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Large and small density approximations to the thermodynamic Bethe ansatz

Fring, A, Korff, C

Published Feb 22, 2000• 2 citations
OpenAlex

Abstract

We provide analytical solutions to the thermodynamic Bethe ansatz equationsin the large and small density approximations. We extend results previouslyobtained for leading order behaviour of the scaling function of affine Todafield theories related to simply laced Lie algebras to the non-simply lacedcase. The comparison with semi-classical methods shows perfect agreement forthe simply laced case. We derive the Y-systems for affine Toda field theorieswith real coupling constant and employ them to improve the large densityapproximations. We test the quality of our analysis explicitly for theSinh-Gordon model and the $(G_2^{(1)},D_4^{(3)})$-affine Toda field theory.

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•Aug 14, 2016•23 citations
We provide analytical solutions to the thermodynamic Bethe ansatz equationsin the large and small density approximations. We extend results previouslyobtained for leading order behaviour of the scaling function of affine Todafield theories related to simply laced Lie algebras to the non-simply lacedcase. The comparison with semi-classical methods shows perfect agreement forthe simply laced case. We derive the Y-systems for affine Toda field theorieswith real coupling constant and employ them to improve the large densityapproximations. We test the quality of our analysis explicitly for theSinh-Gordon model and the (G2(1),D4(3))(G_2^{(1)},D_4^{(3)})(G2(1)​,D4(3)​)-affine Toda field theory.
We investigate properties of the entropy density related to a generalized extensive statistics and derive the thermodynamic Bethe ansatz equation for a system of relativistic particles obeying such a statistics. We investigate the conformal limit of such a system. We also derive a generalized Y-system. The Gentile intermediate statistics and the statistics of gamma-ons are considered in detail. In particular, we observe that certain thermodynamic quantities for the Gentile statistics majorize those for the Haldane-Wu statistics. Specifically, for the effective central charges related to affine Toda models we obtain nontrivial inequalities in terms of dilogarithms.
The Bethe Ansatz is a method that is used in quantum integrable models in order to solve them explicitly. This method is explained here in a general framework, which applies to 1D quantum spin chains, 2D statistical lattice models (vertex models) and relativistic field theories with 1 space dimension and 1 time dimension. The connection with quantum groups is expounded. Several applications are then presented. Finite size corrections are calculated via two methods: The Non-Linear Integral Equations, which are applied to the study of the states of the affine Toda model with imaginary coupling, and their interpolation between the high energy (ultra-violet) and low energy (infra-red) regions; and the Thermodynamic Bethe Ansatz Equations, along with the associated Fusion Equations, which are used to determine the thermodynamic properties of the generalized multi-channel Kondo model. The latter is then studied in more detail, still using the Bethe Ansatz and quantum groups, so as to characterize the spectrum of the low energy excitations.
The Bethe-ansatz (BA) equation for the classical sine-Gordon (SG) thermodynamics is shown to reduce to the one for the classical Toda lattice within two approximations; the phonon-phonon interactions are neglected (the harmonic phonon) and solitons are treated nonrelativistically. Within these approximations the solitons in the SG model correspond to the particles in the Toda lattice. Making use of an analytical solution of the BA equation for the classical Toda lattice due to Opper, we obtain an analytical solution of the approximate SG equation. We propose also a similar approximate BA equation for the ${\ensuremath{\varphi}}^{4}$ model, although the ${\ensuremath{\varphi}}^{4}$ model does not have the BA equation for the classical thermodynamics.
We evaluate the exact energy current and scaled cumulant generating function (related to the large-deviation function) in non-equilibrium steady states with energy flow, in any integrable model of relativistic quantum field theory (IQFT) with diagonal scattering. Our derivations are based on various recent results of D. Bernard and B. Doyon. The steady states are built by connecting homogeneously two infinite halves of the system thermalized at different temperatures TlT_lTl​, TrT_rTr​, and waiting for a long time. We evaluate the current J(Tl,Tr)J(T_l,T_r)J(Tl​,Tr​) using the exact QFT density matrix describing these non-equilibrium steady states and using Al.B. Zamolodchikov's method of the thermodynamic Bethe ansatz (TBA). The scaled cumulant generating function is obtained from the extended fluctuation relations which hold in integrable models. We verify our formula in particular by showing that the conformal field theory (CFT) result is obtained in the high-temperature limit. We analyze numerically our non-equilibrium steady-state TBA equations for three models: the sinh-Gordon model, the roaming trajectories model, and the sine-Gordon model at a particular reflectionless point. Based on the numerics, we conjecture that an infinite family of non-equilibrium ccc-functions, associated to the scaled cumulants, can be defined, which we interpret physically. We study the full scaled distribution function and find that it can be described by a set of independent Poisson processes. Finally, we show that the "additivity" property of the current, which is known to hold in CFT and was proposed to hold more generally, does not hold in general IQFT, that is J(Tl,Tr)J(T_l,T_r)J(Tl​,Tr​) is not of the form f(Tl)−f(Tr)f(T_l)-f(T_r)f(Tl​)−f(Tr​).
We evaluate the exact energy current and scaled cumulant generating function (related to the large-deviation function) in non-equilibrium steady states with energy flow, in any integrable model of relativistic quantum field theory (IQFT) with diagonal scattering. Our derivations are based on various recent results of D. Bernard and B. Doyon. The steady states are built by connecting homogeneously two infinite halves of the system thermalized at different temperatures Tl, Tr, and waiting for a long time. We evaluate the current J(Tl, Tr) using the exact QFT density matrix describing these non-equilibrium steady states and using Al.B. Zamolodchikov’s method of the thermodynamic Bethe ansatz (TBA). The scaled cumulant generating function is obtained from the extended fluctuation relations which hold in integrable models. We verify our formula in particular by showing that the conformal field theory (CFT) result is obtained in the high-temperature limit. We analyze numerically our non-equilibrium steady-state TBA equations for three models: the sinh-Gordon model, the roaming trajectories model, and the sine-Gordon model at a particular reflectionless point. Based on the numerics, we conjecture that an infinite family of non-equilibrium c-functions, associated to the scaled cumulants, can be defined, which we interpret physically. We study the full scaled distribution function and find that it can be described by a set of independent Poisson processes. Finally, we show that the “additivity ” property of the current, which is known to hold in CFT and was proposed to hold more generally, does not hold in general IQFT, that is J(Tl, Tr) is not of the form f(Tl)−f(Tr).