2.1 The index of a pair of pure states

Let $\mathcal {A}$ be a unital C*-algebra. In later sections we shall apply the theory to $\mathcal {A}=\operatorname {CAR}(\mathcal {H})$ for some separable Hilbert space $\mathcal {H}$ but in fact throughout this section no assumptions on $\mathcal {A}$ will be made. We denote by $\mathcal {U}(\mathcal {A})$ the set of unitary elements of $\mathcal {A}$ . A state $\omega : {\mathcal A} \to \mathbb C$ is a positive (i.e., $\omega (a^{*} a)\geq 0$ for all $a\in \mathcal {A}$ ) linear functional on ${\mathcal A}$ with ; the space of all states shall be denoted $\mathcal S({\mathcal A})$ . The condition is equivalent to the normalization $\|\omega \| =1$ , where the norm is given by

$$ \begin{align*} \left\Vert \omega \right\Vert := \sup\left(\{ \left\lvert {\omega(a)} \right\rvert | a\in\mathcal{A} : \left\Vert a \right\Vert\leq 1\}\right). \end{align*} $$

In other words, all states are on the unit sphere of ${{\mathcal A}}^{*}$ and so $\mathcal S({\mathcal A})$ is a weakly*-compact and convex subset of ${\mathcal A}^{*}$ . The extreme points of $\mathcal S({\mathcal A})$ are called pure states and their collection is denoted by ${\mathcal P}({{\mathcal A}})$ .

Definition 2.1 (locally comparable pair of pure states).

Let $\omega _1,\omega _2\in \mathcal {P}(\mathcal {A})$ be a pair of pure states. We say that the pair $(\omega _1,\omega _2)$ is locally comparable iff there exists some $u\in \mathcal {U}(\mathcal {A})$ such that

(2) $$ \begin{align} \omega_2 = \omega_1 \circ \operatorname{Ad}_{u}\!. \end{align} $$

Here and in the sequel, $\operatorname {Ad}_{u}$ denotes the automorphism $\operatorname {Ad}_{u}(a) = u^{*} a u$ . In other words, $\omega _1$ and $\omega _2$ are locally comparable iff they are inner-automorphism equivalent.

Remark 2.2. The intuition we have in mind for two pure states to be locally comparable is that their difference is “compact” in a vague sense. Since

$$ \begin{align*} \omega_2 -\omega_1 = \omega_1\circ \left(\operatorname{Ad}_{u}-\mathrm{id}\right), \end{align*} $$

if $\mathcal {A}=\operatorname {CAR}(\mathcal {H})$ for some Hilbert space $\mathcal {H}$ , then $\mathcal {A}$ may be considered as the operator norm limit of finite-rank (but not necessarily quadratic) observables, and as such, since $u\in \mathcal {U}(\mathcal {A})$ , we should consider and hence also $\operatorname {Ad}_{u}-\mathrm {id}$ to be (the many-body analog of) compact. Then by the ideal property, the whole expression $\omega _1\circ \left (\operatorname {Ad}_{u}-\mathrm {id}\right )$ is.

Back to the case of a general C*-algebra $\mathcal {A}$ , when two pure states $\omega _1,\omega _2$ are locally comparable, we want to measure how different they are, with the expectation that if they are path-connected (in $\mathcal {P}(\mathcal {A})$ ) they should not be different at all. This is the case when $u\in \mathcal {U}_0(\mathcal {A})$ , that is, when u may be continuously deformed to , whence we get a continuous deformation of $\omega _2$ to $\omega _1$ . However, there are C*-algebras where $\pi _0(\mathcal {U}(\mathcal {A}))\neq \{0\}$ (for example $C({\mathbb S}^1)$ ). It will turn out that we are not quite interested in such obstructions. Rather, in many interesting cases (such as UHF algebras), $\pi _0(\mathcal {U}(\mathcal {A}))=\{0\}$ . Then, to gauge a topological obstruction we impose a further symmetry constraint.

Let $\{\rho _t\}_{t\in \mathbb {R}}\subseteq \operatorname {Aut}(\mathcal {A})$ be a strongly continuous one-parameter group of *-automorphisms. Its generator $\delta ^\rho $ , given by

$$ \begin{align*} \delta^\rho := \mathrm{s-lim}_{t\to0^+}\frac{1}{t}\left(\rho_t-\mathrm{id}\right) \end{align*} $$

is a *-derivation. In general $\rho _t$ is not expected to be inner and $\delta ^\rho $ is not expected to be bounded. As such, we must consider $\delta ^\rho $ together with its domain

$$ \begin{align*} \mathcal{D}(\delta^\rho) \equiv \{a\in\mathcal{A} | \lim_{t\to0^+}\frac{1}{t}\left(\rho_t(a)-a\right)\text{ exists in }\mathcal{A}\}. \end{align*} $$

With this, we refine the notion of locally comparable pure states as follows.

Definition 2.3 ( $\rho $ -locally comparable pair of pure states).

Let $\rho _t$ be as above and $(\omega _1,\omega _2)$ be a pair of pure states. We say that this pair is $\rho $ -locally comparable iff there exists some $u\in \mathcal {U}(\mathcal {A})\cap \mathcal {D}(\delta ^\rho )$ such that

$$ \begin{align*} \omega_2 = \omega_1 \circ \operatorname{Ad}_{u}\!. \end{align*} $$

We now have all the definitions to set up our index.

Definition 2.4 (The index of a locally comparable pair of $\rho $ -invariant pure states).

Let $\rho _t$ be as above and $(\omega _1,\omega _2)$ be a pair of $\rho $ -locally comparable pure states. Assume moreover that both $\omega _1$ and $\omega _2$ are $\rho $ -invariant, that is,

$$ \begin{align*} \omega_j \circ \rho_t = \omega_j \qquad (t\in\mathbb{R},j=1,2). \end{align*} $$

Then the index of the pair $(\omega _1,\omega _2)$ is defined as

(3) $$ \begin{align} \mathcal{N}_\rho(\omega_1,\omega_2) := \operatorname{i}\omega_1\left(u^\ast \delta^\rho (u)\right) \end{align} $$

where $u\in \mathcal {D}(\delta ^\rho )$ is any unitary which obeys $\omega _2=\omega _1\circ \operatorname {Ad}_{u}$ .

We note that in general there is no reason for the unitary u above to be $\rho _t$ -invariant; see Remark 3.22(iii) for an explicit example. Indeed, if this happens, then $\delta ^\rho (u) = 0$ and so $\mathcal {N}_\rho (\omega _1,\omega _2)=0$ .

The condition that the pair $(\omega _1,\omega _2)$ must be compatible with $\rho $ in the above sense, namely that they are $\rho $ -locally comparable and invariant, means they are generically not path connected, which is what the index measures.

To clarify further, let us assume temporarily that $\mathcal {A}$ is a uniformly hyperfinite (UHF) algebra. In this case, it is well known (see, e.g., [Reference Bratteli and William Robinson19, Prop 3.2.52]) that one may find a sequence $\{q_n=q_n^\ast \}_n\subseteq \mathcal {A}$ such that

$$ \begin{align*} \delta^\rho = \mathrm{s-lim}_{n\to\infty}\operatorname{i}[q_n,\cdot]. \end{align*} $$

Then it is clear that

(9) $$ \begin{align} \mathcal{N}_\rho(\omega_1,\omega_2) = \lim_{n}\left(\omega_1(q_n)-\omega_2(q_n)\right). \end{align} $$

Since this expression does not involve the unitary u the index is well-defined. The remainder of this subsection does not assume that $\mathcal {A}$ is UHF.

Remark 2.6. If $(\omega _1,\omega _2)$ are $\rho $ -locally comparable and if $\tilde \omega _2 = \omega _2\circ \mathrm {Ad}_v$ for a $v\in {\mathcal U}({{\mathcal A}})$ with $\delta ^\rho (v)=0$ , then $(\omega _1,\tilde \omega _2)$ are also $\rho $ -locally comparable and

(10) $$ \begin{align} {\mathcal N}_\rho(\omega_1,\tilde\omega_2 ) = {\mathcal N}_\rho(\omega_1,\omega_2) \end{align} $$

by additivity (6) and ${\mathcal N}_\rho (\tilde \omega _2, \omega _2) = 0$ .

Proof of Theorem 2.5.

(i) If we have $\omega _2 = \omega _1\circ \operatorname {Ad}_{u} = \omega _1\circ \operatorname {Ad}_{v}$ for two a priori distinct unitaries $u,v\in \mathcal {U}(\mathcal {A})$ then this readily implies $\omega _1 = \omega _1 \circ \operatorname {Ad}_{u^\ast v}$ . Let us define $w := u^\ast v$ whence $\omega _1= \omega _1\circ \operatorname {Ad}_{w}$ . Since $v = uw $ , we have

$$ \begin{align*} v^\ast \delta^\rho(v) = (uw)^\ast \delta^\rho (uw)= w^\ast u^\ast \delta^\rho(u) w + w^\ast \delta^\rho(w). \end{align*} $$

Applying $\omega _1$ to both sides of this and using the w-invariance of $\omega _1$ , it remains to show that $\omega _1(w^\ast \delta ^\rho (w)) =0$ to get the statement. Using Lemma 2.7 below we have that

(11) $$ \begin{align} \left\lvert {\omega_1(w)} \right\rvert =1.\end{align} $$

Defining , we calculate

where we have used that . Now by Cauchy-Schwarz,

(12) $$ \begin{align} \left\lvert {\omega_1(g^\ast a)} \right\rvert ^2\leq\omega_1(|g|^2)\omega_1(|a|^2) = 0 \end{align} $$

for all $a\in \mathcal {A}$ which shows that $\ker \omega _1$ is a two-sided *-ideal within $\mathcal {A}$ . Hence

where we used that $\omega _1\circ \delta ^\rho = 0$ since $\omega _1$ is invariant under $\rho _t$ in the second equality and (12) in the last equality.

(ii) Let $(\mathcal {H}_1,\pi _1,\Omega _1)$ be the GNS triplet associated with $\omega _1$ . Let $Q_1$ be the self-adjoint operator on $\mathcal {H}_1$ such that $\mathrm {e}^{\mathrm {i} t Q_1}$ implements $\rho _t$ ; see Lemma 2.8 right below. By construction, $Q_1\Omega _1 = 0$ .

We claim that $\Omega _2:=\pi _1(u)\Omega _1$ is also an eigenvector of $Q_1$ , namely $Q_1\Omega _2=n\Omega _2$ for some $n\in \mathbb {Z}$ since $Q_1$ has integer spectrum. To see this we proceed as follows. Since $(\omega _1,\omega _2)$ is a locally comparable pair, $\omega _2$ is a vector state in ${\mathcal H}_1$ , namely the vector $\Omega _2 \equiv \pi _1(u) \Omega _1$ is such that

$$ \begin{align*} \langle \Omega_2, \pi_1(a)\Omega_2 \rangle = \omega_2(a)\qquad(a\in\mathcal{A}). \end{align*} $$

Moreover, $\omega _2\circ \rho _t=\omega _2$ implies, after going to the GNS representation and taking the derivative with respect to t at $0$ , that

$$ \begin{align*} \langle \Omega_2, [Q_1,A]\Omega_2 \rangle = 0\qquad(A\in\mathcal{B}(\mathcal{H}_1)), \end{align*} $$

since $\Omega _2$ is in the domain of $Q_1$ . We may take A to be the orthogonal projection onto $\operatorname {span}(\Psi )$ for some $\Psi \in \mathcal {H}_1$ to conclude that

(13) $$ \begin{align} \langle \Omega_2, Q_1\Psi \rangle\langle \Psi, \Omega_ {2} \rangle \in \mathbb{R}\qquad(\Psi\in\mathcal{H}_1). \end{align} $$

Now if $\Omega _2$ is not an eigenvector of $Q_1$ , then there is a nonzero $\Phi \in \operatorname {span}(\Omega _2)^\perp $ such that

$$ \begin{align*} Q_1 \Omega_2 = \lambda \Omega_2 + \Phi \end{align*} $$

for some $\lambda \in \mathbb {C}$ . Actually $\lambda \in \mathbb {R}$ because $Q_1$ is self-adjoint. Now if we invoke (13) with $\Psi =\Omega _2 +\operatorname {i} \Phi $ we obtain

$$ \begin{align*} \mathbb{R} \ni &\:\langle \Omega_2, Q_1\left(\Omega_2+\operatorname{i}\Phi\right) \rangle\langle \Omega_2+\operatorname{i}\Phi, \Omega_2 \rangle = \langle \lambda \Omega_2 + \Phi, \Omega_2+\operatorname{i}\Phi \rangle \left\Vert \Omega_2 \right\Vert{}^2\\ &=\left(\lambda\left\Vert \Omega_2 \right\Vert{}^2+\operatorname{i}\left\Vert \Phi \right\Vert{}^2\right) \left\Vert \Omega_2 \right\Vert{}^2, \end{align*} $$

which is a contradiction unless $\Phi =0$ . Hence $\Omega _2$ is an eigenvector of $Q_1$ , which has integer spectrum, so

(14) $$ \begin{align} \mathcal{N}_\rho(\omega_1,\omega_2) = \langle \Omega_1, Q_1\Omega_1 \rangle -\langle \Omega_2, Q_1\Omega_2 \rangle=-\langle \Omega_2, Q_1\Omega_2 \rangle \in \mathbb{Z}. \end{align} $$

(iii) Using the same GNS notation as above, if $\mathcal {N}_\rho (\omega _1,\omega _2)\neq 0$ then $\Omega _2$ has a nonzero eigenvalue for $Q_1$ . Since $Q_1$ is a self-adjoint operator and $\Omega _1$ is a zero-eigenvalue eigenvector for $Q_1$ , this implies that $\langle \Omega _2, \Omega _1 \rangle =0$ .

