left\lvert K \right\rvert. \end{align} $$
In view of (3.27), (5.1) follows using (5.3) for each one of the eigenvalues
$E_n$
.
Part (ii) follows from Part (i) and (3.27) for
$H^{K^c}$
, since the random variables
$\omega _{K}$
and
$\omega _{K^c}$
are independent.
Let
$E\in \mathbb R$
,
$m>0$
,
$ r\in \mathbb N$
,
$\emptyset \ne K\subset \Lambda $
, and let
$H^\sharp $
denote either
$H^K$
or
$H^{\left ( K^\prime ,K^c \right )}$
. Then the operator
$H^{K^\sharp }$
is said to be
$(m,E,r)$
-regular if
$$ \begin{align}\begin{aligned} &F^{K^\sharp}_E \le \mathrm{e}^{-mr} \quad\text{and}\quad \operatorname{\mathrm{dist}}(E,\sigma (H^{K^\sharp}))>\mathrm{e}^{-mr} ,\\ & \text{where}\quad F^{K^\sharp}_E= \max_{i\in K}F^{K^\sharp}_E(i) \quad\text{with}\quad F^{K^\sharp}_E (i)=\left\lVert \mathcal{N}_i R^{K^\sharp}_E P_+^{[i]_r^K} \right\rVert. \end{aligned}\end{align} $$
In addition, consider the probabilistic event
$$ \begin{align}\begin{aligned} \mathcal{F}_{k}^\Lambda (K,m,r)= \left\{ E\in I_k \implies \;\text{either}\; \ H^{\left( K^\prime,K^c \right)} \;\text{or }\; H^{{K^c}} \;\text{is}\; (m, E,r)\text{-regular} \right\}. \end{aligned}\end{align} $$
Lemma 5.2. Let
$\emptyset \ne K\subsetneq \Lambda $
, and let
$r\in \mathbb N$
,
$r\ge \frac {18} {\theta _k}$
. Then
$$ \begin{align}\begin{aligned} \mathbb P\left\{ \left( \mathcal{F}_{k}^\Lambda (K,\tfrac {\theta_k} 9,r) \right)^c \right\} \le C\left\lvert \Lambda \right\rvert^{\xi^\prime_k} \mathrm{e}^{- \frac {\theta_k} 9 r}. \end{aligned}\end{align} $$
Proof. Let
$\emptyset \ne K\subsetneq \Lambda $
,
$r\ge \tfrac {18} {{\theta _k}}$
, and set
$m= \frac {{\theta _k}} 9$
, so
$\mathrm {e}^{mr}\ge 4$
. Let S denote either the pair
${{K^\prime ,K^c}}$
or
${K^c}$
, and let
$S^\prime =K$
if
$S={{K^\prime ,K^c}} $
, or
$S^\prime =K^c$
if
$S= K^c$
. Consider the (random) energy sets
$$ \begin{align}\begin{aligned} D_S=\left\{ E\in I_k: \ F^{S}_E> \mathrm{e}^{-mr} \right\}\quad\text{and}\quad J_S=\left\{ E\in I_k: \ F^{S}_E> \mathrm{e}^{-2mr} \right\}, \end{aligned}\end{align} $$
and the event
$$ \begin{align}\begin{aligned} \mathcal{J}_S = \left\{ \left\lvert J_S \right\rvert> e^{-5mr} \right\}. \end{aligned}\end{align} $$
Using (2.17), we get
$$ \begin{align}\begin{aligned} \mathbb P\left\{ \mathcal{J}_S \right\} & \le \mathrm{e}^{5mr} \, \mathbb E \left\{ \left\lvert J_S \right\rvert \right\} \le \mathrm{e}^{5mr} \, \mathbb E \left\{ \int_{I_k} \mathrm{e}^{2s mr} \left( F^{S}_E \right)^s \ dE \right\}\\ & \le \mathrm{e}^{7 mr} \int_{I_k} \sum_{i \in S^\prime} \mathbb E\left\{ \left( F^{S}_E(i) \right)^s \right\}\ dE \le C_k \left\lvert \Lambda \right\rvert^{\xi_k+1} e^{-2mr}. \end{aligned}\end{align} $$
We now consider the (random) energy set
$$ \begin{align}\begin{aligned} Y_S= \left\{ E\in I_k: \ \operatorname{\mathrm{dist}}(E,\sigma (H^{S})) \le \mathrm{e}^{-mr} \right\} \end{aligned}\end{align} $$
and claim that
$D_S \subset Y_S$
on the complementary event
$\mathcal {J}_S^c= \left \{ \left \lvert J_S \right \rvert \le e^{-5mr} \right \} $
.
