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arXiv:cond-mat/0212437 · Physical review. B, Condensed matter

Thermodynamics and crossover phenomena in the correlation lengths of the one-dimensionalt-Jmodel

Jesko Sirker, Andreas Klümper

Published Dec 4, 2002• 35 citations
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Abstract

We investigate the thermodynamics of the one-dimensional $t\ensuremath{-}J$ model using transfer matrix renormalization group algorithms and present results for quantities like particle number, specific heat, spin susceptibility, and compressibility. Based on these results we confirm a phase diagram consisting of a Tomonaga-Luttinger liquid (TLL) phase for small $J/t$ and a phase-separated state for $J/t$ large. Close to phase separation we find a spin-gap (Luther-Emery) phase at low densities consistent with predictions by other studies. At the supersymmetric point we compare our results with exact results from the Bethe ansatz and find excellent agreement. In particular we focus on the calculation of correlation lengths and static correlation functions and study the crossover from the nonuniversal high-$T$ lattice into the quantum critical regime. At the supersymmetric point we compare in detail with predictions by conformal field theory and TLL theory and show the importance of logarithmic corrections.

DOI
10.1103/physrevb.66.245102

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We investigate the thermodynamics of the one-dimensional t\ensuremath−Jt\ensuremath{-}Jt\ensuremath−J model using transfer matrix renormalization group algorithms and present results for quantities like particle number, specific heat, spin susceptibility, and compressibility. Based on these results we confirm a phase diagram consisting of a Tomonaga-Luttinger liquid (TLL) phase for small J/tJ/tJ/t and a phase-separated state for J/tJ/tJ/t large. Close to phase separation we find a spin-gap (Luther-Emery) phase at low densities consistent with predictions by other studies. At the supersymmetric point we compare our results with exact results from the Bethe ansatz and find excellent agreement. In particular we focus on the calculation of correlation lengths and static correlation functions and study the crossover from the nonuniversal high-TTT lattice into the quantum critical regime. At the supersymmetric point we compare in detail with predictions by conformal field theory and TLL theory and show the importance of logarithmic corrections.
We describe a modified transfer matrix renormalization group (TMRG) algorithm and apply it to calculate thermodynamic properties of the one-dimensional t − J model. At the supersymmetric point we compare with Bethe ansatz results and make direct connection to conformal field theory (CFT). In particular we study the crossover from the non-universal high- T lattice into the quantum critical regime by calculating various correlation lengths and static correlation functions. Finally, the existence of a spin-gap phase is confirmed.
We investigate the thermodynamics of a one-dimensional Hubbard model with bond-charge interaction X using the transfer-matrix renormalization-group method. Numerical results for various quantities such as spin and charge susceptibilities, particle densities, specific heat, and thermal correlation lengths are presented and discussed. We compare our data also to results for the exactly solvable case X/t=1X/t=1X/t=1 as well as to bosonization results for weak coupling $X/t\ensuremath{\ll}1,$ which shows excellent agreement. We confirm the existence of a Tomonaga-Luttinger and a Luther-Emery liquid phase, in agreement with previous studies at zero temperature. Thermal singlet-pair-correlation lengths are shown to dominate density and spin correlations for finite temperatures in certain parameter regimes.
The phase diagram of the one-dimensional t-J model is investigated by analyzing the results of exact diagonalization and the exact solutions at J/t=0 and 2. Phase separation takes place above a critical value of J around Jc{\mathit{J}}_{\mathit{c}}Jc​/t=2.5--3.5 depending on the electron density. In the small-J region, Tomonaga-Luttinger liquid theory holds and its correlation exponents are calculated as a function of J/t and the electron density. Superconducting correlations become dominant in a region between the solvable case (J/t=2) and phase separation. A spin-gap region is also found at low density.
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.
We study the t-J model in one dimension by numerically projecting the true ground state from a Luttinger-liquid trial wave function. We find the model exhibits Luttinger-liquid behavior for most of the phase diagram in which interaction strength and density are varied. However, at small densities and high interaction strengths a new phase with a gap to spin excitations and enhanced superconducting correlations is found. We show that this phase is a Luther-Emery liquid and study its correlation functions.
The one-dimensional t-J model is investigated by the variational Monte Carlo method. A variational wave function based on the Bethe-ansatz solution is proposed, where the spin-charge separation is realized and a long-range correlation factor of Jastrow-type is included. In most regions of the phase diagram, this wave function provides an excellent description of the ground-state properties characterized as a Tomonaga-Luttinger liquid; both the amplitude and exponent of correlation functions are correctly reproduced. For the spin-gap phase, another trial state of correlated singlet pairs with a Jastrow factor is introduced. This wave function shows generalized Luther-Emery-liquid behavior, exhibiting enhanced superconducting correlations and exponential decay of the spin correlation function. Using these two variational wave functions, the whole phase diagram is determined. In addition, relations between the correlation exponent and variational parameters in the trial functions are derived. \textcopyright{} 1996 The American Physical Society.