On the other hand, since both states $\omega _1,\omega _2$ are represented as vector states on the same Hilbert space, we have $\left \Vert \omega _1-\omega _2 \right \Vert =\sup _{a\in \mathcal {A}:\left \Vert a \right \Vert \leq 1} \left \lvert {\langle \Omega _1, \pi _1(a)\Omega _1 \rangle - \langle \Omega _2, \pi _1(a)\Omega _2 \rangle } \right \rvert $ . Denoting $P_j$ is the orthogonal projection onto the span of $\Omega _{\omega _j}$ for $j=1,2$ , we have

$$ \begin{align*} \left\Vert \omega_1-\omega_2 \right\Vert =\sup_{a\in\mathcal{A}:\left\Vert a \right\Vert\leq 1} \left\lvert {\mathrm{Tr}(\pi_1(a)(P_1-P_2))} \right\rvert = \left\Vert P_1-P_2 \right\Vert{}_1 \end{align*} $$

The second equality is by duality since $\pi _1$ is irreducible, as $\omega _1$ is pure. Since $P_j$ are one-dimensional projections, the trace norm is easily computable and we conclude that

(15) $$ \begin{align} \left\Vert \omega_1-\omega_2 \right\Vert = 2\sqrt{1- \left\lvert {\langle \Omega_2, \Omega_1 \rangle} \right\rvert ^2}. \end{align} $$

Hence,

$$ \begin{align*} \left\lvert {\mathcal{N}_\rho(\omega_1,\omega_2)} \right\rvert>0 \Longrightarrow \left\Vert \omega_1-\omega_2 \right\Vert=2. \end{align*} $$

We note in passing that the reader may want to compare this with a related converse statement about unitary equivalence of the states when $\left \Vert \omega _1-\omega _2 \right \Vert <2$ , see [Reference Glimm and Kadison29, Corollary 9, Remark 10].

(iv) Since $(\omega _2,\omega _1)$ are $\rho $ -locally comparable, there is $u\in {{\mathcal A}}\cap {\mathcal D}(\delta ^\rho )$ such that $\omega _2 = \omega _1\circ \operatorname {Ad}_{u}$ . Therefore,

(16) $$ \begin{align} \omega_2\circ\alpha = \omega_1\circ\operatorname{ad}_{u}\circ\alpha = (\omega_1\circ\alpha)\circ \operatorname{ad}_{\alpha^{-1}(u)} \end{align} $$

so that $(\omega _2\circ \alpha ,\omega _1\circ \alpha )$ are locally comparable. They are in fact $\rho $ -locally comparable since $\alpha $ commutes with $\delta ^\rho $ and so $\alpha ^{-1}({\mathcal D}(\delta ^\rho ))\subset {\mathcal D}(\delta ^\rho )$ . The computation

$$ \begin{align*} -\operatorname{i}\mathcal{N}_\rho(\omega_1\circ\alpha,\omega_2\circ\alpha) = (\omega_1\circ\alpha)\left(\alpha^{-1}(u)^{*}\delta^\rho(\alpha^{-1}(u))\right) = \omega_1 (u^{*}\delta^\rho(u)) = -\operatorname{i}\mathcal{N}_\rho(\omega_1,\omega_2) \end{align*} $$

yields the invariance.

(v) By (i), we can pick , for which $\delta ^\rho (u)=0$ and so the index vanishes.

(vi) To get the additivity of the index, assume that $\omega _2 = \omega _1 \circ \operatorname {Ad}_{u}$ and $\omega _3 = \omega _2 \circ \operatorname {ad}_{v}$ . Hence $\omega _3 = \omega _1 \circ \operatorname {Ad}_{v u}$ so that

$$ \begin{align*} \mathcal{N}_\rho(\omega_1,\omega_3) &= \operatorname{i} \omega_1\left(\left(vu\right)^\ast\delta^\rho\left(vu\right)\right) \\ &= \operatorname{i} \omega_1\left(u^\ast v^\ast\left(\left(\delta^\rho (v)\right)u+v\delta^\rho (u)\right)\right) \\ &= \operatorname{i} \omega_2(v^\ast \delta^\rho (v)) +\operatorname{i} \omega_1(u^\ast \delta^\rho (u))\\ &= \mathcal{N}_\rho(\omega_2,\omega_3) + \mathcal{N}_\rho(\omega_1,\omega_2). \end{align*} $$

Finally, (vii) follows from (vi) with $\omega _3 = \omega _1$ and (v).

(viii) The result follows immediately since $\delta ^{\rho \otimes \widetilde {\rho }} = \delta ^\rho \otimes \mathrm {id}+\mathrm {id}\otimes \delta ^{\widetilde {\rho }}$ .

Proof. Let $(\mathcal {H},\pi ,\Omega )$ be the GNS representation of $\omega $ . Then the invariance condition implies

$$ \begin{align*} \omega(a) = \langle \Omega, \pi(a)\Omega \rangle = \omega(u^{*} a u) = \langle \Omega, \pi(u)^{*} \pi(a) \pi(u)\Omega \rangle.\end{align*} $$

We find that both $\Omega $ and $\pi (u)\Omega $ are cyclic vectors that generate the same state, where $\pi $ is irreducible as $\omega $ is pure. This implies that $\pi (u)\Omega = {\mathrm e}^{\operatorname {i}\theta }\Omega $ for some $\theta \in \mathbb {R}$ and so $\omega (u) = \langle \Omega , \pi (u)\Omega \rangle = {\mathrm e}^{\operatorname {i}\theta }$ as claimed.

Lemma 2.8. Let $\{\rho _t\}_{t\in \mathbb {R}}\subseteq \operatorname {Aut}(\mathcal {A})$ be a one-parameter group of strongly continuous automorphisms such that and let $\omega \in \mathcal {P}(\mathcal {A})$ be $\rho $ -invariant. Then in the GNS representation $(\mathcal {H}_{\omega },\pi _{\omega },\Omega _{\omega })$ of $\omega $ , there exists a self-adjoint operator $Q_{\omega }:\mathcal {H}_{\omega }\to \mathcal {H}_{\omega }$ such that

$$ \begin{align*} \pi_\omega\left(\rho_t \left(a\right) \right) = \mathrm{e}^{-\mathrm{i} t Q_{\omega}} \pi_\omega\left(a\right) \mathrm{e}^{\mathrm{i} t Q_{\omega}} \qquad(t\in\mathbb{R},\quad a\in\mathcal{A}). \end{align*} $$

One may choose $Q_{\omega }$ such that $Q_{\omega }\Omega _\omega = 0$ , in which case $\sigma (Q_{\omega })\subseteq \mathbb {Z}$ .

Of course, the lemma also shows that the period $2\pi $ of the group $\rho _t$ is, at least mathematically, an arbitrary choice: If instead $\rho _{2\pi /\lambda }=\mathrm {id}$ for some $\lambda>0$ (but keeping Q with integer spectrum) then $\widetilde {\mathcal {N}_\rho }(\omega _1,\omega _2):=\lambda ^{-1}\mathcal {N}_\rho (\omega _1,\omega _2)\in \mathbb {Z}$ . Finally, we emphasize again that the requirement that the intertwining unitary u obey $u\in \mathcal {D}(\delta ^\rho )$ is not vacuous with the following example.

Example 2.10. Let $\mathcal {A}=\operatorname {CAR}(\ell ^2(\mathbb {N}))$ with $\{e_j\}_{j\geq 1}$ the canonical position basis of $\ell ^2(\mathbb {N})$ so $a_j\equiv a(e_j)$ are the annihilation operators associated with those basis elements. Then

$$ \begin{align*} q_N := \sum_{j=1}^N a_j^{*} a_j \end{align*} $$

defines a sequence of bounded self-adjoint elements in $\mathcal {A}$ which approximates a derivation $\delta $ as

$$ \begin{align*} \delta(a) \equiv \lim_N \operatorname{i} [q_N,a]\qquad(a\in\mathcal{D}(\delta)) \end{align*} $$

where $\mathcal {D}(\delta )$ is the space of elements a for which the limit exists. Define

$$ \begin{align*} a := \sum_{j=1}^\infty j^{-3/2} a_1^\ast \cdots a_j^\ast \end{align*} $$

which exists since $\left \Vert a \right \Vert \leq \sum _j j^{-3/2} < \infty $ . We may then calculate

$$ \begin{align*} [q_N, a_1^\ast \cdots a_j^\ast] = \min\left(\{N,j\}\right) a_1^\ast \cdots a_j^\ast \end{align*} $$

and hence

$$ \begin{align*} [q_N, a] = \sum_{j=1}^N j^{-1/2} a_1^\ast \cdots a_j^\ast + N \sum_{j=N}^\infty j^{-3/2} a_1^\ast \cdots a_j^\ast. \end{align*} $$

We see that the limit $N\to \infty $ of the above expression does not exist so that $a\notin \mathcal {D}(\delta )$ .

Now take $h := a + a^\ast $ and to get a unitary element which is not in $\mathcal {D}(\delta )$ . As a result, for any $\omega \in \mathcal {P}(\mathcal {A})$ we may define $\widetilde {\omega } := \omega \circ \operatorname {Ad}_{u}$ such that $\omega ,\widetilde \omega $ are locally comparable but not $\rho $ -locally comparable.

2.2 Connection with the index of a pair of projections on a Hilbert space

Let $\mathcal {H}$ be a separable Hilbert space and $P_1,P_2$ be two self-adjoint projections such that

(17) $$ \begin{align}P_1-P_2\in\mathcal{K}(\mathcal{H}).\end{align} $$

In this setting it is well-known that a topological index is associated with the pair $(P_1,P_2)$ (see, e.g., [Reference Avron, Seiler and Simon7] and earlier citations within) given by

(18) $$ \begin{align} \operatorname{index}(P_1,P_2) \equiv \operatorname{index}_{\operatorname{im}(P_1)\to \operatorname{im}(P_2)}\left(P_2P_1\big|_{\operatorname{im}(P_1)}\right) = \operatorname{index}_{\mathcal{H}}\left(P_2P_1 + UP_1^\perp\right) \end{align} $$

where the index on the right-hand sides is the Fredholm index and U is any unitary $U:\operatorname {im}(P_1^\perp )\to \operatorname {im}(P_2^\perp )$ . In the special case that $P-Q\in \mathcal {J}_p({\mathcal H})$ , the ideal of p-Schatten class operators, for some p, we may also write

$$ \begin{align*} \operatorname{index}(P_1,P_2) = \mathrm{tr}\left(\left(P_1-P_2\right)^{2p'+1}\right) \end{align*} $$

for any $p'\in \mathbb {N}$ such that $2p'+1\geq p$ . Finally, if $P_1-P_2\in \mathcal {J}_2({\mathcal H})$ then by [Reference Arveson4, Theorem 3], we may write

(19) $$ \begin{align} \operatorname{index}(P_1,P_2) = \mathrm{tr}\left(P_2\left(P_1-P_2\right)P_2\right) + \mathrm{tr}\left(P_2^\perp \left(P_1-P_2\right)P_2^\perp \right) . \end{align} $$

We turn to the relation of $\operatorname {index}(P_1,P_2)$ with the newly introduced many-body index ${\mathcal N}_\rho (\omega _1,\omega _2)$ .

Remark 2.11. Since any two projections $P_1,P_2$ on an infinite-dimensional Hilbert space with infinite $\ker P_j,\,\operatorname {im} P_j$ admit a unitary U which conjugates them, that is, $P_2 = U^\ast P_1 U$ , one may be tempted to conclude that if $P-Q$ is “small” then so is (so as to have any hope to implement it in $\mathcal {A})$ . However, it is well-known (see, e.g., [Reference Loreaux and Ng41, Theorem 1.2] or [Reference Brown, Douglas and Fillmore21]) that a unitary $U\in \mathcal {U}(\mathcal {H})$ can be found such that both and $P_2=U^\ast P_1 U$ iff

$$ \begin{align*}\operatorname{index}(P_1,P_2)=0.\end{align*} $$

This indicates that to capture a nontrivial index we most likely need to find a unitary in the CAR algebra which is not the second quantization of a Hilbert space one.

As we shall see below, there will be additional obstructions beyond the purely Hilbert space index obstruction.