To see this, suppose
$\left \lvert J_S \right \rvert \le e^{-5mr}$
and
$E\in D_S\setminus Y_S$
. Since
$E\in D_S$
, there exists
$i\in S^\prime $
such that
$F_E^S(i)> e^{-mr}$
. Let
$E^\prime \in I_k$
such that
$\left \lvert E^\prime -E \right \rvert \le 2 e^{-5mr}$
. Using
$E \in Y_S$
, we get
$\operatorname {\mathrm {dist}}(E^\prime , \sigma (H^S)> e^{-mr}- 2 e^{-5mr}\ge \frac 1 2 e^{-mr}$
. Thus, using the resolvent identity and
$r\ge \frac {18} {\theta _k}$
, we have
$$ \begin{align}\begin{aligned} F_{E^\prime}^S(i)\ge F_{E}^S(i)-\left\lvert E^\prime-E \right\rvert\left\lVert R_E^S \right\rVert\left\lVert R_{E^\prime}^S \right\rVert> e^{-mr}- (2e^{-5mr}) e^{mr}(2 e^{mr})\ge e^{-2mr}. \end{aligned}\end{align} $$
It follows that
$[E- 2 e^{-5mr}, E+ 2 e^{-5mr}]\cap I_k \subset J_S$
. Since
$\left \lvert I_k \right \rvert \ge 2 e^{-5mr}$
as
$r\ge \frac {18} {\theta _k}$
, we conclude that
$\left \lvert J_S \right \rvert \ge 2 e^{-5mr}> e^{-5mr}$
, a contradiction.
We proved that
$\left \lvert J_S \right \rvert \le e^{-5mr} $
implies
$ D_S \subset Y_S$
, so
$\widehat Y_S= I_k\setminus Y_S\subset I_k\setminus D_S$
. In particular, outside the event
$\mathcal {J}_S$
,
$E\in \widehat Y_S$
implies that
$H^S$
is
$(m,E,r)$
-regular.
We now consider the event
$$ \begin{align}\begin{aligned} \mathcal{E}_K&= \left\{ I_k\setminus(\widehat Y_{{K^\prime,K^c}} \cup \widehat Y_{K^c})\ne \emptyset \right\}= \left\{ I_k \cap Y_{{K^\prime,K^c}}\cap Y_{K^c}\ne \emptyset \right\} \\ & \subset \left\{ \operatorname{\mathrm{dist}} \left\{ \sigma_{\widehat I_k}(H^{{{K^\prime,K^c}} }), \sigma_{\widehat I_k} (H^{K^c}) \right\}\le 2 e^{-mr} \right\} \end{aligned}\end{align} $$
and note that it follows from Lemma 5.1(ii) that
$$ \begin{align}\begin{aligned} \mathbb P\left\{ \mathcal{E}_K \right\} \le C_k \left\lvert \Lambda \right\rvert^{4k+1} e^{-mr}. \end{aligned}\end{align} $$
Since
$$ \begin{align}\begin{aligned} \mathbb P\left\{ \mathcal{E}_K \cup \mathcal{J}_{{K^\prime,K^c}} \cup \mathcal{J}_{K^c} \right\} \le C_k \left\lvert \Lambda \right\rvert^{4k+1} e^{-mr}+ 2C_k \left\lvert \Lambda \right\rvert^{\xi_k+1} e^{-2mr} \le C\left\lvert \Lambda \right\rvert^{\xi^\prime_k} \mathrm{e}^{-m r}, \end{aligned}\end{align} $$
and on the complementary event, we have
$I_k= \widehat Y_{{K^\prime ,K^c}} \cup \widehat Y_{K^c}$
, so for
$E\in I_k$
, either
$H^{{K^\prime ,K^c}}$
or
$H^{K^c}$
is
$(m,E,r)$
-regular, the lemma is proved.
