Integrable spin chains and the Clifford group

Jones, Nick G.; Linden, Noah

doi:10.1063/5.0095870

We construct new families of spin chain Hamiltonians that are local, integrable, and translationally invariant. To do so, we make use of the Clifford group that arises in quantum information theory. We consider translation invariant Clifford group transformations that can be described by matrix product operators (MPOs). We classify translation invariant Clifford group transformations that consist of a shift operator and an MPO of bond dimension two—this includes transformations that preserve locality of all Hamiltonians and those that lead to non-local images of particular operators but, nevertheless, preserve locality of certain Hamiltonians. We characterize translation invariant Clifford group transformations that take single-site Pauli operators to local operators on at most five sites—examples of Quantum Cellular Automata—leading to a discrete family of Hamiltonians that are equivalent to the canonical XXZ model under such transformations. For spin chains solvable by the algebraic Bethe ansatz, we explain how conjugating by an MPO affects the underlying integrable structure. This allows us to relate our results to the usual classifications of integrable Hamiltonians. We also treat the case of spin chains solvable by free fermions.

I. INTRODUCTION

Integrable models are fundamental tools in our understanding of quantum many-body systems and statistical mechanics.^1–4 While a general definition of quantum integrability is open to debate,⁵ there are many examples of well-established integrable models. Two families of integrable models that are particularly well studied and that are of interest to us are Bethe ansatz solvable models,² including the Heisenberg and XXZ spin chains,^1–3,6,7 and free-fermion models, including the XY and Ising spin chains.⁸ An important topic is the classification of Bethe ansatz solvable models,^9–14 and there are conjectured conditions implying integrability of a Hamiltonian.^13,15 There has, moreover, been significant recent interest in finding new models and deriving conditions where a spin chain can be solved by free-fermion (and free-parafermion) methods.^16–24

Given known integrable Hamiltonians, we can consider transformations applied to them and between them. This can lead to finding new models that can be solved in the same way as known models. Transformations between exactly solvable models are, moreover, important in the field of symmetry-protected topological (SPT) phases; then, it is useful to identify transformations that take one from a simple, exactly solvable model in the trivial phase to an analogous exactly solvable model with non-trivial symmetry properties.^25–27 In this paper, we will construct new families of spin chain Hamiltonians using Clifford group transformations, a family of transformations arising in quantum information theory that we describe below.

To set the scene, consider the following Hamiltonian for a spin-1/2 chain:

H_{0} = - \sum_{n \in s i t e s} (Z_{n - 3} Y_{n - 2} X_{n - 1} Y_{n} Y_{n + 1} X_{n + 2} Y_{n + 3} Z_{n + 4} + Z_{n - 3} Y_{n - 2} Y_{n - 1} Y_{n + 2} Y_{n + 3} Z_{n + 4} + Δ Z_{n - 1} Y_{n} Y_{n + 1} Z_{n + 2})

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with −1 < Δ < 1, where X, Y, and Z are the standard Pauli operators. Perhaps surprisingly, this Hamiltonian is integrable. In particular, it can be analyzed using Bethe ansatz methods, and from that, one can derive highly non-trivial results. For example, letting angle brackets denote the ground state expectation value, one can rigorously prove the following large M asymptotics:

\begin{align} ⟨ Z_{1} X_{2} Z_{3} Z_{M + 1} X_{M + 2} Z_{M + 3} ⟩ = \{\begin{cases} - \frac{1}{8 η^{2} M^{2}} (1 + o (1)), & 0 < η < π / 4, \\ {(- 1)}^{M} \frac{b}{M^{\frac{π}{2 η}}} (1 + o (1)), & π / 4 < η < π / 2, \end{cases}) \end{align}

(2)

where 2η = arccos(Δ) and b is independent of M. As we will explain, this integrability follows by giving a unitary transformation from H₀ to the canonical XXZ spin chain,

H_{XXZ} = - \sum_{n \in s i t e s} (X_{n} X_{n + 1} + Y_{n} Y_{n + 1} + Δ Z_{n} Z_{n + 1}) .

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We can, then, use established results for the ⟨Z₁Z_M+1⟩ correlation function in the XXZ chain.²⁸ Moreover, using conformal field theory (CFT) methods [H₀ has low energy behavior described by a compactified free boson CFT for −1 < Δ < 1, with the SU(2)-point at Δ = −1], one can derive further ground state correlation functions in this model.^2,28,29 We will show that, unlike H_XXZ, H₀ is at a transition between SPT phases^26,30–32 (indeed, unitary transformations preserve the structure of the phase diagram, but not necessarily any symmetry properties).

Now, the fact that, say, the eigenstates of the Hamiltonians H and H′ = UHU^†, for some unitary U, are simply related is elementary. However, a typical unitary will not preserve the locality of the Hamiltonian, by which we mean that the Hamiltonian is a sum of terms, each of which acts non-trivially on a number of sites that is independent of the length of the chain. Moreover, a given local Hamiltonian is typically not integrable. The purpose of this paper is to construct new translationally invariant and local Hamiltonians that are integrable by virtue of being unitarily related to canonical integrable models, such as H_XXZ. We do this through making connections to techniques from quantum information theory. In particular, we study interesting examples from the Clifford group of unitary transformations.

The Clifford group is a discrete subgroup of the full unitary group and consists of those transformations that map products (or strings) of Pauli operators to other products of Pauli operators under conjugation; for example,

\begin{align} X_{1} \to X_{1}, & X_{2} \to Z_{2}, \\ Z_{1} \to Z_{1} Z_{2}, & Z_{2} \to X_{1} X_{2} \end{align}

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is a two-site Clifford group transformation. Since these transformations are unitary, they are constrained by the necessity of preserving the commutation relations between Pauli operators. One reason for considering the Clifford group is that many integrable spin-1/2 chains have a simple form written as a sum of Pauli strings and these transformations preserve that—for example, H_XXZ has three types of interaction term, as does H₀. Symmetries that are Pauli strings, such as the spin-flip ∏_nZ_n, remain of this form after the transformation. This can help in the analysis of symmetries in the context of SPT phases,^33,34 and so this can be used to find examples of integrable models (both gapped and gapless) with non-trivial SPT physics. For the case of H₀, this analysis is given in Appendix A.

Consider, then, a spin-1/2 chain with L sites. One approach to finding Clifford group transformations on such a chain is to start with a simple transformation for each site, such as the following that appears frequently in the literature on SPT phases:^25–27

\begin{align} X_{n} & \to Z_{n - 1} X_{n} Z_{n + 1}, \\ Z_{n} & \to Z_{n} . \end{align}

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This transformation maps the trivial paramagnet with Hamiltonian −∑_nX_n to the SPT cluster model with Hamiltonian −∑_nZ_n−1X_nZ_n+1 (for this reason, it is sometimes referred to as an SPT entangler). One can check that (5) preserves the commutation relations, giving a Clifford group transformation.³⁵ One nice feature of this transformation is that it preserves translation invariance—this is a physically interesting case and is a common, though inessential,^3,28 feature of many integrable models. A key concern of this paper is to identify translation invariant transformations that, in turn, lead to translation invariant Hamiltonians.

Note that transformation (5), taking single-site operators to (at most) three-site operators for each site n, defines a Clifford group transformation on the whole chain. Let us consider, instead, the following partial definition of a transformation taking single-site operators to three-site operators:

\begin{align} X_{1} & \to X_{0} X_{1} Z_{2}, \\ Z_{1} & \to Z_{1} . \end{align}

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There are many ways of extending this to a valid three-site Clifford transformation and, moreover, to a Clifford transformation on the whole chain. However, any way we do so will break translation invariance. In particular, there cannot be a valid Clifford transformation on the whole chain that acts as (6) on each site,

\begin{align} X_{n} & \to_{?} X_{n - 1} X_{n} Z_{n + 1}, \\ Z_{n} & \to_{?} Z_{n} . \end{align}

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Indeed, such a transformation would not be unitary since, say, mapping X₁ → X₀X₁Z₂ and X₃ → X₂X₃Z₄ is inconsistent with the commutation relation [X₁, X₃] = 0. We, thus, see that it is a non-trivial problem to identify those local Clifford transformations that can apply in a translation invariant fashion on the whole chain. We present families of such transformations in Fig. 1.

FIG. 1.

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Translation invariant Clifford group transformations: (a) U_chain = … U_3,4U_2,3U_1,2 and (b) U_chain = … U_3,4,5U_2,3,4U_1,2,3.

General Clifford transformations can be built from a discrete set of one- and two-site operations: CNOT, Hadamard, and Phase^36,37 with matrix forms

(\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{matrix}), \frac{1}{\sqrt{2}} (\begin{matrix} 1 & 1 \\ 1 & - 1 \end{matrix}), a n d (\begin{matrix} 1 & 0 \\ 0 & i \end{matrix}),

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respectively. It is natural, then, to consider how to construct a translation invariant Clifford group transformation out of basic local Clifford transformations. One way to extend a single basic Clifford transformation U to a Clifford transformation U_chain acting on the whole chain is given in Fig. 1 for two- and three-site U, respectively. In fact, transformation (5) takes this form with a basic Clifford transformation that acts by conjugation as

\begin{align} X_{n} \to X_{n} Z_{n + 1}, & Z_{n} \to Z_{n}, \\ X_{n + 1} \to Z_{n} X_{n + 1}, & Z_{n + 1} \to Z_{n + 1} . \end{align}

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While this construction allows us to build translation invariant Clifford transformations from simple basic Clifford transformations U, the transformations U_chain of Fig. 1 have a non-local causal structure and so, for a general choice of U, will not necessarily preserve the locality of the Hamiltonian that we transform. Indeed, consider applying (4) sequentially to pairs of sites, as in Fig. 1(a): this means that U_chain is a product of basic transformations that act by conjugation as

\begin{align} X_{n} \to X_{n}, & X_{n + 1} \to Z_{n + 1}, \\ Z_{n} \to Z_{n} Z_{n + 1}, & Z_{n + 1} \to X_{n} X_{n + 1} . \end{align}

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We can see that on applying U_chain, the operator X₂, for example, will transform into a non-local string: X₂ → Z₂ → Z₂Z₃ → Z₂Z₃Z₄ → ⋯. One part of our analysis is identifying which cases maintain locality for all choices of Hamiltonian. We will call these transformations, which take all local operators to other local operators, locality-preserving transformations (where a local operator acts non-trivially on a number of sites independent of the length of the chain). The other Clifford transformations, which we will call locality-non-preserving, are, however, of real physical interest. In particular, while they take some local operators to infinite strings, they can transform certain local Hamiltonians to other local Hamiltonians (in the bulk). For example, the operator U_chain derived from (10) can be recognized as the Kramers–Wannier duality³⁸ of the Ising model (again, for operators in the bulk; we will discuss this further below; see also Ref. 39).

On a spin-1/2 chain, transformations that map local strings of Pauli operators to local strings of Pauli operators are locality-preserving. In fact, the purely mathematical question that we arrive at by asking which translation invariant Clifford transformations maintain locality for all Pauli operators arises in the field of Clifford Quantum Cellular Automata (QCA). To be precise, a Clifford QCA is a translation invariant, reversible, and local transformation that takes Pauli operators to Pauli operators.^40,41 The motivation in that setting is rather different: a QCA is a discrete time dynamical system, and the purpose of the locality condition is to ensure that the system has a finite propagation speed. There is an established theory of Clifford QCA,^40–42 and certain claims made below are consequences of these more general results—we will point these out as appropriate, but aim to give a self-contained presentation and so provide elementary proofs as needed. Moreover, the condition that locality is preserved for every Hamiltonian is stronger than needed for our purposes. Allowing for locality-non-preserving transformations, which, nevertheless, preserve the locality of certain Hamiltonians, leads to a broader class of transformations than those that define a QCA.

Another important connection to make is to the theory of matrix product operators (MPOs).^43,44 We explain in Sec. II B that the transformation U_chain in Fig. 1 can be understood as an MPO. This is particularly useful because both MPOs and the algebraic Bethe ansatz^1,2 can be formulated in the intuitive graphical notation used in the theory of tensor networks.⁶ This allows us to begin to analyze, for models solved by the algebraic Bethe ansatz, how the integrable structure changes when we conjugate the Hamiltonian by the transformation U_chain.

This paper is organized as follows. In Sec. II, we motivate and analyze certain families of Clifford group transformations. In Secs. II A and II B, we define Clifford group transformations in general, and then the translation invariant Clifford group transformations of interest. We also explain the connection to MPOs. In Secs. III and IV, we, then, demonstrate how these results can be used to construct families of integrable models equivalent to H_XXZ, including H₀ given above. Models related by locality-preserving transformations built from a single two-site basic Clifford transformation are given in Sec. III A, while those related by locality-non-preserving transformations are given in Sec. III B. These are a consequence of a classification, given in Sec. III C, of all the possible transformations that arise from transformations U_chain constructed from a single two-site Clifford transformation. We see that, up to changes of on-site basis, only one new Hamiltonian appears in Sec. III A. In Sec. III A 1, we show that this Hamiltonian has some interesting connections to another integrable model, the folded XXZ chain,^13,45–48 and to the recently introduced concept of a pivot Hamiltonian.^49,50 In Sec. IV, we discuss translation invariant Clifford transformations directly rather than through the U_chain construction. We give, in Sec. IV A, all transformations that map single-site Pauli operators to local operators on up to five sites and show how these can be written using U_chain based on a single two- or three-site Clifford transformation. We give the corresponding families of Hamiltonians related to H_XXZ in Sec. IV B. In Sec. V, we proceed to a more general discussion regarding families of integrable Hamiltonians. We focus our discussion on canonical models that can be solved by the quantum inverse scattering method or algebraic Bethe ansatz^1,2,6 and explain how transformations of the MPO form can be applied to the integrable structures underlying these models. This includes, as a special case, the Clifford transformations that we have analyzed and, thus, gives our results a context by relating them to classifications of integrable models. We find interesting modifications to the underlying integrable structures, but there remain questions to resolve here. We also treat the case of spin chains solvable by transforming to free fermions, making connections to recent papers on families of exactly solvable spin-1/2 chains.^17,19,20,24 Finally, we discuss further avenues of research.

II. CLIFFORD GROUP TRANSFORMATIONS

A. The Clifford group

We now introduce the Clifford group in the spirit of Ref. 36. Let $P = {I, X, Y, Z}$ denote the set of single-site Pauli operators. Consider the set of Pauli strings on N sites; this is given by

P_{N} = \{P_{1} \otimes P_{2} \otimes \dots \otimes (P_{N}| P_{j} \in P\} .

