Thermalization in Kitaev’s quantum double models via tensor network techniques

2.1 Notation and elementary properties

We use Dirac’s bra-ket notation. Vectors in a Hilbert space $\mathcal {H}$ will be represented as “kets” $|{\phi }\rangle $ , and the scalar product between $|{\phi }\rangle $ and $|{\psi }\rangle $ is written as $\langle {\phi }|{\psi }\rangle $ (which is antilinear in the first argument). The linear functional $|{\psi }\rangle \mapsto \langle {\phi }|{\psi }\rangle $ is then denoted as a “bra” $\langle {\phi }|$ . Rank-one linear maps will be written as $\left|{\psi }\right\rangle \!\!\left\langle {\phi }\right|$ .

Let us consider an arbitrary set $\Lambda $ , and associate to every site $x \in \Lambda $ a finite-dimensional Hilbert space $\mathcal {H}_{x} \equiv \mathbb {C}^{d}$ for a prefixed $d \in \mathbb {N}$ . As usual, for a finite subset $X \subset \Lambda $ , we define the corresponding space of states $\mathcal {H}_{X} := \otimes _{x \in X}\mathcal {H}_{x}$ and the space of bounded linear operators (observables) $\mathcal {B}_{X} := \mathcal {B}(\mathcal {H}_{X})$ endowed with the usual operator norm. We denote by the identity. We identify for $X \subset X' \subset \Lambda $ observables $\mathcal {B}_{X} \hookrightarrow \mathcal {B}_{X'}$ via the isometric embedding . Given an operator Q, the minimal region $X\subset \Lambda $ , such that $Q \in \mathcal {B}_X$ is called the support of Q. Let us observe that if $Q \in \mathcal {B}_{X}$ is a self-adjoint element, then and Q have the same eigenvalues $\lambda $ , and the corresponding eigenspaces are related via $V_{\lambda } \mapsto V_{\lambda } \otimes \mathcal {H}_{X' \setminus X}$ . In particular, . We define the spectral gap of such Q, denoted $\operatorname {gap}(Q)$ , as the difference of the two lowest (unequal) eigenvalues of Q. If Q has only one eigenvalue, we set $\operatorname {gap}(Q)=0$ .

A local Hamiltonian is defined in terms of a family of local interactions, that is, a map $\Phi $ that associates to each finite subset $Z \subset \Lambda $ a self-adjoint observable . For each finite $X \subset \Lambda $ , the corresponding Hamiltonian is the self-adjoint operator $H_{X} \in \mathcal {B}_{X}$ given by

$$\begin{align*}H_{X} = \sum_{Z \subset X}{\Phi_{Z}}\,. \end{align*}$$

Next, let us introduce some further conditions on the type of interactions we are going to deal with.

First, we are going to assume the lowest eigenvalue of $\Phi _{Z}$ is zero for each Z. This means that $\Phi _{Z} \geq 0$ , and thus $H_{X} \geq 0$ , for each finite subset $X \subset \Lambda $ . In general, from an arbitrary local interaction $\Phi _{Z}$ , we can always construct a new interaction satisfying this property by shifting each local term $\Phi _{Z}$ to , where $c_{Z}$ is the lowest eigenvalue of $\Phi _{Z}$ . As a consequence, each local Hamiltonian $H_{X}$ is shifted to , and its eigenvalues $\lambda $ are then shifted to $\lambda - (\sum _{Z \subset X} c_{Z}) $ , although the corresponding eigenspaces and the spectral gap are preserved. It should be mentioned that this shifting procedure introduces an energy constraint that can significantly impact certain physical properties of the original system. However, since our sole focus is on studying the spectral gap properties in relation to the orthogonal projectors onto the ground spaces, this argument appears reasonable for reducing the overall problem to this particular setting.

On the other hand, we are also going to assume that the local interaction $\Phi $ is frustration-free, namely, that for every finite subset $X \subset \Lambda $ , it holds that

$$\begin{align*}W_{X}:=\bigcap_{Z \subset X}{\ker(\Phi_{Z})} \neq \{0\} \,. \end{align*}$$

Let $P_{X} \in \mathcal {B}_{X}$ be the orthogonal projector onto $W_{X}$ . As a consequence of

(2.1) $$ \begin{align} \langle{\phi}|{H_X}|{\phi}\rangle = \sum_{Z\subset X} \langle{\phi}|{\Phi_Z}|{\phi}\rangle \ge 0\,, \end{align} $$

we immediately get that $\ker {(H_{X})}= W_{X}$ , and that the frustration-free condition is equivalent to each $H_{X}$ having zero as the lowest eigenvalue. In this case, $W_{X}$ is the ground space of $H_{X}$ , $P_{X}$ is the ground state projector, and the spectral gap of $H_{X}$ , in case the latter is nonzero, can be described as the largest positive constant satisfying that for every $|{\phi }\rangle \in \mathcal {H}_{X}$

(2.2) $$ \begin{align} \langle{\phi}|{P_X^{\perp}}|{\phi}\rangle \cdot \operatorname{gap}(H_{X}) \leq \langle{\phi}|{H_X}|{\phi}\rangle \,, \end{align} $$

where .

We conclude by remarking that for every $X \subset Y \subset \Lambda $ , we have $W_{Y} \subset W_{X}$ , where we are identifying $W_{X} \equiv \mathcal {H}_{Y \setminus X} \otimes W_{X}$ , and therefore $P_{Y} P_{X} = P_{Y}$ .

2.2 Spectral gap of local Hamiltonians

We will now recall a recursive strategy to obtain lower bounds to the spectral gap of frustration-free Hamiltonians described in [Reference Kastoryano and Lucia24], which we will also slightly improve over the original formulation. The main tool will be the following lemma, which is an improved version of [Reference Kastoryano and Lucia24, Lemma 14] in which the constant $(1-2c)$ has been improved to $(1-c)$ . The argument here is different and inspired by [Reference Kitaev, Shen and Vyalyi26, Lemma 14.4].

Lemma 2.1. Let $U, V, W$ be subspaces of a finite-dimensional Hilbert space $\mathcal {H}$ with corresponding orthogonal projectors $\Pi _{U}, \Pi _{V}, \Pi _{W}$ , and assume that $W \subset U \cap V$ . Then,

$$\begin{align*}\Pi_{U}^{\perp} + \Pi_{V}^{\perp} \geq (1 - c) \, \Pi_{W}^{\perp} \quad, \mbox{ where } \quad c:=\| \Pi_{U}\Pi_{V} - \Pi_{W}\|\,. \end{align*}$$

Moreover, $c \in [0,1]$ always holds, and $c \in [0,1)$ if and only if $U \cap V =W$ .

Proof. Let us start by observing that, since $W\subset U$ , it holds that $\Pi _W \Pi _U = \Pi _U \Pi _W = \Pi _W$ , and similarly for V. Thus, the constant c can be rewritten as

(2.3) $$ \begin{align} \begin{aligned} c & = \| (\Pi_{U} - \Pi_{W}) \, (\Pi_{V} - \Pi_{W}) \|\\ & = \| \Pi_{W}^{\perp} \, \Pi_{U} \, \Pi_{V} \, \Pi_{W}^{\perp}\|\\ & = \sup{\{ \, |\langle{a}| \Pi_{W}^{\perp} \, \Pi_{U} \, \Pi_{V} \, \Pi_{W}^{\perp} |{b}\rangle| \, \colon \, \| a\| \leq 1, \, \| b\| \leq 1 \rangle \}}\\ & = \sup{\{ \, | \langle{a}| \Pi_{U} \, \Pi_{V} |{b}\rangle | \, \colon \, a, b \in W^{\perp}, \, \| a\| \leq 1, \, \| b\| \leq 1 \rangle \}}\\ & = \sup{\{ \, | {\langle{a}|{b}\rangle} | \, \colon \, a \in U \cap W^{\perp}, \, b\in V \cap W^{\perp}, \, \| a\| \leq 1, \, \| b\| \leq 1 \rangle \}}. \end{aligned} \end{align} $$

From here, it immediately follows that $c \in [0,1]$ . Moreover, since $\mathcal {H}$ is finite-dimensional, the set over which the supremum is taken is compact, meaning that the supremum is always attained. Therefore, $c=1$ if and only if there exists $a\in U \cap W^{\perp }$ and $b\in V \cap W^{\perp }$ , such that $|\langle {a}| {b}\rangle | = \| a\| \| b\|$ . As the Cauchy–Schwarz inequality is only saturated by vectors which are proportional to each other, $c=1$ is equivalent to the fact that there exists an $a \in (U \cap V \cap W^{\perp }) \setminus \{ 0\}$ , or equivalently, that $W \subsetneq U \cap V$ . The first observation also implies that $\Pi _{U} = \Pi _{W} + \Pi _{W}^{\perp } \, \Pi _{U} \, \Pi _{W}^{\perp }$ and $\Pi _{V} = \Pi _{W} + \Pi _{W}^{\perp } \, \Pi _{V} \, \Pi _{W}^{\perp }$ , and therefore

$$\begin{align*}\Pi_{U}^{\perp} + \Pi_{V}^{\perp} = 2\Pi_W^{\perp} - \Pi_{W}^{\perp} \, (\Pi_{U} + \Pi_{V}) \Pi_{W}^{\perp}.\end{align*}$$

This allows us to reformulate the original inequality we aim to prove as

(2.4) $$ \begin{align} \Pi^{\perp}_{U} + \Pi_{V}^{\perp} \geq (1-c) \, \Pi_{W}^{\perp} \,\, \Leftrightarrow \,\, (1+c) \Pi_{W}^{\perp} \geq \Pi_{W}^{\perp} \, (\Pi_{U} + \Pi_{V}) \Pi_{W}^{\perp}\,. \end{align} $$