Proof of Corollary 2.6
Let
$A\subset B\subset \Lambda $
, A connected in
$\Lambda $
, let
$r=\rho ^\Lambda (A,B)$
, and recall
$\left \lVert P_-^{A}f(H^\Lambda ) P_+^{B} \right \rVert \le \left \lVert P_-^{A}f(H^\Lambda ) P_+^{[A]^\Lambda _r} \right \rVert $
.
We set
$$ \begin{align}\begin{aligned} \Theta^\Lambda(A,r)& =\sup_{\substack{f\in B(I_{\le k}):\\\|f\|_\infty\le1}}\left\lVert P_-^{A}f(H^\Lambda) P_+^{[A]^\Lambda_r} \right\rVert\le 1. \end{aligned}\end{align} $$
To estimate
$\mathbb E\left \{ \Theta ^\Lambda (A,r) \right \}$
, note that
$$ \begin{align}\begin{aligned} \Theta^{\Lambda}(A,r) \le \sum_{E\in \sigma_{I_k}(H^{\Lambda})} \left\lVert {P_-^{A}} P_{\left\{ E \right\}}P_+^{[A]^\Lambda_r} \right\rVert, \quad\text{where}\quad P_{\left\{ E \right\}}= \chi_{\left\{ E \right\}}(H^\Lambda). \end{aligned}\end{align} $$
The spectrum of
$H^\Lambda $
is simple almost surely, as commented in [Reference Elgart, Klein and Stolz19, Section 3], so we assume this on what follows for simplicity. (Otherwise, we just need to label the eigenvalues taking into account multiplicity.) For
$E\in \sigma (H^\Lambda )$
, we let
$\phi _E$
denote the corresponding eigenfunction, and let
$N_E\in {\mathbb {N}}^0$
be given by
$\mathcal {N}_\Lambda \phi _E= N_E \phi _E$
.
For
$E\in I_k$
, we have
$$ \begin{align}\begin{aligned} P_{\left\{ E \right\}}&=\widehat R^{\Lambda}_{k,E}\left( \widehat H^\Lambda_k-E \right)P_{\left\{ E \right\}}=\widehat R^{\Lambda}_{k,E}\left( \widehat H^\Lambda_k-H^\Lambda+(H^\Lambda -E) \right)P_{\left\{ E \right\}}\\&=k\left( 1- \tfrac{1}{\Delta} \right)\widehat R^{\Lambda}_{k,E}\,Q_{\le k}^{\Lambda}P_{\left\{ E \right\}}. \end{aligned}\end{align} $$
Let
$r \ge R_k = 6k(\lceil \frac {18} {\theta _k} \rceil +2)$
. Using (A.7) and (3.23), we obtain
$$ \begin{align}\begin{aligned} {\left\lVert {P_-^{A}}P_{\left\{ E \right\}} P_+^{[A]^\Lambda_r} \right\rVert}&=k \left( 1- \tfrac{1}{\Delta} \right) {\left\lVert {P_-^{A}}\widehat R^{\Lambda}_{k,E}\,Q_{\le k}^{\Lambda}P_{\left\{ E \right\}} P_+^{[A]^\Lambda_r} \right\rVert}\\ & =k \left( 1- \tfrac{1}{\Delta} \right) {\left\lVert {P_-^{A}}\widehat R^{\Lambda}_{k,E}P_-^{[A]_\infty^\Lambda}\,Q_{\le k}^{\Lambda}P_{\left\{ E \right\}} P_+^{[A]^\Lambda_r} \right\rVert} \\ &\le k { \sum_{q=-\left\lvert A \right\rvert}^{\left\lvert \Lambda \right\rvert} } {\left\lVert {P_-^{A}}\widehat R^{\Lambda}_{k,E}P_+^{[A]^\Lambda_q} \right\rVert}{\left\lVert P_-^{]A[^\Lambda_q} Q_{\le k}^{\Lambda}P_{\left\{ E \right\}}P_+^{[A]^\Lambda_r} \right\rVert} \\ &\le C_0{ \sum_{q=-\left\lvert A \right\rvert}^{\left\lvert \Lambda \right\rvert} } \mathrm{e}^{-m_0(q)_+}{\left\lVert P_-^{]A[^\Lambda_q} Q_{\le k}^{\Lambda}P_{\left\{ E \right\}}P_+^{[A]^\Lambda_r} \right\rVert} \\ &\le 2 C_0{ \sum_{q=-\left\lvert A \right\rvert}^{r-1 - R_k}} \mathrm{e}^{-m_0(q)_+} \sum_{u\in ]A[^\Lambda_q} {\left\lVert Q_{\le k}^{\Lambda}\mathcal{N}_uP_{\left\{ E \right\}}P_+^{[u]^\Lambda_{r-q-1}} \right\rVert} + C_k\left\lvert \Lambda \right\rvert \mathrm{e}^{-m_0 r}. \end{aligned}\end{align} $$
Let
$u\in \Lambda $
and
$p\ge R_k$
. If
$8kN_E \ge p$
, it follows from (3.53)–(3.55) that
$$ \begin{align}\begin{aligned} &\left\lVert \chi_{N_E}^\Lambda Q_{\le k}^{\Lambda}\mathcal{N}_uP_{\left\{ E \right\}}P_+^{[u]^\Lambda_{p}} \right\rVert\le \chi_{{{\mathcal B_k^{N_E}}}},\\ &\mathbb P_\Lambda \left( {\mathcal B_k^{N_E}} \right) \le C_k \left\lvert \Lambda \right\rvert^{2k}e^{- c_\mu {N_E}}\le C_k \left\lvert \Lambda \right\rvert^{2k}\mathrm{e}^{- \frac {c_\mu}{8k}p}. \end{aligned}\end{align} $$
If
$p> 8k{N_E}$
, we set (cf. (4.14))
$$ \begin{align}\begin{aligned} K(0)&= [u]^\Lambda _{\frac {3p}4} \quad\text{and}\quad K(a)=[ u]^\Lambda_{a \left\lfloor \frac p {6k}\right \rfloor} \; \ \text{for}\; \ a=1,2,\ldots, 3k-1,\\ S(a)& =[\partial^\Lambda K(a)]^\Lambda_{\left\lfloor \frac p {6k}\right \rfloor-1} \quad\text{for}\quad a=0,1,\ldots, 3k-1. \end{aligned}\end{align} $$
Using Lemma 3.8, we get
$$ \begin{align}\begin{aligned} {\Big\|{ \chi^\Lambda_{N_E} Q_{\le k}^{\Lambda}\mathcal{N}_uP_{\left\{ E \right\}}P_+^{[u]^\Lambda_{p}}}\Big\|} \le \sum_{a=0}^{3k-1} {\Big\| \chi_{N_E}^\Lambda Q_{\le k}^{\Lambda}\mathcal{N}_u Y(a) P_{\left\{ E \right\}}P_+^{[u]^\Lambda_{p}}\Big\|}, \end{aligned}\end{align} $$
where
$Y(0)=P_+^{ \Lambda \setminus [u]^\Lambda _{\frac p2} } $
and
$Y(a)=P_+^{S(a)}P_-^{K(a)}P_-^{K^c(a)}$
for
$a>0$
.
We now consider the event (see (5.5))
$$ \begin{align}\begin{aligned} & \mathcal{J}_k (u,p) =\bigcap_{a=0}^{3k-1} \mathcal{F}_{k}^\Lambda (K(a), {\widehat {\theta_k}} ,\widehat p), \;\text{where}\; \widehat {\theta_k}=\tfrac {{\theta_k}} 9\;\text{and}\;\widehat p= \left\lfloor \tfrac p {6k}\right \rfloor-1\ge \tfrac {18} {{\theta_k}} \end{aligned}\end{align} $$
and note that it follows from Lemma 5.2 that
$$ \begin{align}\begin{aligned} \mathbb P\left\{ \left( \mathcal{J}_k (u,p) \right)^c \right\}\le 3k C\left\lvert \Lambda \right\rvert^{\xi^\prime_k} \mathrm{e}^{-\widehat {\theta_k}\widehat p}. \end{aligned}\end{align} $$
