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Exact thermodynamics and Luttinger liquid properties of the integrable t-J model

Guido Jüttner, Andreas Klümper, Junji Suzuki, Universität Zu Köln

Published Jan 1, 1997• 35 citations
OpenAlex

Abstract

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.

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The present work applies a numerical method, namely the transfer-matrix density-matrix renormalization group (TMRG), to two seemingly different types of models. In a first part the TMRG is used to investigate the thermodynamics of one-dimensional fermionic models. A second part deals with a novel TMRG method for one-dimensional stochastic models, whose development is an integral part of the thesis. First, the traditional TMRG algorithm for quantum systems is outlined in its historical context. Two different variants are presented, following works of Xiang et al. and Sirker and Klümper, respectively. The basic idea of the method is to map the thermodynamics of a one-dimensional quantum model by Trotter-Suzuki decomposition onto a two-dimensional statistical one. The latter is then solved by a transfer-matrix approach combined with the iterative numerical procedure of White's density-matrix renormalization-group (DMRG) algorithm. Thereby precise computations of various thermodynamic properties, such as thermodynamic potentials, susceptibilities, thermal expectation values and correlation functions are possible. As the first part of the thesis deals with fermionic models, we next review some basics about the theory of strongly correlated fermions in one dimension. Thereupon we elucidate the so-called Hirsch model, which recently gained a lot of theoretic interest in respect to high-temperature superconductivity. It extends the well-studied Hubbard model by an off-diagonal bond-charge interaction term. The current state of research is briefly summarized and mainly refers to ground state properties. Showing numerical TMRG results we then investigate and discuss the almost unknown thermodynamics of the Hirsch model. Various phases are identified and characterized in terms of Tomonaga-Luttinger and Luther-Emery liquid properties, in accordance with previous studies of the ground state. As an important result, superconducting singlet-pair correlation lenths are observed to dominate the physics at finite temperatures in a certain spin-gaped phase. Subsequent to our thermodynamic studies, we turn to the second part of the thesis and outline some theoretic basics of stochastic models. Most notably is the important formal analogy of the master equation, that describes the dynamics of the model, to a Schrödinger equation in imaginary time. This analogy is used to construct a stochastic TMRG algorithm almost similar to the quantum case, that facilitates the computation of dynamic properties, e.g. the local density of particles. We intensively focus on interesting mathematical properties of the stochastic transfer-matrix. As an astonishing result it is found, that the temporal evolution of the non-equilibrium process is reflected by a certain causal structure of the stochastic TMRG. But even if this new approach seems to be promising at first glance, severe numerical problems limit significantly its practical use. In order to solve these instabilities we propose a completely new variant of the algorithm, which we call stochastic light-cone corner-transfermatrix DMRG (LCTMRG). As suggested by its name, the LCTMRG makes use of the causal structure mentioned above and combines it with the stochastic TMRG algorithm. Applications of the LCTMRG onto various reaction-diffusion models verify highly precise numerical data and a great improve compared to the old algorithm by several orders of magnitude. Additionally it is stressed, that the newly proposed analysis tool provides some considerable advantages to common simulation techniques.

•Jan 1, 2003•1 citation
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 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.
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.
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/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.
The present work applies a numerical method, namely the transfer-matrix density-matrix renormalization group (TMRG), to two seemingly different types of models. In a first part the TMRG is used to investigate the thermodynamics of one-dimensional fermionic models. A second part deals with a novel TMRG method for one-dimensional stochastic models, whose development is an integral part of the thesis. First, the traditional TMRG algorithm for quantum systems is outlined in its historical context. Two different variants are presented, following works of Xiang et al. and Sirker and Klümper, respectively. The basic idea of the method is to map the thermodynamics of a one-dimensional quantum model by Trotter-Suzuki decomposition onto a two-dimensional statistical one. The latter is then solved by a transfer-matrix approach combined with the iterative numerical procedure of White's density-matrix renormalization-group (DMRG) algorithm. Thereby precise computations of various thermodynamic properties, such as thermodynamic potentials, susceptibilities, thermal expectation values and correlation functions are possible. As the first part of the thesis deals with fermionic models, we next review some basics about the theory of strongly correlated fermions in one dimension. Thereupon we elucidate the so-called Hirsch model, which recently gained a lot of theoretic interest in respect to high-temperature superconductivity. It extends the well-studied Hubbard model by an off-diagonal bond-charge interaction term. The current state of research is briefly summarized and mainly refers to ground state properties. Showing numerical TMRG results we then investigate and discuss the almost unknown thermodynamics of the Hirsch model. Various phases are identified and characterized in terms of Tomonaga-Luttinger and Luther-Emery liquid properties, in accordance with previous studies of the ground state. As an important result, superconducting singlet-pair correlation lenths are observed to dominate the physics at finite temperatures in a certain spin-gaped phase. Subsequent to our thermodynamic studies, we turn to the second part of the thesis and outline some theoretic basics of stochastic models. Most notably is the important formal analogy of the master equation, that describes the dynamics of the model, to a Schrödinger equation in imaginary time. This analogy is used to construct a stochastic TMRG algorithm almost similar to the quantum case, that facilitates the computation of dynamic properties, e.g. the local density of particles. We intensively focus on interesting mathematical properties of the stochastic transfer-matrix. As an astonishing result it is found, that the temporal evolution of the non-equilibrium process is reflected by a certain causal structure of the stochastic TMRG. But even if this new approach seems to be promising at first glance, severe numerical problems limit significantly its practical use. In order to solve these instabilities we propose a completely new variant of the algorithm, which we call stochastic light-cone corner-transfermatrix DMRG (LCTMRG). As suggested by its name, the LCTMRG makes use of the causal structure mentioned above and combines it with the stochastic TMRG algorithm. Applications of the LCTMRG onto various reaction-diffusion models verify highly precise numerical data and a great improve compared to the old algorithm by several orders of magnitude. Additionally it is stressed, that the newly proposed analysis tool provides some considerable advantages to common simulation techniques.