Abstract

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 constructed from solutions of the Yang-Baxter equation, the quantum transfer matrix may be taken as the transfer matrix of an inhomogeneous variant of the underlying vertex model. Its spectrum can then be studied by Bethe Ansatz methods and may exhibit universal features such as the emergence of a conformal subspectrum in the low-temperature regime. Access to the full spectrum of the quantum transfer matrix enables the construction of thermal form factor series representations of the correlation functions of local operators for the spin chain under consideration. These are claims, made by physicists, whose rigorous mathematical justification sets up a long-term research programme. In this work we implement first steps of this programme with the example of the XXZ quantum spin chain in the antiferromagnetic massless parameter regime and in the low-temperature limit. We rigorously establish the existence and uniqueness of the solutions to a set of nonlinear integral equations, that are equivalent to the Bethe Ansatz equations for the quantum transfer matrix of this model, and explicitly characterise the low-temperature form of these solutions. 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 unique eigenvalue of largest modulus and show that those correlations lengths that diverge in the low-temperature limit are, to the leading order in temperature, in one-to-one correspondence with the spectrum of the free boson $c=1$ conformal field theory. Based on two conjectures, that are accepted in the physics literature, but that could so far only be established in the opposite limit of high temperatures, we prove that the eigenvalue of largest modulus in the subspectrum we focus on corresponds, in fact, to the dominant eigenvalue. Its first-order term in temperature is of a universal form conjectured long ago in the physics literature.

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