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As above, we will omit the tensor product symbol and write elements of $P_{N}$ as $\prod_{j = 1}^{N} P_{j}$ ⁠. Furthermore, non-trivial Pauli strings on N sites are elements of

P_{N}^{*} = P_{N} \ {I^{\otimes n}} .

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As mentioned, informally, the Clifford group consists of transformations that take Pauli strings to Pauli strings under conjugation. More carefully, the Clifford group on N sites is defined as the following subgroup of U(2^N):

C_{N} = \{U \in (U (2^{N})| \forall P \in \pm P_{N}^{*}, U P U^{†} \in \pm P_{N}^{*}\} / U (1),

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where $\pm P_{N}^{*}$ consists of ±P for $P \in P_{N}^{*}$ ⁠.

One can specify⁵¹ a Clifford group transformation by its action on each of {X₁, …, X_N, Z₁, …, Z_N}. These operators all mutually commute, apart from operators on the same site where we have the anticommutator {X_j, Z_j} = 0. These commutation relations are preserved under conjugation by unitary transformations and, in particular, by Clifford group transformations. Respecting these commutation relations, one can define a Clifford group transformation sequentially, first specifying, say, X₁ → P, where $P \in \pm P_{N}^{*}$ ⁠; then defining Z₁ → P′, where $P^{'} \in \pm P_{N}^{*}$ and {P, P′} = 0; and so on. It can be shown that this procedure gives a unique element of the Clifford group,³⁶ and so, in this way, one can derive the order of $C_{N}$ ⁠. For example, to define a transformation in $C_{1}$ ⁠, one can choose to take X to any of the six elements of $\pm P_{1}^{*}$ and Z to any of the four elements of $\pm P_{1}^{*}$ that anticommute with that choice, giving $| C_{1} | = 24$ ⁠. The result $| C_{N} | = 2^{N^{2} + 2 N} \prod_{j = 1}^{N} (4^{j} - 1)$ can be derived similarly, as shown in Ref. 36. For our spin chain of length L, we will consider many copies of, say, $P_{2}$ ⁠. These will correspond to each pair (or, more generally, in the case of $P_{N > 2}$ ⁠, set) of neighboring sites. For example, we can view both X₁X₂ and X₂X₃ as elements of $P_{2}$ for different subsets of sites on the chain.

We define a notation that corresponds to translating a Pauli operator by j sites: if $P = P_{1}^{(1)} P_{2}^{(2)} \dots P_{n}^{(n)} \in P_{N}$ ⁠, then let $P (j) = P_{j + 1}^{(1)} P_{j + 2}^{(2)} \dots P_{j + N}^{(n)}$ ⁠, with $P^{(k)} \in P$ fixed. Given a transformation in $C_{N}$ ⁠, we specify the sites it acts on and so we can, then, find the appropriate action on P(j). To illustrate this, let $U \in C_{2}$ be defined by X₁ → X₁X₂, X₂ → Z₁Z₂, Z₁ → Z₁, and Z₂ → X₂. Then, for Q = X₁X₂, we have

\begin{align} U_{12} Q U_{12}^{†} & = - Y_{1} Y_{2}, \\ U_{23} Q U_{23}^{†} & = X_{1} X_{2} X_{3}, \\ U_{j + 1, j + 2} Q (j) U_{j + 1, j + 2}^{†} & = - Y_{j + 1} Y_{j + 2} . \end{align}

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Local Pauli operators are those that are elements of $P_{N}$ ⁠, where N is independent of L, the length of the chain. The translation invariant spin chain Hamiltonians that we consider take the form

H = \sum_{n} \sum_{α} J_{α} P^{α} (n), J_{α} \in R, P^{α} \in P_{N},

(15)

where α labels the different terms that appear and n indexes the sites, where each site has a translated Pauli operator. The couplings J_α are all real since H is Hermitian. For example, H₀ defined in (1) would have α ∈ {1, 2, 3}, $n \in Z$ ⁠, and P¹ = Z₁Y₂X₃Y₄Y₅X₆Y₇Z₈.

B. Translation invariant Clifford transformations and matrix product operators

While we can look at transformations in $C_{L}$ directly, we focus on those Clifford transformations that are of the form U_chain, illustrated in Fig. 1. This is given by

U_{chain} = \prod_{n \in s i t e s} U_{n, n + 1, \dots, n + k - 1}, U \in C_{k},

(16)

where the product is ordered so that we act first with the operator acting on the left-most sites of the chain. Such transformations are illustrated for k = 2 and k = 3 in Fig. 1. For k = 1, this is an on-site change of basis. Note that throughout, we are thinking of a large system size L but will not concern ourselves with any subtleties arising in the thermodynamic limit L → ∞ (see, for example, the rigorous approach in Ref. 40). Now, as mentioned in the Introduction, this transformation has the form of an MPO, and this interpretation will be particularly helpful in Sec. V, where we discuss connections to Bethe ansatz methods. We use the standard tensor network graphical notation below⁵² although we will also write out some formulas explicitly. In Fig. 2, we graphically relate the translation invariant Clifford group transformations of Fig. 1 to MPOs. This particular argument appears, for example, in the theory of matrix product states (MPSs), showing equivalence between MPS and circuits of the form U_chain (not necessarily constructed from basic Clifford transformations).^52–55 A translation invariant MPO on a chain $\otimes_{n = 1}^{L} H_{n}$ is an operator M on the chain of the form

M_{j_{1}, \dots, j_{L}}^{i_{1}, \dots, i_{L}} = t r (V_{j_{1}}^{i_{1}} V_{j_{2}}^{i_{2}} \dots V_{j_{L}}^{i_{L}}) = \sum_{α_{1}, \dots, α_{L}} {(V_{j_{1}}^{i_{1}})}_{α_{2}}^{α_{1}} {(V_{j_{2}}^{i_{2}})}_{α_{3}}^{α_{2}} \dots {(V_{j_{L}}^{i_{L}})}_{α_{1}}^{α_{L}} .

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For fixed i_k, j_k (physical indices corresponding to $H_{k}$ ⁠), each of the $V_{j_{k}}^{i_{k}}$ are χ × χ matrices acting on a “virtual space” as $V_{j_{k}}^{i_{k}} : V \to V$ ⁠, where virtual indices are, then, summed over (i.e., we matrix multiply in the virtual space).^43,44,52 The virtual space $V$ has dimension χ, referred to as the bond dimension.⁵⁶ Alternatively, one can think of V as a four index tensor, with two physical indices and two virtual indices. The trace over the virtual space, illustrated in Fig. 2(f), reflects periodic boundary conditions (i.e., $H_{1}$ and $H_{L}$ are neighboring sites). For open boundary conditions, we do not take the trace and can simply alter the tensors at the boundary (for fixed physical indices, the boundary tensors are vectors rather than matrices)—this is illustrated in Fig. 2(e). To see that U_chain defined in (16) is an MPO, we show in Fig. 2(d) how to find the MPO tensor V from the basic Clifford transformation U. We see that for k = 2, we have⁵⁷ a χ = 4 MPO, and the generalization for k = 3 gives a χ = 16 MPO.

FIG. 2.

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Relating the translation invariant Clifford group transformations of Fig. 1 and MPOs. (a) U_chain = … U_3,4U_2,3U_1,2 and (b) U_chain = … U_3,4,5U_2,3,4U_1,2,3. (c) A periodic analog of (a) given by U_chain = … U_3,4U_2,3U_1,2… . This is interpreted in the main text. In (a)–(c), all lines represent two-dimensional spaces. (d) An MPO tensor V in terms of U that appears in (a) and (c). (e) An MPO with open boundary; the virtual index α has dimension χ. (f) An MPO with periodic boundary; the virtual index α has dimension χ. This shows that (a) and (c) are of the MPO form with χ = 4.

Now, if the operator $M_{j_{1}, \dots, j_{L}}^{i_{1}, \dots, i_{L}}$ defined in (17) is unitary, then it is called a matrix product unitary (MPU)—note that we take the trace, and so this operator is of the periodic boundary condition form. It has been shown that MPUs are equivalent to QCA^58–60 and so will necessarily be locality-preserving.

Let us now relate this equivalence to the MPOs considered in this work. First, using the graphical argument of Fig. 2, our definition of U_chain in (16) is an MPO with open boundary (there is no trace corresponding to summing over a virtual index between site L and site 1). Since the basic Clifford transformation U is unitary, the resulting MPO is unitary and is, moreover, a Clifford transformation in $C_{L}$ ⁠. However, this does not define an MPU since the definition of an MPU is for the periodic boundary condition form of the MPO. Hence, there is no equivalence between this construction⁶¹ and QCA. Indeed, for U_chain derived from (10)—the Kramers–Wannier duality—we saw that this family of MPOs is not necessarily locality-preserving. In fact, this duality is discussed in detail in Ref. 39. In that paper, a periodic MPO form for the Kramers–Wannier duality is given. One can show that this MPO fits the construction we are considering and is built out of a basic Clifford transformation,⁶²

\begin{align} X_{n} \to Z_{n}, & X_{n + 1} \to Z_{n} X_{n + 1}, \\ Z_{n} \to X_{n} Z_{n + 1}, & Z_{n + 1} \to Z_{n + 1} . \end{align}

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Using this basic Clifford to construct U_chain leads to the (bulk) transformation X_n → Z_n−1Z_n and Z_nZ_n+1 → X_n. The corresponding periodic MPO [as in Fig. 2(c)] is more subtle. The analysis in Ref. 39 shows that it is not invertible (and hence not unitary) by analyzing the fusion algebra with the spin-flip operator ∏_nX_n. More generally, if we take a basic Clifford transformation U and use this to construct a periodic boundary MPO [given in Fig. 2(c)], this is not necessarily unitary and so is not necessarily an MPU. However, if U_chain is locality preserving, then conjugation by U_chain acts as a Clifford QCA. This means by the general arguments in Refs. 58 and 59 that there exists a periodic MPU form. Our analysis below allows us, using elementary calculations, to give such a periodic form directly for transformations of interest (see Sec. IV C).

Our calculations focus on the open boundary case, which explicitly breaks translation invariance by fixing a left-most site. However, such transformations preserve bulk translation invariance in the following way. Let us index the sites here so that the first site has n ≃ −L/2 and the last site has n ≃ L/2. Then, given P, a local Pauli operator in the bulk, if U_chain transforms P → Q, where Q is local, then we have that the translated operators transform as P(j) → Q(j) for j independent of L. Note that the causal structure of U_chain means that we can have a locality-non-preserving transformation such that, on conjugating local operators, the non-local images extend to the right. Instead of conjugating by U_chain, we could conjugate by $U_{chain}^{†}$ ⁠, giving locality-non-preserving transformations where the non-local images extend to the left. We do not analyze this case, but analogous calculations would go through.

III. FAMILIES OF SPIN CHAINS FROM THE XXZ CHAIN—U_chain WITH k = 2

In this section, we characterize all local spin chains that are equivalent to H_XXZ under conjugation by any U_chain that is built from a basic two-site Clifford transformation $U \in C_{2}$ ⁠. We first summarize the results for locality-preserving transformations in Sec. III A and then summarize the results for locality-non-preserving transformations in Sec. III B. We, then, prove our claims by classifying all U_chain with k = 2 in Sec. III C. Of course, this classification does not depend on the Hamiltonian, so these results could be easily applied to other canonical models. For example, one could straightforwardly generalize the analysis to the integrable XYZ chain³ simply by allowing independent couplings for each term in (3). Note also that H_XXZ is integrable for all $Δ \in R$ ⁠. For Δ > 1 (Δ < −1), the model is in a gapped (anti-)ferromagnetic phase, while for |Δ| < 1, we have a gapless paramagnetic phase.¹ If we include an external field term, h∑_nZ_n, the model remains integrable. The phase diagram in the Δ − h plane has three regions corresponding to the three phases mentioned above for h = 0; the value of Δ for the boundaries of these regions varies with h.

A. Locality-preserving transformations with k = 2 and the XXZ chain

Consider all locality-preserving U_chain built from a two-site Clifford transformation, as defined in (16) for k = 2 [see also Fig. 1(a)]. In Sec. III C, we will show that acting with any such locality-preserving transformation has the effect on H_XXZ of either (a) performing an on-site change of basis or (b) transforming $H_{XXZ} \to H_{XXZ}^{'}$ ⁠, where

H_{XXZ}^{'} = - \sum_{n \in s i t e s} (J_{1} Z_{n - 1} X_{n} X_{n + 1} Z_{n + 2} + J_{2} Z_{n - 1} Y_{n} Y_{n + 1} Z_{n + 2} + J_{3} Z_{n} Z_{n + 1}),

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up to an on-site change of basis. Here, two of the J_i are equal to 1 and the third is equal to Δ. Note that the transformation acts as (5) for this particular choice of the on-site basis and for J₃ = Δ.

A sum over four-spin interaction terms of the form Z_n−1X_nX_n+1Z_n+2 gives the Hamiltonian of a generalized cluster model.²⁶ Such terms appear in models dual to free-fermion chains^63–68 through a standard Jordan–Wigner transformation (see the discussion in Sec. V B). However, applying this Jordan–Wigner transformation to H_XXZ will lead to four-fermion interactions for Δ ≠ 0. We also note that a model with these interaction terms appears in the literature as a “twisted” XXZ chain.^27,69,70 However, that model is not translation invariant and, moreover, contains additional interactions, meaning that it is not equivalent to $H_{XXZ}^{'}$ ⁠.

