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arXiv:2106.00640 · SciPost Physics

Correlations and commuting transfer matrices in integrable unitary circuits

Pieter W. Claeys, Jonah Herzog-Arbeitman, Austen Lamacraft

Published Jan 6, 2022• 42 citations
arXivHTMLPDF

Abstract

We consider a unitary circuit where the underlying gates are chosen to be \check{R} R ̌ -matrices satisfying the Yang-Baxter equation and correlation functions can be expressed through a transfer matrix formalism. These transfer matrices are no longer Hermitian and differ from the ones guaranteeing local conservation laws, but remain mutually commuting at different values of the spectral parameter defining the circuit. Exact eigenstates can still be constructed as a Bethe ansatz, but while these transfer matrices are diagonalizable in the inhomogeneous case, the homogeneous limit corresponds to an exceptional point where multiple eigenstates coalesce and Jordan blocks appear. Remarkably, the complete set of (generalized) eigenstates is only obtained when taking into account a combinatorial number of nontrivial vacuum states. In all cases, the Bethe equations reduce to those of the integrable spin-1 chain and exhibit a global SU(2) symmetry, significantly reducing the total number of eigenstates required in the calculation of correlation functions. A similar construction is shown to hold for the calculation of out-of-time-order correlations.

DOI
10.21468/scipostphys.12.1.007

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We consider a unitary circuit where the underlying gates are chosen to be \check{R} R ̌ -matrices satisfying the Yang-Baxter equation and correlation functions can be expressed through a transfer matrix formalism. These transfer matrices are no longer Hermitian and differ from the ones guaranteeing local conservation laws, but remain mutually commuting at different values of the spectral parameter defining the circuit. Exact eigenstates can still be constructed as a Bethe ansatz, but while these transfer matrices are diagonalizable in the inhomogeneous case, the homogeneous limit corresponds to an exceptional point where multiple eigenstates coalesce and Jordan blocks appear. Remarkably, the complete set of (generalized) eigenstates is only obtained when taking into account a combinatorial number of nontrivial vacuum states. In all cases, the Bethe equations reduce to those of the integrable spin-1 chain and exhibit a global SU(2) symmetry, significantly reducing the total number of eigenstates required in the calculation of correlation functions. A similar construction is shown to hold for the calculation of out-of-time-order correlations.
One possible approach to studying non-equilibrium dynamics is the so-called influence matrix (IM) formalism. The influence matrix can be viewed as a quantum state that encodes complete information about the non-equilibrium dynamics of a boundary degree of freedom. It has been shown that the IM is the unique stationary point of the temporal transfer matrix. This transfer matrix, however, is non-diagonalizable and exhibits a non-trivial Jordan block structure. In this article, we demonstrate that, in the case of an integrable XXZ spin chain, the temporal transfer matrix itself is integrable and can be embedded into a family of commuting operators. We further {conjecture} the exact expression for the IM as a particular limit of a Bethe wavefunction, with the corresponding Bethe roots given explicitly. We also focus on the special case of the free-fermionic XX chain. In this setting, we uncover additional local integrals of motion, which enable us to analyze the dimensions and structure of the Jordan blocks, as well as the locality properties of the IM. Moreover, we construct a basis of quasi-local creation operators that generate the IM from the vacuum state.
We present a complete classification of integrable Yang-Baxter quantum circuits with open boundary conditions and arbitrary circuit geometries. Starting from the standard transfer-matrix construction with two types of staggered inhomogeneities, we derive a general mapping that determines the arrangement of circuit gates in terms of the inhomogeneities and the system size. We conjecture that time-periodic quantum circuits are integrable whenever the local bulk and boundary gates satisfy the Yang-Baxter equation and the same bulk gate is applied exactly once per period to every nearest-neighbor pair of spins. Our construction also provides an algorithm to detect Yang-Baxter integrability for circuits with arbitrary geometries. Furthermore, we introduce a third type of inhomogeneity, denoted by ρρρ, and demonstrate that the minimum possible circuit depth is four. We show that when these ρρρ-inhomogeneities are placed at the endpoints and in their immediate neighborhood, the resulting boundary gates can be interpreted as single gates acting on multiple sites. Our construction is fully general and applies to regular RRR-matrices, both of difference and non-difference type, together with their associated boundary matrices. As an application, we consider two-qubit gates corresponding to 6- and 8-vertex RRR-matrices of non-difference form satisfying the Yang-Baxter equation, and we construct the associated reflection matrices that generate integrable quantum circuits.
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Solutions of the classical Yang–Baxter equation provide a systematic method to construct integrable quantum systems in an algebraic manner. A Lie algebra can be associated with any solution of the classical Yang–Baxter equation, from which commuting transfer matrices may be constructed. This procedure is reviewed, specifically for solutions without skew-symmetry. A particular solution with an exotic symmetry is identified, which is not obtained as a limiting expansion of the usual Yang–Baxter equation. This solution facilitates the construction of commuting transfer matrices which will be used to establish the integrability of a multi-species boson tunnelling model. The model generalises the well-known two-site Bose–Hubbard model, to which it reduces in the one-species limit. Due to the lack of an apparent reference state, application of the algebraic Bethe Ansatz to solve the model is prohibitive. Instead, the Bethe Ansatz solution is obtained by the use of operator identities and tensor product decompositions.
Matrix-product operators (MPOs) appear throughout the study of integrable lattice models, notably as the transfer matrices. They can also be used as transformations to construct dualities between such models, both invertible (including unitary) and non-invertible (including discrete gauging). We analyse how the local Yang--Baxter integrable structures are modified under such dualities. We see that the Rˇ\check{R}Rˇ-matrix, that appears in the baxterization approach to integrability, transforms in a simple manner. We further argue for a broad class of MPO dualities that the usual Yang--Baxter RRR-matrix should be extended. This extended RRR-matrix satisfies modified algebraic relations, including a modified Yang--Baxter algebra previously identified in the unitary case, that altogether give a local integrable structure underlying the commuting transfer matrices of the dual model. We illustrate these results with two case studies, analysing an invertible unitary MPO and a non-invertible MPO both applied to the canonical XXZ spin chain. The former is the cluster entangler, arising in the study of symmetry-protected topological phases, while the latter is the Kramers--Wannier duality. We show several results for MPOs with exact MPO inverses that are of independent interest.
A new type of quantum transfer matrix, arising as a Cholesky factor for the steady-state density matrix of a dissipative Markovian process associated with the boundary-driven Lindblad equation for the isotropic spin-1/2 Heisenberg (XXX) chain, is presented. The transfer matrix forms a commuting family of non-Hermitian operators depending on the spectral parameter, which is essentially the strength of dissipative coupling at the boundaries. The intertwining of the corresponding Lax and monodromy matrices is performed by an infinitely dimensional Yang–Baxter R-matrix, which we construct explicitly and is essentially different from the standard 4 × 4 XXX R-matrix. We also discuss a possibility to construct Bethe ansatz for the spectrum and eigenstates of the non-equilibrium steady-state density operator. Furthermore, we indicate the existence of a deformed R-matrix in the infinite dimensional auxiliary space for the anisotropic XXZ spin-1/2 chain, which in general provides a sequence of new, possibly quasi-local, conserved quantities of the bulk XXZ dynamics.