Let now $\mathcal {A}=\operatorname {CAR}(\mathcal {H})$ (see Section 3.2 for more details) be the algebra of observables corresponding to systems of many fermions. Any self-adjoint projection P induces a pure state $\omega _{P}\in \mathcal {P}(\mathcal {A})$ , the so-called quasi-free state associated to P, given by the Gaussian formula

(20) $$ \begin{align} \omega_{P}(a(f_1)^\ast \cdots a(f_m)^\ast a(g_n)\cdots a(g_1)) :=\delta_{n,m}\det\left(\{\langle g_i, P f_j \rangle\}_{i,j=1,\ldots,n}\right), \end{align} $$

where $f_i,g_i\in \mathcal {H}$ , and extended linearly to all polynomials and by continuity to all of $\mathcal {A}$ .

The canonical ‘charge’ $U(1)$ *-automorphism $\rho _t$ is defined by

(21) $$ \begin{align} \rho_t(a(f)) = \mathrm{e}^{-\mathrm{i} t} a(f),\qquad \rho_t(a^{*} (f)) = \mathrm{e}^{\mathrm{i} t} a^{*}(f), \end{align} $$

for any $f\in \mathcal {H}$ and $t\in {\mathbb R}$ . Note that . For any orthogonal projection P, the state $\omega _P$ is automatically invariant under $\rho _t$ .

To contextualize the ensuing discussion, let us recall a few basic facts:

1. (i) The Shale-Stinespring condition for quasi-free states on CAR( $\mathcal {H}$ ) [Reference Shale and Forrest Stinespring44]: $\omega _{P_1},\omega _{P_2}$ have unitarily equivalent GNS representations if and only if $P_1-P_2 \in {\mathcal J}_2({\mathcal H})$ .

2. (ii) The Kadison transitivity theorem for general C*-algebras [Reference Glimm and Kadison29, Corollary 8]: If two pure states $\omega ,\widetilde {\omega }\in \mathcal {P}(\mathcal {A})$ have unitarily equivalent GNS representations then there exists some $u\in \mathcal {U}(\mathcal {A})$ such that $\widetilde {\omega } = \omega \circ \operatorname {Ad}_{u}$ .

3. (iii) The condition on Bogoliubov automorphisms of CAR $(\mathcal {H})$ to be inner [Reference Araki2, Theorem 5]: if and only if $\exists {\Gamma }(U)\in \mathcal {U}(\mathcal {A})$ such that $a(Uf) = {\Gamma }(U)^{*} a(f){\Gamma }(U)$ for all $f\in \mathcal {H}$ .

We are now ready to connect the index in (18) with the one defined in (3). For this, given the basic facts stated above, and in particular (iii), we need to strengthen the assumption on $P_1-P_2$ from compact to trace-class.

Theorem 2.12. Let $P_1,P_2$ be self-adjoint projections such that

$$ \begin{align*}P_1-P_2\in \mathcal{J}_1({\mathcal H}).\end{align*} $$

Then $\omega _{P_1}$ and $\omega _{P_2}$ are $\rho $ -locally comparable and

$$ \begin{align*} \operatorname{index}(P_1,P_2) = \mathcal{N}_{\rho}(\omega_{P_1},\omega_{P_2}). \end{align*} $$

This theorem identifies $\mathcal {N}_\rho $ as the many-body analog of the notion of an index of pair of projections, albeit with a certain mathematical gap: if $P_1-P_2\in \mathcal {K}(\mathcal {H})$ but is not Hilbert-Schmidt, then by the above basic facts $\omega _{P_1}$ and $\omega _{P_2}$ cannot be locally comparable: indeed, if they were, they would be GNS unitarily equivalent and hence violate the Shale-Stinespring condition. If $P_1-P_2$ is Hilbert-Schmidt but not trace-class, then by the above there would be a unitary conjugating them in the CAR algebra, but since that unitary is given abstractly by the Kadison transitivity theorem and is not the second quantization of any unitary, we do not know how to establish that that unitary necessarily lies in the domain of $\delta ^\rho $ .

Ultimately, a generalization of the many-body index may exist, which relies on a weaker condition that $(\omega _1,\omega _2)$ be locally comparable, and which would always be defined for quasi-free states $(\omega _{P_1},\omega _{P_2})$ as soon as $P_1-P_2\in \mathcal {K}$ . This problem is not entirely academic since in the interesting case of the integer quantum Hall effect, $P-L^\ast P L\in \mathcal {J}_3\setminus \mathcal {J}_2$ where

(22) $$ \begin{align}L\equiv\mathrm{e}^{\operatorname{i}\arg\left(X_1+\operatorname{i} X_2\right)}\end{align} $$

is the Laughlin unitary [Reference Avron, Seiler and Simon6] implementing one quanta of a magnetic flux insertion. See also Example 2.14 below for a simple example of this type.

Example 2.13 (Nonzero index yet unitary exists thanks to interactions).

The obstruction outlined in Remark 2.11 does not imply that we cannot implement nonzero indices. Indeed, pick $\mathcal {H} := \ell ^2(\mathbb {Z})$ and let P be the multiplication operator by the indicator function $\chi _{\mathbb {N}}$ , namely the projection onto the RHS of space. Let R be the bilateral right shift operator $R:\delta _x\mapsto \delta _{x+1}$ . Then $P_R=R^\ast P R$ is the projection given by $\chi _{\mathbb {N}\cup \{0\}}$ and $P_R-P$ is the finite rank projection given by $\chi _{\{0\}}$ , and $\operatorname {index}(P,P_R) = -1$ . Of course this means $P-P_R \in \mathcal {J}_2({\mathcal H})$ , equivalently, $[R,P]\in \mathcal {J}_2({\mathcal H})$ .

The operator

$$ \begin{align*} u := a_0 + a_0^\ast. \end{align*} $$

is unitary since , such that

$$ \begin{align*} u^{*} a_0 u = a_0^\ast,\quad\text{while}\quad u^{*} a_m u = - a_m \quad\text{whenever } m\neq0. \end{align*} $$

and we claim that

$$ \begin{align*} \omega_{P_R} = \omega_P \circ \operatorname{Ad}_{u}. \end{align*} $$

Indeed, if $n,m\neq 0$ ,

$$ \begin{align*} \omega_P(u^{*} a_m^\ast a_n u) = \omega_P( a_m^\ast a_n ) = \langle \delta_n,P\delta_m\rangle = \delta_{m,n}\chi_{\mathbb{N}}(n) = \omega_{P_R}(a_m^\ast a_n) \end{align*} $$

and if $n=m=0$ then

while both $\omega _P(u^{*} a_m^\ast a_n u)$ and $\omega _{P_R}(a^{*}_m a_n)$ vanish if $n\neq m$ . Since

$$ \begin{align*} \delta^\rho(a_0) = \frac{d}{dt}\rho_t(a_0)\big\vert_{t=0} = -\mathrm{i} a_0 \end{align*} $$

we see that $u\in {\mathcal D}(\delta ^\rho )$ and $-\mathrm {i} \delta ^\rho (u) = -a_0 + a_0^{*}$ , so that

Example 2.14 (Compact but not Hilbert-Schmidt difference of projections so no unitary exists).

Let $\mathcal {H} := \ell ^2(\mathbb {N})\otimes \mathbb {C}^2$ . Let P be given by

$$ \begin{align*} P_{nm} := \delta_{nm} \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}, \end{align*} $$

that is, projection onto the top of each dimer. Let U be the unitary given by

$$ \begin{align*} U_{nm} := \delta_{nm} \begin{bmatrix} \cos(\theta_n) & \sin(\theta_n) \\ -\sin(\theta_n) & \cos(\theta_n) \end{bmatrix} \end{align*} $$

where $\{\theta _n\}_n$ is some sequence of angles to be determined below. If $Q := U^\ast P U$ , then

$$ \begin{align*} P_{nm}-Q_{nm}=\delta_{nm}\begin{bmatrix} \sin^{2}\theta_n & -\cos\theta_n\sin\theta_n\\ -\cos\theta_n\sin\theta_n & -\sin^{2}\theta_n \end{bmatrix}. \end{align*} $$

Since the singular values of $P-Q$ are $\{ \left \lvert {\sin (\theta _n)} \right \rvert \}_{n\geq 1}$ it is clear that if we pick $\theta _n := \arcsin (x_n)$ for some sequence $\{x_n\}_n\subseteq (0,1)$ we can engineer whichever summability we want. For example if we take $x_n = n^{-\beta }$ for any $\beta \in (1/3,1/2]$ then we guarantee that $\{x_n^2\}_n$ is not summable but $\{x_n^3\}_n$ is. Note that as soon as $x_n\to 0$ we have $P-Q\in \mathcal {K}(\mathcal {H})$ .

Now, for $\{x_n\}_n$ in $\ell ^3$ but not $\ell ^2$ there does not exist a unitary $u\in \mathcal {U}(\mathcal {A})$ such that $\omega _Q = \omega _P\circ \operatorname {Ad}_{u}$ . Indeed, the existence of such a unitary would violate the Shale-Stinespring condition [Reference Shale and Forrest Stinespring44].

Before presenting the proof of Theorem 2.12, let us make a final remark. Suppose we knew that $\omega _1$ and $\omega _2$ (with $\omega _j\equiv \omega _{P_j}$ ) are $\rho $ -locally comparable. Then the equivalence of the two indices would be immediate. Indeed, starting from (9) and approximating $\delta ^\rho = \lim _N\operatorname {i}[q_N,\cdot ]$ with $q_N := \sum _{j=1}^N a^{*}(e_j)a(e_j)$ where $\{e_j\}_j$ is any ONB of $\mathcal {H}$ , we obtain using (20)

$$ \begin{align*} \mathcal{N}_\rho(\omega_1,\omega_2) &= \lim_N \left(\omega_1(q_N) - \omega_2(q_N)\right) = \lim_N \sum_{j=1}^N\left(\langle e_j, P_1 e_j \rangle-\langle e_j, P_2e_j \rangle\right) \\ &= \lim_N \mathrm{tr}\left(\left(P_1-P_2\right)\chi_{\{1,\cdots,N\}}(X)\right) \end{align*} $$

where X is the position operator w.r.t. the basis $\{e_j\}_j$ , that is, $X e_j \equiv j e_j$ for any $j\in \mathbb {N}$ . Since we have strongly as $N\to \infty $ , and since $P_1-P_2\in \mathcal J_1$ we are finished, since the product of a strongly convergent sequence with a trace-class operator is a sequence converging in trace-class norm, that is, we find

$$ \begin{align*} \mathcal{N}_\rho(\omega_1,\omega_2) &= \mathrm{tr}\left(P_1-P_2\right) = \operatorname{index}(P_1,P_2). \end{align*} $$

We also note that in the case $P_1-P_2\in {\mathcal J}_2({\mathcal H})$ the proof given (using a particular choice of the ONB $\{e_j\}_j$ ) goes through by invoking the explicit formula (19). In the proof below, we rely on the trace-class assumption to construct the unitary intertwiner between $\omega _1$ and $\omega _2$ , which lies in $\mathcal {D}(\delta ^\rho )$ . So the main difficulty is to establish the $\rho $ -local-comparability of $\omega _1,\omega _2$ , that is, to find the unitary which conjugates them and show it lies in $\mathcal {D}(\delta ^\rho )$ .

The proof of Theorem 2.12 relies on two steps.

1. (i) We first extract the ‘excess states’ out of the difference $P_1-P_2$ which are responsible for a nonzero index, see Lemma 2.15 below. Because of the obstruction in Remark 2.11, the unitary which implements this unfolding is necessarily not the second quantization of a Hilbert space one, but it can be constructed explicitly. In particular, it is in $\mathcal {D}(\delta ^\rho )$ since the charge deficiency is finite. Indeed, it is very much in the spirit of Example 2.13.

2. (ii) As a result, we obtain two projections whose index is zero and whose difference is trace-class. It remains to use the second quantization provided by Lemma 2.16, which does not change the charge deficiency. Since all unitaries are explicit, the equality of indices follows from a computation.

Proof of Theorem 2.12.