Let $|{x}\rangle $ be a norm-one eigenvector of $\Pi _{W}^{\perp } \, (\Pi _{U} + \Pi _{V}) \Pi _{W}^{\perp }$ with corresponding eigenvalue $\lambda>0$ . Note that $\Pi _{W}^{\perp } |{x}\rangle = |{x}\rangle $ necessarily, since eigenvectors with different eigenvalues are orthogonal, and W is contained in the kernel. Thus, to show that the right-hand side inequality of (2.4) holds, it is enough to check that $\lambda \leq 1+c$ necessarily. For that, let us write

$$ \begin{align*} \Pi_{U}|{x}\rangle & = \lambda_{U} \, |{x_{U}}\rangle \quad \mbox{ for some } \quad |{x_{U}}\rangle \in U \cap W^{\perp}\,, \, \langle{x_{U}}|{x_{U}}\rangle = 1\,, \, \lambda_{U} \geq 0\,,\\ \Pi_{V}|{x}\rangle & = \lambda_{V} \, |{x_{V}}\rangle \quad \mbox{ for some } \quad |{x_{V}}\rangle \in V \cap W^{\perp}\,, \, \langle{x_{V}}|{x_{V}}\rangle = 1\,, \, \lambda_{V} \geq 0\,. \end{align*} $$

On the one hand, we have

$$ \begin{align*} \lambda = \langle{x}| \Pi_{U} + \Pi_{V} |{x}\rangle & = \langle{x}| \Pi_{U} |{x}\rangle + \langle{x}| \Pi_{V} |{x}\rangle \\ & \, = \langle{x}| \Pi_{U}^{2} |{x}\rangle + \langle{x}| \Pi_{V}^{2} |{x}\rangle \\ & \, = \, \lambda_{U}^{2} + \lambda_{V}^{2}, \end{align*} $$

and, on the other hand

$$ \begin{align*} \lambda^{2} & = \langle{x}| (\Pi_{U} + \Pi_{V}) \, (\Pi_{U} + \Pi_{V}) |{x}\rangle \\ & = \lambda_{U}^{2} + \lambda_{V}^{2} + \langle{x}|\Pi_{U} \Pi_{V}|{x}\rangle + \langle{x}|\Pi_{V} \Pi_{U}|{x}\rangle \\ & = \lambda_{U}^{2} + \lambda_{V}^{2} + 2 \, \lambda_{U} \, \lambda_{V} \, \operatorname{Re}\langle{x_{U}}|{x_{V}}\rangle \,. \end{align*} $$

Combining both equalities we get, denoting $c_{x}:=|\operatorname {Re}\langle {x_{U}}|{x_{V}}\rangle |$ ,

$$ \begin{align*} (1+c_{x}) \lambda - \lambda^{2} & = c_{x} \left( \lambda_{U}^{2} + \lambda_{V}^{2} \right) - 2 \lambda_{U} \lambda_{V} \operatorname{Re}\langle{x_{U}}|{x_{V}}\rangle \\ & \geq c_{x}(\lambda_{U}^{2} + \lambda_{V}^{2} - 2 \lambda_{U} \lambda_{V}) = c_{x}(\lambda_{U} - \lambda_{V})^{2} \geq 0\,. \end{align*} $$

Since $\lambda>0$ , we conclude that

$$\begin{align*}\lambda \leq 1 + c_{x} \leq 1+c, \end{align*}$$

where the last inequality follows from (2.3). This concludes the argument.

This lemma has important implications for frustration-free local Hamiltonians, when applied to the ground state subspaces $W_X$ and their associated orthogonal projections $P_X$ . The frustration-free condition yields that, for $X, Y \subset \Lambda $ , the ground state subspaces satisfy $ W_{X \cup Y} \subset W_{X} \cap W_{Y}$ . As a consequence of Lemma 2.1,

$$\begin{align*}\| P_{X \cup Y} - P_{X} P_{Y}\| = \| (P_{X \cup Y} - P_{X}) (P_{X \cup Y} - P_{Y})\| \in [0, 1]\,. \end{align*}$$

Moreover, $W_{X \cup Y} = W_{X} \cap W_{Y}\,$ if and only if $\| P_{X \cup Y} - P_{X} P_{Y}\| \in [0, 1)$ . This happens whenever $\Lambda $ is a metric space, $\Phi $ has finite range $r>0$ , namely, that $\Phi _{X}=0$ if the diameter of X is larger than r, and the distance $d(X \setminus Y, Y \setminus X)$ is greater than r, since in this case, every subset $Z \subset X \cup Y$ with $\Phi _{Z} \neq 0$ is either contained in X or Y, so that

$$\begin{align*}W_{X \cup Y} = \bigcap_{Z \subset X \cup Y} \ker{\Phi_{Z}} = \bigcap_{Z \subset X} \ker{\Phi_{X}} \cap \bigcap_{Z \subset Y} \ker{\Phi_{Z}} = W_{X} \cap W_{Y} \,. \end{align*}$$

Definition 2.2 (Spectral gap).

For each finite subset $Y \subset \Lambda $ , let us denote by $\operatorname {gap}(H_{Y})$ , or simply $\operatorname {gap}(Y)$ , the spectral gap of $H_{Y}$ , namely, the difference between the two lowest unequal eigenvalues of $H_{Y}$ . If it has only one eigenvalue, then we define $\operatorname {gap}(Y) = 0$ . Given a family $\mathcal {F}$ of finite subsets of $\Lambda $ , we say that the system of Hamiltonians $(H_{Y})_{Y \in \mathcal {F}}$ is gapped whenever

$$\begin{align*}\operatorname{gap}(\mathcal{F}):=\inf{\{ \operatorname{gap}(Y) \colon Y \in \mathcal{F} \}}> 0\,. \end{align*}$$

Otherwise, it is said to be gapless.

The following result allows to relate the gap of two families. It adapts a result from [Reference Kastoryano and Lucia24, Section 4.2].

Theorem 2.3. Let $\mathcal {F}$ and $\mathcal {F}'$ be two families of finite subsets of $\Lambda $ . Suppose that there are $s \in \mathbb {N}$ and $\delta \in [0,1]$ satisfying the following property: for each $Y \in \mathcal {F}' \setminus \mathcal {F}$ , there exist $(A_{i}, B_{i})_{i=1}^{s}$ pairs of elements in $\mathcal {F}$ , such that:

1. (i) $Y = A_{i} \cup B_{i}$ for each $i=1, \ldots , s$ ,

2. (ii) $(A_{i} \cap B_{i}) \cap (A_{j} \cap B_{j}) = \emptyset $ whenever $i \neq j$ ,

3. (iii) $\|P_{A_{i}}P_{B_{i}} - P_{Y}\| \leq \delta $ for every $i=1, \ldots , s$ .

Then,

$$\begin{align*}\operatorname{gap}(\mathcal{F}') \, \geq \, \frac{1-\delta}{1+ \frac{1}{s}} \, \operatorname{gap}(\mathcal{F}) \, \underset{(\text{if }\delta < 1)}{\geq} \, \exp\left[ -\frac{\delta}{1- \delta} - \frac{1}{s}\right] \, \operatorname{gap}(\mathcal{F}) \,. \end{align*}$$

Proof of Theorem 2.3.

Let $Y \in \mathcal {F}' \setminus \mathcal {F}$ , and let $(A_{i}, B_{i})_{i=1}^{s}$ be the family of pairs satisfying $(i)$ – $(iii)$ provided by the hypothesis. To prove the first inequality, we can assume that $\delta < 1$ and $\operatorname {gap}(\mathcal {F})>0$ , since otherwise, the inequality is obvious. Applying Lemma 2.1, and using (2.2), we can estimate

$$ \begin{align*} \langle{x}| P_{Y}^{\perp} |{x}\rangle \, = \, \frac{1}{s} \sum_{i=1}^{s}{\langle{x}| P_{Y}^{\perp} |{x}\rangle } \,\, & \leq \,\, \frac{1}{s} \sum_{i=1}^{s} \frac{1}{1-\delta} \left(\langle{x}| P_{A_{i}}^{\perp} |{x}\rangle + \langle{x}| P_{B_{i}}^{\perp} |{x}\rangle \right) \\ & \leq \,\, \frac{1}{1-\delta} \, \frac{1}{s} \sum_{i=1}^{s} \frac{1}{\operatorname{gap}(\mathcal{F})} \left(\langle{x}| H_{A_{i}} |{x}\rangle + \langle{x}| H_{B_{i}} |{x}\rangle \right)\\ & \leq \,\, \frac{1}{1-\delta} \, \frac{1}{\operatorname{gap}(\mathcal{F})} \, \frac{1}{s} \sum_{i=1}^{s} \langle{x}| H_{Y} + H_{A_{i} \cap B_{i}} |{x}\rangle \\ & \leq \,\, \frac{1}{1- \delta} \, \frac{1}{\operatorname{gap}(\mathcal{F})} \, \, \langle{x}| H_{Y} + \frac{1}{s} \sum_{i=1}^{s} H_{A_{i} \cap B_{i}} |{x}\rangle \\ & \leq \,\, \frac{1}{1- \delta} \, \frac{1}{\operatorname{gap}(\mathcal{F})} \, \, \left( 1+ \frac{1}{s} \right) \, \langle{x}| H_{Y} |{x}\rangle \,. \end{align*} $$

Notice that in the third line, we have used that $H_{A_{i}} + H_{B_{i}} \leq H_{Y} + H_{A_{i} \cap B_{i}}$ , which holds since all the local interactions are positive semidefinite. Therefore, again by (2.2), it holds that

$$\begin{align*}\operatorname{gap}(Y) \geq \frac{1- \delta}{1+\frac{1}{s}} \, \operatorname{gap}(\mathcal{F})\quad \text{whenever} \quad Y \in \mathcal{F}' \setminus \mathcal{F}\,. \end{align*}$$

On the other hand, if $Y \in \mathcal {F}' \cap \mathcal {F}$ , then $\operatorname {gap}(Y) \geq \operatorname {gap}(\mathcal {F}) $ by definition. Hence, we conclude the that the first inequality holds. To obtain the second inequality, we simply use twice that $(1+x)^{-1} \geq e^{-x}$ for every $x \geq 0$ .