Note that H_XXZ has a U(1) symmetry corresponding to rotations about the Z axis, and so this direction is distinguished. After the transformation, the distinguished direction corresponds to the term with J_i = Δ. In all the models above, we preserve integrability if we add an external field term of the form $\sum_{n} U_{chain}^{} Z_{n} U_{chain}^{†}$ ⁠. For example,

{\tilde{H}}_{XXZ}^{'} = - \sum_{n \in s i t e s} (Δ Z_{n - 1} X_{n} X_{n + 1} Z_{n + 2} + Z_{n - 1} Y_{n} Y_{n + 1} Z_{n + 2} + Z_{n} Z_{n + 1} + h Z_{n - 1} Y_{n} Z_{n + 1})

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is unitarily equivalent to H_XXZ − h∑_nZ_n. To derive this, we conjugate the Hamiltonian by U_chain built from the basic Clifford transformation $U \in C_{2}$ given by

\begin{align} X_{1} \to X_{1} Z_{2}, & X_{2} \to Z_{2}, \\ Z_{1} \to Z_{1}, & Z_{2} \to Z_{1} Y_{2} . \end{align}

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1. The folded XXZ chain and pivot Hamiltonians

As an aside, we note an interesting connection between the model $H_{XXZ}^{'}$ ⁠, the folded XXZ chain,^13,45–48 and the idea of pivot Hamiltonians.^49,50 The folded XXZ chain is an integrable model with many interesting properties, including Hilbert space fragmentation,^71,72 corresponding to exponentially large degeneracies in the spectrum. Pivot Hamiltonians generate SPT entanglers [an example of which is transformation (5)] and have been used to generate webs of dualities between models and to find interesting critical points with enhanced symmetry.^49,50 Indeed, the critical points “half-way between” two models often have the pivot Hamiltonian as an additional U(1) symmetry.⁴⁹ In this section, we show that using two different Ising Hamiltonians as a pivot, starting with H_XXZ, we arrive at the model $H_{XXZ}^{'}$ ⁠. The folded XXZ chain is at the half-way point between these two models, and we see that the two Ising Hamiltonians give two additional U(1) symmetries of the folded XXZ chain. While these symmetries were already known, we hope that this connection will lead to new insights.

Let us define the following commuting charges:

\begin{align} Q_{1} & = \frac{1}{2} \sum_{n \in s i t e s} (1 - Z_{n}), Q_{2} = \frac{1}{2} \sum_{n \in s i t e s} (1 - Z_{n} Z_{n + 1}), \\ Q_{4} & = - \frac{1}{4} \sum_{n \in s i t e s} (X_{n + 1} X_{n + 2} + Y_{n + 1} Y_{n + 2}) (1 + Z_{n} Z_{n + 3}) . \end{align}

(22)

The Hamiltonian, H_F, of the folded XXZ chain is given by

H_{F} = Q_{4} + h Q_{1} + Δ Q_{2} .

(23)

The model H_F also commutes with the operator $Q_{2}^{-} = \sum_{n} {(- 1)}^{n} Z_{n} Z_{n + 1}$ ⁠.⁴⁶ Now, considering the XXZ chain (3) with an external field and the transformed chain (19), we have (up to unimportant constants)

4 H_{F} = \underset{H_{XXZ} (h)}{\underset{︸}{- \sum_{n \in s i t e s} (X_{n} X_{n + 1} + Y_{n} Y_{n + 1} + Δ Z_{n} Z_{n + 1} + h Z_{n})}} + \underset{H_{XXZ}^{'} (h)}{\underset{︸}{U_{\pm}^{†} H_{XXZ} (h) U_{\pm}^{†}}},

(24)

where U_± is a Clifford transformation that can be written as a U_chain with k = 2 and acts by conjugation as

\begin{align} X_{n} & \to \pm Z_{n - 1} X_{n} Z_{n + 1}, \\ Z_{n} & \to Z_{n} . \end{align}

(25)

We see that H_F is at the half-way point on a path of models interpolating between the usual XXZ model (with an external field) and the model $H_{XXZ}^{'}$ discussed above.

To make the connection to pivot Hamiltonians, we can write these choices of U_± as U_±(π), where

U_{\pm} (θ) = e^{- i θ H_{pivot}^{\pm}} f o r H_{pivot}^{+} = \frac{1}{4} \sum_{n \in s i t e s} {(- 1)}^{n} Z_{n} Z_{n + 1}, H_{pivot}^{-} = \frac{1}{4} \sum_{n \in s i t e s} Z_{n} Z_{n + 1} .

(26)

Then, define $H^{\pm} (θ) = U_{\pm} (θ) H_{0} U_{\pm}^{†} (θ)$ ⁠. These are particular examples of the pivot Hamiltonian construction given in Ref. 49. More generally, we define H(θ) in this way for some choice of H_pivot, where there is a further requirement that H(2π) = H(0) and that H(π) is an SPT phase for some symmetry group. An example would be the trivial paramagnet H₀ = −∑_nX_n; then, H⁺(π) is the SPT cluster model. In our case, we will take H₀ = H_XXZ, which is a critical model; the analog of the SPT requirement would, then, be that H^±(π) is a symmetry-enriched gapless model³⁴ for some symmetry group. Analyzing the symmetry properties of these models⁷³ is beyond our scope; we defer it to future work. Understanding this could lead to new perspectives on the folded XXZ model.

One interesting consequence of making this connection is as follows. We can consider a family of models $H_{α}^{\pm} = α H^{\pm} (0) + (1 - α) H^{\pm} (π)$ ⁠. At the half-way point $H_{1 / 2}^{\pm} = \frac{1}{2} (H^{\pm} (0) + H^{\pm} (π))$ ⁠, there is an enhanced symmetry. Indeed, we clearly have $[H_{1 / 2}^{\pm}, U_{\pm} (π)] = 0$ [in our case of interest U_±(π) = U_±, we have U(2π) = 1, and this is a $Z_{2}$ symmetry]. If H₀ is a trivial paramagnet, in many cases, this is enhanced to a U(1) symmetry generated by $H_{pivot}^{\pm}$ ⁠, i.e., $[H_{1 / 2}^{\pm}, H_{pivot}^{\pm}] = 0$ ⁠. This follows from a general result given in Ref. 49 based on a well-chosen Clifford transformation (in this context also called a generalized Kramers–Wannier duality or the gauging map in quantum information theory) that reveals a $Z_{n}$ symmetry and shows that neutrality of the Hamiltonian under this $Z_{n}$ implies neutrality under the full U(1).

Now, taking H₀ = H_XXZ, we have that $H_{1 / 2}^{\pm} = H_{F}$ ⁠. The result given above for trivial H₀ no longer applies, but the property that $[H_{1 / 2}^{\pm}, H_{pivot}^{\pm}] = 0$ in this case corresponds to the commutativity of $Q_{1}, Q_{2}, Q_{2}^{-}$ ⁠, and Q₄. This can be checked straightforwardly in this model and naturally raises the question of how to extend the argument given in Ref. 49 to further families of Hamiltonian H₀, including gapless models.

B. Locality-non-preserving transformations with k = 2 and the XXZ chain

We now consider locality-non-preserving transformations U_chain, again built from a basic Clifford transformation $U \in C_{2}$ ⁠. By considering all locality-non-preserving U_chain with k = 2, we show in Sec. III C that there are three types of non-local Pauli strings that can appear. Two of these types of string will not preserve locality of H_XXZ, while the third does—this includes the Kramers–Wannier duality given in (10). Using this transformation, we have that the following Hamiltonian is unitarily equivalent⁷⁴ to H_XXZ − h∑_nZ_n:

H_{XXZ}^{″} = - \sum_{n \in s i t e s} (Z_{n} + Δ X_{n} X_{n + 2} - X_{n - 1} Z_{n} X_{n + 1} + h X_{n} X_{n + 1}) .

(27)

We conclude that the Kramers–Wannier duality is the only locality-non-preserving transformation (in the class under consideration and up to on-site basis changes) that takes H_XXZ − h∑_nZ_n to a local Hamiltonian. Note that $H_{XXZ}^{^{″}}$ is related to the anisotropic next-nearest-neighbor Ising (ANNNI) model.⁷⁵ Using the Kramers–Wannier duality to map the XXZ chain to this “U(1)-enhanced” ANNNI model is discussed in Ref. 76. The Hamiltonian (27) has also appeared recently in Ref. 13 as an example of a medium-range integrable model (the model that appears is, in fact, the bond-site transformed XYZ chain, rather than XXZ chain; the bond-site transformation corresponds to the Kramers–Wannier transformation used here). There, it was derived as part of a classification of “interaction-round-a-face” models with three-site interaction terms.

We give two other families of locality-non-preserving transformations below. In the Kramers–Wannier case, we have strings of the form X_n → Z_nZ_n+1Z_n+2⋯ and, hence, X_nX_n+1 → Z_n. As mentioned, the other locality-non-preserving transformations have different strings, and these do not cancel when we transform operators such as X_nX_n+1 that appear in H_XXZ. We can, however, use these locality-non-preserving transformations to map between local integrable Hamiltonians. For example, consider $H_{XXZ}^{^{″}}$ with h = 0. Then, the transformation given in (39) transforms this model to

H_{XXZ}^{‴} = - \sum_{n \in s i t e s} (Z_{n} - X_{n} X_{n + 2} + Δ X_{n - 1} Z_{n} X_{n + 1}) .

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C. Classification of U_chain built from a two-site basic Clifford

To prove these results, we now analyze the effect of conjugating once by U_chain on single-site Pauli operators. Using the formula given in Sec. II A, we have that $| C_{2} | = 11 520$ and, hence, can write down that many different U_chain. We will show that these fall into seven classes: three (L1, L2, and L3) that are locality-preserving and four (NL1, NL2, NL3, and NL4) that are locality-non-preserving.

Consider any two anti-commuting single-site Pauli operators $P, Q \in P$ ⁠, we prove in Sec. III D that the locality-preserving transformations are of the following form:

(L1)
On-site change of basis: X_n → ±P_n, Z_n → ±Q_n.
(L2)
On-site change of basis and shift by one site to the left:⁷⁷ X_n → ±P_n−1, Z_n → ±Q_n−1.
(L3)
Decorating transformations²⁵

\begin{align} X_{n} \to \pm S_{n - 1} P_{n} S_{n + 1}, \\ Z_{n} \to \pm S_{n - 1} Q_{n} S_{n + 1}, w h e r e S_{n} ≔ i P_{n} Q_{n} . \end{align}

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Transformation (L3) could, of course, be preceded by an on-site change of basis—this will correspond to U_chain given by a different basic Clifford transformation in $C_{2}$ ⁠. Then, for example, we can also have

\begin{align} X_{n} \to \pm S_{n - 1} P_{n} S_{n + 1}, \\ Y_{n} \to \pm S_{n - 1} Q_{n} S_{n + 1}, w h e r e S_{n} ≔ i P_{n} Q_{n} . \end{align}

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Below, we will take as given that one can always do this initial on-site change of basis.

We now consider locality-non-preserving transformations. These all map at least one Pauli operator to a non-local string of Pauli operators. For a finite system, this string will end at the right-most site (this site has index L). As we are interested in transforming local Hamiltonians to local Hamiltonians, the precise details of the boundary terms are not so important: they must drop out in local expressions—we use S′ and T to represent the Pauli operators that are included in this boundary term. We prove in Appendix B that the locality-non-preserving transformations are of forms (NL1)–(NL4). (Note that there exist transformations U_chain for all choices of anti-commuting $P, Q \in P$ ⁠; the operator appearing in the string, S, is then fixed.)

(NL1)
Transformations leading to a string ∏_jS_n+j,
$\begin{align} X_{n} \to {(- 1)}^{a n + b} S_{n} S_{n + 1} S_{n + 2} S_{n + 3} \dots S_{L - 1} T_{L}, \\ Z_{n} \to {(- 1)}^{a n + c} P_{n - 1} Q_{n} S_{n + 1} S_{n + 2} S_{n + 3} \dots S_{L - 1} T_{L}, \end{align}$
(31)
where S_n ≔ ±iP_nQ_n and a, b, c ∈ {0, 1}.
(NL2)
Transformations leading to a string ∏_jS_n+2j,
$\begin{align} X_{n} \to {(- 1)}^{x (n)} P_{n - 1} S_{n} S_{n + 2} S_{n + 4} S_{n + 6} \dots S_{L - 1}^{'} T_{L}, \\ Z_{n} \to {(- 1)}^{z (n)} Q_{n - 1} S_{n} S_{n + 2} S_{n + 4} S_{n + 6} \dots S_{L - 1}^{'} T_{L}, \end{align}$
(32)
where S_n ≔ ±iP_nQ_n and the oscillatory terms depend on x(n) and z(n) that are discussed in Appendix B.
(NL3)
Transformations leading to a string given by ∏_jS_n+3jS_n+3j+1,
$\begin{align} X_{n} & \to {(- 1)}^{x (n)} P_{n - 1} S_{n} S_{n + 1} S_{n + 3} S_{n + 4} S_{n + 6} S_{n + 7} \dots S_{L - 1}^{'} T_{L}, \\ Z_{n} & \to {(- 1)}^{z (n)} Q_{n - 1} S_{n} S_{n + 1} S_{n + 3} S_{n + 4} S_{n + 6} S_{n + 7} \dots S_{L - 1}^{'} T_{L}, \end{align}$
(33)
where S_n ≔ ±iP_nQ_n and the oscillatory terms depend on x(n) and z(n) that are discussed in Appendix B.
(NL4)
On-site change of basis and shift by one site to the left, with boundary term and oscillatory factor,
$\begin{align} X_{n} \to {(- 1)}^{a n + b} P_{n - 1} T_{L}, \\ Z_{n} \to {(- 1)}^{a n + c} Q_{n - 1} T_{L}, \end{align}$
(34)
where a, b, c ∈ {0, 1}.

As mentioned, S′ and T are the boundary terms—these are unimportant for transforming local Hamiltonians to local Hamiltonians. The operator S′ is either equal to S or to $I$ depending on whether the site L − 1 would appear in the string;⁷⁸ T is not fixed [and, in general, depends on S′, L, and details of the particular Clifford transformation—for example, we show in Appendix B that one can write down a Clifford transformation, leading to (31) with any choice of $T \in P$ ]. Note also that these are bulk transformations, so the transformation will, in general, act differently on sites at the boundary.

D. Derivation of this classification

To prove the claim about locality-preserving transformations, let us define⁷⁹ a two-site Clifford transformation U_n,n+1 acting non-trivially on sites n and n + 1 by

X_{n} \to s_{X} L_{n}^{(X)} R_{n + 1}^{(X)}, Z_{n} \to s_{Z} L_{n}^{(Z)} R_{n + 1}^{(Z)} (\Rightarrow Y_{n} \to s_{X} s_{Z} i L_{n}^{(X)} L_{n}^{(Z)} R_{n + 1}^{(X)} R_{n + 1}^{(Z)}) .