Using Lemma 2.15 below, since $P_1-P_2 \in \mathcal {J}_1(\mathcal {H})$ , we write

$$ \begin{align*}P_2 = \widetilde P + N_+ - N_-,\end{align*} $$

with $\widetilde P=VP_1V^{*}$ and . By Lemma 2.16 below, $\omega _{\widetilde P}$ and $\omega _{P_1}$ are $\rho $ -locally comparable with $\Gamma (V^{*})\in \mathcal U(\mathcal {A}) \cap \mathcal {D}(\delta ^Q)$ and

(23) $$ \begin{align} \omega_{\widetilde P} = \omega_{P_1}\circ\operatorname{Ad}_{\Gamma(V^{*})}. \end{align} $$

since $\omega _{\widetilde P}(a^{*}(f) a(g)) = \langle g, VP_1V^{*} f\rangle = \omega _{P_1}(a^{*}(V^{*} f) a(V^{*} g))$ . Let $n_\pm =\mathrm {rank}(N_\pm )$ and $\{f_i\}$ and $\{g_i\}$ be orthonormal bases of the range of $N_-$ and $N_+$ respectively. We consider

$$ \begin{align*}v_+ = \prod_{i=1}^{n_+} (a(g_i)+a^*(g_i)), \qquad v_-=\prod_{i=1}^{n_-} (a(f_i)+a^*(f_i)). \end{align*} $$

One checks that $v_\pm \in \mathcal {U}({\mathcal A})$ as products of unitary factors, see again Example 2.13. Moreover, $v_\pm \in \mathcal {D}(\delta ^Q)$ as polynomials in creation and annihilation operators. We claim that

$$ \begin{align*}u := v_+v_- \Gamma(V)^{*} \in \mathcal U(\mathcal{A}) \cap \mathcal{D}(\delta^Q). \end{align*} $$

is such that

(24) $$ \begin{align} \omega_{P_2} = \omega_{\widetilde P} \circ \operatorname{Ad}_{v_+v_-} = \omega_{P_1}\circ \operatorname{Ad}_{u}. \end{align} $$

We prove the first equality in (24), the second one is just (23). Since

we see that for $v_\xi = a^{*}(\xi ) + a(\xi )$ ,

$$ \begin{align*} \omega_{\widetilde P}\circ \operatorname{Ad}_{v_\xi}(a^{*} (\psi)a(\phi)) = \langle\phi,(\widetilde P + P_\xi - (\widetilde P P_\xi + P_\xi\widetilde P))\psi\rangle \end{align*} $$

where $P_\xi = \vert \xi \rangle \langle \xi \vert $ . In particular,

$$ \begin{align*} \begin{cases} \omega_{\widetilde P}\circ \operatorname{Ad}_{v_\xi}(a^{*} (\psi)a(\phi)) = \langle\phi,(\widetilde P - P_\xi )\psi\rangle &\text{if}\quad \widetilde P P_\xi = P_\xi, \\ \omega_{\widetilde P}\circ \operatorname{Ad}_{v_\xi}(a^{*} (\psi)a(\phi)) = \langle\phi,(\widetilde P + P_\xi )\psi\rangle & \text{if}\quad \widetilde P P_\xi = 0. \end{cases} \end{align*} $$

Applying this recursively with $\xi $ running through the basis of $N_+$ and then the basis of $N_-$ and recalling that $\widetilde P N_- = N_-$ and $\widetilde P N_+ = 0$ yields

$$ \begin{align*} \omega_{\widetilde P}\circ \operatorname{Ad}_{v_+v_-}(a^{*} (\psi)a(\phi)) = \langle\phi,(\widetilde P + N_+ - N_-)\psi\rangle =\omega_{P_2}(a^{*} (\psi)a(\phi)) \end{align*} $$

which proves (24). Thus $\omega _{P_2}$ and $\omega _{P_1}$ are $\rho $ -locally comparable.

For completeness let us show the equivalence of the indices using the intrinsic decomposition just exhibited. We compute ${\mathcal N}_{\rho }(\omega _{P_1},\omega _{P_2})=\mathrm {i}\omega _1(u^{*}\delta ^\rho (u))$ . Since $\delta ^\rho (\Gamma (V))=0$ , we focus on $v_-^{*} v_+^{*} \delta ^\rho (v_+)v_- + v_-^{*} v_+^{*} v_+\delta ^\rho (v_-)$ . Since

$$ \begin{align*} \delta^\rho(a^{*}(f) + a(f)) = \mathrm{i}(a^{*}(f) - a(f)) \end{align*} $$

for any normalized f, and therefore

we conclude that $v_+^{*}\delta ^\rho (v_+) = n_+ - 2Q_+$ and so

$$ \begin{align*} \mathrm{i} v_-^{*} v_+^{*} \delta(v_+)v_- + \mathrm{i} v_-^{*} v_+^{*} v_+\delta^\rho(v_-) = -(n_- - 2 Q_-) - (n_+ - 2 Q_+). \end{align*} $$

Here $Q_- = \sum _{i=1}^{n_-}a^{*}(f_i) a(f_i)$ and $Q_+ = \sum _{i=1}^{n_+}a^{*}(g_i) a(g_i)$ . Hence

$$ \begin{align*} {\mathcal N}_\rho(\omega_{P_1},\omega_{P_2}) &= \mathrm{i} \omega_{P_1}\left(\Gamma(V)((n_- - 2 Q_-) + (n_+ - 2 Q_+))\Gamma(V^{*})\right) \\ &=-n_- - n_+ + 2\omega_{\widetilde P}\left(Q_-+ Q_+\right) = n_- - n_+ \end{align*} $$

since $\omega _{\widetilde P}(Q_-) = n_-$ while $\omega _{\widetilde P}(Q_+)=0$ by (20).

Lemma 2.15. If $P_1,P_2$ are two orthogonal projections on a Hilbert space $\mathcal {H}$ such that $P_1-P_2\in \mathcal J_p(\cal H)$ for $p\in \mathbb N^*$ , there exists finite-rank orthogonal projections $N_+,N_-$ with $N_+N_-=0$ and a unitary V with such that

(25) $$ \begin{align} P_2 = V P_1 V^* + N_+ - N_- \end{align} $$

Moreover $\mathrm {index}(P_1,P_2) = \mathrm {tr}(N_+-N_-)$ , and for $\tilde P=VP_1V^*$ one has $\widetilde P N_+ = 0$ and $\widetilde P N_-=N_-$ .

The proof of this lemma can be found in [Reference Bachmann, Bols and Rahnama10, Proof of Prop. 5.2], see also [Reference Bols and Cedzich18, Sec 4.2.1] or [Reference Cedzich, Geib, Alberto Grünbaum, Stahl, Velázquez, Werner and Werner22, Sec VII.B].

Lemma 2.16. If V is a unitary operator on ${\mathcal H}$ such that , the *-automorphism defined by $a(f)\mapsto a(V f)$ is unitarily implementable by $\Gamma (V)\in {{\mathcal A}}$ , namely

$$ \begin{align*} a(V f) = \Gamma(V)^{*} a(f)\Gamma(V) \end{align*} $$

for all $f\in {\mathcal H}$ . Moreover, $\delta ^\rho (\Gamma (V)) = 0$ .

This is a known result up to possibly the last property, see, e.g., [Reference Araki2, Theorem 5]. For convenience of the reader we include its proof here.

Proof of Lemma 2.16.

We recall the explicit construction of the implementer $\Gamma (V)\in {{\mathcal A}}$ given in [Reference Avron, Bachmann, Michele Graf and Klich5] whenever is of finite rank. For rank-one operators $A_i=|f_i\rangle \langle g_i|$ , $(i=1,\ldots ,n)$ , we set

(26) $$ \begin{align} \,\mathrm{d}\Gamma(A_1,\,\ldots,\,A_n)= a^{*}(f_n)\cdots a^{*}(f_1)\,a(g_1)\cdots a(g_n). \end{align} $$

The definition is extended by multilinearity to operators $A_i$ of finite rank. The result is independent of the particular decomposition into rank one operators. Then if has finite rank, we set

(27)

The sum is finite, because the terms with vanish by the CAR.

A calculation, see [Reference Avron, Bachmann, Michele Graf and Klich5], then shows that $\Gamma (U)$ is indeed an implementer

(28) $$ \begin{align} \Gamma(V)a^{*}(f) =a^{*}(Vf)\Gamma(V) \end{align} $$

and that $\Gamma (V_1)\Gamma (V_2) = \Gamma (V_1V_2)$ . In particular, $\Gamma (V)$ is unitary if V is. Moreover, the definition (26) immediately implies that $\rho _t(\,\mathrm {d}\Gamma (A_1,\,\ldots ,\,A_n)) = \,\mathrm {d}\Gamma (A_1,\,\ldots ,\,A_n)$ , where $\rho _t$ is the $U(1)$ -automorphism defined in ((21)). This yields in turn that $\rho _t(\Gamma (V)) = \Gamma (V)$ by the definition (27). Hence $\delta ^\rho (\Gamma (V))=0$ .

It remains to extend the map $\Gamma $ to unitaries such that is trace class. For rank-one operators, (26) and the CAR yield

$$ \begin{align*} \Vert \,\mathrm{d}\Gamma(A_1,\,\ldots,\,A_n)\Vert \leq \prod_{i=1}^n\Vert f_i\Vert\Vert g_i\Vert \end{align*} $$

and so if $A_i$ is a rank $r_i$ operator, we have that

$$ \begin{align*} \Vert \,\mathrm{d}\Gamma(A_1,\,\ldots,\,A_n)\Vert \leq \inf_{\{f\},\{g\}}\sum_{i_1=1}^{r_1} \Vert f_{i_1}\Vert\Vert g_{i_1}\Vert \cdots \sum_{i_n=1}^{r_n} \Vert f_{i_n}\Vert\Vert g_{n_1}\Vert = \prod_{i=1}^n \Vert A_i\Vert_1 \end{align*} $$

where the infimum is over all decomposition of the operators $A_j = \sum _{i_j=1}^{r_j}\vert f_{i_j}\rangle \langle g_{i_j}\vert $ . Hence $\,\mathrm {d}\Gamma $ is a bounded linear transformation from finite-rank operators equipped with the trace norm to the algebra, and it extends to a bounded linear transformation $\,\mathrm {d}\Gamma :{\mathcal J}_1({\mathcal H})^{\otimes n}\to {{\mathcal A}}$ , with $\Vert \,\mathrm {d}\Gamma \Vert = 1$ for all $n\in \mathbb {N}$ . This now implies that the series in (27) is convergent since

In fact, we have that .

3.1 Construction overview and comparison with previous work

The construction being rather long, we summarize its main steps. First, we set in Section 3.2 the algebraic framework by defining concretely the CAR algebra ${\mathcal A}$ and its local structure, $0$ -chains and their associated dynamics (locally generated automorphism or LGA), and the charge operator Q associated to $U(1)$ -gauge transformations $\rho $ . This allows us to define Q-symmetric and short-ranged-entangled (SRE) states, which we aim to classify up to homotopy. The SRE property encodes the locality property of a state $\omega $ . In particular, the existence of a parent Hamiltonian H, for which $\omega $ is a gapped ground state, is a direct consequence of the SRE property. Moreover, in contrast to single particle systems where electric charge is the identity operator, its many-body counterpart Q is not always preserved, so we focus on the class of Q-symmetric states to study the integer quantum Hall effect.

Then, a central step is the construction of the flux insertion automorphism from Section 3.4, which starts from an SRE state $\omega $ and ends with the defect state $\omega ^D$ , where a unit of magnetic flux has been inserted at the origin; see Figure 1. Flux insertion relies on gauge transformation, quasi-adiabatic flow and truncation. The main goal is to localize the operations to a half-line, and mimics the half-line gauge flux insertion from single particle picture [Reference De Nittis and Schulz-Baldes24]. At one quantum of flux $\phi =2\pi $ , the two states $\omega $ and $\omega ^D$ only differ near the origin, allowing the existence of a unitary $u\in \mathcal {A}$ relating the two. Namely, they are $\rho $ -locally comparable and the formalism of Definition 2.4 applies for $\mathcal N_\rho (\omega ,\omega ^D)$ . This construction may look convoluted at first, but it is to our current knowledge the only way to construct a defect state which is $\rho $ -locally comparable to $\omega $ (see also the discussion at the beginning of Section 3.4). An analogous version is presented in [Reference Bachmann, Bols and Rahnama10], on which we rely.

Figure 1 Overview of the construction from the SRE state $\omega $ to the defect state $\omega ^D$ via magnetic flux insertion. The $0$ -chains are progressively localized near the half-line. At $\phi =2\pi $ , the two states only differ near the origin, which allows for the existence of a unitary u so that they are $\rho ^Q$ -locally comparable.

However, it is actually not possible to provide a quantum Hall model for which this index on SRE states is nonzero. Indeed, for a translation-invariant system of noninteracting electrons, the (stable) SRE condition is equivalent to the triviality of the bundle, hence the vanishing of the Hall conductance and accordingly the vanishing of our index. To model nontrivial Hall states, in Section 3.5 we extend the construction to invertible states. Those are states $\omega $ for which there exists some auxiliary state $\omega '$ over ${\mathcal A}'$ such that $\omega \hat \otimes \omega '$ is itself SRE on ${\mathcal A}\hat \otimes {\mathcal A}'$ . ${\mathcal A}'$ is another CAR algebra which implements extra degrees of freedom on each lattice site, and $\hat \otimes $ is the stacking operation. In the noninteracting case, this amounts to considering the direct sum of two bundles having opposite Chern numbers; the Chern number is additive under direct sums. We then reproduce the previous construction, with the main subtlety of having two charge operators Q and $Q'$ for each layer. The flux insertion is performed on the two layers with $Q + Q'$ , but the symmetry appearing in our index is Q, acting on the first layer only. Physically, this amounts to a bona fide flux insertion across both layers, which is essential to have $\rho $ -locally comparable states at the end of the process, but a measurement of charge transport only on the first, physical one. The construction is summarized in Figure 2. It leads to Definition 3.30 and Theorem 3.31, main result of Section 3.5.

