Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical review. B, Condensed matter

Thermodynamics and correlations of the quantum Toda lattice

Alessandro Cuccoli, M. Spicci, V. Tognetti, Ruggero Vaia

Published May 1, 1992• 6 citations
Publisher

Abstract

The aim of the explicit evaluation of the equilibrium thermodynamic properties of the quantum Toda lattice is reached here by a method that reduces the evaluation of quantum-statistical averages to a classical configurational integral. This is done by constructing an effective potential of the same form as the original one, whose parameters are renormalized by the quantum fluctuations. In addition to quantum specific heat (in agreement with Bethe-ansatz calculations), explicit results for meaningful static correlation functions are reported.

DOI
10.1103/physrevb.45.10127

Related papers

Thermodynamics of the quantum Toda lattice

A Matsuyama

Thermodynamics of the quantum Toda lattice is studied based on Gutzwiller's quantization condition and Yang - Yang's thermodynamic formulation. It is shown that Gutzwiller's quantization condition becomes identical to the Bethe ansatz equation in the thermodynamic limit. We have calculated the thermodynamic averages of the energy, higher-order conserved quantities and the specific heat. The average energy and the specific heat become close to those of the classical Toda lattice in the high temperature limit, while they are similar to those of a harmonic chain in the low temperature limit.

Journal of Physics A Mathematical and General•Aug 21, 1996•2 citations

THERMODYNAMICS OF THE TODA CHAIN

Alessandro Cuccoli, Roberto Livi, M. Spicci +2 more

A review of the classical and quantum thermodynamic properties of the Toda chain is provided, together with a survey of the techniques that have been used to work them out, i.e., the bilateral Laplace transform method for the classical system, the Bethe–Ansatz and the variational path-integral approach for the quantum one. For the classical Toda chain we also recall some of the main dynamical features, i.e. the integrability of the model and the soliton-like solutions of the equations of motion. In the quantum case a comparison between the Bethe–Ansatz and the variational method is made. In particular it is shown as the latter offers the possibility of also evaluating static correlation functions.

International Journal of Modern Physics B•Aug 15, 1994•12 citations

Classical Thermodynamics of the Toda Lattice: --As a Classical Limit of the Two-Component Bethe Ansatz Scheme--

Hajime Takayama, M. Ishikawa

In our previous work thermodynamics of two-componet (soliton and phonon) gas is constructed from the Yang-Yang equation for one-component integrable quantum gas. We call this the two-component Bethe ansatz (TCBA) scheme. The method is applied to the Toda lattice and its classical limit is analyzed. The key parameter kF which distinguishes solitons from phonons within the TCBA scheme is chosen such that the resultant equations for effective energies of the modes coincide with those expected from the simple-minded ideal-gas phenomenology. The soliton specific heat exhibits a peculiar temperature dependence at low temperatures. It is however compensated with the anharmonic part of phonons.

Progress of Theoretical Physics•Oct 1, 1986•17 citations

The Quantum Toda Lattice Revisited

Rahul Siddharthan, B. Sriram Shastry

In this work we study the quantum Toda lattice, developing the asymptotic Bethe ansatz method first used by Sutherland. Despite its known limitations we find, on comparing with Gutzwiller's exact method, that it works well in this particular problem and in fact becomes exact as $\hbar$ grows large. We calculate ground state and excitation energies for finite sized lattices, identify excitations as phonons and solitons on the basis of their quantum numbers, and find their dispersions. These are similar to the classical dispersions for small $\hbar$, and remain similar all the way up to $\hbar=1$, but then deviate substantially as we go further into the quantum regime. On comparing the sound velocities for various $\hbar$ obtained thus with that predicted by conformal theory we conclude that the Bethe ansatz gives the energies per particle accurate to O($1/N^2$). On that assumption we can find correlation functions. Thus the Bethe ansatz method can be used to yield much more than the thermodynamic properties which previous authors have calculated.

arXiv:cond-mat/9611221•Nov 27, 1996•2 citations

Quantizing the Toda lattice

Rahul Siddharthan, B. Sriram Shastry

In this work we study the quantum Toda lattice, developing the asymptotic Bethe ansatz method first used by Sutherland. Despite its known limitations we find, on comparing with Gutzwiller's exact method, that it works well in this particular problem and in fact becomes exact as \ensuremath{\hbar} grows large. We calculate ground state and excitation energies for finite-sized lattices, identify excitations as phonons and solitons on the basis of their quantum numbers, and find their dispersions. These are similar to the classical dispersions for small \ensuremath{\hbar}, and remain similar all the way up to \ensuremath{\hbar}=1, but then deviate substantially as we go farther into the quantum regime. On comparing the sound velocities for various \ensuremath{\hbar} obtained thus with that predicted by conformal theory we conclude that the Bethe ansatz gives the energies per particle accurate to O(1/${\mathrm{N}}^{2}$). On that assumption we can find correlation functions. Thus the Bethe ansatz method can be used to yield much more than the thermodynamic properties which previous authors have calculated.