For
$\omega \in \mathcal {J}_k (u,p) $
and
$a\in \left \{ 0,1,\ldots , 3k-1 \right \}$
, either
$H^{\left ( K(a), K^c(a) \right ) }$
or
$H^{{K^c}(a)}$
is
$({\widehat {\theta _k}} ,E,\widehat p)$
-regular (
${K^c}(a)=\left ( K(a) \right )^c$
). If
$H^{{K^c}(a)}$
is
$({\widehat {\theta _k}} ,E,\widehat p)$
-regular, we note that
$$ \begin{align}\begin{aligned} P_{\left\{ E \right\}}P_+^{[u]^\Lambda_{p}} &=P_{\left\{ E \right\}}\left( H^{K(a)}+H^{{K^c}(a)}-E \right) R^{{K^c}(a)}_EP_+^{[u]^\Lambda_{p}}\\ & = -P_{\left\{ E \right\}}\Gamma^{\left( K(a),K^c(a) \right)}P_-^{\partial^\Lambda_{ex} K(a)}P_+^{K(a)} R^{{K^c}(a)}_EP_+^{[u]^\Lambda_{p}} , \end{aligned}\end{align} $$
where we have used
$R^{{K^c}(a)}_EP_+^{[u]^\Lambda _{p}}=P_+^{K(a)}R^{{K^c}(a)}_EP_+^{[u]^\Lambda _{p}}$
due to
$K(a)\subset [u]^\Lambda _{p}$
. We deduce that
$$ \begin{align}\begin{aligned} {\left\lVert \chi_{N_E}^\Lambda Q_{\le k}^{\Lambda}\mathcal{N}_u Y(a) P_{\left\{ E \right\}}P_+^{[u]^\Lambda_{p}} \right\rVert}\le \left\lVert P_{\left\{ E \right\}}P_+^{[u]^\Lambda_{p}} \right\rVert\le \tfrac 1 \Delta \left\lVert P_-^{\partial_{ex}^\Lambda K(a)} R^{{K^c}(a)}_EP_+^{[u]^\Lambda_{p}\cap ({K^c}(a))} \right\rVert\le \tfrac {2\mathrm{e}^{-\widehat {\theta_k}\widehat p}} \Delta , \end{aligned}\end{align} $$
using (A.3), (5.4) and the definition of
$K(a)$
. If
$H^{\left ( K(a),K^c(a) \right )}$
is
$({\widehat {\theta _k}} ,E,\widehat p)$
-regular, we use
$$ \begin{align}\begin{aligned} \mathcal{N}_{u}P_{\left\{ E \right\}}P_+^{[u]^\Lambda_{p}} &=\mathcal{N}_{u}R^{\left( K(a),K^c(a) \right)}_E\left( H^{\left( K(a),K^c(a) \right)}-E \right)P_{\left\{ E \right\}}P_+^{[u]^\Lambda_{p}} \\ &=-\mathcal{N}_{u}R^{\left( K(a),K^c(a) \right)}_E P_-^{\partial^\Lambda K(a)}\Gamma^{\left( K(a),K^c(a) \right)}P_{\left\{ E \right\}}P_+^{[u]^\Lambda_{p}}. \end{aligned}\end{align} $$
Thus,
$$ \begin{align}\begin{aligned} {\left\lVert \chi_{N_E}^\Lambda Q_{\le k}^{\Lambda}\mathcal{N}_u Y(a) P_{\left\{ E \right\}}P_+^{[u]^\Lambda_{p}} \right\rVert}&\le \left\lVert \mathcal{N}_{u} Y(a)P_{\left\{ E \right\}}P_+^{[u]^\Lambda_{p}} \right\rVert\le \tfrac 1 \Delta \left\lVert \mathcal{N}_{u} Y(a)R^{\left( K(a),K^c(a) \right)}_E P_-^{\partial^\Lambda K(a)} \right\rVert\\ &\le \tfrac 1 \Delta \left\lVert P_+^{S(a)}R^{\left( K(a),K^c(a) \right)}_E P_-^{\partial^\Lambda K(a)} \right\rVert\le \tfrac 2 \Delta \mathrm{e}^{-\widehat {\theta_k}\widehat p}, \end{aligned}\end{align} $$
using (A.3), (5.4) and the definition of
$S(a)$
.