(35)

L and R are single-site Pauli operators on the left and right sites⁸⁰ and s_X, s_Z ∈ {±1}. It is enough to consider how the transformation U_chain acts left to right since that is the direction where U_chain has a non-local causal structure. There are three cases to consider:

First, if $R^{(X)} = R^{(Z)} = I$ ⁠, then this is a single-site Clifford transformation and we simply have an on-site change of basis: the transformations of the form (L1).
The second case is where R^(X) ≠ R^(Z) and both are non-trivial, implying R^(Z) ≠ R^(X)R^(Z) ≠ R^(X). Hence, all three single-site Pauli operators appear in the R-terms on site n + 1 in (35). This means that X_n, Y_n, and Z_n will keep growing into non-local strings of operators as we continue to apply the transformations making up U_chain. The only exception to this is when, in addition, $L^{(X)} = L^{(Z)} = I$ —this does keep bulk operators local, corresponding to an on-site change of basis and a shift by one site to the left.⁸¹ Hence, the locality-preserving transformations in this case give the transformations of the form (L2).
The final case is where one of R^(X), R^(Z), and R^(X)R^(Z) is equal to identity (and then, the other two must be equal). Let us consider a non-trivial R^(X) = R^(Z) = R. If R = Y, then we have a locality-preserving U_chain. This follows since $R^{(Y)} = I$ ⁠, and so
$\begin{align} U_{n + 1, n + 2}^{†} (U_{n, n + 1}^{†} X_{n}^{†} U_{n, n + 1}^{†}) U_{n + 1, n + 2}^{†} & = s_{X} U_{n + 1, n + 2}^{†} (L_{n}^{(X)} Y_{n + 1}^{†}) U_{n + 1, n + 2}^{†} \\ = s_{Z} L_{n}^{(X)} (i L_{n + 1}^{(X)} L_{n + 1}^{(Z)}) . \end{align}$
(36)
Furthermore, two-site Clifford transformations making up U_chain act on sites to the right of n + 1, so we see that it transforms X_n to a local operator. An analogous calculation holds for transforming Z_n, and so for R = Y, we have a locality-preserving transformation. If R ≠ Y, then we will have non-local strings by iterating the transformation. To see this explicitly,
$\begin{align} U_{n + 1, n + 2}^{†} (U_{n, n + 1}^{†} X_{n}^{†} U_{n, n + 1}^{†}) U_{n + 1, n + 2}^{†} & = s_{X} U_{n + 1, n + 2}^{†} (L_{n}^{(X)} R_{n + 1}^{†}) U_{n + 1, n + 2}^{†} \\ = s_{X} s_{R} L_{n}^{(X)} L_{n + 1}^{(R)} R_{n + 2}, \end{align}$
(37)
where further two-site Clifford transformations making up U_chain will continue to act non-trivially. We can make an analogous argument for the case that $R^{(X)} = I$ and also for the case that $R^{(Z)} = I$ ⁠. For example, if R^(Y) = R^(Z) = X, we have locality-preserving transformations, while if R^(Y) = R^(Z) = Y, we will not.
In (36), we saw that if R = Y, then the transformation is locality-preserving. We now consider how U_chain acts in this case. First, R = Y means that in (35), we must have L^(X) ≠ L^(Z). We can, then, consider how U_n,n+1 in (35) acts on X_n+1 and Z_n+1. One can show that,⁸² up to signs, the images of X_n+1 and Z_n+1 are any two distinct elements of ${Y_{n + 1}, (i L_{n}^{(X)} L_{n}^{(Z)}) X_{n + 1}, (i L_{n}^{(X)} L_{n}^{(Z)}) Z_{n + 1}}$ ⁠. Note that the choice of which operator goes to which operator is simply the choice of the initial basis for our transformation U_chain. Let us define the product S ≔ iL^(X)L^(Z). We see that, without loss of generality, conjugating by U_chain acts as
$\begin{align} X_{n} \to \pm S_{n - 1} L_{n}^{(X)} S_{n + 1}, \\ Z_{n} \to \pm S_{n - 1} L_{n}^{(Z)} S_{n + 1} . \end{align}$
(38)
In this way, we reach the locality-preserving transformations (L3).

To prove the claim about locality-non-preserving transformations, we simply have to consider all the cases we excluded in the above discussion. In particular, we need to consider the cases where R^(X) = R^(Z) ≠ Y and where R^(X) ≠ R^(Z) and both are non-trivial. These different cases are analyzed in Appendix B. Here, we give examples of a basic Clifford transformation that generates each type.

(NL1)
Up to an on-site basis change, this is the Kramers–Wannier duality with basic Clifford transformation (10).
(NL2)
Putting P = X, Q = Y, and S = Z, we have the basic Clifford transformation,
$\begin{align} X_{n} \to - Z_{n} X_{n + 1}, & X_{n + 1} \to X_{n} Z_{n + 1}, \\ Z_{n} \to Z_{n} Y_{n + 1}, & Z_{n + 1} \to Y_{n} Z_{n + 1} . \end{align}$
(39)
This example has x(n) = z(n) = 0, so the string is not oscillatory.
(NL3)
Putting P = Z, Q = Y, and S = X, we have the basic Clifford transformation,
$\begin{align} X_{n} \to X_{n} Z_{n + 1}, & X_{n + 1} \to Z_{n} X_{n + 1}, \\ Z_{n} \to X_{n} Y_{n + 1}, & Z_{n + 1} \to Y_{n} X_{n + 1} . \end{align}$
(40)
This example has x(n) = z(n) = 0, so the string is not oscillatory.
(NL4)
Putting P = X and Q = Z, we have
$\begin{align} X_{n} \to - Y_{n} X_{n + 1}, & X_{n + 1} \to X_{n} Y_{n + 1}, \\ Z_{n} \to Y_{n} Z_{n + 1}, & Z_{n + 1} \to Z_{n} Y_{n + 1} . \end{align}$
(41)
Since Y_n → −Y_n+1, we have an oscillatory term ±(−1)ⁿ, and moreover, T = Y.

We note that the results for locality-preserving transformations given above are consistent with the theory of Clifford QCA,^40,41 as they should be. However, to prove our results using that theory, one would first need to show that the operators under consideration have images on up to three neighboring sites.

IV. GENERALIZATIONS—FURTHER FAMILIES RELATED TO THE XXZ MODEL

In this section, we give a classification of all translation invariant Clifford transformations on the spin chain that map X_n and Z_n to Pauli strings on at most five neighboring sites (i.e., each image is of the form $P_{m - 2}^{(1)} P_{m - 1}^{(2)} P_{m}^{(3)} P_{m + 1}^{(4)} P_{m + 2}^{(5)}$ ⁠). This is the same mathematical problem as classifying Clifford QCA where the images of X_n and Z_n are non-trivial on at most five neighboring sites.⁸³ We, furthermore, show that all these transformations are of the form U_chain for some basic Clifford $U \in C_{2}$ or $U \in C_{3}$ ⁠. We, then, use these results to write down a form for all Hamiltonians that are equivalent to H_XXZ under translation invariant Clifford group transformations that take X_n and Z_n to local operators on at most five sites.

In Sec. III, we started by considering basic Clifford transformations and showed that certain families of U_chain were locality-preserving. An alternative approach is to consider locality-preserving transformations directly. As discussed in Sec. II A, unitary transformations preserve commutation relations. If our unitary is a translation invariant Clifford group transformation, this gives a map X_n → P and Z_n → Q, with P and Q being some Pauli strings. Commutators such as [X_n, X_n+m] = 0 ⇔ [P, P(m)] = 0 give us non-trivial constraints on P and Q [recall that P(m) is defined in Sec. II A as a translation of P by m sites]. In Appendix C, we analyze these constraints in the case that P and Q are Pauli strings on up to five sites. The outcome is that if we map any single-site Pauli operator to a two- or four-site operator, then it must map the other on-site Pauli operators to non-local strings (this follows from the theory of Clifford QCA—see Ref. 83). We also prove the results for locality-preserving transformations given in Subsection IV A.

A. Locality-preserving transformations with images on at most five neighboring sites

If we map to a three- or five-site operator, we show below that we have the transformations that we already found using basic Clifford transformations in $C_{2}$ ⁠, along with the following additional transformations:

(L4)
Decorating transformation,
$\begin{align} X_{n} \to \pm A_{n - 2} A_{n - 1} C_{n} A_{n + 1} A_{n + 2}, \\ Z_{n} \to \pm A_{n - 2} A_{n - 1} C_{n}^{'} A_{n + 1} A_{n + 2}, \end{align}$
(42)
with $A, C, C^{'} \in P \ {I}$ ⁠, C ≠ C′, and A = iCC′.
(L5)
Decorating transformation,
$\begin{align} X_{n} \to \pm A_{n - 2} C_{n} A_{n + 2}, \\ Z_{n} \to \pm A_{n - 2} C_{n}^{'} A_{n + 2}, \end{align}$
(43)
with $A, C, C^{'} \in P \ {I}$ ⁠, C ≠ C′, and A = iCC′.
(L6)
Decorating transformation,
$\begin{align} X_{n} \to \pm A_{n - 2} B_{n - 1} A_{n} B_{n + 1} A_{n + 2}, \\ Z_{n} \to \pm A_{n - 2} C_{n - 1} C_{n} C_{n + 1} A_{n + 2}, \end{align}$
(44)
with $A, B, C \in P \ {I}$ ⁠, A ≠ B, and C = iAB.

As above, we can always precede these transformations by an on-site change of basis, equivalent to taking a different U_chain. Furthermore, one can show that (L4)–(L6) can all be derived using U_chain built from $U \in C_{3}$ ⁠. This is demonstrated⁸⁴ in Table I. Finally, note that by considering only the constraints from translation invariance and the commutation relations, we can include an arbitrary translation (or shift operator). We have fixed this translation so that the images are symmetric about site n—transformations U_chain do not include such an arbitrary translation.

TABLE I.

Transformations U_chain of the type shown in Fig. 1 that take H_XXZ to H_i by $H_{i} = {\tilde{U}}_{i} H_{XXZ} {\tilde{U}}_{i}^{†}$ ⁠.

		Basic Clifford	(U₁₂ or U₁₂₃)	Clifford group transformation (U_chain)
${\tilde{U}}_{1}^{†}$	X₁ → X₁Z₂	X₂ → Z₁Y₂		X_n → Z_n−1Y_nZ_n+1
${\tilde{U}}_{1}^{†}$	Z₁ → Z₁	Z₂ → Z₂		Z_n → Z_n
${\tilde{U}}_{2}^{†}$	X₁ → Y₁Z₂Z₃	X₂ → Z₁Y₂	X₃ → Z₁Y₃	X_n → −Z_n−2Z_n−1Y_nZ_n+1Z_n+2
${\tilde{U}}_{2}^{†}$	Z₁ → Z₁	Z₂ → Z₂	Z₃ → Z₃	Z_n → Z_n
${\tilde{U}}_{3}^{†}$	X₁ → X₁Z₂Z₃	X₂ → Z₁X₂Z₃	X₃ → Z₁Z₂X₃	X_n → Z_n−2X_nZ_n+2
${\tilde{U}}_{3}^{†}$	Z₁ → Z₁	Z₂ → Z₂	Z₃ → Z₃	Z_n → Z_n
${\tilde{U}}_{4}^{†}$	X₁ → X₁X₂X₃	X₂ → Z₁Y₂X₃	X₃ → Z₁X₂Y₃	X_n → −Z_n−2Y_n−1Y_nY_n+1Z_n+2
${\tilde{U}}_{4}^{†}$	Z₁ → Z₁	Z₂ → X₂	Z₃ → X₃	Z_n → −Z_n−1X_nZ_n+1

		Basic Clifford	(U₁₂ or U₁₂₃)	Clifford group transformation (U_chain)
${\tilde{U}}_{1}^{†}$	X₁ → X₁Z₂	X₂ → Z₁Y₂		X_n → Z_n−1Y_nZ_n+1
${\tilde{U}}_{1}^{†}$	Z₁ → Z₁	Z₂ → Z₂		Z_n → Z_n
${\tilde{U}}_{2}^{†}$	X₁ → Y₁Z₂Z₃	X₂ → Z₁Y₂	X₃ → Z₁Y₃	X_n → −Z_n−2Z_n−1Y_nZ_n+1Z_n+2
${\tilde{U}}_{2}^{†}$	Z₁ → Z₁	Z₂ → Z₂	Z₃ → Z₃	Z_n → Z_n
${\tilde{U}}_{3}^{†}$	X₁ → X₁Z₂Z₃	X₂ → Z₁X₂Z₃	X₃ → Z₁Z₂X₃	X_n → Z_n−2X_nZ_n+2
${\tilde{U}}_{3}^{†}$	Z₁ → Z₁	Z₂ → Z₂	Z₃ → Z₃	Z_n → Z_n
${\tilde{U}}_{4}^{†}$	X₁ → X₁X₂X₃	X₂ → Z₁Y₂X₃	X₃ → Z₁X₂Y₃	X_n → −Z_n−2Y_n−1Y_nY_n+1Z_n+2
${\tilde{U}}_{4}^{†}$	Z₁ → Z₁	Z₂ → X₂	Z₃ → X₃	Z_n → −Z_n−1X_nZ_n+1

B. Families of models equivalent to the XXZ chain under locality-preserving transformations

We can summarize some of the above results as follows: We have that the spin chains,

H_{1} = - \sum_{n \in s i t e s} (J_{1} Z_{n - 1} X_{n} X_{n + 1} Z_{n + 2} + J_{2} Z_{n - 1} Y_{n} Y_{n + 1} Z_{n + 2} + J_{3} Z_{n} Z_{n + 1}),

(45)

H_{2} = - \sum_{n \in s i t e s} (J_{1} Z_{n - 2} X_{n} X_{n + 1} Z_{n + 3} + J_{2} Z_{n - 2} Y_{n} Y_{n + 1} Z_{n + 3} + J_{3} Z_{n} Z_{n + 1}),

(46)

H_{3} = - \sum_{n \in s i t e s} (J_{1} Z_{n - 2} Z_{n - 1} X_{n} X_{n + 1} Z_{n + 2} Z_{n + 3} + J_{2} Z_{n - 2} Z_{n - 1} Y_{n} Y_{n + 1} Z_{n + 2} Z_{n + 3} + J_{3} Z_{n} Z_{n + 1}),

(47)

H_{4} = - \sum_{n \in s i t e s} (J_{1} Z_{n - 2} X_{n - 1} X_{n + 2} Z_{n + 3} + J_{2} Z_{n - 2} Y_{n - 1} Y_{n} Y_{n + 1} Y_{n + 2} Z_{n + 3} + J_{3} Z_{n - 1} Y_{n} Y_{n + 1} Z_{n + 2}),

(48)

where two of the J_i are equal to 1 and the third is equal to Δ, are equivalent to the XXZ chain under Clifford group transformations. Furthermore, these are all unitarily related to H_XXZ by a transformation of the form given in Fig. 1. Moreover, up to an on-site change of basis, this is a complete list of Hamiltonians that are equivalent to H_XXZ under translation invariant Clifford group transformations that take X_n and Z_n to local operators on at most five sites—in Table I, we give the relevant transformations for J₃ = Δ. For the cases with J₁ or J₂ = Δ, we do an on-site change of basis to H_XXZ and then use a transformation from Table I.

