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Finite size effects in the spin-1 XXZ and supersymmetric sine-Gordon models

Changrim Ahn, Rafael I. Nepomechie, Junji Suzuki

Published Nov 13, 2006• 0 citations
OpenAlex

Abstract

Starting from the Bethe Ansatz solution of the open integrable spin-1 XXZ quantum spin chain with diagonal boundary terms, we derive a set of nonlinear integral equations (NLIEs), which we propose to describe the boundary supersymmetric sine-Gordon model BSSG${}^{+}$ with Dirichlet boundary conditions on a finite interval. We compute the corresponding boundary $S$ matrix, and find that it coincides with the one proposed by Bajnok, Palla and Tak\\'acs for the Dirichlet BSSG${}^{+}$ model. We derive a relation between the (UV) parameters in the boundary conditions and the (IR) parameters in the boundary $S$ matrix. By computing the boundary vacuum energy, we determine a previously unknown parameter in the scattering theory. We solve the NLIEs numerically for intermediate values of the interval length, and find agreement with our analytical result for the effective central charge in the UV limit and with boundary conformal perturbation theory.

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Starting from the Bethe Ansatz solution of the open integrable spin-1 XXZ quantum spin chain with diagonal boundary terms, we derive a set of nonlinear integral equations (NLIEs), which we propose to describe the boundary supersymmetric sine-Gordon model BSSG+{}^{+}+ with Dirichlet boundary conditions on a finite interval. We compute the corresponding boundary SSS matrix, and find that it coincides with the one proposed by Bajnok, Palla and Tak\'acs for the Dirichlet BSSG+{}^{+}+ model. We derive a relation between the (UV) parameters in the boundary conditions and the (IR) parameters in the boundary SSS matrix. By computing the boundary vacuum energy, we determine a previously unknown parameter in the scattering theory. We solve the NLIEs numerically for intermediate values of the interval length, and find agreement with our analytical result for the effective central charge in the UV limit and with boundary conformal perturbation theory.
Abstract We investigate the free-Fermion point of a boundary sine-Gordon model with nondiagonal boundary interactions for the ground state using auxiliary functions obtained from T − Q equations of a corresponding inhomogeneous open spin- 1 2 XXZ chain with nondiagonal boundary terms. In particular, we obtain the Casimir energy. Our result for the Casimir energy is shown to agree with the result from the TBA approach. The analytical result for the effective central charge in the ultraviolet (UV) limit is also verified from the plots of effective central charge for intermediate values of volume.
The XXX spin- \frac{1}{2} 1 2 Heisenberg chain with non-diagonal boundary fields represents a cornerstone model in the study of integrable systems with open boundaries. Despite its significance, solving this model exactly has remained a formidable challenge due to the breaking of U(1) U ( 1 ) symmetry. Building on the off-diagonal Bethe Ansatz (ODBA), we derive a set of nonlinear integral equations (NLIEs) that encapsulate the exact spectrum of the model. For U(1) U ( 1 ) symmetric spin- \frac{1}{2} 1 2 chains such NLIEs involve two functions a(x) a ( x ) and \bar{a}(x) a ‾ ( x ) coupled by an integration kernel with short-ranged elements. The solution functions show characteristic features for arguments at some length scale which grows logarithmically with system size N N . In the case considered here the U(1) U ( 1 ) symmetry is broken by the non-diagonal boundary fields and the equations involve a novel third function c(x) c ( x ) , which captures the inhomogeneous contributions to the T T - Q Q relation in the ODBA. The kernel elements coupling this function to the standard ones are long-ranged and lead for the ground-state to a winding phenomenon. In \log(1+a(x)) log ( 1 + a ( x ) ) and \log(1+\bar a(x)) log ( 1 + a ‾ ( x ) )
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.