In the next section, we will apply Theorem 2.3 in order to bound the spectral gap of a quantum spin Hamiltonian on an arbitrarily large torus (with periodic boundary conditions) in terms of the spectral gap of the same model on a finite family of rectangles with open boundary conditions. This is reminiscent to the bounds on the spectral gap based on the local gap thresholds [Reference Anshu3, Reference Gosset and Mozgunov18, Reference Lemm and Mozgunov20, Reference Lemm21, Reference Knabe27]. The reason for which we are following a different approach is that the constants appearing in the spectral gap thresholds are highly dependent on the specific shape and range of the interactions of the Hamiltonian, and in our case, we will have to consider $\beta $ -dependent interaction length. Therefore, it will be unfeasible to verify the local gap threshold conditions for our models. The connection between the control of quantities of the type ${\lVert }{P_{X\cup Y} - P_X P_Y}{\rVert }$ and spectral gap estimates originated in the seminal work on finitely correlated states [Reference Fannes, Nachtergaele and Werner16], later extended to more general spin models [Reference Nachtergaele32]. The main difference between that approach and the one of [Reference Kastoryano and Lucia24], which we are following here, is due to the way in which we are growing the lattice: while the methods of [Reference Fannes, Nachtergaele and Werner16, Reference Nachtergaele32] consider a single increasing sequence of regions, Theorem 2.3 permits more rich families of subsets, thus allowing us to keep the shape of their intersections more well-behaved (i.e., they will always be rectangles or cylinders).

2.3 Periodic boundary conditions on a torus

To describe the periodic boundary conditions case, we have to introduce further notation. For each natural N, let us denote by $\mathbb {S}_{N}$ the quotient $\mathbb {R}/\sim $ , where we relate $x \sim x+N$ for every $x \in \mathbb {R}$ . Note that we can identify $\mathbb {S}_{N} \equiv [0, N)$ .

We will take as $\Lambda _{N}$ , the set where the spins of the system are located, the set of midpoints of the edges $\mathcal {E}_{N}$ of the square lattice on the torus $\mathbb {S}_{N} \times \mathbb {S}_{N}$ (as this is the setting in which the quantum double models are defined). We will identify each point of $\Lambda _{N}$ with the corresponding edge from $\mathcal {E}_{N}$ , so that we will indistinctly use $\Lambda _{N}$ or $\mathcal {E}_{N}$ (see Figure 1).

Figure 1 The square lattice on the torus (left), and the quantum spin system with spins on the midpoints of the edges (right). The marks on the borders of both squares represent the pairwise identification of edges, a standard depiction of the torus in topology. A similar notation will be used for the cylinder, where only one pair of edges is identified.

Let us recall the notion of a (closed) interval in $\mathbb {S}_{N}$ . Given $x,y \in \mathbb {S}_{N}$ , we denote by $d_{+}(x,y)$ the unique $0 \leq c < N$ , such that $x+c \sim y$ . Then, we define the interval $[a,b]$ as the set $\{ x \in \mathbb {S}_{N} \colon d_{+}(a, x) \leq d_{+}(a,b) \}$ . We are only going to consider intervals with integer endpoints, that is, $a,b \in \mathbb {Z}_{N}$ . A proper rectangle $\mathcal {R}$ in $\mathbb {R}^{2}$ or $\mathbb {S}_{N} \times \mathbb {S}_{N}$ is a Cartesian product of intervals $\mathcal {R}=[a_{1}, b_{1}] \times [a_{2}, b_{2}]$ (with integer endpoints). Its number of plaquettes per row is then $d_{+}(a_{1},b_1)$ and per column is $d_{+}(a_{2},b_2)$ . Shortly, we say that $\mathcal {R}$ has dimensions $d_{+}(a_{1},b_1)$ and $d_{+}(a_{2},b_2)$ . A cylinder is a Cartesian product of the form $\mathbb {S}_{N} \times [a,b]$ or $[a,b] \times \mathbb {S}_{N}$ . We will refer simply as rectangles to proper rectangles, cylinders, and the whole torus $\mathbb {S}_{N} \times \mathbb {S}_{N}$ . In an abuse of notation, we will identify $\mathcal {R}$ with $\mathcal {R} \cap \Lambda $ and often write $\mathcal {R} \subset \Lambda $ and $\mathcal {H}_{\mathcal {R}}$ to denote the associated Hilbert space (see Figure 2).

Figure 2 Examples of rectangles as a subset of the square lattice $\Lambda _{N} \equiv \mathcal {E}_{N}$ . Above, two examples of proper rectangles, whereas below, two examples of cylinders. In each case, we present two pictures: on the right, we highlight the spins belonging to the region, while on the left, we highlight the edges. We will use this latter representation in the forthcoming pictures.

We are going to define, for every $N,r \in \mathbb {N}$ with $N \geq r \geq 2$ , the following sets of rectangular regions in $\mathcal {E}_{N}$ :

• $\triangleright $ $\mathcal {F}_{N}$ is the set of all rectangular regions having at least two plaquettes per row and per column.

• $\triangleright $ $\mathcal {F}_{N}^{torus}$ is the family consisting of only one element, the whole torus.

• $\triangleright $ $\mathcal {F}_{N}^{cylin}$ is the family of all cylinders having at least two plaquettes per row and per column.

• $\triangleright $ $\mathcal {F}_{N, r}^{rect}$ is the family of all proper rectangles having at least two and at most r plaquettes per row and per column.

Martingale condition on local projectors

For each $X \in \mathcal {F}_{N}$ , let $\Pi _{X}$ be an orthogonal projector onto a subspace of $\mathcal {H}_{X}$ , such that $(\Pi _{X})_{X}$ satisfies the frustration-free condition: $\Pi _{X} \Pi _{Y}=\Pi _{Y}\Pi _{X} =\Pi _{Y}$ for every pair of rectangular regions $X \subset Y$ . Notice that the family of projectors $(P_{X})_{X}$ associated to a frustration-free Hamiltonian, as it was defined in Section 2.1, satisfies this condition.

Estimating the gap from below

Let us fix a local interaction $\Phi $ on the torus $\mathcal {E}_{N}$ defining a frustration-free Hamiltonian. For each rectangular region $X \subset \mathcal {E}_{N}$ , let us denote by $P_{X}$ the orthogonal projector onto the ground space of $H_{X}$ . Let us, moreover, assume that the family of projectors $(P_{X})_{X}$ satisfies the martingale condition for a decay function $\delta (\ell )$ as in Definition 2.4.

Theorem 2.5. If $\delta (\lfloor N/2-1 \rfloor )<1/2$ , then

$$\begin{align*}\operatorname{gap}(\mathcal{F}_{N}^{torus}) \, \geq \, \frac{1}{4} \operatorname{gap}(\mathcal{F}_{N}^{cylin}) \, \geq \, \frac{1}{16} \operatorname{gap}(\mathcal{F}_{N,N}^{rect})\,.\end{align*}$$

Proof. Let us first compare $\operatorname {gap}(\mathcal {F}_{N}^{torus})$ and $\operatorname {gap}(\mathcal {F}_{N}^{cylin})$ . We consider a decomposition of the torus $\Lambda _{N}$ into four regions $A,B,C,B'$ as in (2.8), so that $\mathcal {C}_{1}:=B'AB$ and $\mathcal {C}_{2}:=BCB'$ are cylinders having dimensions $N-1$ and N belonging to $\mathcal {F}_{N}^{cylin}$ and whose intersection consists of two cylinders B and $B'$ , each having N plaquettes per column and at least $\lfloor N/2-1\rfloor $ plaquettes per row.

(2.8)

Applying the martingale condition from Definition 2.4 (iii), we deduce that

(2.9) $$ \begin{align} \| P_{\mathcal{C}_{1}} P_{\mathcal{C}_{2}} - P_{\Lambda_{N}} \| \, \leq \, \delta(\lfloor N/2-1 \rfloor) < \frac{1}{2}\,, \end{align} $$

and so, by Theorem 2.3 with $s=1$ and $\delta =1/2$ , we can estimate

$$\begin{align*}\operatorname{gap}(\mathcal{F}_{N}^{torus}) \, \geq \, \frac{1}{4} \, \operatorname{gap}(\mathcal{F}_{N}^{cylin}) \,. \end{align*}$$

Next, we compare $\mathcal {F}_{N}^{cylin}$ and $\mathcal {F}_{N,N}^{rect}$ . Given a cylinder $\mathcal {C} \in \mathcal {F}_{N}^{cylin}$ , we can split it along the wrapping direction into four regions $A,B,C,B'$ as in (2.10), so that $\mathcal {R}_{1}:=B'AB$ and $\mathcal {R}_{2}:=BCB'$ are proper rectangles having N plaquettes along the splitting direction, belonging to $\mathcal {F}_{N}^{rect}$ , and whose intersection consists of two proper rectangles B and $B'$ having at least $\lfloor N/2-1 \rfloor $ plaquettes along the splitting direction.

