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Physical Review Letters

Thermodynamical properties of quantum many-body systems

Eduardo R. Gagliano, S. Bacci

Published Mar 6, 1989• 5 citations
Publisher

Abstract

We present a method to compute exactly the partition function of quantum many-body systems. This approach involves the evaluation of a set of coefficients for the continued-fraction expansion of a resolventlike operator. The main advantage of this method is that this set of coefficients is unique and allows one to calculate the partition function at any temperature. As an illustration, the method is applied to a simple one-dimensional spinless model.

DOI
10.1103/physrevlett.62.1154

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We present a method to compute exactly the partition function of quantum many-body systems. This approach involves the evaluation of a set of coefficients for the continued-fraction expansion of a resolventlike operator. The main advantage of this method is that this set of coefficients is unique and allows one to calculate the partition function at any temperature. As an illustration, the method is applied to a simple one-dimensional spinless model.
A simple Monte Carlo high-temperature expansion method to evaluate the partition function of quantum spin systems which was proposed in the previous paper l) is improved so that all of the needed expansion-coefficients are obtained from interpolating points of less than ten with an interpolation formula. By this improvement, this method is applicable to the finite spin-systems of rather large spin size regardless of dimensionality of the system. As an example, the method is applied to the one-dimensional spin-l/2 isotropic Heisenberg ferromagnetic system.
We present a method to compute dynamical correlation functions in quantum many-body systems at zero temperature. The ground state of a finite system is evaluated exactly by a modified Lanczos method and from it the real-frequency correlation functions are obtained by a projective technique for the memory-function formalism. We apply the method to the density-density correlation function of a one-dimensional spinless fermion system. Exact results are recovered in the noninteracting limit; in the strong-coupling limit our results are compared with Monte Carlo simulation and analytical approximations.
We present a simple numerical approach to compute dynamic correlation functions in quantum many-body systems at zero temperature. The method is based on a projective technique for the memory-function formalism, and requires as a starting point an independent evaluation of the ground-state energy and wave function. We illustrate the method with a one-dimensional model of spinless fermions and calculate the density-density, current-current, and single-particle correlation functions. The obtained spectral functions show different types of elementary excitations: particle-hole, soliton-antisoliton, exciton, etc. We compare our results with analytical results, and with those obtained through different approaches available in the literature.
We present a method for calculating the Yang-Lee partition function zeros of a translationally invariant model of lattice fermions, exemplified by the Hubbard model. The method rests on a theorem involving the usual single-electron self-energy ${\mathrm{\ensuremath{\Sigma}}}_{\ensuremath{\sigma}}(\stackrel{P\vec}{k},i{\ensuremath{\omega}}_{n}|\ensuremath{\mu})$ with chemical potential $\ensuremath{\mu}$, in the imaginary time Matsubara formulation. The theorem maps the Yang-Lee zeros to a set of wave vector and spin labeled virtual energies ${\ensuremath{\xi}}_{\stackrel{P\vec}{k}\ensuremath{\sigma}}$. These, thermodynamically derived virtual energies, are solutions of the set of equations ${\ensuremath{\xi}}_{k\ensuremath{\sigma}}={\ensuremath{\varepsilon}}_{\ensuremath{\sigma}}(\stackrel{P\vec}{k})\ensuremath{-}\frac{1}{2}U+{\mathrm{\ensuremath{\Sigma}}}_{\ensuremath{\sigma}}(\stackrel{P\vec}{k},i\frac{\ensuremath{\pi}}{\ensuremath{\beta}}|{\ensuremath{\xi}}_{k\ensuremath{\sigma}}+\frac{1}{2}U\ensuremath{-}i\frac{\ensuremath{\pi}}{\ensuremath{\beta}})=0$. Examples of the method in simplified situations are provided.
The partition function, a fundamental tenet in statistical thermodynamics, contains in principle all thermodynamic information about a system. It encapsulates both microscopic information through the quantum energy levels and statistical information from the partitioning of the particles among the available energy levels. For identical particles, this statistical accounting is complicated by the symmetry requirements of the allowed quantum states. In particular, for Fermi systems, the enforcement of the Pauli principle is typically a numerically demanding task, responsible for much of the cost of the calculations. The interplay of these three elements---the structure of the many-body spectrum, the statistical partitioning of the NNN particles among the available levels, and the enforcement of the Pauli principle---drives the behavior of mesoscopic and macroscopic Fermi systems. In this paper, we develop an approach for the determination of the partition function, a numerically difficult task, for systems of strongly interacting identical fermions and apply it to a model system of harmonically confined, harmonically interacting fermions. This approach uses a recently introduced many-body method that is an extension of the symmetry-invariant perturbation method (SPT) originally developed for bosons. It uses group theory and graphical techniques to avoid the heavy computational demands of conventional many-body methods which typically scale exponentially with the number of particles. The SPT application of the Pauli principle is trivial to implement since it is done ``on paper'' by imposing restrictions on the normal-mode quantum numbers at first order in the perturbation. The method is applied through first order and represents an extension of the SPT method to excited states. Our method of determining the partition function and various thermodynamic quantities is accurate and efficient and has the potential to yield interesting insight into the role played by the Pauli principle and the influence of large degeneracies on the emergence of the thermodynamic behavior of large-NNN systems.