It is simple to generate further models by applying a sequence of such transformations (including single-site unitaries). This is illustrated in Fig. 3(a) for the case of two transformations $U_{chain}^{'} U_{chain}$ ⁠. The transformations ${\tilde{U}}_{i}$ defined in Table I are such that $H_{i} = {\tilde{U}}_{i} H_{XXZ} {\tilde{U}}_{i}^{†}$ ⁠. Then, $H_{0} = {\tilde{U}}_{4} {\tilde{U}}_{2} H_{XXZ} {\tilde{U}}_{2}^{†} {\tilde{U}}_{4}^{†}$ is the Hamiltonian given in (1). In Appendix A, we use this unitary transformation to understand the nearby phases to H₀ and to show that it is at a phase transition between SPT phases.

FIG. 3.

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(a) Applying U_chain followed by $U_{chain}^{'}$ generates further transformations. (b) A depth-four local unitary quantum circuit; Fig. 1(a) is of this form for U^⋆ given in (49). (c) An MPU tensor corresponding to (b) (note that we block neighboring spins, so the physical dimension is four). (d) Reordering $U_{chain}^{⋆}$ built from U^⋆ that commute on different pairs of sites. (e) A depth-eight circuit that has the same effect as transforming with ${\tilde{U}}_{4}$ ⁠.

C. Commuting Clifford transformations and the MPU form

Note that the two-site Clifford $U_{12}^{⋆}$ ⁠,

\begin{align} X_{1} \to X_{1} Y_{2}, & X_{2} \to \pm Y_{1} X_{2}, \\ Z_{1} \to Z_{1} Y_{2}, & Z_{2} \to \pm Y_{1} Z_{2}, \end{align}

(49)

satisfies $[U_{n, n + 1}^{⋆}, U_{n + 1, n + 2}^{⋆}] = 0$ and the corresponding transformation $U_{chain}^{⋆}$ takes X_n → ±Y_n−1X_nY_n+1 and Z_n → ±Y_n−1Z_nY_n+1. Reordering the U^⋆ as in Fig. 3(d), we, then, have a depth-two local unitary quantum circuit for $U_{chain}^{⋆}$ —illustrated in Fig. 3(b) (on setting $U^{'} = U^{^{″}} = I$ ⁠). By using on-site basis transformations beforehand and afterward, we can, then, do any of the transformations (L3). This means that for any decorating transformation of the form (L3), we have a depth-four local unitary quantum circuit, as shown in Fig. 3(b). (Note that translation invariance is retained due to the commutativity of the $U_{n, n + 1}^{⋆}$ ⁠.) This tells us, in particular, that the transformation ${\tilde{U}}_{1}$ appearing in Table I can be written in this way.

We can similarly find local unitary circuits of depth at most eight for the remaining transformations of Table I. One can check that the basic Clifford transformations that appear in ${\tilde{U}}_{2}$ and ${\tilde{U}}_{3}$ commute when applied to different sites. This means that, by analogy with Fig. 3(d), ${\tilde{U}}_{2}$ and ${\tilde{U}}_{3}$ can be written as depth-three local unitary quantum circuits. The basic Clifford transformation appearing in ${\tilde{U}}_{4}$ does not commute when applied to different sites; however, note that applying two consecutive two-site decorating transformations gives the same effect. In particular, let V₁ take X_n → Z_n−1Y_nZ_n+1, Y_n → Z_n−1X_nZ_n+1 and V₂ take X_n → Z_n−1X_nZ_n+1, Z_n → Z_n−1Y_nZ_n+1. Then, conjugation by ${\tilde{U}}_{4}$ is the same as conjugation by V₂V₁. This means that ${\tilde{U}}_{4}$ can be written as a local unitary circuit of depth at most eight—this is illustrated in Fig. 3(e), where U′, U″ and V′, V″ are the appropriate basis changes for the decorating transformations V₁ and V₂, respectively.

This leads to the following conclusion: any U_chain that takes each X_n and each Z_n to a local operator on at most five sites can be written as a local unitary quantum circuit of depth at most eight. (We include on-site changes of basis before and after applying transformations in Table I. For ${\tilde{U}}_{4}$ ⁠, we can change U′ and V″ to give the correct basis.) Moreover, in this form, one can see how to explicitly make this transformation periodic. Then, as in going from Fig. 3(b) to Fig. 3(c), one can find an MPU tensor. Having such an MPU tensor is helpful in understanding how the integrable structure of the XXZ model transforms, as we will see in Sec. V.

V. INTEGRABLE MODELS

Thus far, we have considered translation invariant Clifford transformations and used them to construct families of Hamiltonians that are unitarily equivalent to H_XXZ. The XXZ model, defined in (3), is an integrable model, and this integrability is related to some underlying structure. It is intuitive that this integrable structure will be roughly unchanged under Clifford transformations or, indeed, other unitary transformations. It is the purpose of this section to begin to address in what sense the transformed XXZ chains constructed above share this structure and to see how this work fits more generally into the classification of integrable models. We will not comment on the precise meaning of the term quantum integrable model; rather, we refer the interested reader to the discussion in Ref. 5. Pragmatically, if one knows something useful about the states of some Hamiltonian, the ideas introduced above can be used to understand some other Hamiltonian. Two classes about which we can say a lot are free models and models solvable by the Bethe ansatz, such as the XXZ model. Free models are those where there is no quasiparticle scattering, while Bethe ansatz solvable models have the property that many-quasiparticle scattering can be factorized into two-body scattering.

We will first analyze how conjugation by a general MPU affects the integrable structure of a Bethe ansatz solvable model. Then, we will explain what this means for the Clifford transformations that we consider. Finally, we will discuss free models.

A. Models solvable by algebraic Bethe ansatz

1. Generalities

The algebraic Bethe ansatz is a well-developed method that allows us, by solving a system of “Bethe equations,” to find the eigenstates and eigenvalues of an integrable Hamiltonian. It is a further non-trivial step to derive correlation functions in these eigenstates; however, there are many known results.^1–3,6,28 We point out that certain results apply at finite system sizes, while others apply only on taking the thermodynamic limit. The central object in this method is the $R$ -matrix. This is an operator that solves the Yang–Baxter equation, depicted graphically⁸⁵ in Fig. 4(a). Using such a solution to the Yang–Baxter equation, one can generate a set of local conserved charges, and one of these is the Hamiltonian of the corresponding fundamental integrable model. There is no prescription to go from an integrable Hamiltonian (which may or may not be fundamental) to the $R$ -matrix¹ although there are interesting tests for integrability of spin chain Hamiltonians.^13,15 The XXZ spin chain is a Hamiltonian that is the fundamental model derived from the XXZ $R$ -matrix and can be analyzed using the algebraic Bethe ansatz.

FIG. 4.

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(a) Yang–Baxter equation for $R$ -matrices. (b) Yang–Baxter algebra for $L$ -operators.

2. The fixed-point of an MPU

Before proceeding, we briefly review some further technical results about MPUs that are needed in the following discussion. These results and corresponding proofs are found in Refs. 58 and 59. All of the results are given diagrammatically in Fig. 5.

FIG. 5.

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(a) Blocking an MPO; the two shaded tensors V are combined (by contraction of a virtual index) to give a single tensor W. (b) and (c) Definition of the vectors ⟨l| and |r⟩ as left- and right-eigenvectors of a matrix formed from the tensors V and $\bar{V}$ ⁠; d is the dimension of the physical index. (d) The first MPU fixed-point equation. (e) The second MPU fixed-point equation. (f) and (g) The “pulling through conditions,” a consequence of the fixed-point equations.

We can combine tensors by blocking physical indices; this simply involves viewing multiple physical sites as one physical site with a larger on-site dimension. This is illustrated in Fig. 5(a); if i₁ and i₂ correspond to a two-dimensional spin-1/2 site, then ${\tilde{i}}_{1}$ will be a four-dimensional index corresponding to the tensor product. An important result is that if we have an MPU with bond dimension (dimension of the virtual index) equal to χ, then by blocking at most χ⁴ sites, we have a fixed-point (or “simple”) form for the MPU.^58,59

Now, let us suppose that we have blocked sites such that the MPU is in the fixed-point form. From the corresponding blocked tensors V and $\bar{V}$ ⁠, we can form a “transfer matrix.” This transfer matrix has just one non-zero eigenvalue, and this is equal to one.⁵⁸ Then, we have two vectors ⟨l| and |r⟩, the unique left- and right-eigenvectors⁸⁶ with eigenvalue one—see Figs. 5(c) and 5(d). The blocked tensor V satisfies two fixed-point equations, given in Figs. 5(d) and 5(e). These equations imply two further equations called pulling through conditions, given in Figs. 5(f) and 5(g).

3. Integrable models transformed by an MPU

Recall that in Sec. IV C we showed that for a given locality-preserving U_chain given in Table I, we can find an MPU tensor V such that conjugating by the MPU $M_{j_{1}, \dots, j_{L}}^{i_{1}, \dots, i_{L}} = t r (V_{j_{1}}^{i_{1}} \dots V_{j_{L}}^{i_{L}})$ gives the same transformation as conjugating by U_chain. In the discussion below, we show that, after blocking sites, if we conjugate our integrable spin chain by any MPU, then given the $R$ -matrix for the original integrable model, we can find another $R$ -matrix depending on the MPU that satisfies the Yang–Baxter equation. The new $R$ -matrix contains the original $R$ -matrix acting on a subspace of an enlarged virtual Hilbert space—see Fig. 6. On the other part of the enlarged space, there is a projection $I \otimes | r 〉〈 l | \otimes I$ constructed from the eigenvectors of the MPU transfer matrix described in Sec. II. This projection means that this new $R$ -matrix does not satisfy a technical condition called regularity and is, moreover, not invertible. However, we will show below that one can still deduce certain features of the model using this modified $R$ -matrix,⁸⁷ including the conserved charges. Thus, we see that the underlying integrability is modified in a relatively straightforward way depending on the original $R$ -matrix and the MPU. The Hamiltonian itself is, in fact, more closely related to another object called the $L$ -operator, introduced below. We will see that the $L$ -operator is acted on by the operators V that form the MPU, illustrated in Fig. 7.

FIG. 6.

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(a) The $R^{'}$ -matrix in the enlarged virtual space (the virtual space of the original $R$ -matrix and an additional virtual space). (b) The Yang–Baxter equation in the additional virtual space. This identity holds for any vectors |r⟩ and ⟨l| and implies that $R^{'} (λ, μ)$ satisfies the Yang–Baxter equation if $R (λ, μ)$ does. Note that this is not a regular solution of the Yang–Baxter equation.

FIG. 7.

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The monodromy matrix (a) before and (b) after conjugation by an MPU M constructed from tensors ${(V_{j}^{i})}_{β}^{α}$ ⁠. In (b), the shaded tensor on the left is $T^{'} (λ)$ ⁠, while the shaded tensor on the right is $L^{'} (λ)$ ⁠.

4. Classification of integrable models

In order to explore these ideas in greater depth, let us review some further details about the algebraic Bethe ansatz. As mentioned, along with the $R$ -matrix, we have the $L$ -operator. The $L$ -operator satisfies the Yang–Baxter algebra depicted in Fig. 4(b). Given an $L$ -operator that satisfies this algebra, one can generate further $L$ -operators satisfying the algebra. In this way, one generates the Hamiltonian and other conserved charges.⁶

Now, the fundamental solution to the Yang–Baxter algebra is given by a “rotated” $R$ -matrix. In particular, if we identify the horizontal (virtual) Hilbert space and the vertical (physical) Hilbert space, then identifying the upper left and lower right indices of the $R$ -matrix as physical indices (rotating the picture by π/4) gives an $L$ -operator that solves Fig. 4(b). This is, however, not a unique solution to the Yang–Baxter algebra.

In general, the $L$ -operator is a tensor acting on a single site, with two physical indices and two uncontracted virtual indices. From $L$ -operators, $L (λ)$ ⁠, one can form the monodromy matrix,⁸⁸

T {(λ)}_{β}^{α} = \sum_{α_{2}, \dots, α_{L}} L_{1} {(λ)}_{α_{2}}^{α} L_{2} {(λ)}_{α_{3}}^{α_{2}} \dots L_{L} {(λ)}_{β}^{α_{L}} .

(50)

This is an MPO on L sites—depicted in Fig. 7(a)—and is a tensor formed by matrix multiplication of the virtual indices of the $L$ operators. $T (λ)$ acts as a matrix on physical indices and has two uncontracted virtual indices: by taking the trace, we reach the transfer matrix T(λ), where T(λ) has physical indices only. Transfer matrices at different values of the spectral parameter commute, [T(λ), T(μ)] = 0, and this means that the transfer matrix is the generating function of the conserved charges. In particular, the canonical Hamiltonian of the integrable model is the logarithmic derivative of the transfer matrix at λ = 0.¹

Now, let us address the question of classifying integrable models. One approach is to classify solutions to the Yang–Baxter equation, or $R$ -matrices, of some type.⁹ For example, the case corresponding to nearest-neighbor Hamiltonians was studied recently in Refs. 11 and 12 for on-site Hilbert space $C^{2}$ and $C^{4}$ ⁠, respectively (although it remains to apply Bethe ansatz techniques to find the spectrum and eigenstates of some of these examples). By the discussion above, this would classify fundamental integrable models. In those works, there is discussion of the fact that there are whole classes of equivalent spin chains that can be found by applying local basis transformations to the $R$ -matrix—this is equivalent to applying single-site basis transformations to the fundamental spin chain Hamiltonian. In Ref. 10, the following analogy with representation theory is drawn: fixing a particular $R$ -matrix is similar to fixing a particular group, while enumerating all monodromy matrices for a given $R$ -matrix is analogous to constructing representations of the group. Representations are reducible, in general, so one can try to find the analog of irreducible representations—these are the $L$ -operators studied in Ref. 10.

In Figs. 6 and 7(b), we demonstrate the action of conjugation by an MPU on the various operators. First, in our conjugated model, we can define a new $L$ -operator, $L^{'}$ ⁠, that includes the V and $\bar{V}$ tensors that constitute the MPU [see the shaded tensor in Fig. 7(b)]. The dimension of the physical indices of $L^{'}$ (corresponding to the dimension of the site of the spin chain) is unchanged, while the bond dimension of $L^{'}$ is the bond dimension of $L$ multiplied by χ², where χ is the bond dimension of the tensor V. We also define a blocked operator $\tilde{L^{'}}$ ⁠, where we block k physical sites so that the MPU reaches its fixed point. In Fig. 8, we show that $\tilde{L^{'}}$ satisfies a modified Yang–Baxter algebra, with a modified $R$ -matrix: $R^{'}$ ⁠; this is itself defined in Fig. 6. In Fig. 9, we show that despite this modification of the algebra, we can still use it to show the commutativity of the corresponding transfer matrices.