(2.10)

Thus, applying the martingale condition from Definition 2.4(ii), we deduce that

$$\begin{align*}\| P_{\mathcal{R}_{1}} P_{\mathcal{R}_{2}} - P_{\mathcal{C}} \| \, \leq \, \delta(\lfloor N/2-1 \rfloor) < \frac{1}{2}\,, \end{align*}$$

and so, by Theorem 2.3 with $s=1$ and $\delta =1/2$ , we can estimate

(2.11) $$ \begin{align} \operatorname{gap}(\mathcal{F}_{N}^{cylin}) \, \geq \, \frac{1}{4} \, \operatorname{gap}(\mathcal{F}_{N,N}^{rect})\,. \end{align} $$

Combining (2.9) and (2.11), we conclude the result.

Theorem 2.6. For fixed integers $N \geq r \geq 16$ , let us denote $\delta _{k}:= \delta (\lfloor \tfrac {r}{4}(\sqrt {9/8}\,)^{k}\rfloor \,)$ and $s_k:=\lfloor (\sqrt {4/3}\,)^{k} \rfloor $ for each integer $k \geq 0$ . Then, we can estimate from below

$$\begin{align*}\operatorname{gap}(\mathcal{F}_{N,N}^{rect}) \, \geq \, \left(\, \prod_{k=0}^{\infty} \frac{1-\delta_{k}}{1+\frac{1}{s_{k}}}\right) \, \operatorname{gap}(\mathcal{F}^{rect}_{N,r})\,. \end{align*}$$

Notice that the infinite product that appears in the previous expression is convergent to a positive value whenever the decay function $\delta (\ell )$ decays polynomially fast, namely, $\delta (\ell ) = O(\ell ^{-\alpha })$ for some $\alpha>0$ .

Proof. The idea of the proof is inspired by [Reference Cesi9]. Fix $N,r$ as above, for each $k \geq 0$ , let $\mathcal {G}_{k}$ be the family of all proper rectangles in $\mathcal {F}_{N,N}^{rect}$ of dimensions a and b satisfying

$$\begin{align*}a,b \geq 2 \quad \mbox{ and } \quad a \cdot b \leq r (3/2)^{k}\,.\end{align*}$$

Observe that $\mathcal {G}_{k} \subset \mathcal {G}_{k+1}$ for every $k \geq 0$ and that $\mathcal {G}_{k} = \mathcal {F}_{N,N}^{rect}$ for k large enough. Next, let $Y \in \mathcal {G}_{k+1} \setminus \mathcal {G}_{k}$ of dimensions a and b. Then

$$\begin{align*}r (3/2)^{k} < a \cdot b \leq r (3/2)^{k+1}\,. \end{align*}$$

We can assume without loss of generality that $a \leq b$ , so using the previous inequalities, we can estimate from below (recall that $r \geq 16$ )

(2.12) $$ \begin{align} 4 \leq \sqrt{r} (\sqrt{3/2}\,)^{k} \leq b\,. \end{align} $$

Let us define

$$\begin{align*}s_{k}:= \lfloor (\sqrt{4/3}\,)^{k} \rfloor \quad \mbox{and} \quad \ell_{k} := \left\lfloor b/(4 s_{k}) \right\rfloor \,. \end{align*}$$

Observe that by (2.12)

(2.13) $$ \begin{align} \ell_{k} \geq \left\lfloor\frac{b}{4s_{k}}\right\rfloor \geq \left\lfloor \frac{\sqrt{r}}{4} (\sqrt{3/2}\,)^{k} (\sqrt{3/4}\,)^{k} \right\rfloor = \left\lfloor\frac{\sqrt{r}}{4} (\sqrt{9/8}\,)^{k}\right\rfloor \geq 1\,, \end{align} $$

and for every $j = 0, \ldots , s_{k}-1$

(2.14) $$ \begin{align} (2j+1) \ell_{k} \leq 2 s_{k} \ell_{k} -\ell_{k} \leq \frac{b}{4}-1\,. \end{align} $$

Next, let us identify our rectangle Y with $[0,b] \times [0,a]$ , and consider the subrectangles (see Figure 3)

$$ \begin{align*} A_{j} := \left[0, \lceil b/3 \rceil + (2j+1) \ell_k \, \right] \times [0,a] \quad ,\quad B_{j} := \left[\lceil b/3 \rceil + 2j \ell_k, b \, \right] \times [0,a] \,. \end{align*} $$

Note that, applying (2.14), we deduce that $A_{j}$ is contained in a rectangle of dimensions a and $\lceil b/3\rceil +b/4-1 \leq 2b/3$ . Similarly, $B_{j}$ is contained in a rectangle of dimensions a and $b-\lceil b/3 \rceil \leq 2b/3$ . Since $a \geq 2$ , $2b/3 \geq 2$ , and $a \cdot (2b/3) \leq r(3/2)^{k}$ , we conclude that $A_{j}, B_{j} \in \mathcal {G}_{k}$ by definition. Moreover, the intersections

$$\begin{align*}A_{j} \cap B_{j} = \left[ \, \lceil b/3 \rceil + 2j \ell_k, \lceil b/3 \rceil + (2j+1) \ell_k \, \right] \times [0,a] \, \end{align*}$$

are disjoint, that is $A_{j} \cap B_{j} \cap A_{j'} \cap B_{j'} = \emptyset $ whenever $j \neq j'$ , and have $\ell _{k}$ plaquettes per row and a plaquettes per column. Using the martingale condition from Definition 2.4(i) and (2.13)

$$\begin{align*}\| P_{A_{j}} P_{B_{j}} - P_{Y} \| \leq \delta(\ell_{k}) \leq \delta(\lfloor\tfrac{\sqrt{r}}{4}(\sqrt{9/8})^{k}\rfloor)\,.\end{align*}$$

Figure 3 Split of the rectangle Y in the proof of Theorem 2.6. The thick subrectangles correspond to the intersections $A_{i} \cap B_{i}$ , being disjoint and having width equal to $\ell _{k}$ .

Applying now Theorem 2.3, we deduce that

$$\begin{align*}\operatorname{gap}(\mathcal{G}_{k+1}) \geq \frac{1-\delta_{k}}{1+\frac{1}{s_{k}}} \operatorname{gap}(\mathcal{G}_{k}) \geq \ldots \geq \, \left(\prod_{j=0}^{k} \frac{1-\delta_{j}}{1+\frac{1}{s_{j}}}\right) \,\, \operatorname{gap}(\mathcal{G}_{0})\,. \end{align*}$$

Finally, noticing that $\mathcal {G}_{0} \subset \mathcal {F}_{r,N}^{rect}$ , we conclude the result.

3.1 Tensor notation

Let us denote $[d]:=\{ 0,1,\ldots , d-1\}$ for each $d \in \mathbb {N}$ . Recall that a tensor with n indices is simply an element $T \in \mathbb {C}^{[d_{1}] \times \ldots \times [d_{n}]} $ , where each $d_{j} \in \mathbb {N}$ is called the dimension of the j-th index. We will employ the usual notation

$$\begin{align*}T_{\alpha} = T_{\alpha_{1} \ldots \alpha_{n}} \quad , \quad \alpha = (\alpha_{1}, \ldots, \alpha_{n}) \in [d_{1}] \times \ldots \times [d_{n}]\,. \end{align*}$$

By definition, a tensor with zero indices will be a scalar $T \in \mathbb {C}$ . We can represent tensors in the form of a ball with a leg for each index:

Let us describe the basic operations that we will perform with tensors. Given a tensor with (at least) two indices, say $\alpha _{1}$ and $\alpha _{2}$ with dimensions $d_{1}$ and $d_{2}$ , respectively, we can combine them into one index $\gamma $ of dimension $d_{1}\cdot d_{2}$ , so that the resulting tensor has one index less. This process can be iterated to combine several indices into one index. Graphically, we have the following example:

Given two tensors $T=(T_{\alpha })$ and $S=(S_{\beta })$ with m and n indices, respectively, we define its tensor product as the unique tensor with $n+m$ indices given by $T \otimes S :=(T_{\alpha } \cdot S_{\beta })$ . For instance:

If a tensor $(T_{\alpha })$ has two indices with the same dimension, say $\alpha _{1}$ and $\alpha _{2}$ , then we can contract them, resulting in a tensor with $n-2$ indices $S_{\alpha _{3} \ldots \alpha _{n}} = \sum _{j} T_{jj\alpha _{3}\ldots \alpha _{n}}$ . Combining this operation with the tensor product of tensors, we define the contraction of tensors: given two tensors $(T_{\alpha })$ and $(S_{\beta })$ having both an index with the same dimension, say $\alpha _{1}$ and $\beta _{1}$ , we can contract these indices to generate a new tensor $(\sum _{j}T_{j\alpha _{2}\ldots \alpha _{n}} \cdot S_{j\beta _{2}\ldots \beta _{m}})$ . Graphically, we have the following example:

If two tensors have the same indices with the same dimensions $(T_{\alpha })_{\alpha }$ and $(S_{\alpha })_{\alpha }$ , we can define their tensor sum $T+S:=(T_{\alpha }+S_{\alpha })_{\alpha }$ . Given a tensor $(T_{\alpha })_{\alpha }$ and a scalar $\lambda \in \mathbb {C}$ , we define $\lambda T := (\lambda T_{\alpha })_{\alpha }$ .

