We introduce an integrable stochastic process associated with the D2 quantum group, which can be decomposed into two symmetric simple exclusion processes. We establish the integrability of the model under three types of boundary conditions (periodic, twisted, and open boundaries), and present its exact solution, including the spectrum, eigenstates, and some observables. This integrable model can be generalized to the asymmetric case, decomposing into two asymmetric simple exclusion processes, and its exact solutions are also studied.

We investigate the structure of Bethe-root configurations in the spin-1/2 XXZ chain with degenerate open boundaries. The physical solutions of the Bethe Ansatz equations are classified into three types in the fully degenerate case and two types in the partially degenerate case, for which we propose counting formulas for the number of physical solution sets of each type. Furthermore, we observe spectral degeneracies in the fully degenerate case and show that they can be naturally explained by the presence of specific phantom strings in the Bethe roots. The classification and resulting spectral degeneracies in the diagonal limit are also discussed.

We develop a new method to study the ground state energy of the one-dimensional supersymmetric t-J model with open boundary conditions. The eigenvalues of the nested transfer matrix are characterized by the zero roots of corresponding polynomials instead of the T-Q relation and Bethe roots. The distribution of zero roots at the ground state is studied. We find that the zero roots form two-string pairs, finite pure real and pure imaginary boundary strings. Based on the distribution of zero roots, we obtain the ground state energy of the system in the thermodynamic limit.

In this work we obtain the exact solution of quantum integrable system associated with the Lie superalgebra gl(1|1), both for periodic and for generic open boundary conditions. By means of the fusion technique we derive a closed set of operator identities among the fused transfer matrices. These identities allow us to determine the complete energy spectrum and the corresponding Bethe Ansatz equations of the model. Our approach furnishes a systematic framework for studying the spectra of quantum integrable models based on Lie superalgebras, in particular when the U(1) symmetry is broken. The derivation of the Bethe states from the exact spectrum is also addressed.

A quantum integrable spin chain model associated with the G2 exceptional Lie algebra is studied. By using the fusion technique, the closed recursive relations among the fused transfer matrices are obtained. These identities allow us to derive the exact energy spectrum and Bethe ansatz equations of the system based on polynomial analysis. The present method provides a unified treatment to investigate the Bethe ansatz solutions for both the periodic and the non-diagonal open boundary conditions associated with exceptional Lie algebras.

The ground state degeneracy of topologically ordered gapped Hamiltonians is the bedrock for self-correcting quantum memories, which are unfortunately not stable away from equilibrium even at zero temperature. This plague precludes practical robust self-correction since stability at zero temperature is a prerequisite for finite-temperature robustness. In this work, we show that the emergence of a bounded light cone renders the unitary time evolution a quasi-adiabatic continuation that preserves topological order, with the initial ground space retaining its macroscopic distance at all times as a quantum code. We also show how bounded light cones can emerge through suitable perturbations in Kitaev's toric code and honeycomb model. Our results suggest that topological orders and self-correcting quantum memories can be dynamically robust at zero temperature.

Abstract We study the quantum integrable spin chain model associated with the twisted D 2 2 $$ {D}_2^{(2)} $$ algebra (or simply the D 2 2 $$ {D}_2^{(2)} $$ model) under generic open boundary conditions. The Hamiltonian of this model can be factorized into the sum of two staggered XXZ spin chains. Applying the t-W method, we derive the homogeneous Bethe ansatz equations for the zeros of the transfer matrix eigenvalues and the patterns of the corresponding zeros of the staggered XXZ spin chain with generic integrable boundaries. Based on these results, we analytically compute the surface energies and excitation energies of the D 2 2 $$ {D}_2^{(2)} $$ model in different regimes of boundary parameters.

Abstract The Izergin-Korepin model is an integrable model with the simplest twisted quantum affine algebra U q ( A 2 2 $$ {A}_2^{(2)} $$ ) symmetry. Applying the t-W method, we derive the homogeneous zeroes Bethe ansatz equations and the corresponding zeroes patterns of the Izergin-Korepin model with generic integrable boundaries. Based on these results, we analytically compute the surface energies and boundary excitations in different regimes of boundary parameters of the model. It is shown that in some regimes, correlation effect appears between two boundary fields.

We study the quantum integrable spin chain model associated with the twisted D_2^(2) algebra (or simply the D_2^(2) model) under generic open boundary conditions. The Hamiltonian of this model can be factorized into the sum of two staggered XXZ spin chains. Applying the t-W method, we derive the homogeneous Bethe ansatz equations for the zeros of the transfer matrix eigenvalues and the patterns of the corresponding zeros of the staggered XXZ spin chain with generic integrable boundaries. Based on these results, we analytically compute the surface energies and excitation energies of the D_2^(2) model in different regimes of boundary parameters.

The string hypothesis for Bethe roots represents a cornerstone in the study of quantum integrable systems, providing access to physical quantities such as the ground-state energy and the finite-temperature free energy. While the t-W scheme and the inhomogeneous T-Q relation have enabled significant methodological advances for systems with broken U(1) symmetry, the underlying physics induced by symmetry breaking remains largely unexplored, due to the previously unknown distributions of the transfer-matrix roots. In this paper, we propose a new approach to determining the patterns of zero roots and Bethe roots for the Λ-θ and inhomogeneous Bethe ansatz equations using tensor-network algorithms. As an explicit example, we consider the isotropic Heisenberg spin chain with non-diagonal boundary conditions. The exact structures of both zero roots and Bethe roots are obtained in the ground state for large system sizes, up to (N≃ 60 and 100). We find that even in the absence of U(1) symmetry, the Bethe and zero roots still exhibit a highly structured pattern. The zero roots organize into bulk strings, boundary strings, and additional roots, forming two dominant lines with boundary-string attachments. Correspondingly, the Bethe roots can be classified into four distinct types: regular roots, line roots, arc roots, and paired-line roots. These structures are associated with a real-axis line, a vertical line, characteristic arcs in the complex plane, and boundary-induced conjugate pairs. Comparative analysis reveals that the t-W scheme generates significantly simpler root topologies than those obtained via off-diagonal Bethe Ansatz.

