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Journal of the Physical Society of Japan

Bethe Ansatz Cluster Expansion Method for Quantum Integrable Particle Systems

Go Kato, Miki Wadati

Published May 15, 2004• 5 citations
Publisher

Abstract

Presented is a method, which we call Bethe ansatz cluster expansion (BACE) method, to investigate statistical mechanics of quantum integrable systems. By use of only the Bethe ansatz equation, n -particle partition functions are exactly calculated. From the expression for the partition function, the n -particle cluster integrals are derived. The results completely agree with those of the thermal Bethe ansatz (TBA) by Yang and Yang. Thus, the BACE method directly proves the validity of the TBA. Partition theory and graphical representation are used to make the presentation simple and transparent.

DOI
10.1143/jpsj.73.1171

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Presented is a method, which we call Bethe ansatz cluster expansion (BACE) method, to investigate statistical mechanics of quantum integrable systems. By use of only the Bethe ansatz equation, n -particle partition functions are exactly calculated. From the expression for the partition function, the n -particle cluster integrals are derived. The results completely agree with those of the thermal Bethe ansatz (TBA) by Yang and Yang. Thus, the BACE method directly proves the validity of the TBA. Partition theory and graphical representation are used to make the presentation simple and transparent.
Recent developments in statistical mechanics of quantum integrable systems are reviewed. Those studies are fundamental and have a renewed interest related to newly developing fields such as atomic Bose-Einstein condensations, photonic crystals and quantum computations. After a brief summary of the basic concepts and methods, the following three topics are discussed. First, by the thermal Bethe ansatz (TBA), a hard-core Bose gas is exactly solved. The model includes fully the effect of excluded volume and is identified to be a c=1 conformal field theory. Second, the cluster expansion method based on the periodic boundary condition for the Bethe wave function, which we call the Bethe ansatz cluster expansion (BACE) method, is developed for a δ-function gas and the XXX Heisenberg chain. This directly proves the TBA and reveals intrinsic properties of quantum integrable systems. Third, for a δ-function gas, the integral equations for the distribution functions of the quasi-momentum and the quasi-particle energy are solved in the form of power series. In the weak coupling case, the results reproduce those of Bogoliubov theory.
We propose an exact summation method to compute thermodynamic observables in integrable quantum field theories. The key idea is to use the matrix-tree theorem to write the Gaudin determinants that appear in the cluster expansion as a sum over graphs. For theories with a diagonal S-matrix, this method is more powerful than the standard Thermodynamic Bethe Ansatz (TBA) technique as it is exact to all orders of powers in inverse volume. We have obtained using this method the TBA equation, the excited state energies in finite volume, the Leclair-Mussardo formula for one point functions, the finite-temperature boundary entropy and cumulants of conserved charges in Generalized Gibbs Ensembles. Moreover, the graph expansion can also be regarded as an alternative to algebraic manipulations involving Gaudin determinants. We have applied this idea to derive the equations of state and other transport properties in Generalized Hydrodynamics. For theories with a non-diagonal S-matrix, the description of a complete set of states is more involved and it is not known how a cluster expansion can be implemented. It is nevertheless possible to apply the direct summation method in reverse and interpret known TBA equations with complex strings in terms of diagrams.