Non-diagonal Boundary Terms Research Articles

NLIE and finite size effects of the spin-1/2 XXZ and sine-Gordon models with two boundaries revisited

Starting from the T-Q equation of an open integrable \( {\text{spin}} - \frac{1}{2} \) XXZ quantum spin chain with nondiagonal boundary terms, we derive a nonlinear integral equation (NLIE) of the sine-Gordon model on a finite interval. We compute the boundary energy and the Casimir energy for the sine-Gordon model with both left and right boundaries. A relation between the boundary parameters of the continuum model and the lattice model is given. We also present numerical results for the effective central charge of an open \( {\text{spin}} - \frac{1}{2} \) XXZ quantum spin chain which find agreement with our analytical result for the central charge of the sine-Gordon model in the ultraviolet (UV) limit.

NLIE and finite size effects of the spin-1/2 XXZ and sine-Gordon models with two boundaries revisited

Starting from the T-Q equation of an open integrable spin-1/2 XXZ quantum spin chain with nondiagonal boundary terms, we derive a nonlinear integral equation (NLIE) of the sine-Gordon model on a finite interval. We compute the boundary energy and the Casimir energy for the sine-Gordon model with both left and right boundaries. A relation between the boundary parameters of the continuum model and the lattice model is given. We also present numerical results for the effective central charge of an open spin-1/2 XXZ quantum spin chain which find agreement with our analytical result for the central charge of the sine-Gordon model in the ultraviolet (UV) limit.

Drinfeld twists of the open XXZ chain with non-diagonal boundary terms

The Drinfeld twists or factorizing F-matrices for the open XXZ spin chain with non-diagonal boundary terms are constructed. It is shown that in the F-basis the two sets of pseudo-particle creation operators simultaneously take completely symmetric and polarization free form. The explicit and completely symmetric expressions of the two sets of Bethe states of the model are obtained.

Bethe ansatz of the open spin-sXXZ chain with nondiagonal boundary terms

We consider the open spin-s XXZ quantum spin chain with nondiagonal boundary terms. By exploiting certain functional relations at roots of unity, we propose the Bethe ansatz solution for the transfer matrix eigenvalues for cases where atmost two of the boundary parameters are set to be arbitrary and the bulk anisotropy parameter has values \eta = i \pi/3, i \pi/5,... We present numerical evidence to demonstrate completeness of the Bethe ansatz solutions derived for s = 1/2 and s = 1.

Multiple reference states and complete spectrum of the [formula omitted] Belavin model with open boundaries

The multiple reference state structure of the Z n Belavin model with non-diagonal boundary terms is discovered. It is found that there exist n reference states, each of them yields a set of eigenvalues and Bethe Ansatz equations of the transfer matrix. These n sets of eigenvalues together constitute the complete spectrum of the model. In the quasi-classic limit, they give the complete spectrum of the corresponding Gaudin model.

Finite-size correction and bulk hole-excitations for special case of an open XXZ chain with nondiagonal boundary terms at roots of unity

Using our solution for the open spin-1/2 XXZ quantum spin chain with N spins and two arbitrary boundary parameters at roots of unity, the central charge and the conformal dimensions for bulk hole excitations are derived from the 1/N correction to the energy (Casimir energy).

On the second reference state and complete eigenstates of the open XXZ chain

The second reference state of the open XXZ spin chain with non-diagonal boundary terms is studied. The associated Bethe states exactly yield the second set of eigenvalues proposed recently by functional Bethe Ansatz. In the quasi-classical limit, two sets of Bethe states give the complete eigenstates of the associated Gaudin model.

Spectrum of the supersymmetric [formula omitted] model with non-diagonal open boundaries

In this work we diagonalize the double-row transfer matrix of the supersymmetric t – J model with non-diagonal boundary terms by means of the algebraic Bethe ansatz. The corresponding reflection equations are studied and two distinct classes of solutions are found, one diagonal solution and other non-diagonal. Except for the case of two non-diagonal boundaries, the eigenvalues are given for arbitrary values of the boundary parameters.

Boundary Energy of the Open XXZ Chain from New Exact Solutions

Bethe Ansatz solutions of the open spin-\(\frac{1}{2}\) integrable XXZ quantum spin chain at roots of unity with nondiagonal boundary terms containing two free boundary parameters have recently been proposed.We use these solutions to compute the boundary energy (surface energy) in the thermodynamic limit.

Formula omitted] relation and exact solution for the XYZ chain with general non-diagonal boundary terms

We propose that the Baxter's Q-operator for the quantum XYZ spin chain with open boundary conditions is given by the j → ∞ limit of the corresponding transfer matrix with spin- j (i.e., ( 2 j + 1 ) -dimensional) auxiliary space. The associated T – Q relation is derived from the fusion hierarchy of the model. We use this relation to determine the Bethe Ansatz solution of the eigenvalues of the fundamental transfer matrix. The solution yields the complete spectrum of the Hamiltonian.