FIG. 8.

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A variant of the Yang–Baxter algebra for the $R^{'}$ matrix. Physical sites are blocked (k times) so that the MPU satisfies the fixed-point equations. We can interpret the left-hand side equaling the right-hand side as the usual Yang–Baxter algebra holding whenever we contract virtual indices with additional (sufficiently blocked) MPU tensors such that we can use the pulling through condition. Indeed, the first equality is an application of the pulling through condition for the MPU, the second equality follows from the usual Yang–Baxter algebra for the original $R$ -matrix, and then the third equality uses the pulling through condition again to give the analog of the Yang–Baxter algebra.

FIG. 9.

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A graphical proof that the transfer matrices commute for different values of the spectral parameter. The dashed lines indicate tracing over the virtual indices. In writing the first equality, we assume that the (original) $R$ -matrix is invertible, and then, we use the fixed-point MPU equations to insert the projector |r⟩⟨l|. The modified Yang–Baxter algebra and the fixed-point MPU equation give the second equality.

To summarize, we have shown that by applying MPU transformations to integrable models, we find a modified form of the underlying integrability. We have derived an underlying $R$ -matrix, where the dependence on the spectral parameter is through the $R$ -matrix of the original model only. This $R$ -matrix of the transformed model is not regular and can be written as a tensor product of the original (assumed regular) $R$ -matrix and a trivial (but not regular) solution of the Yang–Baxter equation that is itself derived from the MPU. Moreover, we find a transformed $L$ -operator that satisfies a modified Yang–Baxter algebra. These transformed models fall outside the usual equivalence under basis transformations to the indices of the $R$ -matrix; indeed, this must be the case since the Hamiltonian is no longer nearest neighbor, in general. We, hence, conclude that the essential features of the underlying integrability are unchanged, but our transformed models are not equivalent to the original model in the usual classification scheme. It would be most interesting to take this analysis further, for example, by showing the existence of an $R$ -matrix and $L$ -operator that give the transformed Hamilton and satisfy the unmodified Yang–Baxter algebra. One interesting point is whether such an $L$ -operator would have a blocked physical index, and if so, how this blocking relates to the range of the transformed Hamiltonian.

5. Relation to translation invariant Clifford transformations

We argued above that the Clifford transformations U_chain in Table I be written in an MPU form. This means that we can directly use the above analysis to see that the integrable spin chains in Sec. IV B correspond to Bethe ansatz solvable models. These models have an $R$ -matrix that is simply related to the XXZ $R$ -matrix, and the $L$ -operator is not fundamental, but is related to the fundamental $L$ -operator of the XXZ chain by conjugating with the appropriate MPU tensor. These operators do not satisfy the usual Yang–Baxter algebra, but a variant of this algebra that, nevertheless, allows us to prove commutativity of transfer matrices in the usual way.

The locality-non-preserving transformations do not fit directly into this analysis, as we do not have an MPU form. However, we can consider the system on open boundaries. Then, the formalism of the algebraic Bethe ansatz is more involved, but can still be described in terms of tensor networks.⁶ Moreover, the locality-non-preserving transformations that we consider can be written as unitary MPOs. It would be interesting to generalize the above analysis to the open boundary condition case to better understand the locality-non-preserving transformations discussed in Sec. III B. We also note that the $L$ -operator and (implicitly) the $R$ -matrix for the bond-site transformed XYZ model [see also model (27)] are given in Ref. 13. It would be interesting to connect their results with the methods of this paper.

B. Free-fermion chains

We now turn to free models. In this section, we introduce a standard spin chain that can be analyzed by transforming to a free-fermion model. Then, we point out a construction, using Clifford group transformations, of further spin chain Hamiltonians that can be solved by free fermions.

Now, focusing on translation invariant spin-1/2 chains, if we can find a transformation into fermionic modes such that the resulting Hamiltonian is quadratic, then this is a free-fermion chain.^8,89 An example is the Ising model,

H_{Ising} = \frac{1}{2} \sum_{n \in s i t e s} (h Z_{n} - X_{n} X_{n + 1}),

(51)

and the Jordan–Wigner transformation takes us to a quadratic fermion model. In particular, define

γ_{n} = \prod_{m = 1}^{n - 1} Z_{m} X_{n}, {\tilde{γ}}_{n} = \prod_{m = 1}^{n - 1} Z_{m} Y_{n},

(52)

where $c_{n} = (γ_{n} + i {\tilde{γ}}_{n}) / 2$ is a canonical spinless fermion at site n. Then, $H_{Ising} = \frac{i}{2} \sum_{n} ({\tilde{γ}}_{n} γ_{n + 1} + h {\tilde{γ}}_{n} γ_{n})$ ⁠. This is a nearest neighbor model. An interesting class of longer-range spin chains are the generalized cluster models,^26,63,64

\begin{align} H_{free} = \frac{1}{2} \sum_{n \in s i t e s} & (t_{0} Z_{n} - t_{1} X_{n} X_{n + 1} - t_{2} X_{n} Z_{n + 1} X_{n + 2} - t_{3} X_{n} Z_{n + 1} Z_{n + 2} X_{n + 3} - \dots) \\ - (t_{- 1} Y_{n} Y_{n + 1} - t_{- 2} Y_{n} Z_{n + 1} Y_{n + 2} - t_{- 3} Y_{n} Z_{n + 1} Z_{n + 2} Y_{n + 3} - \dots), \end{align}

(53)

where we can extend to longer range terms by including longer strings of Z operators. Using the same Jordan–Wigner transformation (52), this can be mapped into a quadratic fermion Hamiltonian with terms coupling sites beyond nearest neighbor: $H_{fermion} = i / 2 \sum_{n, α} t_{α} {\tilde{γ}}_{n} γ_{n + α}$ ⁠. This can be solved similarly to the Ising model, giving the spectrum and eigenstates. Moreover, one can derive many results for the phase diagram, correlations, and topological edge modes.^66,68,90,91 Note that these models have fermionic couplings only of the form $i {\tilde{γ}}_{n} γ_{m}$ ⁠; more generally, one could also include couplings $i {\tilde{γ}}_{n} {\tilde{γ}}_{m}$ ⁠.

Now, one can apply Clifford group transformations U_chain to these spin chains and the transformed model will remain solvable by transforming to free fermions, and moreover, we retain all the mentioned results (appropriately transformed). The transformation ${\tilde{U}}_{4}$ of Table I, for example, leads to the following transformations of terms from (53) up to four-sites (one- and two-site terms appear already in Sec. IV B):

\begin{align} X_{n} Z_{n + 1} Z_{n + 2} X_{n + 3} & \to Z_{n - 2} Y_{n - 1} X_{n} Z_{n + 1} Z_{n + 2} X_{n + 3} Y_{n + 4} Z_{n + 5}, \\ X_{n} Z_{n + 1} X_{n + 2} & \to Z_{n - 2} Y_{n - 1} Y_{n} X_{n + 1} Y_{n + 2} Y_{n + 3} Z_{n + 4}, \\ Y_{n} Z_{n + 1} Y_{n + 2} & \to - Z_{n - 2} X_{n - 1} Z_{n} X_{n + 1} Z_{n + 2} X_{n + 3} Z_{n + 4}, \\ Y_{n} Z_{n + 1} Z_{n + 2} Y_{n + 3} & \to Z_{n - 2} X_{n - 1} X_{n + 4} Z_{n + 5} . \end{align}

(54)

One way to approach these free-fermion models is to transform the definitions of the Majorana fermions (52),

\begin{align} γ_{n} = \prod_{m = 1}^{n - 1} (U Z_{m} U^{†}) (U X_{n} U^{†}), \\ {\tilde{γ}}_{n} = \prod_{m = 1}^{n - 1} (U Z_{m} U^{†}) (U Y_{n} U^{†}), \end{align}

(55)

and then insert these definitions into the Hamiltonian H_fermion. A related approach is explored in Ref. 17—there, the starting point is any two Pauli operators that obey the commutation algebra of the Ising model terms Z_n and X_nX_n+1 and then building a Jordan–Wigner transformation from them. By applying U_chain to the operators Z_n and X_nX_n+1, we preserve this algebra, so our results give a different perspective on that work. We also mention that the authors of Refs. 19, 20, and 24 discussed the question of whether a particular spin chain Hamiltonian can be transformed into a free-fermion form. In these papers, the commutation relations between the terms in the Hamiltonian define a graph, and then, graph-theoretic conditions for the existence of a free-fermion solution are given (see also earlier work on bond algebras⁹²). Note that this graph depends on the basis of Pauli operators with which we expand the Hamiltonian. As pointed out in Ref. 19, this graph is invariant under Clifford transformations, and so given a Hamiltonian of the form (15), the transformations discussed in this paper map between families of spin chains with the same underlying graph.

VI. DISCUSSION

In this paper, we have explained how to generate new families of integrable Hamiltonians using translation invariant Clifford transformations. As an example, we classified certain families of transformed XXZ models. These families were derived from a classification of translation invariant Clifford transformations that are built from $U \in C_{2}$ and a classification of translation invariant Clifford transformations that take single-site Pauli operators to local operators on at most five sites. Finally, we put these results in context by discussing how they relate to usual classifications of integrable Hamiltonians.

This is, however, far from a complete picture. We do not know how to determine whether or not two given Hamiltonians are related by a Clifford transformation. Understanding how to do this is an avenue of future research. As mentioned, there do exist interesting (conjectural) tests for integrability of spin chains.^13,15 These conjectures are based on the existence of a conserved charge. The models we have considered that are related to the XXZ chain will naturally have a conserved charge corresponding to a certain operator in the XXZ chain. However, the second conjecture of Ref. 15 relies on deriving this charge from a “boost operator,” and since the models in Sec. IV B are not nearest neighbor, the transformed boost operator will not be of the standard form given there. A generalization to the case of medium-range models was given recently in Ref. 13, and it would be interesting to understand whether our models satisfy the conjectures given in that paper. Furthermore, there are unresolved questions regarding the underlying integrable structures for the transformed models considered above. Finally, as discussed above, recent papers^19,20,24 analyze the question of when a spin chain can be solved by transforming to a free-fermion model. It would certainly be of interest to find related results for integrable models beyond free fermions, such as those we have considered in this paper.

In terms of applications, we have already pointed out the appearance of Clifford transformations in the literature on SPT phases and have given a discussion of H₀ as a transition between SPT phases. As another example, one can take the critical Ising model,

H_{Ising} = - \sum_{n \in s i t e s} (Z_{n} Z_{n + 1} + X_{n}),

(56)

and using transformation (5), one can map it to

H_{Cluster - Ising} = - \sum_{n \in s i t e s} (Z_{n} Z_{n + 1} + Z_{n - 1} X_{n} Z_{n + 1}) .

(57)

One notable feature of this model is that its low energy physics is described by a symmetry-enriched Ising CFT.³⁴ The symmetry in this case is $Z_{2} \times Z_{2}^{T}$ ⁠, generated by the spin-flip ∏_nX_n and an antiunitary symmetry T that acts as complex-conjugation (in particular, X_n and Z_n are real, while Y_n is imaginary). In general, Clifford transformations will not preserve the symmetry of the Hamiltonian. A key point is that for a Clifford transformation on the whole chain that takes us from one symmetric Hamiltonian to another, the basic Clifford transformation that we use to construct it need not be symmetric. This property allows us to map between distinct symmetric phases as in the example above. Our methods give a general family of transformations that we could use to explore mapping standard integrable models to symmetry-enriched examples in a similar way. Conversely, given a Hamiltonian and a certain choice of symmetry group, one could classify the Clifford transformations considered above according to how they affect these symmetry properties. In Sec. I, we made connections to a recent study⁴⁹ on SPT entanglers and pivot Hamiltonians. It would be interesting to understand more deeply these connections. For example, understanding the integrability properties and symmetry properties of pivoted integrable models would be most interesting. As mentioned, this could give new perspectives on the folded XXZ model.

In this work, we focus on integrable spin chains, including spin chains that can be solved by mapping into free-fermion chains. One may be directly interested in fermionic integrable models, for example, the Hubbard chain.^2,93 In that case, it would be natural to consider generalizing our discussion above to include fermionic transformations. This would include Majorana translations γ_n → γ_n+1 that do not have an analog in the spin chain.⁹⁴ Moreover, in the fermionic case, the equivalence between MPUs and QCA that we discuss above no longer holds.⁹⁵

Finally, while we have focused our discussion on integrable models, the Clifford transformations U_chain could, of course, be applied to any Hamiltonian. A motivation for restricting to integrable models was that we could use the many results about integrable models to analyze the resulting transformed model. For general models, it may still be of interest to understand which families of models are equivalent under the transformations that we have analyzed here.

ACKNOWLEDGMENTS

We thank R. Verresen for a number of stimulating conversations and for insightful comments on the manuscript. We are also grateful to P. Fendley, S. Piddock, L. Piroli, B. Pozsgay, D. Shepherd, and R. Thorngren for helpful discussions and correspondence. We thank Y. Miao for valuable discussions and for identifying an issue with Sec. V A of the first version of this manuscript. N.L. acknowledges support from the UK Engineering and Physical Sciences Research Council through Grant Nos. EP/R043957/1, EP/S005021/1, and EP/T001062/1.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Nick G. Jones: Conceptualization (equal); Formal analysis (equal); Writing – original draft (lead); Writing – review & editing (equal). Noah Linden: Conceptualization (equal); Formal analysis (equal); Writing – original draft (supporting); Writing – review & editing (equal).

DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

APPENDIX A: H₀ IS A TRANSITION BETWEEN SPT PHASES

In this appendix, we will discuss nearby phases of the integrable model given in (1), which, for convenience, we repeat here, allowing for arbitrary (real) coupling constants,

\begin{align} H_{0} = - \sum_{n \in s i t e s} (J_{1} Z_{n - 3} Y_{n - 2} X_{n - 1} Y_{n} Y_{n + 1} X_{n + 2} Y_{n + 3} Z_{n + 4}) \\ + (J_{2} Z_{n - 3} Y_{n - 2} Y_{n - 1} Y_{n + 2} Y_{n + 3} Z_{n + 4} + J_{3} Z_{n - 1} Y_{n} Y_{n + 1} Z_{n + 2}) . \end{align}

(A1)

Recall also that $H_{0} = U_{0} H_{XYZ} U_{0}^{†}$ ⁠, where $U_{0} = {\tilde{U}}_{4} {\tilde{U}}_{2}$ are defined in Table I and H_XYZ is given by

H_{XYZ} = - \sum_{n \in s i t e s} (J_{1} Y_{n} Y_{n + 1} + J_{2} X_{n} X_{n + 1} + J_{3} Z_{n} Z_{n + 1}) .