If we identify each index j with a Hilbert space $\mathbb {C}^{d_{j}}$ , we can interpret a tensor T as the coefficients of a ket $|{\psi }\rangle $ in the computational basis

$$\begin{align*}|{\psi}\rangle \in \otimes_{j=1}^{n} \mathbb{C}^{d_{1}} \quad , \quad |{\psi}\rangle = \sum_{\alpha_{1}, \ldots, \alpha_{n}} T_{\alpha_{1} \ldots \alpha_{n}} |{\alpha_{1} \ldots \alpha_{n}}\rangle\,. \end{align*}$$

More generally, if we split the set of indices into two subsets A and B called input and output indices, respectively, we can then associate to T the operator

$$\begin{align*}T: \otimes_{j \in A} \mathbb{C}^{d_{j}} \longrightarrow \otimes_{j \in B}\mathbb{C}^{d_{j}} \quad , \quad \sum_{\alpha} T_{\alpha} |{\alpha_{B}}\rangle \langle{\alpha_{A}}| \,, \end{align*}$$

where $|{\alpha _{J}}\rangle = \otimes _{j \in J} |{\alpha _{j}}\rangle $ for any subset of indices J. Following the graphical description, we will represent input (respectively, output) indices with arrows that will point at (away from) the ball. For instance, in the case of a tensor with five legs $(T_{\alpha _{1} \ldots \alpha _{5}})$ , we can consider

We will take advantage of these multiple interpretations to find easy descriptions of tensors. For instance, the tensor T with four indices of dimension two given by $T_{0 0 0 0} = T_{1 1 1 1} =1, T_{0 1 1 0} = T_{1 0 0 1}=-1$ and $T_{\alpha _{1} \alpha _{2} \alpha _{3} \alpha _{4}}=0$ , otherwise, can be represented as

Simple descriptions of a tensor can also be obtained by taking linear combinations of tensors having the same number of indices and the same dimensions. For instance, the tensor T with four indices of dimension two given by $T_{0 0 0 0} = T_{1 1 1 1} =1$ and $T_{\alpha _{1} \alpha _{2} \alpha _{3} \alpha _{4}}=0$ , otherwise, can be represented as

Finally, we can also represent the previous tensor as the contraction of two tensors, namely

(3.1)

3.2 PEPS

Let us recall the notation and main concepts for PEPS [Reference Cirac, Pérez-García, Schuch and Verstraete10, Reference Cirac, Garre-Rubio and Pérez-García11]. Let us consider a finite graph consisting of a finite set of vertices $\Lambda $ and a set of edges E. At each vertex $x \in \Lambda $ , consider a tensor in the form of an operator

Here, $\mathbb {C}^{d}$ is the physical space associated with x and each $\mathbb {C}^{D}$ is the virtual space corresponding to an edge $e \in \partial x$ , the set of edges incident to x (in the figure, we have shown the case ${|}{\partial x}{|} =4$ ).

For a finite region $X \subset \Lambda $ , we can assign the tensors $V_{x}$ to each site $x \in X$ and perform contractions between pairs of sites $x, y \in X$ that share an edge e. Specifically, we contract the two virtual indices from $V_{x}$ and $V_{y}$ that are associated with the same edge e. The contraction of the PEPS tensors gives a linear map from the virtual edges $\partial X$ connecting X with its complement to the bulk physical Hilbert space, which we denote as

$$\begin{align*}V_X : \mathcal{H}_{\partial X} \longrightarrow \mathcal{H}_{X}\,, \end{align*}$$

where we can identify $\mathcal {H}_{\partial X} \equiv (\mathbb {C}^{D})^{\otimes \partial X}$ and $\mathcal {H}_{X} \equiv (\mathbb {C}^{d})^{\otimes X}$ . The image of $V_X$ is the space of physical states which can be represented by the PEPS with an appropriate choice of boundary condition.

The boundary state on region X is defined as

(3.2)

If $\rho _{\partial X}$ has full rank, we say that the PEPS is injective on region X. If the PEPS is injective on every sufficiently large region X, we will simply say that it is injective. We will denote by $J_{\partial X}$ the orthogonal projector onto the support of $\rho _{\partial X}$ .

If we replace the physical space $\mathbb {C}^d$ with a space of operators $\mathcal {B}(\mathbb {C}^d)$ , then we talk of Projected Entangled-Pair Operators (PEPOs) instead. If we fix a basis of matrix units $\{ \left|{i}\right\rangle \!\left\langle {j}\right| \}_{i,j=1}^{d}$ for $\mathcal {B}(\mathbb {C}^d)$ (although we will find it convenient to sometimes work with different bases), then the single-site tensor $V_x$ takes the form:

In the formula, we have separated the physical indices, given by the matrix units $\left|{i}\right\rangle \!\left\langle {j}\right|$ , from the virtual indices. In the figures, we add arrows on the physical indices to indicate which spaces correspond to the “ket” and “bra” part of $\left|{i}\right\rangle \!\left\langle {j}\right|$ , see, for example, (3.1). A particularly simple case of a PEPO is when the lattice is simply a 1D ring of n sites, in which case, it is also called an MPO. These will be the building blocks for our PEPO constructions. An MPO is any operator that can be written as

$$\begin{align*}\sum_{i_1,\dots,i_n=1}^{d^2} \operatorname{Tr} \left[ M^{(1)}_{i_1} \cdots M^{(n)}_{i_n} \right] B^{(1)}_{i_1} \otimes \cdots \otimes B^{(n)}_{i_n}, \end{align*}$$

where, for each site $k=1,\dots , n$ , we have fixed a basis $\{ B^{(k)}_i \}_{i=1}^{d^2}$ of $\mathcal {B}(\mathbb {C}^d)$ and a set of $d^2$ matrices $\{ M^{(k)}_i \}_{i=1}^{d^2}$ of dimension $D\times D$ . To pass from an MPO to a PEPO representation, it is sufficient to check that the single site tensor

$$\begin{align*}V_{k} = \sum_{i=1}^{d^2} B^{(k)}_i \otimes \sum_{\alpha, \beta = 1}^D (M^{(k)}_{i})_{\alpha,\beta} \langle{\alpha, \beta}|, \end{align*}$$

where $(M^{(k)}_{i})_{\alpha ,\beta }$ denotes the matrix units of $M^{(k)}_{i}$ , represents the same operator. It will be convenient to have a “hybrid” representation of a PEPO, in which the virtual level is still represented as a matrix, that is by writing

$$\begin{align*}V_{k} = \sum_{i=1}^{d^2} B^{(k)}_i \otimes M^{(k)}_{i}, \end{align*}$$

as this allows a more direct calculation of the tensor contractions.

Note that we can always represent a PEPO as a PEPS with physical space $\mathbb {C}^d \otimes \mathbb {C}^d$ , by choosing an orthonormal basis of $\mathcal {B}(\mathbb {C}^d)$ in order to identify that space with $\mathbb {C}^d \otimes \mathbb {C}^d$ . If we choose the basis given by matrix units $\{ \left|{i}\right\rangle \!\left\langle {j}\right|\}_{i,j=1}^d$ , then this identification can be done as follows. Let $|{\Psi }\rangle = (\sum _{i=1}^d |{i,i}\rangle )^{\otimes |{\mathcal {R}} |}$ be a maximally entangled state on $\mathcal {H}_{\mathcal {R}} \otimes \mathcal {H}_{\mathcal {R}}$ . Each operator $Q \in \mathcal {B}(\mathcal {H}_{\mathcal {R}})$ can be represented in “vector form” by

The map $Q \mapsto |{Q}\rangle $ is an isometry between $\mathcal {B}(\mathcal {H}_{\mathcal {R}})$ with the Hilbert-Schmidt scalar product and $\mathcal {H}_{\mathcal {R}} \otimes \mathcal {H}_{\mathcal {R}}$ . We should mention that, if $\rho \in \mathcal {B}(\mathcal {H}_{\mathcal {R}})$ is a positive PEPO (in the sense that it is a positive operator in the range of the map $V_{\mathcal {R}}$ for some region $\mathcal {R}$ ), it is not always true that it admits a local purification, in the sense that there exists a PEPS on a doubled physical space $\mathcal {H}_{\mathcal {R}} \otimes \mathcal {H}_{\mathcal {R}}$ , such that we recover $\rho $ when we trace out one of the copies of $\mathcal {H}_{\mathcal {R}}$ . On the other hand, in the case when $\rho ^{1/2}$ is a PEPO, then $|{\rho ^{1/2}}\rangle $ is a PEPS and a purification of $\rho $ . Therefore, when studying the case in which $\rho _{\beta }$ is the Gibbs state of a local, commuting Hamiltonian at inverse temperature $\beta $ , we will write a PEPO representation for $\rho _{\beta }^{1/2}$ (which is proportional to $\rho _{\beta /2}$ up to normalization), and from it, we will obtain the PEPS representation for the thermofield double state $|{\rho _{\beta }^{1/2}}\rangle $ .

Let us briefly discuss some characteristics that the PEPS description of the thermofield double state will have. It will be convenient for us to consider a more general definition of PEPS in which the underlying graph $(\Lambda , E)$ can be a multigraph. This means that there might be multiple edges joining the same pair of different vertices of $\Lambda $ . Of course, we might combine virtual indices joining the same pair of sites into only one virtual index, adhering to the original definition of PEPS on graphs, as we represent in the next picture:

However, as we will see in Section 4, when finding the PEPS description of the thermofield double state, it is more natural and useful using the representation with the multigraph, especially when applying results, such as Theorem 3.5, that require a suitable arrangement of the set of all virtual indices of $V_{X}$ for a region X into several subsets. In the setting of the quantum double model, recall that $\Lambda _{N}$ is the set of edges $\mathcal {E}_{N}$ of the squared lattice on the torus. It is important not to confuse this set of edges with the set of edges $E_{N}$ joining the sites in $\Lambda _{N}$ and define the PEPS. In this case, every site $x \in \Lambda _{N}$ will be connected to the other four sites via two edges of $E_{N}$ as in the next picture:

If G is the finite group from which the quantum double model will be constructed, then the individual tensor $V_{x}$ will have as physical space $\mathbb {C}^{{|{G}|}}$ , and each virtual space will correspond to $\mathbb {C}^{|G|}$ :

3.3 Parent Hamiltonian

A parent Hamiltonian of a PEPS is a local, frustration-free Hamiltonian whose local ground state spaces $W_{\mathcal {R}}$ coincide with the range of $V_{\mathcal {R}}$ for all sufficiently large regions $\mathcal {R}$ .