Figure 2 Extending the construction to invertible states.

We consider here exclusively the ‘bulk picture’ of the quantum Hall effect. As pointed out in the introduction, the technicalities of the construction we provide in this section are heavily inspired by [Reference Bachmann, Bols and Rahnama10]. There, the additional assumption of time reversal invariance implies that there is no net charge transport and the interest is on the remaining $\mathbb {Z}_2$ -valued index which can be nontrivial even for SRE states. In the infinite planar geometry we consider here, the construction of a flux insertion automorphism for invertible states goes back to [Reference Kapustin and Sopenko33], which in turn was inspired by the flux threading procedure of [Reference Bachmann, Bols, De Roeck and Fraas9, Reference Bachmann, Bols, De Roeck and Fraas8] on large but finite tori. From the point of view of these works, the goal of this section is merely to point out that they provide a concrete physical setting in which the general abstract index of the previous section is realized. Still in the many-body, interacting setting, the Hall conductance and its quantization has also been described as a Thouless pump and [Reference Bachmann, De Roeck, Fraas and Jappens13, Reference Artymowicz, Kapustin and Sopenko3] provide a different, albeit related picture of its quantization. For weakly interacting fermions, [Reference Giuliani, Mastropietro and Porta28] provides a rigorous renormalization group approach to the quantization of the quantum Hall conductance: while the previous works rely on the assumption of a spectral gap, this result proves the stability of the gap in the perturbative regime. Recently, [Reference Teufel and Wesle46] have extended the strongly interacting framework to some cases where the gap closes. This is in fact closely related to [Reference Hastings and Michalakis31], which initiated the present line of work. While these works are all concerned with microscopic lattice systems, the quantum Hall effect was analyzed earlier from a field theoretic point of view, see, for example, [Reference Fröhlich and Zee27] or the reviews [Reference Fröhlich, Studer and Thiran25, Reference Fröhlich, Studer and Thiran26]. Finally, Section 4 will establish the equality of the Hall index defined in this work for invertible states with the Hall index for a pair of projections, and through it with the many indices long known to be equivalent to it, whenever they can be compared: spectral flow, Chern number or K-theoretic indices. In particular, we will prove that the trace-class assumption of the general Theorem 2.12 follows from the locality of the Hamiltonian and the gap assumption.

3.2 Algebraic framework

Fermionic observables are elements of the Canonical Anti-commutation Relation (CAR) algebra ${\mathcal A}$ over the one-particle Hilbert space ${{\mathcal H}} = \ell ^2(\mathbb Z^2)\otimes \mathbb C^n$ where $n \in \mathbb N$ takes into account internal degrees of freedom such as spin. The algebra ${\mathcal A}$ is the unital $C^*$ -algebra generated by an identity element and the annihilation operators $\{a(f):f\in {{\mathcal H}}\}$ , which satisfy the CAR

(29)

for all $f,g\in {h}$ , and where $a^{*}(f) = (a(f))^{*}$ are the creation operators. By picking the orthonormal basis $\{\delta _x\otimes e_i:x\in {\mathbb Z}^2,i\in \{1,\ldots ,n\}\}$ of ${h}$ , the algebra is also generated by $a_{x,i},a^{*}_{y,j}$ for $x,y \in \mathbb Z^2$ and $i,j \in \{1,\ldots , n\}$ , where $a_{x,i}^\sharp = a^\sharp (\delta _x\otimes e_i)$ . Here and in the following $a^\sharp $ stands for either a or $a^{*}$ .

For any (possibly infinite) $\Lambda \subset \mathbb Z^2$ we denote by ${\mathcal A}_\Lambda $ the unital $C^*$ -subalgebra of ${\mathcal A}$ generated by and the $a_{x,i}$ for $x\in \Lambda $ only. When $\Lambda = \{x\}$ , we denote ${{\mathcal A}}_x = {{\mathcal A}}_{\{x\}}$ . There is a natural hierarchy of inclusions: if $\tilde {\Lambda }\subseteq \Lambda $ then ${\mathcal A}_{\tilde {\Lambda }}$ is identified with the subalgebra of of ${\mathcal A}_\Lambda $ . Let $\mathcal F$ be the set of finite subsets of $\mathbb Z^2$ . An operator $A \in {\mathcal A}_\Lambda $ with $\Lambda \in \mathcal F$ is called local. The smallest (in the sense of set inclusion) $\Lambda \in \mathcal F$ such that $A \in {\mathcal A}_\Lambda $ is called the support of A, denoted by $\mathrm {supp}(A)$ . The $*$ -subalgebra

$$ \begin{align*}{\mathcal A}^{\mathrm{loc}} = \bigcup_{\Lambda \in \mathcal F} {\mathcal A}_\Lambda \end{align*} $$

is called the algebra of local observables. By the definition of ${\mathcal A}$ , ${\mathcal A}\equiv \overline {\mathcal A}^{\mathrm {loc}}$ , in the $\| \cdot \|$ -topology.

Almost local observables.

While ${\mathcal A}^{\mathrm {loc}}$ is extremely convenient for keeping track of where in space observables act, as we shall recall below, it is in general not invariant under the Schrödinger time evolution. For that reason, we work with a standard invariant algebra, originally introduced in [Reference Bachmann, Dybalski and Naaijkens14], which still keeps track of the “support center” of observables. It can be defined as follows. Let

$$ \begin{align*}\mathcal L = \{f : [0,\infty)\to (0,\infty)\,|\, f\text{ is bounded, nonincreasing, and } \forall p>0,\, \lim_{ r\to \infty} r^p f(r)=0\}. \end{align*} $$

For $x \in \mathbb Z^2$ and $r \in \mathbb N$ we denote by $B_x(r)$ the ball of center x and radius r within $\mathbb {Z}^2$ . An observable $A \in {\mathcal A}$ called f-localized near $x \in \mathbb Z^2$ if there exists $f \in \mathcal L$ , a sequence $A_n \in {\mathcal A}_{B_x(n)}$ such that

$$ \begin{align*}\| A - A_n\| \leq f( n ) \|A\| \end{align*} $$

for all $n \in \mathbb N$ . An observable is almost local if it is f-localized for some function $f\in \mathcal {L}$ and some $x \in \mathbb Z^2$ . We denote by ${\mathcal A}^{\mathrm {al}}$ the $*$ -algebra of almost local observables. Since ${\mathcal A}^{\mathrm {loc}} \subset {\mathcal A}^{\mathrm {al}}$ , the latter is also dense in ${\mathcal A}$ .

$0$ -chains and autonomous dynamics.

Let $f\in \mathcal {L}$ be given. An f-local 0-chain F [Reference Kapustin and Sopenko34] is a sequence $(F_x)_{x\in \mathbb Z^2}\subseteq {\mathcal A}^{\mathrm {al}}\cap {\mathcal A}^+$ such that

• • For all $x \in \mathbb Z^2$ , $F_x$ is self-adjoint and f-localized near x.

• • $\displaystyle \sup _{x\in \mathbb Z^2} \| F_x \| < \infty $ .

Proposition 3.1. Let F be an f-local 0-chain. Then the densely defined map $\delta ^F:{\mathcal A}^{\mathrm {al}}\to {\mathcal A}^{\mathrm {al}}$ given by

$$ \begin{align*}A \mapsto \mathrm{i}[F,A]:= \sum_{x\in \mathbb Z^2} \mathrm{i}[F_x,A] \end{align*} $$

is a *-derivation on ${\mathcal A}$ .

In general, $\delta ^F$ is an unbounded operator on ${{\mathcal A}}$ .

Proof. Let A be $f_A\in \mathcal {L}$ localized near some $x_A\in \mathbb {Z}^2$ and let $C := \sup _x \left \Vert F_x \right \Vert $ . We claim that $\delta ^F(A)$ is g-localized near $x_A$ for some $g\in {\mathcal L}$ . Let $n\geq 2\left \Vert x-x_A \right \Vert $ then

$$ \begin{align*} [F_x,A] = [F_{x,n/2},A_{n}]+[F_{x},A-A_{n}]+[F_x-F_{x,n/2},A] \end{align*} $$

with the first term being supported on $B_{x_A}(n)$ and both remaining terms are bounded above by $C\Vert A\Vert g(n)$ for some $g\in {\mathcal L}$ , where g can be chosen to be independent of x. Hence $[F_x,A]$ is g-localized near $x_A$ . It follows that $\sum _{x\in \mathbb Z^2} [F_x,A]$ is summable and defines an element in ${\mathcal A}^{\mathrm {al}}$ .

For a f-local 0-chain F, the derivation $\delta ^F$ generates a strongly continuous one-parameter group of $*$ -automorphisms $\mathbb R\ni t\mapsto \alpha _t^F$ on ${\mathcal A}$ which is defined by $\alpha _t^F := \exp (t \delta ^F)$ , that is, as the solution to

$$ \begin{align*}\alpha^F_0(A) = A, \qquad \dfrac{\,\mathrm{d} }{\,\mathrm{d} t}\alpha_t^F(A) = \alpha_t^F(\delta^F(A)) \end{align*} $$

on ${\mathcal A}^{\mathrm {al}}$ and extends by continuity to all of ${{\mathcal A}}$ . Furthermore, it satisfies the semigroup property $\alpha ^F_{t} \circ \alpha ^F_s = \alpha ^F_{t+s}$ .

Nonautonomous setting.

The use of quasi-adiabatic flow and parallel transport below requires that we extend the formalism beyond the autonomous setting. A time-dependent 0-chain is a family $(F(s))_{s \in \mathbb R}$ such that:

• • For any $s_0 \in \mathbb R$ there exists $f\in {\mathcal L}$ such that $F(s)$ is an f-localized 0-chain for all $|s|<s_0$ .

• • $\displaystyle \sup _{x\in \mathbb Z^2, s \in \mathbb R} \|F_x(s)\| < \infty $ .

• • $s \mapsto F_x(s)$ is norm-continuous for all $x \in \mathbb Z^2$ .

As in the time-independent case, a time-dependent $0$ -chain generates a nonautonomous time evolution $\alpha _{s\to t}^F$ for $s,t \in \mathbb R$ on ${{\mathcal A}}$ defined by

$$ \begin{align*}\alpha^F_{s\to s}(A) = A, \qquad \dfrac{\,\mathrm{d} }{\,\mathrm{d} t}\alpha_{s\to t}^F(A) = \alpha_{s\to t}^F(\delta^{F(t)}(A))\qquad(A\in{\mathcal A}^{\mathrm{al}}). \end{align*} $$

The semigroup property is replaced by the following cocycle property $\alpha _{t'\to t} \circ \alpha _{s\to t'}=\alpha _{s\to t}$ .

Remark 3.2. From now on, unless stated, all the $0$ -chains that we consider are time-dependent. For convenience we shall slightly abuse notation and denote $\alpha ^F_{t}:=\alpha ^F_{0\to t}$ .

The next result is a consequence of the Lieb-Robinson bound (see for example [Reference Teufel and Wessel47] for a strong version that applies in our setting, and the references therein) and it is the main reason for the introduction of the algebra ${\mathcal A}^{\mathrm {al}}$ .

Proposition 3.3. Let F be a $0$ -chain. Then, for all $t,s \in \mathbb R$ ,

$$ \begin{align*}\alpha_{s\to t}^F({\mathcal A}^{\mathrm{al}}) \subset {\mathcal A}^{\mathrm{al}}. \end{align*} $$

Moreover, $\alpha _{s\to t}^F$ is homogeneous in the sense that $\alpha _t^F \circ \theta = \theta \circ \alpha _t^F$ .

Definition 3.4 (LGA).

An automorphism $\alpha $ is called a locally generated automorphism (LGA) if there exists a $0$ -chain F and some $s\in [0,\infty )$ such that $\alpha = \alpha _s^F$ .

Charge and symmetry.

The relevant symmetry to consider in the quantum Hall effect is charge conservation, which was already defined in (21) but we give here a more concrete description. The charge operator at any site $x\in {\mathbb Z}^2$ is given by

(30) $$ \begin{align} Q_x = \sum_{i=1}^n a^*_{x,i} a_{x,i}, \end{align} $$

which accounts for internal degrees of freedom. Since $Q_x$ is supported on the single site x, the collection $\{Q_x:x\in {\mathbb Z}^2\}$ defines a (time-independent) $0$ -chain which is trivially f-localized. For $\Lambda \in {\mathcal F}$ the local charge $Q^\Lambda = \sum _{x \in \Lambda } Q_x$ is supported in $\Lambda $ and it has integer spectrum. For $\phi \in \mathbb R$ we denote by $\rho _\phi = \alpha _\phi ^Q$ and $\rho _\phi ^\Lambda = \alpha _\phi ^{Q^\Lambda }$ the LGA respectively associated to Q and $Q^\Lambda $ .