The thermodynamic limits of the XYZ spin chain with periodic or twisted boundary conditions are studied. By using the technique of characterizing the eigenvalue of the transfer matrix by the T - Q relation and by the zeros of the associated polynomial, we obtain the constraints of the Bethe roots and the zeros for the eigenvalues. With the help of structure of Bethe roots, we obtain the distribution patterns of zeros. Based on them, the physical quantities such as the surface energy and excitation energy are calculated. We find that both of them depend on the parity of sites number due to the topological long-range Neel order on the Mobius manifold in the spin space. We also check our results with those obtaining by the density matrix renormalization group. The method provided in this paper can be applied to study the thermodynamic properties at the thermal equilibrium state with finite temperature.

We study the Izergin-Korepin Gaudin models with both periodic and open integrable boundary conditions, which describe quantum systems exhibiting novel long-range interactions. Using the Bethe ansatz approach, we derive the eigenvalues of the Gaudin operators and the corresponding Bethe ansatz equations.

An exactly solvable one-dimensional Hubbard model with a single Anderson impurity embedded at the boundary is constructed in the framework of the quantum inverse scattering method. The model is solved exactly by the nested Bethe ansatz method. We identify the boundary bound states and determine the ground state phase diagram. By analyzing the impurity contributions to the magnetization density and magnetic susceptibility, we demonstrate that a local moment is formed at the impurity site and is screened by the host electrons, consistent with Kondo physics.

In this paper, we studied the exact solution of the C2(1) invariant quantum spin chain with off-diagonal open boundary condition. We obtain a solution of the reflection equation where the all matrix element of reflection matrix are nonzeros. By using the technique of fusion, we construct the fused transfer matrix and find the closed recursive relations among the transfer matrices. Based on the algebraic analysis, we obtain the eigenvalue of the system and express it as the inhomogeneous T−Q relation.

We study the exact solution of the C (1) 2 -invariant quantum integrable systems associated with off -diagonal open boundary condition. The boundary reflections break the U (1) -symmetry of the model. We find that the fusion relations among the fused transfer matrices can be closed at the inhomogeneous points. Based on the algebra analysis instead of constructing the eigenstates, we obtain the eigenvalues of conserved quantities including the Hamiltonian. The inhomogeneous T - Q relations and the related Bethe ansatz equations are given explicitly.

Abstract We study the thermodynamic limit of the anisotropic XYZ spin chain with non-diagonal integrable open boundary conditions. Although the U(1)-symmetry is broken, by using the new parametrization scheme, we exactly obtain the surface energy and the excitation energy of the system, which has solved the difficulty in the inhomogeneous T − Q relation. With the boundary parameters in the regions making the Hamiltonian Hermitian, we have obtained the distribution patterns of the zeros of the eigenvalue of the transfer matrix for the ground state and the excited ones. We find that the surface and excitation energies depend on the parities of sites number N, due to the long-range Neel order in the bulk. The easy-axis and thermodynamic limit for all the regions of boundary parameters are studied. We also obtain the physical quantities in the thermodynamic limit of boundary XXZ model by taking the trigonometric limit.

We investigate the thermodynamic limit and exact surface energy of the isotropic spin-1 Heisenberg chain with integrable generic open boundary conditions by a novel Bethe ansatz method. We obtain the homogeneous (or two-term) Bethe ansatz like equations for the zero roots of the transfer matrix. Based on the patterns of the zero roots, we analytical calculate the densities of zero roots and the surface energies of the model in all regimes of the boundary parameters.

A one-dimensional Bose-Hubbard model with unidirectional hopping is shown to be exactly solvable. Applying the algebraic Bethe ansatz method, we prove the integrability of the model and derive the Bethe ansatz equations. The exact eigenvalue spectrum can be obtained by solving these equations. The distribution of Bethe roots reveals the presence of a superfluid-Mott insulator transition at the ground state, and the critical point is determined. By adjusting the boundary parameter, we demonstrate the existence of a non-Hermitian skin effect even in the presence of interaction, but it is completely suppressed for the Mott insulator state in the thermodynamical limit. Our result represents a new class of exactly solvable non-Hermitian many-body systems, which has no Hermitian correspondence and can be used as a benchmark for various numerical techniques developed for non-Hermitian many-body systems.

We study the thermodynamics of the antiperiodic XXZ chain with anisotropy parameter {\eta}=i{\pi}/3 by means of the t-W method. We parameterize the eigenvalues of both the transfer matrix and the corresponding fused transfer matrix by their zero points instead of Bethe roots. Based on the patterns of the zero points distribution and the reconstructed entropy, we obtain the nonlinear integral equations (NLIEs) describing the thermodynamics of the model and compute its free energy at a finite temperature.

In this study, we explore the precise physical quantities in the thermodynamic limit of the one-dimensional Hubbard model with nonparallel boundary magnetic fields based on the off-diagonal Bethe ansatz solution. A particular emphasis is placed on the half-filling condition to investigate the distinct patterns of Bethe roots in the reduced Bethe ansatz equations for different boundary parameters. The ground state of the system can be divided into five regions according to the distribution of Bethe roots. By analyzing these patterns, we calculate the densities of states, ground-state energy density, and surface energy. The results reveal the existence of stable- boundary bound states, which are dependent on specific constraints regarding the boundary magnetic fields.