There are well-known conditions that can be imposed on the local PEPS tensors $V_x$ that ensure that a parent Hamiltonian exists (in which case, it will not be unique), one of which is injectivity [Reference Pérez-García, Verstraete, Cirac and Wolf36], and which can be generalized to G-injectivity and MPO-injectivity. Since the PEPS we will consider will not satisfy any of them, we omit further details in this direction, and we will prove directly the existence of a parent Hamiltonian in Section 4.5.

For a given PEPS, there is a canonical construction of a local and frustration-free Hamiltonian whose local ground spaces contain the range of $V_{\mathcal {R}}$ . For every finite subset X of $\Lambda $ , let $P_{X}:\mathcal {H}_{X} \longrightarrow \mathcal {H}_{X}$ be the orthogonal projector onto $\operatorname {Im} (V_{X})$ . Note that if $X \subset Y$ , then $\operatorname {Im} (V_{Y}) \subset Im (V_{X}) \otimes \mathcal {H}_{Y \setminus X}$ . Thus, the projectors satisfy the frustration-free condition $P_{X}P_{Y}=P_{Y}P_{X}=P_{Y}$ , or also, $P_{X}^{\perp } \geq P_{Y}^{\perp }$ .

Next, let us fix a certain family $\mathcal {X}$ of subsets of $\Lambda $ having small range, for example, rectangular regions in $\Lambda = \mathbb {Z}^{2}$ of dimensions $r \times r$ for a fixed value r. Then, consider the local interaction defined by the operators . Note that for each finite region R, the local Hamiltonian

$$\begin{align*}H_{\mathcal{R}} = \sum_{X \in \mathcal{X} \, , \, X \subset \mathcal{R}} P_{X}^{\perp}\end{align*}$$

satisfies $\operatorname {Im} (V_{R}) \subset \ker {H_{R}}$ , so that the PEPS is in the ground space of this Hamiltonian. For this to be a parent Hamiltonian, we need a condition ensuring that this is indeed an equality for large enough regions. We will now show that such condition can be obtained starting from the martingale condition (see Definition 2.4). As the result is not specific to PEPS, we will state it here in its full generality.

Let us consider the setting of a quantum spin system over a finite set $\Lambda $ that we presented at the beginning of Section 2.3. Let us assume that for each finite subset X, we have an orthogonal projector $P_{X}$ onto a subspace of $\mathcal {H}_{X}$ satisfying the frustration-free condition $P_{X}P_{Y} = P_{Y}P_{X}=P_{Y}$ for every pair of subsets $X \subset Y$ . In the case of a PEPS, these projections will be the ones onto $\operatorname {Im} (V_{X})$ .

Lemma 3.1. Under the setting just described, fix a family $\mathcal {X}$ of finite subsets of $\Lambda $ , and consider the local interactions , $X \in \mathcal {X}$ defining for each finite $\mathcal {R} \subset \Lambda $ the local Hamiltonian

$$\begin{align*}H_{\mathcal{R}} = \sum_{\substack{X \in \mathcal{X}\, , \, X \subset \mathcal{R}}} P_{X}^{\perp}\,. \end{align*}$$

If $\mathcal {R}_{1}, \mathcal {R}_{2} \subset \Lambda $ satisfy the following properties:

1. (i) $\ker (H_{\mathcal {R}_{i}}) = \Im (P_{\mathcal {R}_{i}})$ for $i=1,2$ ,

2. (ii) $\|P_{\mathcal {R}_{1} \cup \mathcal {R}_{2}} - P_{\mathcal {R}_{1}} P_{\mathcal {R}_{2}}\| < 1$ ,

3. (iii) for every $X \in \mathcal {X}$ with $X \subset \mathcal {R}_{1} \cup \mathcal {R}_{2}$ , we have that $X \subset \mathcal {R}_{1}$ or $X \subset \mathcal {R}_{2}$ ,

then $\ker (H_{\mathcal {R}_{1} \cup \mathcal {R}_{2}}) = \Im (P_{\mathcal {R}_{1} \cup \mathcal {R}_{2}})$ .

Proof. Let us denote $\mathcal {R}:= \mathcal {R}_{1} \cup \mathcal {R}_{2}$ . Applying $(iii)$ , we get

Combining the previous equality with the frustration-free condition $P_{\mathcal {R}} P_{\mathcal {R}_{i}} = P_{\mathcal {R}}$ and with $(i)$ , we obtain that

$$ \begin{align*} \operatorname{Im}(P_{\mathcal{R}}) \subset \ker (H_{\mathcal{R}}) = \operatorname{Im}(P_{\mathcal{R}_{1}}) \cap \operatorname{Im}(P_{\mathcal{R}_{2}})\,. \end{align*} $$

It remains to prove that this is indeed a chain of equalities, which is a consequence of the last statement of Lemma 2.1 together with $(ii)$ .

Let us now restrict to the case in which $\Lambda $ is the torus $\Lambda _{N}$ and the family of projectors $P_X$ satisfies the martingale condition from Definition 2.4 with decay function $\delta (\ell )$ . Let $3 \leq r \in \mathbb {N}$ with $N \geq 2(1+r)$ and such that $\delta (\ell ) <1/2$ for every $\ell \geq r-2$ . Then, let us consider the family $\mathcal {X} = \mathcal {F}_{N,r}^{rect}$ of all proper rectangles in $\mathcal {F}_{N}$ having at most r plaquettes per row and per column

and the set of local interactions $(P_{X}^{\perp })_{X \in \mathcal {X}}$ , where

.

Proposition 3.2. Under the previous hypothesis, for every rectangular region $\mathcal {R} \in \mathcal {F}_{N}$ containing at least r plaquettes per row and per column, we have that the associated Hamiltonian

$$\begin{align*}H_{\mathcal{R}} = \sum_{X \in \mathcal{X} , X \subset \mathcal{R}} P_{X}^{\perp} \quad \quad \mbox{satisfies} \quad \quad \ker(H_{\mathcal{R}}) = \operatorname{Im}(P_{\mathcal{R}})\,. \end{align*}$$

In other words, $P_{\mathcal {R}}$ is the orthogonal projector onto the ground state space of $H_{\mathcal {R}}$ .

Proof. We are going to prove that every rectangle $\mathcal {R}$ having a plaquettes per row and b per column with $a,b \geq r$ satisfies $\ker {(H_{\mathcal {R}})} = \operatorname {Im}(P_{\mathcal {R}})$ arguing by induction on $a+b$ . The first case is $a+b=2r$ , for which we necessarily have $a=b=r$ and so $\mathcal {R} \in \mathcal {X}$ . In this case, the frustration-free condition of the projectors and the fact that $P_{\mathcal {R}}^{\perp }$ is one of the summands of $H_{\mathcal {R}}$ immediately yields the equality.

Let us assume that $a+b>2r$ and that the claim holds for all rectangular regions $\mathcal {R'}$ with dimensions $a',b'$ satisfying $a'+b' < a+b$ . We claim that there exist rectangular subregions $\mathcal {R}_{1}, \mathcal {R}_{2} \subset \mathcal {R}$ , such that:

1. (i) $\mathcal {R} = \mathcal {R}_{1} \cup \mathcal {R}_{2}$ ,

2. (ii) $\mathcal {R}_{j}$ has dimensions $a_{j}, b_{j}$ satisfying $a_{j} + b_{j} < a+b$ for each $j=1,2$ ,

3. (iii) If $X \in \mathcal {X}$ is contained in $\mathcal {R}$ , then $X \subset \mathcal {R}_{1}$ or $X \subset \mathcal {R}_{2}$ ,

4. (iv) $\| P_{\mathcal {R}_{1}} P_{\mathcal {R}_{2}} - P_{\mathcal {R}} \| < 1$ .

If this claim holds, then by Lemma 3.1, we immediately conclude that $\ker (H_{\mathcal {R}}) = \Im (P_{\mathcal {R}})$ , and so the proof is finished. Let us the show the validity of the claim, distinguishing three possible cases according to whether the region $\mathcal {R}$ is a proper rectangle, a cylinder, or the whole torus.

Case 1: If $\mathcal {R}$ is a proper rectangle, we can assume without loss of generality that $b>r$ , where we recall that b is the number of plaquettes of each row. Then, we split $\mathcal {R}$ along the rows into three disjoint parts $A,B,C$ , as in the next picture

so that $\mathcal {R}_{1}=AB$ and $\mathcal {R}_{2} = BC$ are proper rectangles of dimensions a and $b-1$ , and thus they satisfy (i)–(ii). They also satisfy (iii), since if $X \subset \mathcal {R}$ is a rectangle contained neither in $\mathcal {R}_{1}$ nor in $\mathcal {R}_{2}$ , it must have b plaquettes per row, but $b>r$ , so X cannot belong to $\mathcal {X}$ . Since $\mathcal {R}_{1} \cap \mathcal {R}_{2} = B$ is, again, a rectangle of dimensions a and $b-2$ , we deduce from the martingale condition that

$$\begin{align*}\| P_{\mathcal{R}_{1}} P_{\mathcal{R}_{2}} - P_{\mathcal{R}_{1} \cup \mathcal{R}_{2}} \| \, \leq \, \delta(b-2) < 1 \,. \end{align*}$$

Case 2: If $\mathcal {R}$ is a cylinder, we can assume without loss of generality that its border lies on the horizontal sides, so that it contains N plaquettes per row. We then split $\mathcal {R}$ along the rows into four disjoint regions $A, B, C, B'$ , as in the next picture, where A and C correspond to columns of horizontal edges, and $B, B'$ have horizontal dimension greater than $\lfloor N/2-1 \rfloor $

Taking $\mathcal {R}_{1}=B'AB$ and $\mathcal {R}_{2}=BCB'$ , we immediately get that these are proper rectangles satisfying (i) and (ii). They also satisfy (iii), since $\lfloor N/2-1\rfloor \geq r$ by the hypothesis. Property (iv) follows from the martingale condition, since it yields that

$$\begin{align*}\| P_{B'AB} P_{BCB'} - P_{ABCB'} \| \, \leq \, \delta(\lfloor N / 2-1 \rfloor) < 1\,. \end{align*}$$

Case 3: Assume $\mathcal {R}$ is the whole torus $\Lambda _{N}$ . Then, we can split it into four regions $A,B,C,B'$ , as in the next picture, where B and $B'$ have dimensions $a = N$ and $b \geq \lfloor N/2-1\rfloor $

Taking $\mathcal {R}_{1} = B'AB$ and $\mathcal {R}_{2}=BCB'$ as rectangular subregions, we can argue analogously to the previous cases to deduce that they satisfy (i)–(iii) and

$$ \begin{align*} \| P_{\mathcal{R}_{1}} P_{\mathcal{R}_{2}} - P_{\Lambda_{N}} \| \, &\leq \, \delta(\lfloor N / 2-1 \rfloor) < 1\,. \end{align*} $$

This concludes the proof of the claim.