Proposition 3.5. For any $\Lambda \in \mathcal F,\phi \in \mathbb {R}$ and $A\in {{\mathcal A}}$ , one has $ \rho _\phi ^\Lambda (A) = {\mathrm e}^{{\mathrm i} \phi Q^\Lambda } A {\mathrm e}^{-{\mathrm i} \phi Q^\Lambda }$ . In particular, $\rho ^\Lambda _{\phi +2\pi }=\rho ^\Lambda _{\phi }$ . Moreover,

$$ \begin{align*} \rho_\phi(A) = \lim_{\Lambda \to \mathbb Z^2} {\mathrm e}^{{\mathrm i} \phi Q^\Lambda} A {\mathrm e}^{-{\mathrm i} \phi Q^\Lambda} \end{align*} $$

for all $\phi \in \mathbb {R}$ and $\rho _{\phi +2\pi }=\rho _{\phi }$ .

Proof. The periodicity follows from the integrality of the spectrum of $Q^\Lambda $ and the group property of ${\mathrm e}^{{\mathrm i} \phi Q^\Lambda }$ . Existence of the limit is immediate for any $A\in {\mathcal A}^{\mathrm {loc}}$ since the sequence is eventually constant. The general case follows by density.

Notice that the same limiting procedure yields an automorphism $\rho ^\Lambda _{\phi }$ for any other infinite subset of ${\mathbb Z}^2$ . Below, we shall in particular consider $\Lambda $ to be a half-plane.

The LGA $\rho _\phi $ is the $U(1)$ - (equivalently $\mathbb {S}^1$ -) transformation associated with charge conservation. A $0$ -chain F is called Q-preserving (or $U(1)$ -symmetric) if $\rho _\phi (F_x(s)) = F_x(s)$ for all $\phi \in \mathbb R$ , $x \in \mathbb Z^2$ and $s \in \mathbb R$ . If F is a Q-preserving $0$ -chain, then $\alpha _{s\to t}^F$ such that $\rho _\phi \circ \alpha _{s\to t}^F = \alpha _{s\to t}^F\circ \rho _\phi $ for all $s,t,\phi \in \mathbb R$ . Consequently, an LGA is called Q-preserving if there exists a Q-preserving $0$ -chain which generates it.

3.3 Symmetric SRE states

A state $\omega _0 \in \mathcal S({\mathcal A})$ is called a product state iff

$$ \begin{align*}\forall X,Y \subset \mathbb Z^d, \quad X\cap Y = \emptyset, \quad A\in{{\mathcal A}}_X,B\in{{\mathcal A}}_Y, \qquad \omega_0(A B) = \omega_0(A) \omega_0(B). \end{align*} $$

Note that a SRE state is necessarily pure, since automorphisms preserve purity.

We remark that, with Q defined as in (30), a Q-symmetric state is necessarily homogeneous, namely $\omega \circ \theta = \omega $ since $\theta = \mathrm {e}^{\mathrm {i}\pi Q}$ .

Definition 3.10. Let $\omega $ be a pure state and H be a 0-chain. We say that $\omega $ is a ground state of H iff $\omega (A^*\delta ^H(A))\geq 0$ for all $A \in {\mathcal A}^{\mathrm {loc}}$ . Moreover we say that $\omega $ is a locally unique gapped ground state of H with gap $\Delta>0$ if

$$ \begin{align*} \omega(A^*\delta^H(A)) \geq \Delta \omega(A^*A) \end{align*} $$

for all $A \in {\mathcal A}^{\mathrm {loc}}$ such that $\omega (A)=0$ .

If $\omega $ is a pure state, we say that a 0-chain H is a parent Hamiltonian for $\omega $ if $\omega $ is a locally unique gapped ground state of H. An essential, although simple, result of this section is the following lemma, whose proof may be found in [Reference Bachmann, Bols and Rahnama10, Lemma 3.3].

Last but not least, we introduce the relevant notion of ‘deformation’ of states, which is used to define the topological phases. We remind the reader of the overall symmetry assumption introduced above.

3.4 Magnetic flux insertion at the origin

To probe the Hall conductivity associated with a state $\omega $ , we follow the Laughlin picture and focus on charge deficiency arising from a magnetic flux insertion in our system. We will show, see also [Reference Bachmann, Bols, De Roeck and Fraas9, Reference Kapustin and Sopenko33], that the Hall conductivity of $\omega $ is equal to $\mathcal {N}_\rho (\omega ,\omega ^D)$ where $\omega ^D$ is obtained from $\omega $ by inserting one unit of magnetic flux.

Before we proceed with the technical construction, we make a few comments to motivate what is ahead. Following the work of [Reference Avron, Seiler and Simon6], in the many-body setting we would like to create $\omega ^D$ out of $\omega $ by analogy to how $L^\ast P L$ is obtained out of the Fermi projection P, where the operator L is the Laughlin unitary given in (22). For quasi-free states, this lifting is done by considering the outer automorphism $\ell :\mathcal {A}\to \mathcal {A}$ via

$$ \begin{align*} a(f) \mapsto a(Lf)\qquad (f\in {\mathcal H}) \end{align*} $$

and extend linearly, and recalling that $\omega _P\circ \ell =\omega _{L^\ast P L}$ . Hence, also for non-quasi-free-states, the natural choice is then $\omega ^D := \omega \circ \ell $ with which one may hope to show that, under suitable assumptions on the initial state $\omega $ , the pair $\omega ,\omega ^D$ is $\rho $ -locally comparable and thus define the charge deficiency index of $\omega $ as $\mathcal {N}_\rho (\omega ,\omega \circ \ell )$ .

As proposed, this is impossible. Indeed, $P-L^\ast P L \in {\mathcal J}_3({\mathcal H})$ but the analysis in Section 2.2 indicates that if $P-Q$ is not Hilbert-Schmidt then $(\omega _P,\omega _Q)$ cannot be $\rho $ -locally comparable. In fact we made the even stronger assumption $P-Q\in {\mathcal J}_1({\mathcal H})$ .

This is remedied by considering another gauge (the ‘half-line gauge’) for the same flux insertion procedure. For noninteracting Fermions, this was studied in detail in [Reference De Nittis and Schulz-Baldes24]. In the interacting setting, this was carried out in [Reference Bachmann, Bols and Rahnama10], following the closely related [Reference Kapustin and Sopenko33], where it is shown that the corresponding automorphism applied to an initial SRE state yields a unitarily equivalent state. This extends to so-called invertible states, see Definition 3.26 below. Interestingly, in general the unitary depends on the initial state (unlike the universal L), and we shall see that continuity properties of ${\mathcal N}(\omega ,\omega ^D)$ must be dealt with carefully. In particular, there is no inner automorphism that directly implements a half-line gauge.

Fundamentally, the implementability of the flux insertion procedure for invertible states is due to the triviality of their superselection sectors [Reference Bachmann, Getz, Naaijkens and Wray15]. Beyond the invertible setting, one expects flux insertion to produce states that are truly inequivalent to the initial state in the sense that they carry anyonic quasi-particles [Reference Bachmann, Corbelli, Fraas and Ogata11].

For the sake of clarity, we consider the upper half-plane $\mathbb {P}^\uparrow := \mathbb Z \times \mathbb N$ , but in principle one could consider any half-plane, see, for example, [Reference Bachmann, Bols and Rahnama10].

$U(1)$ -transformations on half-planes.

We consider the $U(1)$ -transformation $\rho ^\uparrow := \rho ^{\mathbb {P}^\uparrow }$ associated with the upper half-plane. Let H be a Q-preserving $0$ -chain. For $\phi \in \mathbb R$ we consider the 0-chain $H^\uparrow (\phi )$ defined by

$$ \begin{align*}\forall x \in \mathbb Z^2,\qquad H^\uparrow(\phi)_x = \rho^\uparrow_{-\phi} (H_x),\end{align*} $$

Notice that $H^\uparrow (\phi )$ is Q-preserving for all $\phi $ .

Lemma 3.15. Let $\omega $ be an SRE state and let H be a Q-preserving parent Hamiltonian. Then $H^\uparrow (\phi )$ is a Q-preserving parent Hamiltonian of $\omega ^\uparrow _\phi = \omega \circ \rho ^\uparrow _\phi $ for all $\phi \in \mathbb R$ .

Note that if the state is ‘rotated forward’ while the Hamiltonian is ‘rotated backwards’, see also the proof in [Reference Bachmann, Bols and Rahnama10, Section 3.1.3].

Quasi-adiabatic continuation

The fact that $\omega $ is invariant under $U(1)$ -transformation indicates that $\omega ^\uparrow _\phi $ differs from $\omega $ only along the boundary of the upper half-plane. This intuition can be made precise by using the fact that $\omega $ has a gapped Hamiltonian: This is the role of the spectral flow [Reference Hastings and Wen32, Reference Bachmann, Michalakis, Nachtergaele and Sims16].

Proposition 3.16. There exists a $0$ -chain $K=(K(\phi ))_{\phi \in \mathbb R}$ such that

(31) $$ \begin{align} \omega^\uparrow_\phi = \omega \circ \alpha_\phi^K. \end{align} $$

The generator satisfies $K_x(\phi ) = \theta (K_x(\phi )) = \rho _{\phi '}(K_x(\phi ))$ for all $x\in \mathbb Z^2$ and $\phi ,\phi ' \in \mathbb R$ . Moreover, there exists a function $g \in \mathcal L$ such that

(32) $$ \begin{align} \forall x=(x_1,x_2) \in \mathbb Z^2,\quad \forall \phi \in \mathbb R, \qquad \| K_x(\phi) \| \leq g(|x_2|). \end{align} $$

The proof is in [Reference Bachmann, Bols and Rahnama10, Section 3.1.3], but we recall here that

(33) $$ \begin{align} K_x(\phi) := \int_{\mathbb R} \,\mathrm{d} t W(t) \alpha_t^{H^\uparrow(\phi)} \left(\dfrac{\,\mathrm{d} }{\,\mathrm{d} \phi} [H^\uparrow(\phi)_x]\right) \end{align} $$

where $W : \mathbb R \to \mathbb R$ is an odd bounded function decaying faster than any power at $t\to \pm \infty $ and with Fourier transform $\hat W(E)$ equals $-{\mathrm i}/E$ for $|E|>1$ . Notice that for a given $\phi \in \mathbb R$ , $\alpha _t^{H^\uparrow (\phi )}$ is the LGA generated by the t-independent generator $H^\uparrow (\phi )$ .

This proposition is essential: equation (31) states that $U(1)$ -transformations on half-spaces can be implemented on the state $\omega $ by another LGA, which is nonautonomous but acts nontrivially only along the boundary of the upper half-space, see (32).

Definition 3.17 (Magnetic flux insertion).

Let K be as in (33). For $\phi \in \mathbb R$ and $x=(x_1,x_2) \in \mathbb Z^2$ we define

The associated LGA, which we denote $\gamma _\phi = \alpha ^D_\phi $ , is called the flux insertion automorphism.

Remark 3.18. By construction $D(\phi )$ is g-localized near the half-line $\{x_1 <0, x_2=0\}$ . It corresponds to a defect line (or branch cut) of vector potential associated with a $\phi $ -flux insertion at the origin. The dynamics $\gamma _\phi $ is a caricature of the adiabatic switching of a magnetic flux from $0$ to $\phi $ .

Definition 3.19 (Defect state).

Let $\omega $ be an SRE state. The state $\omega ^D_\phi := \omega \circ \gamma _\phi $ is called the defect state at flux $\phi $ . At $\phi = 2\pi $ , we denote $\omega ^D = \omega ^D_{2\pi }$ .

We recall that the construction from the symmetric SRE state $\omega $ to the defect state $\omega ^D$ is summarized in Figure 1.

Remark 3.20. Since K is even and Q-preserving, so is D, and so $ \gamma _\phi \circ \rho _{\phi '} = \rho _{\phi '} \circ \gamma _\phi $ as well as $\gamma _\phi \circ \theta = \theta \circ \gamma _\phi $ . This implies immediately that $\omega ^D_\phi $ is Q-symmetric.

The two states $\omega ,\omega ^D_\phi $ differ from each other along the half-line $\{x_1 <0, x_2=0\}$ . At $\phi = 2\pi $ , the intrinsic locality of the state, namely the SRE assumption, implies that, in fact, $\omega ,\omega ^D$ differ from each other only around the flux insertion point. Indeed, far along the half-line, $\omega ^D \simeq \omega \circ \rho _{2\pi } = \omega $ because $\rho $ is a $U(1)$ -transformation. This implies that $(\omega ,\omega ^D)$ are locally comparable. In fact, they are even $\rho $ -locally comparable because the unitary can be shown to be almost localized near the origin, see Definition 2.1 and Definition 2.3. This is the content of the following proposition.

Proposition 3.21. Let $\omega \in \mathcal S_{\mathrm {sym}}^Q({\mathcal A})$ be SRE and let $\omega ^D$ be the associated defect state with flux $2\pi $ . Then there exists a homogeneous unitary $u\in {\mathcal A}^{\mathrm {al}}$ such that $\omega ^D = \omega \circ \mathrm {Ad}_u$ . Moreover, $u\in {\mathcal D}(\delta ^Q)$ .