(A2)

For J₁ = J₂ = 1, J₃ = Δ, this is the XXZ model.

Let us first discuss H_XYZ setting J₃ = 0 for convenience: then, we have the XY model.⁹⁰ This model has a $Z_{2} \times Z_{2}$ symmetry generated by P_X = ∏_nX_n and P_Y = ∏_nY_n (we also have that P_Z = ∏_nZ_n = P_XP_Y is a symmetry, where we assume that the chain has length a multiple of four, for convenience) and an anti-unitary $Z_{2}^{T}$ time-reversal symmetry, T, that corresponds to complex conjugation. Now, for J₁ > J₂, the ground state spontaneously breaks P_X and T with order parameter ⟨Y_n⟩. Similarly, for J₂ > J₁, the ground state spontaneously breaks P_Y with order parameter ⟨X_n⟩. More generally, for |J₃| < J₂ < J₁, the nearby phases are the same.⁹⁶ Hence, the XXZ model is a transition between an X-ordered ferromagnetic phase and a Y-ordered ferromagnetic phase.

We can make the same analysis for H₀. This model has the same symmetries as H_XYZ. First, for J₁ > J₂ > |J₃|, by applying the unitary U₀, we have symmetry-breaking with order parameter ⟨Z_n−3X_n−2Y_nX_n+2Z_n+3⟩. Hence, the ground state spontaneously breaks P_X and T but preserves P_Y and P_ZT. In the fixed-point limit J₁ → ∞, we have the following:

| 〈 Z_{n - 3} X_{n - 2} Y_{n} X_{n + 2} Z_{n + 3} 〉 | = 1 \Rightarrow | 〈 Z_{1} Y_{2} Y_{3} X_{4} (\prod_{n = 5}^{M} Y_{n}) X_{M + 1} Y_{M + 2} Y_{M + 3} Z_{M + 4} 〉 | = 1 .

(A3)

Hence, this correlator is a string-order parameter⁹⁷ for the symmetry string P_Y, and the end-point operator (X_M+1Y_M+2Y_M+3Z_M+4) has a non-trivial charge under the unbroken symmetry P_ZT. This means that it is a non-trivial SPT phase, and this is a stable property for all J₁ > J₂ > |J₃|.

Second, for J₂ > J₁ > |J₃|, we have symmetry-breaking with order parameter

⟨ Z_{n - 3} X_{n - 2} Z_{n - 1} Z_{n} Z_{n + 1} X_{n + 2} Z_{n + 3} ⟩ .

(A4)

The ground state spontaneously breaks P_X and P_Y but preserves P_Z and T. Again, considering the fixed-point limit, we have that

\begin{align} | 〈 Z_{n - 3} X_{n - 2} Z_{n - 1} Z_{n} Z_{n + 1} X_{n + 2} Z_{n + 3} 〉 | & = 1 \Rightarrow \\ | 〈 Z_{1} Y_{2} X_{3} Y_{4} X_{5} & (\prod_{n = 6}^{M} Z_{n}) X_{M + 1} Y_{M + 2} X_{M + 3} Y_{M + 4} Z_{M + 5} 〉 | = 1, \end{align}

(A5)

giving a string order parameter for the phase. The end-point operator (X_M+1Y_M+2X_M+3Y_M+4Z_M+5) has trivial charge under the unbroken symmetry P_ZT. Hence, we see that H₀ (J₁ = J₂ > |J₃|) is at a transition between two different SPT phases.

Note that for the XXZ chain, the analogous string orders are simply P_X and P_Y, which have trivial end-points, and hence, no non-trivial charges appear.

APPENDIX B: LOCALITY-NON-PRESERVING TRANSFORMATIONS

Here, we prove the claims about the different locality-non-preserving transformations arising from two-site Clifford transformations, given in Sec. III C. Recall that we use the following notation for images of X_n and Z_n when conjugating by U_n,n+1:

X_{n} \to s_{X} L_{n}^{(X)} R_{n + 1}^{(X)}, Z_{n} \to s_{Z} L_{n}^{(Z)} R_{n + 1}^{(Z)} (\Rightarrow Y_{n} \to s_{X} s_{Z} i L_{n}^{(X)} L_{n}^{(Z)} R_{n + 1}^{(X)} R_{n + 1}^{(Z)}),

(B1)

where s_X, s_Z ∈ {±1}. Above, we considered the case R^(X) = R^(Z) = R, which gave rise to local transformations for R = Y. If, instead, R = X or R = Z, then we reach a transformation of type (NL1) given in (31). More precisely, suppose R = X. Then, we must have (up to signs)

\begin{align} X_{n} & \to S_{n} X_{n + 1}, {\tilde{P}}_{n + 1} \to P_{n} Y_{n + 1}, \\ Z_{n} & \to Q_{n} X_{n + 1}, {\tilde{Q}}_{n + 1} \to P_{n} Z_{n + 1}, \end{align}

(B2)

where P and Q are anti-commuting single-site Pauli operators and S ≔ iPQ. $\tilde{P}$ and $\tilde{Q}$ are also anti-commuting single-site Pauli operators. By applying U_chain, we can fix them to be $\tilde{P} = Y$ and $\tilde{Q} = Z$ by an initial choice of on-site basis. The string comes from repeated transformations of X. We see that under conjugation by U_chain, we have (31) with boundary term T = X. Now, suppose that in (B2) we take X_n → −S_nX_n−1. Then, we will have an oscillatory sign, depending on site-index: (−1)ⁿ. Other signs in (B2) will change overall signs, but not give this oscillatory behavior, which comes from repeated transformations of X. Note that if we take R = Z, we would have T = Z, while, for completeness, the two-site Clifford,

\begin{align} Y_{n} & \to S_{n} Y_{n + 1}, Y_{n + 1} \to P_{n} X_{n + 1}, \\ Z_{n} & \to Q_{n} Y_{n + 1}, Z_{n + 1} \to P_{n} Z_{n + 1}, \end{align}

(B3)

will give the same transformation (31), but with T = Y.

We, then, need to analyze the cases where R^(X) ≠ R^(Z) and both are non-trivial, and so neither are equal to R^(Y) = iR^(X)R^(Z). Now, if $L^{(X)} = L^{(Z)} = I$ ⁠, then U_chain is a swap transformation (up to an on-site change of basis) that is local in the bulk. Without loss of generality, we, then, have $L^{(X)} = L^{(Z)} \neq I$ ⁠. Let us first consider L^(X) = L^(Z) = R^(Y). This implies that the basic Clifford acts as

\begin{align} X_{n} & \to R_{n}^{(Y)} R_{n + 1}^{(X)}, {\tilde{P}}_{n + 1} \to R_{n}^{(X)} R_{n + 1}^{(Y)}, \\ Z_{n} & \to R_{n}^{(Y)} R_{n + 1}^{(Z)}, {\tilde{Q}}_{n + 1} \to R_{n}^{(Z)} R_{n + 1}^{(Y)}, \end{align}

(B4)

where $\tilde{P} \neq \tilde{Q} \in P \ {I}$ ⁠. We, then, have three different cases that can be understood by⁹⁸

(N L 2) X_{n} \to s_{X} Z_{n} X_{n + 1}, Z_{n} \to s_{Z} Z_{n} Y_{n + 1},

(B5)

(N L 3) X_{n} \to s_{X} X_{n} Z_{n + 1}, Z_{n} \to s_{Z} X_{n} Y_{n + 1},

(B6)

(N L 4) X_{n} \to s_{X} Y_{n} X_{n + 1}, Z_{n} \to s_{Z} Y_{n} Z_{n + 1} .

(B7)

The choice of signs s_X, s_Z gives different oscillatory terms (see Table II). The three different cases correspond to the remaining transformations given in Sec. III C. In particular, (B5) leads to (32), (B6) leads to (33), and (B7) leads to (34). The oscillatory terms are determined by explicit calculation.

TABLE II.

Oscillatory terms that appear in non-local transformations depending on the signs (s_X, s_Z). For the image of X_n (Z_n), the oscillation will be (−1)^x(n) [(−1)^z(n)], where x(n) [z(n)] is the appropriate entry. We have b ∈ {0, 1}, where b may be different for x(n) and z(n) and is fixed by the other signs in the basic Clifford transformation.

(s_X, s_Z)

(1, 1)

(1, −1)

(−1, 1)

(−1, −1)

Transformation ()

⌊n/2⌋ + b

⌊n/2⌋ + n + b

b

n + b

Transformation ()

b

⌊n/3⌋ + n + b

⌊n/3⌋ + ⌊2n/3⌋ + n + b

⌊2n/3⌋ + b

Transformation ()

b

n + b

b

Finally, we consider the other possibility, which is without loss of generality, that L^(X) = L^(Z) = R^(X). Then,

\begin{align} X_{n} & \to R_{n}^{(X)} R_{n + 1}^{(X)}, {\tilde{P}}_{n + 1} \to R_{n}^{(Y)} R_{n + 1}^{(Y)}, \\ Z_{n} & \to R_{n}^{(X)} R_{n + 1}^{(Z)}, {\tilde{Q}}_{n + 1} \to R_{n}^{(Z)} R_{n + 1}^{(Y)} . \end{align}

(B8)

This is similar to the previous case. The string depends on the choices of R^(X) and R^(Z), and we are again led to the three cases (NL2)–(NL4). One can take, for example,

(N L 2) X_{n} \to s_{X} X_{n} X_{n + 1}, Z_{n} \to s_{Z} X_{n} Y_{n + 1},

(B9)

(N L 3) X_{n} \to s_{X} Z_{n} Z_{n + 1}, Z_{n} \to s_{Z} Z_{n} Y_{n + 1},

(B10)

(N L 4) X_{n} \to s_{X} X_{n} X_{n + 1}, Z_{n} \to s_{Z} X_{n} Z_{n + 1} .

(B11)

The oscillatory factors for these transformations are also given in Table II—the relevant entry depends on the corresponding transformation [(B5)–(B7)] according to the type of the non-local string [(NL2)–(NL4)].

Note that this discussion shows that for all choices of basic Clifford transformation that is not one of the locality-preserving cases, we have a transformation of form (NL1)–(NL4). We also have all transformations of this form (ignoring boundary terms) since we can apply an on-site basis change after the transformation—this is equivalent to taking a different basic Clifford transformation.

APPENDIX C: CONSTRAINTS FROM COMMUTATION RELATIONS

In this appendix, we prove certain results about translation invariant Clifford transformations. As explained above, the results for two- and four-site images follow directly from results in the literature on Clifford QCA; we find it helpful to have a self-contained presentation here. In particular, we see explicitly the general rule that the images of X_n and Z_n must be symmetric under reflection about a site.^40,41

1. Two-site

Suppose that we have a translation invariant Clifford group transformation that maps X_n to a two-site Pauli operator, without loss of generality, say, sites n − 1 and n. Then, we must have that X_n → ±A_n−1A_n and that $Z_{n} \to \pm \prod_{j} Q_{j}^{(j)}$ where this string is non-local.

To show this, let X_n → ±A_n−1B_n. Then, since [X_n, X_n+1] = 0, we see that [A_n, B_n] = 0. Since we have a genuine two-site Pauli operator, we must have that A_n = B_n, i.e., X_i → ±A_n−1A_n. Now, consider $Z_{n} \to \prod_{j} Q_{j}^{(j)}$ ⁠; this must anticommute with A_n−1A_n; hence, exactly one of Q⁽ⁿ⁻¹⁾ and Q⁽ⁿ⁾ must be a non-trivial Pauli, different from A. Let us say that Q⁽ⁿ⁾ anticommutes with A. Now,

[Z_{n - 1}, X_{n}] = 0 \Leftrightarrow [Q_{n - 1}^{(n)} Q_{n}^{(n + 1)}, A_{n - 1} A_{n}] = 0,

(C1)

and so we must have that Q⁽ⁿ⁺¹⁾ anticommutes with A. By repeating this reasoning, we end up with a non-local string of non-trivial Pauli operators to the right. If we, instead, have that Q⁽ⁿ⁻¹⁾ anticommutes with A, we would get a non-local string to the left.

2. Three-site

Suppose that we have a translation invariant Clifford group transformation that takes

\begin{align} X_{n} \to A_{n - 1} B_{n} C_{n + 1}, \\ Z_{n} \to A_{m - 1}^{'} B_{m}^{'} C_{m + 1}^{'} . \end{align}

(C2)

Since [X_n, X_n+2] = 0, we must have [C, A] = 0. Hence, either X_n → A_n−1B_nA_n+1 or X_n → B_n. Similarly, [Z_n, Z_n+2] = 0 means $Z_{n} \to A_{m - 1}^{'} B_{m} A_{m + 1}^{'}$ or $Z_{n} \to B_{m}^{'}$ ⁠.

If X_n → B_n and $Z_{n} \to B_{m}^{'}$ ⁠, then clearly m = n and B ≠ B′.
If X_n → A_n−1B_nA_n+1 and $Z_{n} \to B_{m}^{'}$ ⁠, then m = n. (Since the image of Z_n anticommutes with the image of X_n, we must have n − 1 ≤ m ≤ n + 1. If, say, m = n + 1, then since the image of Z_n anticommutes with the image of X_n, it would also anticommute with the image of X_n+2, which is a contradiction.) Then, we have B′ ≠ B so that X_n → A_n−1B_nA_n+1, $Z_{n} \to B_{n}^{'}$ ⁠, and $Y_{n} \to \pm A_{n - 1} B_{n}^{^{″}} A_{n + 1}$ ⁠. Since [Y_n+1, X_n] = 0, we must have that A ≠ B and A ≠ B′.
If X_n → A_n−1B_nA_n+1 and $Z_{n} \to A_{m - 1}^{'} B_{m}^{'} A_{m + 2}^{'}$ ⁠, then again m = n. Any other option would mean that the image of Z_n and the image of Z_n′ both anticommute with X_n, where n′ ∈ {n ± 2, n ± 3}. Then, [Z_n+2, X_n] = 0 implies that A′ = A. This is now the same as the previous case with Z and Y swapped.

