Inhomogeneous Transfer Matrix Research Articles

Exact densities of loops in O(1) dense loop model and of clusters in critical percolation on a cylinder: II. Rotated lattice

This work continues the study started in Povolotsky (2021 J. Phys. A: Math. Theor. 54 22LT01), where the exact densities of loops in the O(1) dense loop model on an infinite strip of the square lattice with periodic boundary conditions were obtained. These densities are also equal to the densities of critical percolation clusters on the 45∘ rotated square lattice rolled into a cylinder. Here, we extend those results to the square lattice with a tilt. This in particular allows us to obtain the densities of critical percolation clusters on the cylinder of the square lattice of standard orientation extensively studied before. We obtain exact densities of contractible and non-contractible loops or equivalently the densities of critical percolation clusters, which do not and do wrap around the cylinder, respectively. The solution uses the mapping of O(1) dense loop model to the six-vertex model in the Razumov–Stroganov point, while the effective tilt is introduced via the inhomogeneous transfer matrix proposed by Fujimoto. The further solution is based on the Bethe ansatz and Fridkin–Stroganov–Zagier’s solution of Baxter’s T–Q equation. The results are represented in terms of the solution of two explicit systems of linear algebraic equations, which can be performed either analytically for small circumferences of the cylinder or numerically for larger ones. We present exact rational values of the densities on the cylinders of small circumferences and several lattice orientations and use the results of high precision numerical calculations to study the finite-size corrections to the densities, in particular their dependence on the tilt of the lattice.

Integrable nonunitary open quantum circuits

We explicitly construct an integrable and strongly interacting dissipative quantum circuit via a trotterization of the Hubbard model with imaginary interaction strength. To prove integrability, we build an inhomogeneous transfer matrix, from which conserved superoperator charges can be derived, in particular, the circuit's dynamical generator. After showing the trace preservation and complete positivity of local maps, we reinterpret them as the Kraus representation of the local dynamics of free fermions with single-site dephasing. The integrability of the map is broken by adding interactions to the local coherent dynamics or by removing the dephasing. In particular, even circuits built from convex combinations of local free-fermion unitaries are nonintegrable. Moreover, the construction allows us to explicitly build circuits belonging to different non-Hermitian symmetry classes, which are characterized by the behavior under transposition instead of complex conjugation. We confirm all our analytical results by using complex spacing ratios to examine the spectral statistics of the dissipative circuits.

Integrable Trotterization: Local Conservation Laws and Boundary Driving.

We discuss a general procedure to construct an integrable real-time Trotterization of interacting lattice models. As an illustrative example, we consider a spin-1/2 chain, with continuous time dynamics described by the isotropic (XXX) Heisenberg Hamiltonian. For periodic boundary conditions, local conservation laws are derived from an inhomogeneous transfer matrix, and a boost operator is constructed. In the continuous time limit, these local charges reduce to the known integrals of motion of the Heisenberg chain. In a simple Kraus representation, we also examine the nonequilibrium setting, where our integrable cellular automaton is driven by stochastic processes at the boundaries. We show explicitly how an exact nonequilibrium steady-state density matrix can be written in terms of a staggered matrix product ansatz, and we propose quasilocal conservation laws for the model with periodic boundary conditions. This simple Trotterization scheme, in particular in the open system framework, could prove to be a useful tool for experimental simulations of the lattice models in terms of trapped ion and atom optics setups.

