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arXiv:hep-th/0410004 · Journal of Statistical Mechanics Theory and Experiment

The diluteALmodels and the Φ1,2perturbation of unitary minimal CFTs

J Suzuki

Published Jan 21, 2005• 4 citations
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Abstract

Motivated by recent studies by Dorey, Pocklington and Tateo for unitary minimal models perturbed by ϕ 1,2 , we examine the thermodynamics of one-dimensional quantum systems, whose counterparts in the two-dimensional classical model are the dilute A L models in regime 2. The functional relations for arbitrary values of L are established. Guided by numerical evidence, we obtain a set of coupled integral equations from the established relations, which yields the evaluation of the free energy at arbitrary temperature. In the scaling limit, the integral equations coincide with the thermodynamic Bethe ansatz equations (TBA) proposed by Dorey, Pocklington and Tateo, thereby supporting their results. The new fermionic representations of the Virasoro characters are remarked upon briefly.

DOI
10.1088/1742-5468/2005/01/p01004

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Motivated by recent studies by Dorey, Pocklington and Tateo for unitary minimal models perturbed by ϕ 1,2 , we examine the thermodynamics of one-dimensional quantum systems, whose counterparts in the two-dimensional classical model are the dilute A L models in regime 2. The functional relations for arbitrary values of L are established. Guided by numerical evidence, we obtain a set of coupled integral equations from the established relations, which yields the evaluation of the free energy at arbitrary temperature. In the scaling limit, the integral equations coincide with the thermodynamic Bethe ansatz equations (TBA) proposed by Dorey, Pocklington and Tateo, thereby supporting their results. The new fermionic representations of the Virasoro characters are remarked upon briefly.
RSOS models and their associated RZMS scattering QFT models, this is certainly not the case, and here the corresponding TBA equations have been derived only in a few special cases.This shortcoming provides the primary motivation for the work contained in this thesis.After presenting a brief review of the various techniques for deriving the TBA equations associated with the ground state energy for the 0 lj3 perturbations of the minimal CFT models, attempts are made at extending these techniques to the derivation of the TBA equations associated with the ground state energy of the 0 i;2 and perturbations of the minimal CFT models.More recently, the functional relation approach for constructing the ground state TBA equations was extended to the derivation of TBA-like integral equations which correspond to excited state energy levels.In principle, this approach can be used to determine the complete energy spectrum of the perturbed CFT, but in practice this is a very difficult undertaking and only the lowest few excited states are rela tively easily to obtain.In the final chapter of this thesis, this approach is used to determine the TBA-like equations corresponding to the lowest three energy levels of a particular simple perturbed CFT, notably the non-unitary minimal CFT perturbed by its operator <^2,i-L ist o f P u b lic a tio n sThe bulk of the original research content of this thesis is contained in Chapters 4, 5 and 6, with some numerical results tabulated in Appendices B and C. Various parts of this research are, or are soon to be, published as follows.The ground state TBA results of Chapter 5, appear in the paper:• R. M. Ellem, and V. V. Bazhanov: Thermodynamic Bethe Ansatz for the subleading magnetic perturbation of the tricritical Ising Model, Nucl.Phys.B512 (1998), 563-580.The ground state and excited state TBA results of Sections 4.1, 5.1, 5.4 and Chapter 6, and the transfer matrix fusion relations in Section 4.2 and Chapter 6, appear in the paper:• R. M. Ellem, and V. V. Bazhanov: Excited State TBA for the < /> 2,i Perturbed A43;5 Model.To be submitted.The general 5-matrix amplitudes and bootstrap properties of Section 4.3, will ap pear in the paper:• R. M. Ellem: Scattering Structure of the (f)\i2 and (f)2,i Perturbed Minimal Conformal Field Theory Models. Currently in preparation.An extension of the integral equation techniques introduced in Chapter 6, to the fusion equation for the regime 1+ dilute A 2 model described in Section 4.2.1, will appear in the paper:• R. M. Ellem: Excited State TBA for the 02jl Perturbed M .%,4Model.Currently in preparation.
The hidden E 7 ( E 6 ) structure has been conjectured for the minimal model M4.5(M6.7){{\mathcal{M}}_{{4.5}}}({{\mathcal{M}}_{{6.7}}})M4.5​(M6.7​) perturbed by Φ 1.2 in the context of conformal field theory (CFT). Motivated by this, we examine the dilute A 4.6 models, which are expected to be corresponding lattice models. Thermodynamics of the equivalent one-dimensional quantum systems is analyzed via the quantum transfer matrix approach. Appropriate auxiliary functions, related to kinks in the theory, play a role in constructing functional relations among transfer matrices. We successfully recover the universal Y -systems and thereby the thermodynamic Bethe ansatz equations for E 6.7 from the dilute A 6.4 model, respectively. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Employing factorized versions of characters as products of quantum dilogarithms corresponding to irreducible representations of the Virasoro algebra, we obtain character formulae which admit an anyonic quasi-particle interpretation in the context of minimal models in conformal field theories. We propose anyonic thermodynamic Bethe ansatz equations, together with their corresponding equation for the Virasoro central charge, on the base of an analysis of the classical limit for the characters and the requirement that the scattering matrices are asymptotically phaseless.
Inspired by recent results in the context of AdS/CFT integrability, we reconsider the Thermodynamic Bethe Ansatz equations describing the 1D fermionic Hubbard model at finite temperature. We prove that the infinite set of TBA equations are equivalent to a simple nonlinear Riemann-Hilbert problem for a finite number of unknown functions. The latter can be transformed into a set of three coupled nonlinear integral equations defined over a finite support, which can be easily solved numerically. We discuss the emergence of an exact Bethe Ansatz and the link between the TBA approach and the results by Jüttner, Klümper and Suzuki based on the Quantum Transfer Matrix method. We also comment on the analytic continuation mechanism leading to excited states and on the mirror equations describing the finite-size Hubbard model with twisted boundary conditions.
Abstract We give a pedagogical introduction to the thermodynamic Bethe ansatz, a method that allows us to describe the thermodynamics of integrable models whose spectrum is found via the (asymptotic) Bethe ansatz. We set the stage by deriving the Fermi–Dirac distribution and associated free energy of free electrons, and then in a similar though technically more complicated fashion treat the thermodynamics of integrable models, focusing first on the one-dimensional Bose gas with delta function interaction as a clean pedagogical example, secondly the XXX spin chain as an elementary (lattice) model with prototypical complicating features in the form of bound states, and finally the SU ( 2 ) chiral Gross–Neveu model as a field theory example. Throughout this discussion we emphasize the central role of particle and hole densities, whose relations determine the model under consideration. We then discuss tricks that allow us to use the same methods to describe the exact spectra of integrable field theories on a circle, in particular the chiral Gross–Neveu model. We moreover discuss the simplification of TBA equations to Y systems, including the transition back to integral equations given sufficient analyticity data, in simple examples.