Combining (5.21), (5.25) and (5.27), we conclude that for
$p>8kN_E$
and
$\omega \in \mathcal {J}_k (u,p) $
, we have
$$ \begin{align}\begin{aligned} {\left\lVert \chi^\Lambda_{N_E} Q_{\le k}^{\Lambda}\mathcal{N}_uP_{\left\{ E \right\}}P_+^{[u]^\Lambda_{p}} \right\rVert} \le \tfrac {12k} \Delta \mathrm{e}^{-\widehat {\theta_k}\widehat p}. \end{aligned}\end{align} $$
Since
$ {\left \lVert \chi ^\Lambda _{N_E} Q_{\le k}^{\Lambda }\mathcal {N}_uP_{\left \{ E \right \}}P_+^{[u]^\Lambda _{p}} \right \rVert } \le 1$
, it follows that for
$p>8kN_E $
, we have
$$ \begin{align}\begin{aligned} {\left\lVert \chi^\Lambda_{N_E} Q_{\le k}^{\Lambda}\mathcal{N}_uP_{\left\{ E \right\}}P_+^{[u]^\Lambda_{p}} \right\rVert} \le \tfrac {12k} \Delta \mathrm{e}^{-\widehat {\theta_k}\widehat p} + \chi_{ {\mathcal{J}_k (u,p)}^c}. \end{aligned}\end{align} $$
It follows that for
$u\in \Lambda $
and
$p\ge R_k$
, using (5.19), (5.29) and (3.27), we conclude that
$$ \begin{align}\begin{aligned} \mathbb E \Big({ \sum_{E\in \sigma_{I_k}(H^{\Lambda})}\Big\|{ Q_{\le k}^{\Lambda}\mathcal{N}_uP_{\left\{ E \right\}}P_+^{[u]^\Lambda_{p}}} }\Big\|\Big) \le C_k \left\lvert \Lambda \right\rvert^{\xi^{\prime}_k}\mathrm{e}^{- \theta^\prime_k p}. \end{aligned}\end{align} $$
Combining with (5.16), (5.18) and (3.27), we obtain
$$ \begin{align}\begin{aligned} \mathbb E\left\{ \Theta^\Lambda (A,r) \right\}\le C_k \left\lvert \Lambda \right\rvert^{ \xi_k^\prime} \mathrm{e}^{- {\theta_k^\prime} r}. \end{aligned}\end{align} $$
The estimate (5.31) holds for
$r\ge R_k$
. Since
$\mathbb E\left \{ \Theta ^\Lambda (A,r) \right \}\le 1$
for all
$r\ge 0$
, it holds for all
$r\ge 0$
if the constant
$C_k$
is replaced by the constant
$\widetilde C_k= C_k \mathrm {e}^{ {\theta _k^\prime } \widehat R_k}$
.
A Useful identities
In this appendix, we list some useful identities. Their derivations are straightforward, so we leave out the proofs.
We fix
$\Lambda \subset \mathbb Z$
finite,
-
○ For all
$i,j\in \Lambda $
, we have (recall (2.15)) (A.1)
$$ \begin{align}\begin{aligned} P_-^{\left\{ i \right\}}&=\mathcal{N}_i, \\ P_-^{\left\{ i,j \right\}}&=\mathcal{N}_i +\mathcal{N}_j -\mathcal{N}_i \mathcal{N}_j= \mathcal{N}_i\left( 1-\mathcal{N}_j \right) + \mathcal{N}_j= P_+^{\left(\left\{ j \right\}\right)}\mathcal{N}_i+ \mathcal{N}_j. \end{aligned}\end{align} $$
-
○ Consider the self-adjoint operator
${h}_{i,i+1}$
(recall (2.8)) on the four-dimensional Hilbert space
$\mathcal {H}_{\left \{ i,i+1 \right \}}=\mathbb C_i^2\otimes \mathbb C_{i+1}^2$
. An explicit calculation shows that
${h}_{i,i+1}$
has eigenvalues
$-1,0, \pm \frac 1 \Delta $
. It follows that if
$\left \{ i,i+1 \right \}\subset \Lambda $
, we have (A.2)
