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arXiv:hep-th/9909104 · Birkhäuser Boston eBooks

Hidden E-Type Structures in Dilute A Models

Junji Suzuki

Published Jan 1, 2000• 4 citations
arXivHTMLPDF

Abstract

The hidden E 7 ( E 6 ) structure has been conjectured for the minimal model \({{\mathcal{M}}_{{4.5}}}({{\mathcal{M}}_{{6.7}}})\) 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.

DOI
10.1007/978-1-4612-1378-9_7

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The hidden E 7 ( E 6 ) structure has been conjectured for the minimal model ({{\mathcal{M}}{{4.5}}}({{\mathcal{M}}{{6.7}}})) 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.
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We generalize the results of [Comm.Math.Phys.299 (2010), 825-866] (hidden Grassmann structure IV) to the case of excited states of the transfer matrix of the sixvertex model acting in the so-called Matsubara direction.We establish an equivalence between a scaling limit of the partition function of the six-vertex model on a cylinder with quasi-local operators inserted and special boundary conditions, corresponding to particlehole excitations, on the one hand, and certain three-point correlation functions of conformal field theory (CFT) on the other hand.As in hidden Grassmann structure IV, the fermionic basis developed in previous papers and its conformal limit are used for a description of the quasi-local operators.In paper IV we claimed that in the conformal limit the fermionic creation operators generate a basis equivalent to the basis of the descendant states in the conformal field theory modulo integrals of motion suggested by A. Zamolodchikov (1987).Here we argue that, in order to completely determine the transformation between the above fermionic basis and the basis of descendants in the CFT, we need to involve excitations.On the side of the lattice model we use the excited-state TBA approach.We consider in detail the case of the descendant at level 8.
We derive the fusion hierarchy of functional equations for critical A-D-E lattice models related to the sℓ ( 2 ) unitary minimal models, the parafermionic models and the supersymmetric models of conformal field theory and deduce the related TBA functional equations. The derivation uses fusion projectors and applies in the presence of all known integrable boundary conditions on the torus and cylinder. The resulting TBA functional equations are universal in the sense that they depend only on the Coxeter number of the A-D-E graph and are independent of the particular integrable boundary conditions. We conjecture generally that TBA functional equations are universal for all integrable lattice models associated with rational CFTs and their integrable perturbations. 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.
The free energy per lattice site of a quantum spin chain in the thermodynamic limit is determined by a single `dominant' Eigenvalue of an associated quantum transfer matrix in the infinite Trotter number limit. For integrable quantum spin chains, related with solutions of the Yang-Baxter equation, an appropriate choice of the quantum transfer matrix enables to study its spectrum, e.g.\ by means of the algebraic Bethe Ansatz. In its turn, the knowledge of the full spectrum allows one to study its universality properties such as the appearance of a conformal spectrum in the low-temperature regime. More generally, accessing the full spectrum is a necessary step for deriving thermal form factor series representations of the correlation functions of local operators for the spin chain under consideration. These are statements that have been established by physicists on a heuristic level and that are calling for a rigorous mathematical justification. In this work we implement certain aspects of this programme with the example of the XXZ quantum spin chain in the antiferromagnetic massless regime and in the low-temperature limit. We rigorously establish the existence, uniqueness and characterise the form of the solutions to the non-linear integral equations that are equivalent to the Bethe Ansatz equations for the quantum transfer matrix of this model. This allows us to describe that part of the quantum transfer matrix spectrum that is related to the Bethe Ansatz and that does not collapse to zero in the infinite Trotter number limit. Within the considered part of the spectrum we rigorously identify the dominant Eigenvalue and show that those correlations lengths that diverge in the low-temperature limit are given, to the leading order, by the spectrum of the free Boson c=1c=1c=1 conformal field theory. This rigorously establishes a long-standing conjecture present in the physics literature.