Symmetry classes of alternating sign matrices in a nineteen-vertex model

The nineteen-vertex model of Fateev and Zamolodchikov on a periodic lattice with an anti-diagonal twist is investigated. Its inhomogeneous transfer matrix is shown to have a simple eigenvalue, with the corresponding eigenstate displaying intriguing combinatorial features. Similar results were previously found for the same model with a diagonal twist. The eigenstate for the anti-diagonal twist is explicitly constructed using the quantum separation of variables technique. A number of sum rules and special components are computed and expressed in terms of Kuperberg’s determinants for partition functions of the inhomogeneous six-vertex model. The computations of some components of the special eigenstate for the diagonal twist are also presented. In the homogeneous limit, the special eigenstates become eigenvectors of the Hamiltonians of the integrable spin-one XXZ chain with twisted boundary conditions. Their sum rules and special components for both twists are expressed in terms of generating functions arising in the weighted enumeration of various symmetry classes of alternating sign matrices (ASMs). These include half-turn symmetric ASMs, quarter-turn symmetric ASMs, vertically symmetric ASMs, vertically and horizontally perverse ASMs and double U-turn ASMs. As side results, new determinant and pfaffian formulas for the weighted enumeration of various symmetry classes of alternating sign matrices are obtained.

Polynomial solutions of qKZ equation and ground state of XXZ spin chain at Δ = −1/2

Integral formulae for polynomial solutions of the quantum Knizhnik–Zamolodchikov equations associated with the R-matrix of the six-vertex model are considered. It is proved that when the deformation parameter q is equal to and the number of vertical lines of the lattice is odd, the solution under consideration is an eigenvector of the inhomogeneous transfer matrix of the six-vertex model. In the homogeneous limit, it is a ground-state eigenvector of the antiferromagnetic XXZ spin chain with the anisotropy parameter Δ equal to −1/2 and an odd number of sites. The obtained integral representations for the components of this eigenvector allow us to prove some conjectures on its properties formulated earlier. A new statement relating the ground-state components of XXZ spin chains and Temperley–Lieb loop models is formulated and proved.

Branches in the Spectral Flow of the Inhomogeneous Transfer Matrix for the XXZ Spin Chain

Branches such as bifurcations or multiple branches are observed in the flow of eigenvalues for the inhomogeneous transfer matrix of the XXZ spin chain. The spectral flow for the XXZ anisotropic parameter Δ is obtained by directly diagonalizing the transfer matrix in the regime: -1 ≤Δ≤1, where the inhomogeneous transfer matrix is real and symmetric. The branches are considered as novel spectral behaviors, which are not decomposed into simple superpositions of level crossings. We may call them level bifurcations . We also observe that no branches appear in the homogeneous case. The appearance of some of the branches are confirmed through the Bethe ansatz method.

Coordinate Bethe ansatz for the one-dimensional SU(n) Hubbard model with open boundary conditions

The one-dimensional (1D) SU(n) Hubbard model with open boundary condition is solved by using the coordinate Bethe ansatz method. The energy and integrable boundary conditions are obtained. At the same time, the corresponding Bethe ansatz equations are achieved by diagonalizing the inhomogeneous transfer matrix of the open SU(2n−2) XXX vertex model. When n=2, our result comes back to that of the 1D Hubbard model given by Deguchi and Yue (con-mat/9704138).

The exact solution of the SU(3) Hubbard model

The Bethe ansatz equations of the 1-D SU(3) Hubbard model are systematically derived by diagonalizing the inhomogeneous transfer matrix of the XXX model. We first derive the scattering matrix of the SU(3) Hubbard model through the coordinate Bethe ansatz method. Then, with the help of the quantum inverse scattering method we solve the nested transfer matrix and give the eigenvalues, the eigenvectors and the Bethe ansatz equations. Finally, we obtain the exactly analytic solution for the ground state.

Thermodynamics and form factors of supersymmetric integrable field theories

We study on-shell and off-shell properties of the supersymmetric sinh-Gordon and perturbed SUSY Yang-Lee models using the thermodynamic Bethe ansatz and form factors. Identifying the supersymmetric models with the Eight Vertex Free Fermion Model, we derive an inversion relation for the inhomogeneous transfer matrix and TBA equations and get correct UV results. We obtain two-point form factors of the trace of the energy-momentum tensor using the Watson equations and their SUSY transformations. As an application, we compute the UV central charge using these form factors and the spectral representation of the C-theorem.