The proof of the existence of u with all claimed properties except for the last one can be found in [Reference Bachmann, Bols and Rahnama10, Proposition 3.8], with the replacements of $\omega _{-\pi }$ , respectively $\omega _{\pi }$ , there by $\omega $ , respectively $\omega ^D$ , here. See also [Reference Artymowicz, Kapustin and Sopenko3, Appendix A]. Finally, $u\in {\mathcal A}^{\mathrm {al}}$ implies that it lies in the domain of $\delta ^Q$ , see Proposition 3.1.

Remark 3.22. (i) u is the many-body analog of the Laughlin unitary discussed above. It exists only for $\phi =2\pi $ (and could also be constructed for other integer fluxes). (ii) If $\omega $ is a symmetric product state, then one can take . (iii) The fact that u implements the Q-preserving transformation $\gamma _{2\pi }$ on the symmetric state $\omega $ does not imply that $\delta ^Q(u) = 0$ . In Example 2.13, the bilateral shift reads in second quantization $\alpha ^R(a_x) = a_{x+1}$ and for all $x\in {\mathbb Z}$ , while $\rho _\phi (a_x) = \mathrm {e}^{-\mathrm {i}\phi }a_x$ . Clearly, we have that $\alpha ^R\circ \rho _\phi = \rho _\phi \circ \alpha ^R$ . However, with $u = a_0 + a_0^*$ given there, we see that

$$ \begin{align*} -\mathrm{i} \delta^Q(u) = -\mathrm{i} \frac{d}{d\phi}\rho_\phi(u)\big\vert_{\phi=0} = -a_0 + a_0^*. \end{align*} $$

Proposition 3.23. Let $\omega \in \mathcal S_{\mathrm {sym}}^Q({\mathcal A})$ be SRE and let $u\in {\mathcal A}^{\mathrm {al}}$ be the unitary given by Proposition 3.21. Let

(34) $$ \begin{align} i(\omega,Q) := \mathrm{i}\omega(u^{*} \delta^Q(u)) \end{align} $$

If $\beta $ is a Q-preserving LGA then $i(\omega ,Q) =i(\omega \circ \beta ,Q)$ . Moreover, $i(\omega ,Q)$ is independent of the choice of symmetric parent Hamiltonian H for $\omega $ .

Proof. The definition (34) is nothing else than the abstract index (3) applied in the present specific context, namely

$$ \begin{align*} i(\omega,Q) = {\mathcal N}_{\rho}(\omega,\omega^D). \end{align*} $$

An LGA deformation of the initial state $\omega (s) = \omega \circ \beta _s$ yields a deformation of the flux insertion automorphism $\gamma _s$ and hence a family of states $\omega ^D(s) = \omega (s) \circ \gamma _s$ . If $\omega $ is a symmetric SRE state, so is $\omega (s)$ and so there is a family $u_s$ such that $\omega ^D(s) = \omega (s)\circ \mathrm {Ad}_{u_s}$ . In particular, $(\omega ^D(s), \omega (s))$ are $\rho $ -locally comparable for all s. Note that, in general, $\omega $ and $\omega (s)$ (as well as their defect counterparts) are not locally comparable, even for infinitesimal s. However, the state

$$ \begin{align*} \widetilde\omega(s) = \omega\circ\beta_s\circ\mathrm{Ad}_{u_s}\circ\beta_s^{-1} \end{align*} $$

can be expressed as $\widetilde \omega (s) = \omega \circ \mathrm {Ad}_{\beta _s(u_s)}$ , so that $(\widetilde \omega (s), \omega )$ are $\rho $ -locally comparable for all s and $i(\omega ,Q) = {\mathcal N}_{\rho }(\omega ,\tilde \omega (0))$ . The family of normalized vectors $\Omega _{\widetilde \omega (s)} = \pi _{\omega }(\beta _s(u_s))\Omega _{\omega }$ are vector representatives of the state $\widetilde \omega (s)$ in the GNS Hilbert space ${\mathcal H}_\omega $ of $\omega $ . If we again let $Q_\omega $ be the charge operator in the GNS representation, then $\Omega _{\widetilde \omega (s)}$ are all eigenvectors of $Q_\omega $ . We claim that $s\mapsto \Omega _{\widetilde \omega (s)}$ is a norm-continuous family of vectors in ${\mathcal H}_\omega $ . Since orthogonal unit vectors $\Psi ,\Phi $ have $\Vert \Psi - \Phi \Vert ^2 =2$ , we conclude that $\Omega _{\widetilde \omega (s)}$ are in a constant eigenspace. This in turn means that ${\mathcal N}_{\rho }(\omega ,\tilde \omega (s))$ is constant. We conclude that

$$ \begin{align*} i(\omega,Q) = {\mathcal N}_{\rho}(\omega,\tilde\omega(0)) = {\mathcal N}_{\rho}(\omega,\tilde\omega(s)) = {\mathcal N}_{\rho}(\omega(s),\omega^D(s)) \end{align*} $$

where the last equality is by Theorem 2.5(iv).

It remains to prove the continuity of $\Omega _{\widetilde \omega (s)}$ . We first claim that $s\mapsto \gamma _s$ is strongly continuous. If $\omega = \omega _0\circ \alpha $ , the family $\omega (s)$ is also SRE, with entangler $\beta _s\circ \alpha $ . Recalling the summary ??, we see that the parent Hamiltonian $H_s$ is simply given by $H_{s,x} = \beta _s^{-1} (H_x)$ and the interaction terms are continuous. Also, the family $H_s$ has constant spectral gap. It is now a consequence of the Lieb-Robinson bound, see, for example, [Reference Bratteli and William Robinson20, Theorem 6.2.11], that the dynamics $\alpha _t^{H^\uparrow _s(\phi )}$ is strongly continuous with respect to $H^\uparrow $ . In particular, the interaction terms defined by

$$ \begin{align*}K_{s,x}(\phi) := \int_{\mathbb R} \,\mathrm{d} t W(t) \alpha_t^{H_s^\uparrow(\phi)} \left(\dfrac{\,\mathrm{d} }{\,\mathrm{d} \phi} [H^\uparrow(\phi)_{s,x}]\right) \end{align*} $$

are continuous as a function of s. So is restriction $D_s$ of the $0$ -chain $K_s$ to the half-line, and the claimed strong continuity of $s\mapsto \gamma _{s,\phi }$ follows again by the Lieb-Robinson bound. With this, the formula

$$ \begin{align*} \widetilde\omega_2(s) = \omega_1\circ\beta_s\circ\gamma_{s,2\pi}\circ\beta_s^{-1} \end{align*} $$

implies immediately the weak-* continuity of $s\mapsto \widetilde \omega _2(s)$ . Hence $s\mapsto \Omega _{\widetilde \omega _2(s)}$ is weakly continuous (in ${\mathcal H}_\omega $ ) and since this is a family of constant norm, it is in fact norm continuous, concluding that part of the proof.

We turn to the invariance under the choice of parent Hamiltonian. For all $\phi \in [0,2\pi ]$ , let $\omega _\phi ^{D,j} = \omega \circ \gamma ^{j}_\phi , j=1,2$ be the defect states generated by the Hamiltonians $H_j$ . We claim that, for all $A\in {{\mathcal A}}$ ,

(35) $$ \begin{align} \vert \omega_0(A) - \tilde \omega^{D}_\phi(A)\vert\leq f(r)\Vert A \Vert \end{align} $$

where $\tilde \omega ^D_\phi = \omega _0\circ \alpha \circ \gamma ^1_\phi \circ (\gamma _\phi ^2)^{-1}\circ \alpha ^{-1}$ . Indeed, by the locality of $\alpha $ , we can replace $\gamma _{\phi }^{1} \circ \big ( \gamma _{\phi }^{2} \big )^{-1}$ by $\alpha _{\phi }^{K_1} \circ \big ( \alpha _{\phi }^{K_2} \big )^{-1}$ for all $\phi \in [0,2\pi ]$ and all A supported in $(\Lambda \cap B_0(r))\cup (\Lambda ^c\cap B_0(r))$ , where $\Lambda $ is the left half-plane. This is because, far away from the origin, the effect of the truncation $K\to D$ is irrelevant in the left half-plane and similarly on the right half-plane simply because both $\gamma _\phi ^D$ and $\alpha _\phi ^K$ act as identity, see [Reference Bachmann, Bols and Rahnama10, Lemma 3.7]. This, and the identity

$$ \begin{align*} \omega\circ\alpha_{\phi}^{K_1} \circ \big( \alpha_{\phi}^{K_2} \big)^{-1} = \omega_\phi^{\uparrow} \circ \big( \alpha_{\phi}^{K_2} \big)^{-1} = \omega, \end{align*} $$

implies (35) for all $\phi \in [0,2\pi ]$ and all A supported in $(\Lambda \cap B_0(r))\cup (\Lambda ^c\cap B_0(r))$ . That this can be lifted to all $A\in {{\mathcal A}}$ follows from [Reference Bachmann, Bols and Rahnama10, Appendix A], and in turn there is a unitary $w_\phi $ such that

$$ \begin{align*} \tilde \omega^D_\phi = \omega_0\circ\mathrm{Ad}_{w_{\phi}}. \end{align*} $$

The family of Q-symmetric states $\tilde \omega ^D_\phi $ is weakly-* continuous and so

$$ \begin{align*} {\mathcal N}_{\rho}(\omega_0, \tilde \omega^D_\phi) = {\mathcal N}_{\rho}(\omega_0,\omega_0) = 0 \end{align*} $$

by the continuity argument in the first part of the proof, with $\omega \to \omega _0$ and $\tilde \omega (s)\to \tilde \omega ^D_\phi $ , and Theorem 2.5(v). Evaluating this at $\phi = 2\pi $ , we conclude by Theorem 2.5(iv) that

$$ \begin{align*} 0 = {\mathcal N}_{\rho}(\omega_0, \tilde \omega^D_{2\pi}) = {\mathcal N}_{\rho}(\omega^{D,2}, \omega^{D,1}), \end{align*} $$

and by additivity

$$ \begin{align*} {\mathcal N}_{\rho}(\omega,\omega^{D,1}) ={\mathcal N}_{\rho}(\omega,\omega^{D,2}) + {\mathcal N}_{\rho}(\omega^{D,2},\omega^{D,1}) ={\mathcal N}_{\rho}(\omega,\omega^{D,2}) \end{align*} $$

which is what we had set to prove.

Remark 3.24. If $\omega $ is a symmetric product state, then the Q-preserving parent Hamiltonian of Lemma 3.11 can be chosen to be purely on-site. Since the $U(1)$ -transformation is also on-site, the Hamiltonian is invariant under $\rho ^{\uparrow }$ . Hence the quasi-adiabatic generator K of (33) is uniformly equal to $0$ . It follows that $\omega ^D = \omega $ , so that u can be taken to be the identity and we conclude that $i(\omega ,Q) = 0$ for an initial product state, as it should.

As we shall see in Section 4 below, the problem with Proposition 3.23 is that $i(\omega ,Q)=0$ for any quasi-free SRE state $\omega $ describing the (noninteracting) quantum Hall effect. In the next section, we extend the definition of the index to the larger class of invertible states which covers all topological ground states of noninteracting models. In principle, other physical models might provide examples of many-body SRE states with a nontrivial index, without stacking with an auxiliary system, but we are not aware of them.

3.5 Many-body index

So far, we have shown how to associate an index to an SRE state in the presence of a $U(1)$ -symmetry. The SRE assumption is used in two distinct steps: first to ensure that the state has a gapped parent Hamiltonian, and second to show that the associated defect state is $\alpha $ -locally comparable with the initial state. In both cases, the SRE assumption is a sufficient condition but in no way necessary. In this section, we extend the index to invertible states.

Stacking.

In the following we shall make use of stacking which is a procedure to increase the number of internal degrees of freedom. For this, let ${\mathcal A}^1$ and ${\mathcal A}^2$ respectively be the CAR algebras over $\ell ^2(\mathbb Z^2)\otimes \mathbb C^n$ and $\ell ^2(\mathbb Z^2)\otimes \mathbb C^m$ for $n,m \in \mathbb N$ . We denote by ${\mathcal A}^1 \hat \otimes {\mathcal A}^2$ the CAR algebra over $\ell ^2(\mathbb Z^2)\otimes \mathbb C^{n+m}$ .

Remark 3.25. A word about $\hat {\otimes }$ : The standard construction of the tensor product ${\mathcal A}^1\otimes {\mathcal A}^2$ is also a CAR algebra, but to see that one has to reshuffle its generators. Indeed, consider the simple example of space consisting of a single point and $n=m=1$ . Then $a,\tilde {a}$ , the two generators of ${\mathcal A}^1,{\mathcal A}^2\cong \mathrm {Mat}_2({\mathbb C})$ respectively, embed naturally into ${\mathcal A}^1\otimes {\mathcal A}^2$ as respectively, which commute rather than anti-commute. A choice that realizes the CAR is and $a_2 = (-1)^{a^{*} a}\otimes \tilde a$ . We use the symbol $\hat {\otimes }$ to denote this reshuffling of the generators (though the underlying C*-algebraic structure is, of course, identical).

