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Physical Review A

Effect of boundary conditions on the quantum inverse problem for Alfvén waves and finite-size corrections

N Dasgupta, Asesh Roy Chowdhury

Published Jun 1, 1992• 3 citations
Publisher

Abstract

We have extended the previous analysis of the quantum inverse problem for the Alfv\'en wave propogation by incorporating boundary conditions other than the periodic ones. Our approach is that of Sklyanin [Func. Ana. Appl. 21, 164 (1986)], which allows one to introduce different boundary conditions at the two ends. A generalized Bethe ansatz is used to deduce the eigenvalues of the Hamiltonian. An immediate outcome of our analysis is the effect of finite-size corrections, which is essential to study scale invariance and conformal properties of the model.

DOI
10.1103/physreva.45.8776

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We have extended the previous analysis of the quantum inverse problem for the Alfv'en wave propogation by incorporating boundary conditions other than the periodic ones. Our approach is that of Sklyanin [Func. Ana. Appl. 21, 164 (1986)], which allows one to introduce different boundary conditions at the two ends. A generalized Bethe ansatz is used to deduce the eigenvalues of the Hamiltonian. An immediate outcome of our analysis is the effect of finite-size corrections, which is essential to study scale invariance and conformal properties of the model.
We have formulated the quantum inverse problem for the unstable nonlinear Schr\"odinger equation that forms the basis of nonlinear optics, following the modified approach of Sklyanin. Due to the nonexistence of the vacuum for the corresponding Lax L operator, the algebraic Bethe ansatz cannot be formulated. This difficulty is removed by recourse to the functional Bethe ansatz. The whole analysis is performed by considering different boundary conditions at the two ends of the interval on the x axis. Both signs of the nonlinearity are treated, corresponding to ``bright'' and ``dark'' solitons.
Numerical solutions of the Bethe-Ansatz equations for the eigenenergies of XXZ Hamiltonian on very large chains are used to identify, via conformal invariance, the scaling dimensions of various two-dimensional models. With periodic boundary conditions, eight-vertex and Gaussian model operators are found. The scaling dimensions of the Ashkin-Teller and Potts models are obtained by the exact relating of eigenstates of their quantum Hamiltonians to those of the XXZ chain with modified boundary conditions. The irrelevant operators governing the dominant finite-size corrections are also identified.
On Bäcklund transformations and boundary conditions associated with the quantum inverse problem for a discrete nonlinear integrable system and its connection to Baxter’s QQQ-operator
A discrete nonlinear system is analysed in case of open chain boundary conditions at the ends.It is shown that the integrability of the system remains intact, by obtaining a modified set of Lax equations which automatically take care of the boundary conditions.The same Lax pair also conforms to the conditions stipulated by Sklyanin [5].The quantum inverse problem is set up and the diagonalisation is carried out by the method of sparation of variables.Backlund transformations are then derived under the modified boundary conditions using the classical r-matrix .Finally by quantising the Backlund transformation it is possible to identify the relation satisfied by the eigenvalue of Baxter's Q-operator even for the quasi periodic situation.
We develop the finite-size scaling (FSS) theory at quantum transitions. We consider various boundary conditions, such as open and periodic boundary conditions, and characterize the corrections to the leading FSS behavior. Using renormalization-group (RG) theory, we generalize the classical scaling ansatz to describe FSS in the quantum case, classifying the different sources of scaling corrections. We identify nonanalytic corrections due to irrelevant (bulk and boundary) RG perturbations and analytic contributions due to regular backgrounds and analytic expansions of the nonlinear scaling fields. To check the general predictions, we consider the quantum XYXYXY chain in a transverse field. For this model exact or numerically accurate results can be obtained by exploiting its fermionic quadratic representation. We study the FSS of several observables, such as the free energy, the energy differences between low-energy levels, correlation functions of the order parameter, etc., confirming the general predictions in all cases. Moreover, we consider bipartite entanglement entropies, which are characterized by the presence of additional scaling corrections, as predicted by conformal field theory.
Quantum integrability is established for the one-dimensional supersymmetric UUU model with boundary terms by means of the quantum inverse-scattering method. The boundary supersymmetric UUU chain is solved by using the coordinate-space Bethe-ansatz technique and Bethe-ansatz equations are derived. This provides us with a basis for computing the finite-size corrections to the low-lying energies in the system.