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arXiv:1711.00961 · Journal of Statistical Mechanics Theory and Experiment

A note on a boundary sine-Gordon model at the free-Fermion point

Rajan Murgan

Published Feb 1, 2018• 0 citations
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Abstract

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.

DOI
10.1088/1742-5468/aaa8f1

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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.
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.
RG flow of central charge ceffc_{\rm eff}ceff​ is investigated for the two boundary sine-Gordon model at the free Fermi limit. Thermodynamic Bethe ansatz approach is used to check the non-monotonic decreasing properties of ceffc_{\rm eff}ceff​, its resonance, and the modification of ceffc_{\rm eff}ceff​ due to the mismatch of the periodicity of a Lagrangian parameter and that of the boundary scattering parameter. The detailed analysis uses the singularity structure of the system on the complex rapidity plane.
We study the ground-state energy of integrable 1 + 1 quantum field theories with boundaries (the genuine Casimir effect). In the scalar case, this is done by introducing a new “R-channel TBA”, where the boundary is represented by a boundary state, and the thermodynamics involves evaluating scalar products of boundary states with all the states of the theory. In the non-scalar, sine-Gordon case, this is done by generalizing the method of Destri and De Vega. The two approaches are compared. Miscellaneous other results are obtained, in particular formulas for the overall normalization and scalar products of boundary states, exact partition functions for the critical Ising model in a boundary magnetic field, and also results for the energy, excited states and boundary S-matrix of O(n) and minimal models.
We derived the corresponding boundary condition on Fermi fields to the spin-1/2 Heisenberg chain with boundary magnetic fields. In order to obtain the correct boundary condition from the variation of the action at the edges, we carefully treat the oscillating terms which emerge as a result of the chiral decomposition of fermions and do not contribute to the bulk Lagrangian. The obtained result is checked by compared with the exact result derived from the Bethe ansatz, by considering the mode expansion of fermions on the light-cone coordinates. We also give the spin-wave interpretation to the emergence of boundary bound states.
In this report we compute the boundary states (including the boundary entropy) for the boundary sine-Gordon theory. From the boundary states, we derive both correlation and partition functions. Through the partition function, we show that boundary sine-Gordon maps onto a doubled boundary Ising model. With the current-current correlators, we calculate for finite system size the ac-conductance of tunneling quantum wires with dimensionless free conductance 1/2 (or, alternatively interacting quantum Hall edges at filling fraction 1/2). In the dc limit, the results of C. Kane and M. Fisher, Phys. Rev. B46 (1992) 15233, are reproduced.
The N site open XXZ quantum spin chain with a right non-diagonal boundary and special diagonal left boundary is considered. The boundary non-local charges originally obtained from a field theoretical viewpoint, for the sine–Gordon model on the half line, are recovered from the spin chain point of view. Furthermore, the symmetry of the open spin chain is exhibited. More specifically, we show that certain non-local charges commute with the transfer matrix of the open spin chain, depending on the choice of boundary conditions. In addition, we show explicitly that for a special choice of the left boundary one of the non-local charges, in a particular representation, commutes with each one of the generators of the blob algebra, and hence with the corresponding local Hamiltonian.