3.4 Spectral gap of a parent Hamiltonian

We will now recall the relationship established in [Reference Kastoryano, Lucia and Perez-Garcia25] between boundary states and the spectral gap of a parent Hamiltonian of a PEPS.

3.4.1 Boundary states and approximate factorization

Let us consider three (connected) regions $A,B,C \subset \Lambda $ and assume that B shields A from C, so that there is no edge joining vertices from A and C (see Figure 4). Let us consider the boundary states $\rho _{\partial ABC}, \rho _{\partial AB}, \rho _{\partial BC}$ , and $\rho _{\partial B}$ . In the case, where they are all full rank, the approximate factorization condition is defined as follows.

Figure 4 An example of three regions A, B, C.

Definition 3.3 (Approximate factorization for injective PEPS [Reference Kastoryano, Lucia and Perez-Garcia25]).

Let $\varepsilon>0$ . We will say that the boundary states are $\varepsilon $ -approximately factorizable, if we can divide the regions

and find invertible matrices $\Delta _{az}$ , $\Delta _{zc}$ , $\Omega _{\alpha z}$ , $\Omega _{z \gamma }$ with support in the regions indicated by the respective subindices, such that the boundary observables

approximate the boundary states

The approximate factorization of the boundary states implies a small norm of the overlaps of ground space projections.

Theorem 3.4 [Reference Kastoryano, Lucia and Perez-Garcia25, Theorem 10].

If the boundary states are $\varepsilon $ -approximately factorizable for some $\varepsilon \leq 1$ , then

$$\begin{align*}\| P_{AB}P_{BC} - P_{ABC} \| < 8 \varepsilon\,. \end{align*}$$

3.4.2 Approximate factorization for locally noninjective PEPS

In [Reference Kastoryano, Lucia and Perez-Garcia25], the approximate factorization condition was extended to noninjective PEPS satisfying what is known as the pulling through condition, which holds in the case of G-injective and MPO-injective PEPS. Unfortunately, the PEPS representing the thermofield double state $|{\rho _{\beta }^{1/2}}\rangle $ will neither be injective nor satisfy such a condition. At the same time, it will turn out to have some stronger property which will make up for the lack of it: it can be well approximated by a tensor product operator. We will now present the necessary modifications to the results of [Reference Kastoryano, Lucia and Perez-Garcia25] required to treat this case.

There are three geometrical cases we need to consider in our decomposition of the torus $\mathbb {Z}_{N}\times \mathbb {Z}_{N}$ into subregions: two cylinders to cover the torus, two rectangles to cover a cylinder, and two rectangles to cover a rectangle. The following theorem is an adaptation of [Reference Kastoryano, Lucia and Perez-Garcia25, Theorem 10] that covers each of these three cases.

Theorem 3.5. Let $A,B,C$ be three disjoint regions of $\Lambda $ , such that A and C do not share mutually contractible boundary indices. Let us, moreover, assume that the (boundary) virtual indices of $ABC$ , $AB$ , $BC$ , and B can be arranged into four sets $a ,c, \alpha , \gamma $ , as in the next picture, so that

Let us also assume that the orthogonal projections $J_{\partial \mathcal {R}}$ onto the support of $\rho _{\partial \mathcal {R}}$ admit a factorization in terms of projections $J_{a}, J_{c}, J_{\alpha }, J_{\gamma }$ (subindices indicate their corresponding support), namely

$$\begin{align*}J_{\partial ABC} = J_{a} \otimes J_{c} \quad , \quad J_{\partial AB} = J_{a} \otimes J_{\gamma} \quad , \quad J_{\partial BC} = J_{\alpha} \otimes J_{c} \quad , \quad J_{\partial B} = J_{\alpha} \otimes J_{\gamma}\,,\end{align*}$$

and there also exist positive semidefinite operators $\sigma _{a}, \sigma _{c}, \sigma _{\alpha }, \sigma _{\gamma }$ with full-rank on $J_{\alpha }, J_{a}, J_{\gamma }, J_{c}$ , such that

$$\begin{align*}\sigma_{\partial ABC}:= \sigma_{a} \otimes \sigma_{c} \quad , \quad \sigma_{\partial AB}:= \sigma_{a} \otimes \sigma_{\gamma} \quad , \quad \sigma_{\partial BC}:= \sigma_{\alpha} \otimes \sigma_{c} \quad , \quad \sigma_{\partial B}:= \sigma_{\alpha} \otimes \sigma_{\gamma} \end{align*}$$

satisfy for some $0 \le \varepsilon \leq 1$

Here, inverses are taken on the corresponding support. Then,

$$\begin{align*}\| P_{AB} P_{BC} - P_{ABC} \| \, \leq \, 8\varepsilon \,.\end{align*}$$

In case the region $\mathcal {R}$ consists of the whole lattice (e.g., torus), then the sets a and c would be empty. Consequently, $\sigma _{a}$ and $\sigma _{c}$ would be simply scalars.

Proof. Let us define the approximate projections

Note that $V_{\mathcal {R}}\rho _{\mathcal {R}}^{-1/2}$ is a partial isometry from the support of $\rho _{\partial \mathcal {R}}$ to $\Im (V_{\mathcal {R}})$ , since

and

where the last equality is a consequence of the fact that

is a self-adjoint projection whose image is exactly $\Im (V_{\mathcal {R}})$ . As a consequence

(3.3)

We are going to denote by $V_{C \rightarrow B}$ the tensor obtained from $V_{C}$ by taking all input indices that connect with B into output indices, so that

$$\begin{align*}V_{ABC} = V_{AB} V_{C \rightarrow B} \quad \text{and} \quad V_{BC} = V_{B} \, V_{C \rightarrow B} \,. \end{align*}$$

Analogously, we define $V_{A \rightarrow B}$ satisfying

$$\begin{align*}V_{ABC} = V_{BC} V_{A \rightarrow B} \quad \text{and} \quad V_{AB} = V_{B} V_{A \rightarrow B}\,. \end{align*}$$

Then, we can rewrite

At this point, we can use the local structure of the projections to write $V_{AB} = V_{AB} J_{\partial AB} = V_{AB} J_{\partial AB} J_{\gamma } = V_{AB} J_{\gamma } = V_{AB} \sigma _{\gamma } \sigma _{\gamma }^{-1}$ . Analogously, $V_{BC}= V_{BC} \sigma _{\alpha } \sigma _{\alpha }^{-1}$ . Inserting both identities above, we can rewrite

Similarly, we handle

To compare the expressions for $Q_{ABC}$ and $Q_{AB} \, Q_{BC}$ , we introduce

It is easy to check that

(3.4)

Since

, we can apply (3.3) to estimate

Analogously,

, and so $\| \Delta _{AB}\| \leq 1 + \varepsilon $ . Combining these inequalities with (3.4), we get

(3.5) $$ \begin{align} \| Q_{ABC} - Q_{AB} Q_{BC} \| \leq (1+ \varepsilon)^2 \, \| J_{\partial B} - (\rho_{\partial B}^{-1/2} \sigma_{\partial B} \rho_{\partial B}^{-1/2}) \, \|\, \leq \varepsilon(1+\varepsilon)^2\,. \end{align} $$

Finally, we combine the previous inequality with (3.3) to conclude

which gives the result.

3.4.3 Gauge invariance of the approximate factorization condition

An interesting observation, omitted in [Reference Kastoryano, Lucia and Perez-Garcia25], is that the property of $\varepsilon $ -approximately factorization is gauge invariant if the transformation does not change the support of the boundary state. Indeed, for every vertex $x \in \Lambda $ and every edge $e \in E$ incident to x, let us fix an invertible matrix $\mathcal {G}(e,x) \in \mathbb {C}^{D} \otimes \mathbb {C}^{D}$ . We assume that for every edge e with vertices $x,y$ , we have

(3.6) $$ \begin{align} \mathcal{G}(e,x) = \mathcal{G}(e,y)^{-1}\,. \end{align} $$

We will simply write $\mathcal {G}(e)$ when the site is clear from the context. Let us assume that we have two PEPS related via this gauge, namely, for every site $x \in \Lambda $ , we have that the local tensors $\widetilde {V}_{x}$ and $V_{x}$ are related via (see Figure 5)

$$\begin{align*}V_{x} = \widetilde{V}_{x} \, \circ \, \mathcal{G}_{x} \quad \quad \mbox{, where } \quad \mathcal{G}_{x}:=\bigotimes_{e \in \partial x}{\mathcal{G}(e,x)} \,. \end{align*}$$

Figure 5 Tensors with (right) and without (left) gauge.

