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arXiv:cond-mat/9606192 · Europhysics Letters (EPL)

Exact calculation of thermodynamical quantities of the integrable t – J model

Guido Jüttner, Andreas Klümper

Published Feb 10, 1997• 34 citations
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Abstract

The specific heat and the compressibility for the integrable t – J model are calculated showing Luttinger liquid behavior for low temperatures. A Trotter-Suzuki mapping and the quantum transfer matrix approach are utilized. Using an algebraic Bethe ansatz, this method permits the exact calculation of the free energy and related quantities. A set of just two non-linear integral equations determining these quantities is studied for various particle densities and temperatures. The structure of the specific heat is discussed in terms of the elementary charge as well as spin excitations.

DOI
10.1209/epl/i1997-00153-2

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The specific heat and the compressibility for the integrable t – J model are calculated showing Luttinger liquid behavior for low temperatures. A Trotter-Suzuki mapping and the quantum transfer matrix approach are utilized. Using an algebraic Bethe ansatz, this method permits the exact calculation of the free energy and related quantities. A set of just two non-linear integral equations determining these quantities is studied for various particle densities and temperatures. The structure of the specific heat is discussed in terms of the elementary charge as well as spin excitations.
A Trotter-Suzuki mapping is used to calculate the finite-temperature properties of the one-dimensional supersymmetric t − J model. This approach allows for the exact calculation of various thermodynamical properties by means of the quantum transfer matrix (QTM). The free energy and other interesting quantities are obtained such as the specific heat and compressibility. For the largest eigenvalue of the QTM leading to the free energy a set of just two non-linear integral equations is presented. These equations are studied analytically and numerically for different particle densities and temperatures. The structure of the specific heat is discussed in terms of the elementary charge as well as spin excitations. Special emphasis is placed on the study of the low-temperature behavior confirming scaling predictions by conformal field theory and Luttinger liquid theory. To our knowledge this is the first complete investigation of a strongly correlated electron system on a lattice at finite temperature.
The supersymmetric tJ model in one dimension is investigated by means of the quantum transfer matrix (QTM). The free energy of the quantum system is given by just the largest eigenvalue of the QTM whereas the next-largest ones determine the correlation lengths at finite temperature. The first step of the analysis consists of an embedding of the QTM into a family of commuting matrices. In the second step the eigenvalues equations are derived by an algebraic Bethe ansatz. Finally, the eigenvalue equations are transformed into a set of nonlinear integral equations. Some numerical results are presented. The specific heat for instance shows the typical Luttinger liquid behavior at low temperatures.
We present a novel treatment of finite temperature properties of the onedimensional Hubbard model. Our approach is based on a Trotter-Suzuki mapping utilizing Shastry’s classical model and a subsequent investigation of the quantum transfer matrix. We derive non-linear integral equations for three auxiliary functions which have a clear physical interpretation of elementary excitations of spin type and charge excitations in lower and upper Hubbard bands. This allows for a transparent analytical study of certain limiting cases as well as for precise numerical investigations. We present data for the specific heat, magnetic and charge susceptibilities for various particle densities and coupling strengths U. The structure exposed by these curves is discussed in terms of the elementary charge and spin excitations. Special emphasis is placed on the study of the low-temperature behavior within our ab initio approach confirming the scaling predictions by Tomonaga-Luttinger liquid theory. In addition we make contact with the “dressed energy ” formalism established for the analysis of ground state properties.
The quantum mechanical analogue of the classical Perk - Schultz model is considered which comprises the Uimin - Sutherland model and the integrable t - J chain. The quantum transfer matrix of these systems is established and the eigenvalue equations are obtained by an algebraic Bethe ansatz. Only the largest eigenvalue is needed for the calculation of the free energy of the quantum chain at finite temperature. The Bethe ansatz equations for the leading eigenvalue are transformed into a set of integral equations for some appropriately defined auxiliary functions. Furthermore, the eigenvalue of the quantum transfer matrix is expressed in terms of these functions. The integral formulation allows for taking the limit of infinite Trotter - Suzuki number analytically. The low-temperature limit of the free energy is obtained analytically and for intermediate temperatures numerical results are presented.
We report on a new approach to the calculation of thermodynamic functions for crossing-invariant models solvable by Bethe Ansatz. In the case of the XXZ Heisenberg chain we derive, for arbitrary values of the anysotropy, a single non-linear integral equation from which the free energy can be exactly calculated.These equations are shown to be equivalent to an infinite set of algebraic equations of Bethe type which provide alternatively the thermodinamics. The high-temperature expansion follows in a sistematic and relatively simple way from our non-linear integral equations. For low temperatures we obtain the correct central charge and predict the analytic structure of the full expansion around T=0. Furthermore, we derive a single non-linear integral equation describing the finite-size ground-state energy of the Sine-Gordon quantum field theory.