Physical review. B, Condensed matter•May 1, 1997•7 citations

Thermodynamics of the quantum Toda lattice

M Hader, Franz G. Mertens

A linear chain with exponential plus linear interactions (Toda lattice) can be mapped by the Bethe ansatz to a Fermi gas. The thermodynamics is treated by analogy to the delta-function Bose gas (Yang and Yang). Analytical solutions are given for the strongly anharmonic regime and low temperatures; the general case is treated numerically. For the case of weak anharmonicity the specific heats are compared with earlier approximate calculations. The anharmonic part of the specific heat is shown to be not very different from the corresponding classical result which can be interpreted as the specific heat of an ideal gas of solitons. For the quantum case the soliton-gas picture needs further investigation.

Journal of Physics A Mathematical and General•Jul 11, 1986•14 citations
The aim of the explicit evaluation of the equilibrium thermodynamic properties of the quantum Toda lattice is reached here by a method that reduces the evaluation of quantum-statistical averages to a classical configurational integral. This is done by constructing an effective potential of the same form as the original one, whose parameters are renormalized by the quantum fluctuations. In addition to quantum specific heat (in agreement with Bethe-ansatz calculations), explicit results for meaningful static correlation functions are reported.
Thermodynamics of the quantum Toda lattice is studied based on Gutzwiller's quantization condition and Yang - Yang's thermodynamic formulation. It is shown that Gutzwiller's quantization condition becomes identical to the Bethe ansatz equation in the thermodynamic limit. We have calculated the thermodynamic averages of the energy, higher-order conserved quantities and the specific heat. The average energy and the specific heat become close to those of the classical Toda lattice in the high temperature limit, while they are similar to those of a harmonic chain in the low temperature limit.
A review of the classical and quantum thermodynamic properties of the Toda chain is provided, together with a survey of the techniques that have been used to work them out, i.e., the bilateral Laplace transform method for the classical system, the Bethe–Ansatz and the variational path-integral approach for the quantum one. For the classical Toda chain we also recall some of the main dynamical features, i.e. the integrability of the model and the soliton-like solutions of the equations of motion. In the quantum case a comparison between the Bethe–Ansatz and the variational method is made. In particular it is shown as the latter offers the possibility of also evaluating static correlation functions.
In our previous work thermodynamics of two-componet (soliton and phonon) gas is constructed from the Yang-Yang equation for one-component integrable quantum gas. We call this the two-component Bethe ansatz (TCBA) scheme. The method is applied to the Toda lattice and its classical limit is analyzed. The key parameter kF which distinguishes solitons from phonons within the TCBA scheme is chosen such that the resultant equations for effective energies of the modes coincide with those expected from the simple-minded ideal-gas phenomenology. The soliton specific heat exhibits a peculiar temperature dependence at low temperatures. It is however compensated with the anharmonic part of phonons.
In this work we study the quantum Toda lattice, developing the asymptotic Bethe ansatz method first used by Sutherland. Despite its known limitations we find, on comparing with Gutzwiller's exact method, that it works well in this particular problem and in fact becomes exact as ℏ\hbarℏ grows large. We calculate ground state and excitation energies for finite sized lattices, identify excitations as phonons and solitons on the basis of their quantum numbers, and find their dispersions. These are similar to the classical dispersions for small ℏ\hbarℏ, and remain similar all the way up to ℏ=1\hbar=1ℏ=1, but then deviate substantially as we go further into the quantum regime. On comparing the sound velocities for various ℏ\hbarℏ obtained thus with that predicted by conformal theory we conclude that the Bethe ansatz gives the energies per particle accurate to O(1/N21/N^21/N2). On that assumption we can find correlation functions. Thus the Bethe ansatz method can be used to yield much more than the thermodynamic properties which previous authors have calculated.
In this work we study the quantum Toda lattice, developing the asymptotic Bethe ansatz method first used by Sutherland. Despite its known limitations we find, on comparing with Gutzwiller's exact method, that it works well in this particular problem and in fact becomes exact as \ensuremath{\hbar} grows large. We calculate ground state and excitation energies for finite-sized lattices, identify excitations as phonons and solitons on the basis of their quantum numbers, and find their dispersions. These are similar to the classical dispersions for small \ensuremath{\hbar}, and remain similar all the way up to \ensuremath{\hbar}=1, but then deviate substantially as we go farther into the quantum regime. On comparing the sound velocities for various \ensuremath{\hbar} obtained thus with that predicted by conformal theory we conclude that the Bethe ansatz gives the energies per particle accurate to O(1/N2{\mathrm{N}}^{2}N2). On that assumption we can find correlation functions. Thus the Bethe ansatz method can be used to yield much more than the thermodynamic properties which previous authors have calculated.
A linear chain with exponential plus linear interactions (Toda lattice) can be mapped by the Bethe ansatz to a Fermi gas. The thermodynamics is treated by analogy to the delta-function Bose gas (Yang and Yang). Analytical solutions are given for the strongly anharmonic regime and low temperatures; the general case is treated numerically. For the case of weak anharmonicity the specific heats are compared with earlier approximate calculations. The anharmonic part of the specific heat is shown to be not very different from the corresponding classical result which can be interpreted as the specific heat of an ideal gas of solitons. For the quantum case the soliton-gas picture needs further investigation.