For a region $\mathcal {R} \subset \Lambda $ , when contracting indices to construct $V_{\mathcal {R}}$ , we have, as a consequence of (3.6), that contracting inner edges of $\mathcal {R}$ cancel the gauge matrices. Thus, $\widetilde {V}_{\mathcal {R}}$ and $V_{\mathcal {R}}$ are related via (see Figure 6):

(3.7) $$ \begin{align} V_{\mathcal{R}} = \widetilde{V}_{\mathcal{R}} \, \circ \, \mathcal{G}_{\partial \mathcal{R}} \quad \quad \mbox{ with } \quad \mathcal{G}_{\partial \mathcal{R}}:=\bigotimes_{e \in \partial \mathcal{R}}{\mathcal{G}(e)} \, .\end{align} $$

Figure 6 Tensor network with (right) and without (left) gauge (physical indices are not shown).

The boundary state after the change of gauge is transformed as

where

.

Proposition 3.6. Assume that $[{J_{\partial \mathcal {R}}},{\mathcal {G}_{\partial \mathcal {R}}}] =0$ . Let $\widetilde \sigma _{\partial \mathcal {R}}$ supported on $J_{\partial \mathcal {R}}$ , and define

Then, ${\sigma }_{\partial \mathcal {R}}$ is also supported on $J_{\partial \mathcal {R}}$ , and it holds that

$$\begin{align*}\left\| \rho_{\partial \mathcal{R}}^{1/2} \sigma_{\partial \mathcal{R}}^{-1} \rho_{\partial \mathcal{R}}^{1/2} - J_{\partial R} \right\| \, = \, \left\| \widetilde{\rho}_{\partial \mathcal{R}}^{1/2} \widetilde{\sigma}_{\partial \mathcal{R}}^{-1} \widetilde{\rho}_{\partial \mathcal{R}}^{1/2} - J_{\partial R} \right\|\,. \end{align*}$$

Proof. Since $J_{\partial \mathcal {R}}$ and $\mathcal G_{\partial \mathcal {R}}$ commute, we have that $\rho _{\partial \mathcal {R}}$ , $\widetilde {\rho }_{\partial \mathcal {R}}$ , $\sigma _{\partial \mathcal {R}}$ , and $\widetilde {\sigma }_{\partial \mathcal {R}}$ all have the same support, namely, $J_{\partial \mathcal {R}}$ , and so

Therefore, if $P_{\mathcal {R}}$ denotes the orthogonal projection onto $\operatorname {Im}(V_{\mathcal {R}}) = \operatorname {Im}(\widetilde {V}_{\mathcal {R}})$ , then we can write

From the definition of $\widetilde {\sigma }_{\partial \mathcal {R}}$ , we similarly see that

The statement then follows from the fact that

where $W_{\mathcal {R}} = V_{\mathcal {R}} \rho _{\partial \mathcal {R}}^{-1/2}$ and $\widetilde {W}_{\mathcal {R}} = \widetilde {V}_{\mathcal {R}} \widetilde {\rho }_{\partial \mathcal {R}}^{-1/2}$ are isometries.

Corollary 3.7. Let $A, B,C$ be three regions of $\Lambda $ , as in the definition of approximate factorization (Definition 3.3), and assume that $[{J_{\partial \mathcal {R}}},{G_{\partial \mathcal {R}}}] = 0$ for $\mathcal {R} \in \{ABC, AB, BC, B\}$ . If the PEPS generated by $\widetilde {V}_{x}$ is $\varepsilon $ -approximately factorizable, then so does the PEPS generated by $V_{x}$ .

Proof. Let us assume then that the PEPS with local tensors $\widetilde {V}_{x}$ is $\varepsilon $ -approximately factorizable. Because of Proposition 3.6, it is sufficient to verify that ${\sigma }_{\partial \mathcal {R}}$ satisfies the necessary locality properties. If $\widetilde J_{\partial \mathcal {R}}$ and $\widetilde {\sigma }_{\partial \mathcal {R}}$ are product operators (as in Theorem 3.5), then so are $J_{\partial \mathcal {R}}$ and $\sigma _{\partial \mathcal {R}}$ , and there is nothing to prove.

Let us now consider the case in which $\widetilde {\sigma }_{\partial \mathcal {R}}$ is not in a tensor product form (as in Definition 3.3). Let $a,\alpha , z, \gamma , c$ be the regions dividing the boundaries $\partial ABC = a z c$ , $\partial AB = az \gamma $ , $\partial BC = \alpha z c$ , $\partial B = \alpha z \gamma $ , and let $\widetilde {\Delta }_{az}, \widetilde {\Delta }_{zc}, \widetilde {\Omega }_{\alpha z}, \widetilde {\Omega }_{z \gamma }$ be the corresponding matrices. Note that the gauge matrices $\mathcal {G}_{\partial \mathcal {R}}$ can be rearranged according to the boundary subregions, for example

$$\begin{align*}\mathcal{G}_{\partial ABC} = \mathcal{G}_{a zc} = \mathcal{G}_{az} \mathcal{G}_{c} = \mathcal{G}_{a} \mathcal{G}_{zc}\,. \end{align*}$$

If we define

(3.8)

then, we can directly check that $\sigma _{\partial \mathcal {R}}$ satisfies

$$ \begin{align*} & \sigma_{\partial AB} := \Omega_{z \gamma} \Delta_{az}, & \sigma_{\partial BC} := \Delta_{z c} \Omega_{\alpha z},\\ & \sigma_{\partial ABC} := \Delta_{z c} \Delta_{az}, & \sigma_{\partial B} := \Omega_{z \gamma} \Omega_{\alpha z}. \end{align*} $$

This finishes the proof.

4.1 Quantum double models

Let us begin by recalling the definition of the quantum double models. They are defined on the lattice $\Lambda _{N}$ consisting of midpoints of the edges of the square lattice $\mathbb {Z}_N \times \mathbb {Z}_{N}$ (see Section 2.3). Let us denote by $\mathcal {V}=\mathcal {V}_{N}$ the set of vertices, and by $\mathcal {E}=\mathcal {E}_{N}$ the set of edges of $\mathbb {Z}_N \times \mathbb {Z}_{N}$ . Each edge is given an orientation: for simplicity, we will assume that all horizontal edges point to the left, while vertical edges point downwards.

Let us fix an arbitrary finite group G, and let $\ell _2(G)$ be the complex finite dimensional Hilbert space with orthonormal basis given by $\{ |{g}\rangle \mid g \in G\}$ . At each edge $e \in \mathcal {E}$ , we have a local Hilbert space $\mathcal {H}_e$ and a space of observables $\mathcal {B}_e$ defined as

$$\begin{align*}\mathcal{H}_{e} = \ell_{2}(G) \quad \text{and} \quad \mathcal{B}_{e} = \mathcal{B}(\mathcal{H}_{e}) = \mathcal{M}_{|G|}(\mathbb{C}). \end{align*}$$

We will use the alternative notation $\mathcal {H}_{\Lambda } = \mathcal {H}_{\mathcal {E}}$ and $\mathcal {B}_{\Lambda } = \mathcal {B}_{\mathcal {E}}$ .

Given $g \in G$ , we define operators on $\ell _2(G)$ by

(4.1) $$ \begin{align} L^{g} := \sum_{h \in G} |{gh}\rangle\!\langle{h}|. \end{align} $$

Then $g\mapsto L^g$ is a representation of the group G, known as the left regular representation.

For each finite group G, the quantum double model on $\Lambda $ is defined by a Hamiltonian $H_{\Lambda }^{\text {syst}}$ of the form

$$\begin{align*}H_{\Lambda}^{\text{syst}} = - \sum_{v {\tiny \, \mathrm{vertex}}}{A(v)} \,\, - \sum_{p {\tiny \, \mathrm{plaquette} }} B(p)\,;\end{align*}$$

where the terms $A(v)$ are star operators, supported on the four incident edges of v, which we will denote as $\partial v$ , while $B(p)$ are plaquette operators, supported on the four edges forming the plaquette p. Both terms are projections, and they commute, namely

$$\begin{align*}[A(v), A(v')] \,\, = \,\, [B(p), B(p')] \,\, = \,\, [A(v), B(p)] \,\, = \,\, 0 \end{align*}$$

for all vertices $v,v'$ and plaquettes $p,p'$ . We will now explicitly define these terms, and a straightforward calculation will show that they satisfy these properties.

Let v be a vertex and e an edge incident to v. For each $g \in G$ , we define the operator $T^{g}(v,e)$ acting on $\mathcal {H}_{e}$ according to the orientation given to e as

In other words, the operator $T^{g}(v,e)$ acts on the basis vector of $\mathcal {H}_{e}$ by taking h into $g h$ (respectively, $hg^{-1}$ ) if the oriented edge e points away from (respectively, to) v. It is easily checked that

(4.2)

for every $g,h \in G$ .

With this definition, the vertex operator $A(v)$ is given by

Using (4.2), it is easy to verify that $A(v)$ is a projection.

The plaquette operator $B(p)$ is defined as follows. Let us enumerate the four edges of p as $e_{1}, e_{2}, e_{3}, e_{4}$ following counterclockwise order starting from the upper horizontal edge. The plaquette operator on p acts on $\otimes _{j=1}^{4}{\mathcal {H}_{e_{j}}}$ and is defined as the orthogonal projection $B(p)$ onto the subspace spanned by basis vectors of the form $|{g_{1} g_{2}g_{3}g_{4}}\rangle $ with $\sigma _{p}(g_{1}) \sigma _{p}(g_{2}) \sigma _{p}(g_{3}) \sigma _{p}(g_{4}) = 1$ . Here,