$$ \begin{align} \left\lVert {h}_{i,i+1} \right\rVert=1 \quad\text{on}\quad \mathcal{H}_\Lambda. \end{align} $$
-
○ The following identities hold on
$\mathcal {H}_\Lambda $
for
$\left \{ i,i+1 \right \}\subset \Lambda $
: (A.3)In particular, the first identity above implies
$$ \begin{align}\begin{aligned} &{h}_{i,i+1}P_+^{\left\{ i,i+1 \right\}}=P_+^{\left\{ i,i+1 \right\}}{h}_{i,i+1}=0,\\ &\left\lVert P_+^{\left\{ i \right\}}{h}_{i,i+1} \right\rVert=\left\lVert P_+^{\left\{ i+1 \right\}}{h}_{i,i+1} \right\rVert={ \tfrac 1{2\Delta}},\\ &P_+^{\left(\left\{ i \right\}\right)}{h}_{i,i+1}P_+^{\left(\left\{ i \right\}\right)}=P_+^{\left(\left\{ i+1 \right\}\right)}{h}_{i,i+1}P_+^{\left(\left\{ i+1 \right\}\right)}=0,\\ &{h}_{i,i+1} \mathcal{N}_i\mathcal{N}_{i+1} = \mathcal{N}_i\mathcal{N}_{i+1} {h}_{i,i+1}= \mathcal{N}_i\mathcal{N}_{i+1}{h}_{i,i+1} \mathcal{N}_i\mathcal{N}_{i+1}. \end{aligned}\end{align} $$
(A.4)
$$ \begin{align} {h}_{i,i+1}= {h}_{i,i+1}P_-^{\left\{ i,i+1 \right\}}=P_-^{\left\{ i,i+1 \right\}}{h}_{i,i+1}=P_-^{\left\{ i,i+1 \right\}}{h}_{i,i+1}P_-^{\left\{ i,i+1 \right\}}. \end{align} $$
-
○ Let
$K\subset \Lambda $
, and recall (3.10). It follows from (A.4) that (A.5)If K is connected in
$$ \begin{align} \Gamma^K= P_-^{ \partial^\Lambda K} \Gamma^K P_-^{ \partial^\Lambda K}. \end{align} $$
$\Lambda $
, it follows from (A.5) that (A.6)
$$ \begin{align} \left\lVert P_+^{K} \Gamma^K \right\rVert\le \tfrac 1 \Delta \quad\text{and}\quad \left\lVert P_+^{K^c} \Gamma^K \right\rVert\le \tfrac 1 \Delta. \end{align} $$
-
○ The following identities hold for any nonempty
$M\subset \Lambda $
(recall (2.16)): (A.7)
$$ \begin{align}\begin{aligned} P_-^{[M]_\infty} P_+^M & = \sum_{q=0}^{\left\lvert \Lambda \right\rvert} P_+^{[M]^\Lambda_{q}}P_-^{]M[^\Lambda_q}= \sum_{q=0}^{\left\lvert \Lambda \right\rvert} P_+^{[M]^\Lambda_{q}} P_-^{{\partial_{ex}^\Lambda{[M]_{q}}}} ,\\ P_-^{ M}&= \sum_{q=-\left\lvert M \right\rvert}^{-1}P_+^{[M]^\Lambda_{q}} P_-^{]M[^\Lambda_q}=\sum_{q=-\left\lvert M \right\rvert}^{-1}P_+^{[M]^\Lambda_{q}} P_-^{{\partial_{in }^\Lambda{[M]_{q+1}}}} , \\P_-^{[M]^\Lambda_\infty} &= \sum_{q=-\left\lvert M \right\rvert}^{\left\lvert \Lambda \right\rvert}P_+^{[M]^\Lambda_{q}} P_-^{]M[^\Lambda_{q}}. \end{aligned}\end{align} $$
B Many-body quasi-locality
In this appendix, we prove (1.6). Recall we only consider finite subsets of
$\mathbb Z$
. We fix
$\Lambda \subset \mathbb Z$
and consider the Hilbert space
$\mathcal {H}_\Lambda $
.
Lemma B.1. Suppose that
$H\in \mathcal A_{\Lambda }$
satisfies
-
1. For all
$K \subset {\Lambda } $
, we have
$[P_-^{K},H]P_+^{[K]_1^{\Lambda }}=0$
. -
2. For all connected
$K \subset {\Lambda } $
, we have
$\left \lVert [P_-^{K},H] \right \rVert \le \gamma $
.