The stacking operation on algebras naturally extends to states: for $\omega _1 \in \mathcal S({\mathcal A}^1)$ and $\omega _2 \in \mathcal S({\mathcal A}^2)$ we define $\omega _1 \hat \otimes \omega _2 \in \mathcal S({\mathcal A}^1 \hat \otimes {\mathcal A}^2)$ by

$$ \begin{align*}(\omega_1 \hat \otimes \omega_2)(A_1 \hat \otimes A_2) :=\omega_1(A_1)\omega_2(A_2)\qquad(A_1\in{\mathcal A}^1,A_2\in{\mathcal A}^2)\end{align*} $$

and extended beyond simple tensors by linearity.

Definition 3.26 (Invertible and stably SRE).

A state $\omega \in \mathcal S({\mathcal A})$ is invertible iff there is an auxiliary system ${\mathcal A}'$ (namely a CAR algebra over $\ell ^2(\mathbb Z^2) \otimes \mathbb C^m$ for some $m\in {\mathbb N}$ ) and a state $ \omega ' \in \mathcal S({\mathcal A}')$ such that $\omega \hat \otimes \omega '$ is SRE on ${\mathcal A} \hat \otimes {\mathcal A}'$ . If $\omega '$ can be taken to be a product state, then $\omega $ is called stably SRE.

Notice that a stably SRE state is necessarily invertible. The converse is true in one dimension [Reference Kapustin, Sopenko and Yang35]. That these two classes are not equal in two dimensions is a corollary of the results presented below, see Section 4.

Remark 3.27. (i) The relevance of invertible states lies in the observation that they correspond to integer quantum Hall systems. Any quasi-free state is invertible and so all noninteracting models have invertible ground states. However, it is believed that a nonvanishing Hall conductance is an obstruction to being stably SRE. Moreover, as Theorem 3.31 below shows, they do not cover fractional quantum Hall states. In fact, it can be shown that invertibility guarantees the absence of anyons, see [Reference Bachmann, Getz, Naaijkens and Wray15].

(ii) The stacking with product states, which appears in the notion of stably SRE, is analogous to the adding of some trivial bands in the single-particle setting. It also parallels a common procedure in K-theory.

Let $\omega $ be a state on ${{\mathcal A}}$ which is invertible, with inverse $\omega '$ on auxiliary algebra ${\mathcal A}'$ . We denote by $\hat \omega = \omega \hat \otimes \omega '$ and $\hat {\mathcal A} = {\mathcal A} \hat \otimes {\mathcal A}'$ . The algebra ${\mathcal A}'$ is equipped with a charge Q as well, so we are left with 3 distinct charges on $\hat {\mathcal A}$ :

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Notice that these three charges commute.

Definition 3.28. We say that $\omega $ is symmetric invertible if it is invertible with inverse $\omega '$ and if $\hat \omega = \omega \hat \otimes \omega '$ is both Q and $Q'$ -symmetric.

In particular if $\omega $ is symmetric invertible then $\hat \omega $ is also $\hat Q$ -symmetric. The parent Hamiltonian $\hat H$ provided by Lemma 3.11 is $\hat Q$ -symmetric. Defining then

$$ \begin{align*} \tilde H_x = \frac{1}{4\pi^2}\int_0^{2\pi}\int_0^{2\pi} \rho^{Q'}_{\phi_2}\circ\rho_{\phi_1}^Q(\hat H_x) \,\mathrm{d}\phi_1\,\mathrm{d}\phi_2 \end{align*} $$

for all $x\in {\mathbb Z}^2$ yields a parent Hamiltonian that is both Q- and $Q'$ - preserving – a fortiori still $\hat Q$ -preserving. Note that the commutativity of $Q,Q',\hat Q$ is essential here.

With this, we can carry out the construction described in Section 3.4 and define the flux insertion automorphism $\hat \gamma _\phi $ and a defect state $\hat \omega _\phi ^D:= \hat \omega \circ \hat \gamma _\phi $ , with respect to the total charge $\hat Q$ . By inspecting the formula (33), we conclude that $\hat \gamma _\phi $ is invariant under both $Q,Q'$ , and so the defect state $\hat \omega _\phi ^D$ is $Q,Q',\hat Q$ -symmetric. Hence we have the following

Proposition 3.29. A symmetric invertible state $\hat \omega $ has a symmetric parent Hamiltonian which is both Q- and $Q'$ -symmetric. Moreover the defect state $\hat \omega ^D$ , generated by $\hat Q$ , remains Q- and $Q'$ -symmetric.

Continuing as in Section 3.4, Proposition 3.21 yields $\hat u \in \hat {\mathcal A}^{\mathrm {al}}$ such that $\hat \omega ^D = \hat \omega \circ \mathrm {Ad}_{\hat u}$ . Since $\hat u\in \hat {{\mathcal A}}^{\mathrm {al}}$ , we further have that $\hat u\in {\mathcal D}(\delta ^{ Q})$ , namely $(\hat \omega ^D,\hat \omega )$ are $\rho ^Q$ -locally comparable.

Definition 3.30. Let $\omega $ be an invertible state as above. The index of $\omega $ is defined by

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We display the second expression to emphasize that the charge used in this definition is on the first subsystem ${{\mathcal A}}$ , on which $\omega $ is defined, although the flux insertion was carried out using the full charge $\hat Q$ .

With these preparations, we can now state our main result.

Theorem 3.31. Let $\omega $ be a symmetric invertible state. The index $\mathcal I(\omega )$ of Definition 3.30 has the following properties:

1. (i) Integrality: $\mathcal I(\omega ) \in \mathbb Z$ .

2. (ii) Well-definedness: $\mathcal I(\omega )$ is independent from the choice of symmetric parent Hamiltonian for $\hat \omega $ .

3. (iii) Continuity: If $\tilde \omega $ is Q-equivalent to $\omega $ , then $\tilde \omega $ is symmetric invertible and $ \mathcal I(\omega ) = \mathcal I(\tilde \omega )$ .

4. (iv) Additivity: If $\omega _1$ and $\omega _2$ are symmetric invertible states on ${\mathcal A}^1$ and ${\mathcal A}^2$ then $\omega _1 \hat \otimes \omega _2$ is a symmetric invertible state on ${\mathcal A}^1 \hat \otimes {\mathcal A}^2$ and $\mathcal I(\omega _1 \hat \otimes \omega _2) = \mathcal I(\omega _1)+\mathcal I(\omega _2)$ .

5. (v) Surjectivity: For all $n \in \mathbb Z$ , there exists an invertible state $\omega $ such that $\mathcal I(\omega ) = n$ .

Remark 3.32. (i) If the stacked state $\omega \hat \otimes \omega '$ is not only SRE, but it is $\hat Q$ -equivalent to a pure product state, then

$$ \begin{align*} \mathcal I(\omega')=-\mathcal I(\omega) \end{align*} $$

by (iii) and (iv) of Theorem 3.31.

(ii) At this point, the injectivity of the index remains an open question, namely whether two invertible states with the same index are necessarily Q-equivalent.

Proof. Since

the theorem is a direct consequence of Theorem 2.5 for the abstract index and Proposition 3.23 on $\hat {\mathcal A}$ . We check that the assumptions hold in the context of invertible states.

(i) The states $\hat \omega $ and $\hat \omega ^D = \hat \omega \circ \mathrm {Ad}_{\hat U}$ are $\hat \rho $ -locally comparable since $\hat U \in \hat {\mathcal A}^{\mathrm {al}}$ and is a $0$ -chain of $\hat {\mathcal A}$ . The state $\hat \omega = \omega \hat \otimes \omega '$ is Q-symmetric since $\omega $ is symmetric invertible. Moreover, $\hat \omega ^D$ is Q-symmetric as well by Proposition 3.29. Thus $\mathcal I(\omega ) = \mathcal N_{\rho ^Q}(\hat \omega , \hat \omega ^D) \in \mathbb Z$ by Theorem 2.5(ii).

(ii) This follows as in the proof of Proposition 3.23

(iii) Let $\tilde \omega $ be Q-equivalent to $\omega $ . Let $\beta $ be a Q-preserving LGA on ${\mathcal A}$ such that $\tilde \omega = \omega \circ \beta $ . Since $\omega $ is invertible, let $ \omega '$ be its inverse. One has $\omega \hat \otimes \omega ' = \hat \omega _0 \circ \hat \alpha $ with an $LGA \hat \alpha $ and a pure product state $\hat \omega _0$ on $\hat {\mathcal A}$ . Thus we have

Thus $\tilde \omega $ is invertible. Moreover it is Q-symmetric and since $\hat \omega $ is symmetric invertible then $ \omega '$ is $Q'$ -symmetric. Thus $\tilde \omega $ is symmetric invertible, so that $\mathcal I(\tilde \omega )$ is well defined. The invariance of the index $\mathcal I(\tilde \omega ) = \mathcal I(\omega )$ follows from Theorem 2.5(iv) on $\hat {\mathcal A}$ and goes along the same lines of the proof of Proposition 3.23 but on $\hat {\mathcal A}$ .

(iv) If $\omega _1$ and $\omega _2$ are symmetric invertible with inverses $ \omega ^{\prime }_1$ and $ \omega ^{\prime }_2$ , then $\omega _1 \hat \otimes \omega _2$ is symmetric invertible on ${\mathcal A}^1\hat \otimes {\mathcal A}^2$ with inverse $ \omega ^{\prime }_1\hat \otimes \omega ^{\prime }_2$ . The defect state construction factors on each algebra. In particular, up to the reshuffling

$$ \begin{align*}{\mathcal A}^1 \hat\otimes {\mathcal A}^2 \hat \otimes ({\mathcal A}^1)' \hat \otimes ({\mathcal A}^2)' \cong {\mathcal A}^1 \hat\otimes ({\mathcal A}^1)' \hat \otimes {\mathcal A}^2 \hat \otimes ({\mathcal A}^2)' = \hat {\mathcal A}^1 \hat \otimes \hat {\mathcal A}^2, \end{align*} $$

one has $(\hat \omega _1 \hat \otimes \hat \omega _2)^D = (\hat \omega _1)^D \hat \otimes (\hat \omega _2)^D$ so that, by Theorem 2.5(viii),

$$ \begin{align*} \mathcal I(\omega_1 \hat\otimes \omega_2) &= \mathcal N_{\hat \rho^{Q_1}\hat\otimes \hat \rho^{Q_2}}(\omega_1 \hat\otimes \omega_2, (\hat \omega_1)^D \hat \otimes (\hat \omega_2)^D) \cr &= \mathcal N_{\hat \rho^{Q_1}}(\omega_1, (\hat \omega_1)^D) + \mathcal N_{\hat \rho^{Q_2}}(\omega_2, (\hat \omega_2)^D) \cr &= \mathcal I(\omega_1)+\mathcal I(\omega_2). \end{align*} $$

(v) This part is an immediate consequence of the quasi-free case, see Theorem 4.3 and Proposition Proposition 4.4 below.

Remark 3.33. One may wonder if the abstract index could also find some relevance for the fractional quantum Hall effect. We believe that it may be the case in following situation, which is similar to the quantization argument in [Reference Bachmann, Corbelli, Fraas and Ogata11]: Even if the initial state is not invertible, it may be that the flux insertion operation becomes inner after q units of quantum flux have been inserted. If this is the case, the arguments above yield a fractional Hall conductance of the form $\frac {n}{q}$ . It should be noted that in such a framework there is no prediction about the possible fractions one may obtain, see [Reference Fröhlich and Zee27].

4.1 Beyond quasi-free states

We now have a complete picture of the relationship between the many-body index of a pair of pure states, the index of a pair of projections, the spectral flow associated with flux insertion and the spectral flow in a noninteracting picture. We point out that the stability of the many-body index and many-body perturbation theory allow one to go beyond the free setting. The proposition below is phrased in a translation-invariant setting to be able to use Proposition 4.1. This assumption can be dropped if Conjecture 4.2 is true.

Note that a locally unique gapped ground state is necessarily pure, see [Reference Tasaki45, Theorem A.3].

Proof. Since $H_\lambda $ has a uniform gapped ground state then the spectral flow [Reference Bachmann, Michalakis, Nachtergaele and Sims16], generated by a $0$ -chain $K_\lambda $ , is such that $\omega _{\lambda }= \omega _0\circ \alpha _{0\to \lambda }^K$ . Then since $\omega _0$ is quasi-free then it admits an inverse $\omega ^{\prime }_0$ on ${\mathcal A}'$ . Now,

and since $\omega _0 \hat \otimes \omega ^{\prime }_0$ is stably SRE, this shows that $\omega _1$ is invertible. Finally, since $H_\lambda $ is Q-preserving, then $\alpha ^K$ is Q-preserving. Thus $\omega _1$ is Q-equivalent to $\omega _0$ and so $\mathcal I(\omega _{1})=\mathcal I (\omega _0)$ by Theorem 3.31(iii).