Additional signs on the right-hand side of (C2) are analyzed in the same way. Hence, we can conclude that the most general translation invariant Clifford group transformation that takes us to strings of length at most three is of the form

\begin{align} P_{n} \to \pm S_{n - 1}^{'} P_{n}^{'} S_{n + 1}^{'}, \\ Q_{n} \to \pm S_{n - 1}^{'} Q_{n}^{'} S_{n + 1}^{'}, S_{n}^{'} = i P_{n}^{'} Q_{n}^{'} . \end{align}

(C3)

3. Four-site

Suppose that we have a translation invariant Clifford group transformation that takes

X_{n} \to A_{n - 1} B_{n} C_{n + 1} D_{n + 2} .

(C4)

Then, we must have that

X_{n} \to A_{n - 1} B_{n} B_{n + 1} A_{n + 2},

(C5)

and the image of Z_n is infinite.

To derive the possible images of X_n, note that [X_n, X_n+3] = 0 means that A = D. Then, [X_n, X_n+1] = [X_n, X_n+2] = 0 gives us that B = C. To, then, show that the image of Z_n is infinite, there are two cases to consider, [B, A] = 0 and {B, A} = 0. Let us consider the first case. Then, either B = A or $B = I$ ⁠. If X_n → A_n−1A_n+2, then we can proceed as in the two-site case. The image of Z_n must contain a Pauli operator either on site n − 1 or on site n + 2 that anticommutes with A. Then, since the image of Z_n commutes with A_m−1A_m+2 for all m ≠ n, the image of Z_n must be an infinite string to either the left or the right. If X_n → A_n−1A_nA_n+1A_n+2, letting $Z_{n} = \dots P_{n - 1}^{(n - 1)} P_{n}^{(n)} P_{n + 1}^{(n + 1)} P_{n + 2}^{(n + 2)} \dots$ ⁠, we have that either one or three of {P⁽ⁿ⁻¹⁾, …, P⁽ⁿ⁺²⁾} anticommute with A. In the case that only P⁽ⁿ⁻¹⁾ (P⁽ⁿ⁺²⁾) anticommutes with A, then the image of Z_n is infinite to the left (right); otherwise, it is infinite in both directions. This follows easily from the commutativity with images of X_m for m ≠ n.

For the case {B, A} = 0, we use a different approach. In that case,

X_{n} \to A_{n - 1} B_{n} B_{n + 1} A_{n + 2} .

(C6)

Let us call our four-site Clifford transformation U₄; we know from Appendix C 2 that we can find a translation invariant Clifford group transformation U₃ such that A_n → A_n and B_n → A_n−1B_nA_n+1. Now, conjugating by U₃U₄ will take X_n → B_nB_n+1 and will take $Z_{n} \to U_{3} U_{4} Z_{n} U_{4}^{†} U_{3}^{†}$ ⁠. Since U₃ maps single-site operators to at most three-site operators, if $U_{4} Z_{n} U_{4}^{†}$ is a finite string, then U₃U₄ contradicts the result of Appendix C 1. Hence, the image of Z_n must be an infinite string.

4. Five-site

Suppose that we have a translation invariant Clifford group transformation that takes

\begin{align} X_{n} \to A_{n - 2} B_{n - 1} C_{n} D_{n + 1} E_{n + 2}, \\ Z_{n} \to A_{m - 2}^{'} B_{m - 1}^{'} C_{m}^{'} D_{m + 1}^{'} E_{m + 2}^{'} . \end{align}

(C7)

Let us first consider the case where none of these operators are the identity. Then, [X_n, X_n+4] = [Z_n, Z_n+4] = 0 gives A = E and A′ = E′. That is,

\begin{align} X_{n} \to A_{n - 2} B_{n - 1} C_{n} D_{n + 1} A_{n + 2}, \\ Z_{n} \to A_{m - 2}^{'} B_{m - 1}^{'} C_{m}^{'} D_{m + 1}^{'} A_{m + 2}^{'} . \end{align}

(C8)

Now, [X_n, X_n+3] = 0 tell us that either B = D = A or that both B and D are distinct from A. In the second case, [X_n, X_n+2] = 0 gives us that B = D. The same considerations apply to the image of Z. Since both images are symmetric about the middle site, if m ≠ n, we have an inconsistency: if m = n + k, then the image of Z_n−2k will also anticommute with the image of X_n. Hence, we have that the image of X_n is either A_n−2B_n−1C_nB_n+1A_n+2 or A_n−2A_n−1C_nA_n+1A_n+2, while the image of Z_n is either $A_{n - 2}^{'} B_{n - 1}^{'} C_{n}^{'} B_{n + 1}^{'} A_{n + 2}^{'}$ or $A_{n - 2}^{'} A_{n - 1}^{'} C_{n}^{'} A_{n + 1}^{'} A_{n + 2}^{'}$ ⁠. In all cases, [X_n, Z_n+4] = 0 implies A′ = A. Considering also [X_n, Z_n+3] = 0, [X_n, Z_n+2] = 0, and {X_n, Z_n} = 0 gives us two options.

The first case is
$\begin{align} X_{n} \to A_{n - 2} A_{n - 1} C_{n} A_{n + 1} A_{n + 2}, \\ Z_{n} \to A_{n - 2} A_{n - 1} C_{n}^{'} A_{n + 1} A_{n + 2}, \end{align}$
(C9)
where $C_{n} \neq C_{n}^{'} a n d A_{n} = i C_{n} C_{n}^{'}$ ⁠.
The second case is
$\begin{align} X_{n} \to A_{n - 2} B_{n - 1} C_{n} B_{n + 1} A_{n + 2}, \\ Z_{n} \to A_{n - 2} B_{n - 1}^{'} C_{n}^{'} B_{n + 1}^{'} A_{n + 2}, \end{align}$
(C10)
with $A, B, C, C^{'} \in P$ such that B ≠ A, B′ ≠ A, C ≠ C′. Now, if B′ = B, then not only C ≠ A and C′ ≠ A but also C ≠ B and C′ ≠ B, which is inconsistent. Hence, B′ ≠ B, and either C = A or C′ = A. If C = A, then C′ = B′. Otherwise, if C′ = A, then C = B.

An example of this case is

\begin{align} X_{n} \to X_{n - 2} Z_{n - 1} Z_{n} Z_{n + 1} X_{n + 2}, \\ Z_{n} \to X_{n - 2} Y_{n - 1} X_{n} Y_{n + 1} X_{n + 2}, \\ Y_{n} \to X_{n - 1} Y_{n} X_{n - 2} . \end{align}

(C11)

Now, let us consider the five-site case where we allow some operators to be the identity. Due to Appendixes C 1 and C 3, we can take that X_n maps to a five-site, three-site, or one-site operator. First, let

X_{n} \to A_{n - 2} B_{n - 1} C_{n} D_{n + 1} A_{n + 2},

where $A \neq I$ ⁠. We can immediately exclude cases where each of $B, C, D \in {I, A}$ since, similarly to Appendix C 3, one could, then, argue that the image of Z_n is infinite. Since the images of different X_n commute, one can show that there are the following possibilities that include at least one identity and one operator not equal to A:

X_{n} \to A_{n - 2} C_{n} A_{n + 2}, C \neq A,

(C12)

X_{n} \to A_{n - 2} B_{n - 1} B_{n + 1} A_{n + 2}, B \neq A .

(C13)

Let U₅ be any transformation that acts as (C13), and using U₃ of Appendix C 3, we have that X_i → B_n−1B_n+1 under conjugation by U₃U₅. Clearly, any image of Z_n that anticommutes with that image and commutes with the images of all other X_m will be infinite, and so the image of Z_n under U₅ must be infinite. We can, hence, ignore transformations of the form (C13).

For the other possibility, (C12), we can find images of Z_n over five sites. First, the image of Z_n must be either one-site, three-site, or five-site according to Appendixes C 1 and C 3. If the image of Z_n is one site, the only consistent possibility is Z_n → A_n. If the image of Z_n is three-site, then we must have $Z_{n} \to A_{m - 1}^{'} B_{m}^{'} A_{m + 1}^{'}$ ⁠. Then, the form of (C12) means that the only consistent choice is $A^{'} = I$ ⁠. Finally, if the image of Z_n is five site, we have $Z_{n} \to A_{m - 2}^{'} B_{m - 1}^{'} C_{m}^{'} B_{m + 1}^{'} A_{m + 2}^{'}$ ⁠. Since only one image of Z_n can anticommute with (C12), we must have A′ = A and that $B^{'} \in {A, I}$ ⁠. Then, by symmetry, we must have m = n with {C′, C} = 0 and C′ ≠ A. By considering [X_n±1, Z_n] = 0, we have that $B^{'} = I$ ⁠. Note that in this final case, Y_n → ±A_n, so all five-site possibilities involving identity reduce to the same case.

For completeness, we also consider the cases where the image of X_n is a three-site or a one-site operator. Excluding cases treated in earlier appendices, we may assume that the image of Z_n is a five-site operator. We now show that all such cases reduce to previously considered five-site transformations up to some relabeling on the left-hand side. If X_n → P_n, a one-site operator, then it is easy to see from the commutation relations that Z_n → A_n−2B_n−1C_nB_n+1A_n+2, where C ≠ A and where [B, A] = 0. These possibilities have already appeared above. If the image of X_n is a three-site operator, then we have either X_n → A_n−1A_n+1 or X_n → A_n−1B_nA_n+1. The first option leads to an infinite image of Z_n. The second option must have B ≠ A. Otherwise, again, we have an infinite image of Z_n. In particular, we have that A ≠ B, $A, B \neq I$ ⁠, and $Z_{n} \to A_{n - 2} B_{n - 1}^{'} C_{n} B_{n + 1}^{'} A_{n + 2}$ ⁠. We have already seen that $B^{'} = I \Rightarrow B = I$ ⁠. If $B^{'} \neq I$ ⁠, then in all consistent cases, we reach an example of the form (C10).

In conclusion, we have proved that all locality-preserving translation-invariant Clifford transformations that take single-site Pauli operators to at most five-site Pauli operators must be of the form (L1)–(L6) given in Secs. III and IV.

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Note that this implies that the images of X_n and Z_n are restricted to a single set of five neighbouring sites. This follows from the general result for Clifford QCA that the images of X_n and Z_n must be symmetric under reflection about a site and that this is the same site for both images⁴¹ (we give a derivation for the case under consideration in Appendix C). This means that the image of Y_n is also symmetric about the same site, with support restricted to the support of the images of X_n and Z_n.

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Note that a change of basis after the transformation is needed to reach the general transformations given above. To find the corresponding basic Clifford transformation, take the transformation from Table I and then transform the Pauli operators on the first site on the right-hand side of the transformation.

85.

The $R$ -matrix acts on virtual Hilbert spaces $V_{i} \otimes V_{j} \to V_{i} \otimes V_{j}$ and depends on two “spectral parameters” or “rapidities” λ and μ: i.e., it is a tensor of the form $R {(λ, μ)}_{α_{i}, β_{j}}^{γ_{i}, δ_{j}}$ ⁠, where each index ranges from 1 to χ (the bond dimension). These indices are always contracted in calculating physical quantities. We prefer the graphical notation and refer the reader to Ref. 6 for a detailed exposition of this.

86.

We have $〈 l | = \sum_{n = 1}^{χ} 〈 n, n |$ if we gauge-fix the tensor V such that it is in a canonical form,⁵⁸ but we will not need the explicit form of these vectors for our purposes.

87.

Note that we do not exclude the possibility that there is some other natural modification of the $R$ -matrix.

88.

Typically, one includes on-site rapidities ξ_j, i.e., $T {(λ; ξ_{1}, \dots, ξ_{L})}_{β}^{α} = L_{1} {(λ - ξ_{1})}_{α_{2}}^{α} L_{2} {(λ - ξ_{2})}_{α_{3}}^{α_{2}} \dots L_{L} {(λ - ξ_{L})}_{β}^{α_{L}}$ ⁠. These are helpful in certain intermediate calculations. Then, one can restore translation invariance by putting ξ_j = ξ₀ at the end.¹ For the purposes of our discussion, we suppress these variables. In any case, our intention is to use the transformation to carry over known results rather than to rederive them from scratch.

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Other transformations fall into these three cases depending on whether the Pauli operator on site n + 1 in the image is the same as the Pauli operator before the transformation. For example, X_n → Z_nY_n+1 and Z_n → Z_nX_n+1 is of type (NL3) since the Pauli operator on site n + 1 in each image is different from the initial Pauli operator on site n. X_n → X_nY_n+1 and Z_n → X_nZ_n+1 are of type (NL2) since exactly one Pauli operator (Z) has the same Pauli operator on site n + 1 in its image.

2022

Author(s)

Integrable spin chains and the Clifford group

I. INTRODUCTION

II. CLIFFORD GROUP TRANSFORMATIONS

A. The Clifford group

B. Translation invariant Clifford transformations and matrix product operators

III. FAMILIES OF SPIN CHAINS FROM THE XXZ CHAIN—Uchain WITH k = 2

A. Locality-preserving transformations with k = 2 and the XXZ chain

1. The folded XXZ chain and pivot Hamiltonians

B. Locality-non-preserving transformations with k = 2 and the XXZ chain

C. Classification of Uchain built from a two-site basic Clifford

D. Derivation of this classification

IV. GENERALIZATIONS—FURTHER FAMILIES RELATED TO THE XXZ MODEL

A. Locality-preserving transformations with images on at most five neighboring sites

B. Families of models equivalent to the XXZ chain under locality-preserving transformations

C. Commuting Clifford transformations and the MPU form

V. INTEGRABLE MODELS

A. Models solvable by algebraic Bethe ansatz

1. Generalities

2. The fixed-point of an MPU

3. Integrable models transformed by an MPU

4. Classification of integrable models

5. Relation to translation invariant Clifford transformations

B. Free-fermion chains

VI. DISCUSSION

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

APPENDIX A: H0 IS A TRANSITION BETWEEN SPT PHASES

APPENDIX B: LOCALITY-NON-PRESERVING TRANSFORMATIONS

APPENDIX C: CONSTRAINTS FROM COMMUTATION RELATIONS

1. Two-site

2. Three-site

3. Four-site

4. Five-site

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III. FAMILIES OF SPIN CHAINS FROM THE XXZ CHAIN—U_chain WITH k = 2

C. Classification of U_chain built from a two-site basic Clifford

APPENDIX A: H₀ IS A TRANSITION BETWEEN SPT PHASES