Addendum to Completeness of the Bethe Ansatz solution of the open XXZ chain with nondiagonal boundary terms

Addendum to Completeness of the Bethe Ansatz solution of the open XXZ chain with nondiagonal boundary terms

Bethe ansatz solution of the open XXZ chain with nondiagonal boundary terms

We consider the integrable open XX quantum spin chain with non-diagonal boundary terms. We derive an exact inversion identity, by which we obtain the eigenvalues of the transfer matrix and the Bethe ansatz equations. For generic values of the boundary parameters, the Bethe ansatz solution is formulated in terms of the Jacobian elliptic functions.

Completeness of the Bethe Ansatz solution of the open XXZ chain with nondiagonal boundary terms

A Bethe Ansatz solution of the open spin- XXZ quantum spin chain with nondiagonal boundary terms has recently been proposed. Using a numerical procedure developed by McCoy et al, we find significant evidence that this solution can yield the complete set of eigenvalues for generic values of the bulk and boundary parameters satisfying one linear relation. Moreover, our results suggest that this solution is practical for investigating the ground state of this model in the thermodynamic limit.

Functional Relations and Bethe Ansatz for the XXZ Chain

There is an approach due to Bazhanov and Reshetikhin for solving integrable RSOS models which consists of solving the functional relations which result from the truncation of the fusion hierarchy. We demonstrate that this is also an effective means of solving integrable vertex models. Indeed, we use this method to recover the known Bethe Ansatz solutions of both the closed and open XXZ quantum spin chains with U(1) symmetry. Moreover, since this method does not rely on the existence of a pseudovacuum state, we also use this method to solve a special case of the open XXZ chain with nondiagonal boundary terms.

Solving the open XXZ spin chain with nondiagonal boundary terms at roots of unity

We consider the open XXZ quantum spin chain with nondiagonal boundary terms. For bulk anisotropy value η= iπ p+1 , p=1,2,…, we propose an exact ( p+1)-order functional relation for the transfer matrix, which implies Bethe-ansatz-like equations for the corresponding eigenvalues. The key observation is that the fused spin- p+1 2 transfer matrix can be expressed in terms of a lower-spin transfer matrix, resulting in the truncation of the fusion hierarchy.

Bethe ansatz solution of the openXXspin chain with non-diagonal boundary terms

We consider the integrable open XX quantum spin chain with nondiagonal boundary terms. We derive an exact inversion identity, using which we obtain the eigenvalues of the transfer matrix and the Bethe Ansatz equations. For generic values of the boundary parameters, the Bethe Ansatz solution is formulated in terms of Jacobian elliptic functions.

The `topological' charge for the finiteXXquantum chain

It is shown that an operator (in general non-local) commutes with the Hamiltonian describing the finite XX quantum chain with certain non-diagonal boundary terms. In the infinite volume limit this operator gives the "topological" charge.

N-species stochastic models with boundaries and quadratic algebras

Stationary probability distributions for stochastic processes on linear chains with closed or open ends are obtained using the matrix product Ansatz. The matrices are representations of some quadratic algebras. The algebras and the types of representations considered depend on the boundary conditions. In the language of quantum chains we obtain the ground state of N-state quantum chains with free boundary conditions or with non-diagonal boundary terms at one or both ends. In contrast to problems involving the Bethe Ansatz, we do not have a general framework for arbitrary N which when specialized, gives the known results for N=2; in fact, the N=2 and N>2 cases appear to be very different.

Spectra of non-Hermitian quantum spin chains describing boundary induced phase transitions

The spectrum of the non-hermitian asymmetric XXZ-chain with additional non-diagonal boundary terms is studied. The lowest lying eigenvalues are determined numerically. For the ferromagnetic and completely asymmetric chain that corresponds to a reaction-diffusion model with input and outflow of particles the smallest energy gap which corresponds directly to the inverse of the temporal correlation length shows the same properties as the spatial correlation length of the stationary state. For the antiferromagnetic chain with both boundary terms, we find a conformal invariant spectrum where the partition function corresponds to the one of a Coulomb gas with only magnetic charges shifted by a purely imaginary and a lattice-length dependent constant. Similar results are obtained by studying a toy model that can be diagonalized analytically in terms of free fermions.

Representations of the quadratic algebra and partially asymmetric diffusion with open boundaries

We consider the one-dimensional partially asymmetric exclusion model with open boundaries. The model describes a system of hard-core particles that hop stochastically in both directions with different rates. At both boundaries particles are injected and extracted. By means of the method of Derrida, Evans, Hakim and Pasquier the stationary probability measure can be expressed as a matrix-product state involving two matrices subject to a quadratic algebra. We obtain the representations of this algebra and use the two-dimensional one to derive exact expressions for the density profile and correlation functions. Using the correspondence between the stochastic model and a quantum spin chain, we obtain exact correlation functions for a spin-$\frac{1}{2}$ Heisenberg XXZ chain with non-diagonal boundary terms. Generalizations to other reaction-diffusion models are discussed.