Unified approach to Thermodynamic Bethe Ansatz and finite size corrections for lattice models and field theories

Abstract

We present a unified approach to the Thermodynamic Bethe Ansatz (TBA) for magnetic chains and field theories that includes the finite size (and zero temperature) calculations for lattice BA models. In all cases, the free energy follows by quadratures from the solution of a single non-linear integral equation (NLIE).[A system of NLIE appears for nested BA]. We derive the NLIE for: a) the six-vertex model with twisted boundary conditions; b) the XXZ chain in an external magnetic field h z and c) the sine-Gordonmassive Thirring model (sG-mT) in a periodic box of size β ≡ 1/T using the light-cone approach. This NLIE is solved by iteration in one regime (high T in the XXZ chain and low T in the sG-mT model). In the opposite (conformal) regime, the leading behaviors are obtained in closed form. Higher corrections can be derived from the Riemann-Hilbert form of the NLIE that we present.

Numerical confirmation of analytic predictions for the finite volume mass gap of the XY-model

“…They were originally suggested [12,13] for the ground state of the sine-Gordon or the closely related massive Thirring model, but can be systematically generalized to describe excited states as well; the (common) even charge sector of both models [14,15,16] and the (different) odd charge sector in both sine-Gordon and massive Thirring model [17]. The ground state energy is given by…”

Kink scaling functions in 2D non-integrable quantum field theories

“…and on the range of the values of the couplings λ k . For several integrable quantum field theories in 2D, the above quantities have been exactly computed in finite volume with the so-called Thermodynamics Bethe Ansatz method [5] or Destri-deVega equations [6]. These techniques, however, require the integrability of the model, and cannot be applied to the LG theories (1.1), due to their non-integrable nature.…”

“…The universal thermodynamical properties of statistical systems with multicritical behavior are described, in mean-field approximation, by appropriate Landau-Ginzburg (LG) field theories: Structural (commensurate-incommensurate) phase transitions [1], interface phenomena in ordered and disordered media [2] and phase structure of ferromagnetic systems (see for instance [3]) provide few examples for the applications of the simplest φ 4 and φ 6 LG models to statistical mechanics and condensed matter physics. In two dimensions, the LG potentials (1.1) appear also in the description of the relevant perturbations of Virasoro minimal models of conformal field theory [4], as well as of the renormalization group flows between them.…”

Review of AdS/CFT Integrability, Chapter III.5: Lüscher Corrections

“…Fortunately, for integrable quantum field theories, there exists a technique of Thermodynamic Bethe Ansatz -TBA [5] (and/or nonlinear integral equations -NLIE [6,7]), which allows for finding the exact energy spectrum and thus effectively resuming all these virtual corrections. This is, however, technically very involved, so even for the cases where it is known, Lüscher corrections remain an efficient calculational tool.…”