Then for all
$A\subset B\subset {\Lambda }$
, A connected in
${\Lambda }$
, we have
$$ \begin{align}\begin{aligned} \left\lVert P_-^{A}\,e^{itH}\,P_+^{B} \right\rVert\le \gamma^{r}\frac{ \left\lvert t \right\rvert^r}{r!}, \quad\text{where}\quad r= \operatorname{\mathrm{dist}}_{\Lambda} \left( A,B^c \right)\ge 1. \end{aligned}\end{align} $$
Proof. We note that
$[ A]^{\Lambda }_{s} \subset B$
for
$s=0,1,\ldots ,r-1$
. We have
$$ \begin{align}\begin{aligned} P_-^{ A}\,e^{itH}\,P_+^{ B} =ie^{itH}\int_0^tK(s)\,P_+^{B}ds, \end{aligned}\end{align} $$
where
$K(s)=e^{-isH}\,[P_-^{ A},H]\,e^{isH}$
. If
$r\ge 2$
, condition (i) of the Lemma yields
$K(s)=e^{-isH}\,[P_-^{ A},H] P_-^{ [A]^{\Lambda }_1}\,e^{isH}$
. Proceeding recursively, we get
$$ \begin{align}\begin{aligned} P_-^{ A}\,e^{itH}\,P_+^{ B}&=i^{r}\int_0^t\int_0^{s_1}\ldots\int_0^{s_{r-1}}\prod_{j=1}^{r} K_{j-1}(s_j)ds_j \,P_+^{B},\\ K_j(s)&=e^{-isH}\,[P_-^{ [A]_j},H] \,e^{isH}. \end{aligned}\end{align} $$
Using assumption (ii), we get
$$ \begin{align}\begin{aligned} \left\lVert P_-^{ A}\,e^{itH}\,P_+^{ B} \right\rVert\le \gamma^{r}\frac{ \left\lvert t \right\rvert^r}{r!}. \end{aligned}\end{align} $$
Lemma B.2. Let
$f\in C_0^n$
(i.e., f is compactly supported and n times differentiable function on
$\mathbb R$
(with
$n\ge 2$
)). Then for
$A,B,H$
as in Lemma B.1 and
$r=\operatorname {\mathrm {dist}}_\Lambda \left ( A,B^c \right )$
, we have
$$ \begin{align}\begin{aligned} \left\lVert P_-^A\ f(H)\, P_+^B \right\rVert\le \widetilde C(f,n)r^{-(n-1)\min(1,\frac r n)} \le \widetilde C(f,n)r^{-n}. \end{aligned}\end{align} $$
Proof. Let
$\hat f$
denote the Fourier transform of f. Then we have
$\left \lvert \hat f(t) \right \rvert \le C(f,n)\langle t\rangle ^{-n}$
for
$t\in \mathbb R$
(we recall that
$\langle t\rangle :=\sqrt {1+t^2}$
). We can bound
$$ \begin{align}\begin{aligned} \left\lVert P_-^A\ f(H)\, P_+^B \right\rVert\le \int_{\mathcal R}\left\lVert P_-^A\ e^{itH}\, P_+^B \right\rVert\left\lvert \hat f(t) \right\rvert dt+\int_{\mathcal R^c}\left\lvert \hat f(t) \right\rvert dt, \end{aligned}\end{align} $$
where
$\mathcal R:=[-R,R]$
, where
$R>0$
will be chosen later.
We can bound the first integral on the right-hand side of (B.6) using (B.1) as
$$ \begin{align}\begin{aligned} \int_{\mathcal R}\left\lVert P_-^A\ e^{itH}\, P_+^B \right\rVert\left\lvert \hat f(t) \right\rvert dt & \le C(f,n)\frac{\gamma^{r}}{r!}\int_{\mathcal R} \left\lvert t \right\rvert^r\langle t\rangle^{-n}dt\le C_n C(f,n)\frac{\gamma^{r}{R}^{1+(r-n)_+}}{r!}\\ & \le C'_n C(f,n)\left( \frac {\mathrm{e} \gamma }r \right)^r{R}^{1+(r-n)_+}, \end{aligned}\end{align} $$
where we used
$r!\ge e^{1-r}r^r$
.
We can bound the second integral in (B.6) as
$$ \begin{align}\begin{aligned} \int_{\mathcal R^c}\left\lvert \hat f(t) \right\rvert dt\le C(f,n)\int_{\mathcal R^c} \langle t\rangle^{-n}dt\le C_n C(f,n)\left( 1+R \right)^{1-n}\le C_n C(f,n){R}^{1-n}. \end{aligned}\end{align} $$
Choosing
$R=\left ( \frac r{e\gamma } \right )^{\frac {r}{n+ (r-n)_+}}$
, we get (B.5).
Competing interest
The authors have no competing interest to declare.
Funding statement
A.E. was supported in part by the NSF under grants DMS-1907435 and DMS-2307093 and by the Simons Fellowship in Mathematics Grant 522404.
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