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Zeitschrift für Naturforschung A

All-Temperature Partition Function from Variationally Optimized Canonical Density Matrix

R. Baltin

Published Dec 1, 1983• 0 citations
Publisher

Abstract

Abstract For the canonical density matrix C(r, r0,β) a variational ansatz C̄̄f = (1 - f̄) Ccl + f̄ Cgr is made where Ccl and Cgr are the classical and the ground state expressions which are exact in the high temperature (β → 0) and in the low-temperature limits (β → + ∞), respectively, and f̄ is a trial function subject to the restriction that f̄ → 0 for β → 0 and f̄ → 1 for β → ∞. With the approximation that f̄ be dependent only upon β, not upon spatial variables, the mean square error arising when Cf is inserted into the Bloch equation is made a minimum. The Euler equation for this variational problem is an ordinary second order differential equation for f̄=f(β) to be solved numerically. The method is tested for the exactly solvable case of the one dimensional harmonic oscillator.

DOI
10.1515/zna-1983-1215

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Abstract For the canonical density matrix C(r, r0,β) a variational ansatz C̄̄f = (1 - f̄) Ccl + f̄ Cgr is made where Ccl and Cgr are the classical and the ground state expressions which are exact in the high temperature (β → 0) and in the low-temperature limits (β → + ∞), respectively, and f̄ is a trial function subject to the restriction that f̄ → 0 for β → 0 and f̄ → 1 for β → ∞. With the approximation that f̄ be dependent only upon β, not upon spatial variables, the mean square error arising when Cf is inserted into the Bloch equation is made a minimum. The Euler equation for this variational problem is an ordinary second order differential equation for f̄=f(β) to be solved numerically. The method is tested for the exactly solvable case of the one dimensional harmonic oscillator.
The variational density-matrix approach introduced by Campbell, K\"urten, Ristig, and Senger [Phys. Rev. B 30, 3728 (1984)], to study finite-temperature Bose fluids is reformulated to avoid dealing directly with the entropy of the trial density matrix. We obtain exact expressions for the Euler-Lagrange equations for a finite-temperature trial statistical density matrix, which is the simplest finite-temperature generalization of a Jastrow trial ground-state wave function. A multicomponent hypernetted-chain method is developed to solve approximately these Euler-Lagrange equations at low temperatures. In leading order the results are shown to be equivalent to those obtained by Campbell et al. The current formalism provides a scheme to improve results of the previous approach.
The one-dimensional equation of motion for the canonical or Bloch density matrix is solved by the method of separation of variables for independent particles moving in a potential V(x). The solution is exact when third and higher derivatives of V(x) are zero, embracing thereby the examples of (a) harmonic potentials and (b) a constant electric field of arbitrary strength. The form of the Slater sum obtained involves then two functions of temperature only, to be determined. One obvious relation between these two functions is via the partition function. A further relation comes from the orthonormality of the wave functions appearing in the Slater sum: the analog of idempotency for the ground-state Dirac density matrix. These two conditions again are shown to lead back to the exact result for harmonic potentials. The results presented generalize the one-dimensional Thomas-Fermi theory to apply to a temperature range that is significantly wider than for the original method.
Numerical evaluation of functional integrals usually involves a finite ( L -slice) discretization of the imaginary-time axis. In the auxiliary-field method, the L -slice approximant to the density matrix can be evaluated as a function of inverse temperature at any finite L as L (β) = [ 1 (β/ L )] L , if the density matrix 1 (β) in the static approximation is known. We investigate the convergence of the partition function Z L (β) ≡ Tr L (β), the internal energy and the density of states g L ( E ) (the inverse Laplace transform of Z L ), as L → ∞ . For the simple harmonic oscillator, g L ( E ) is a normalized truncated Fourier series for the exact density of states. When the auxiliary-field approach is applied to spin systems, approximants to the density of states and heat capacity can be negative. Approximants to the density matrix for a spin-½ dimer are found in closed form for all L by appending a self-interaction to the divergent Gaussian integral and analytically continuing to zero self-interaction. Because of this continuation, the coefficient of the singlet projector in the approximate density matrix can be negative. For a spin dimer, Z L is an even function of the coupling constant for L < 3: ferromagnetic and antiferromagnetic coupling can be distinguished only for L ⩾3, where a Berry phase appears in the functional integral. At any non-zero temperature, the exact partition function is recovered as L → ∞ .
We exploit the grassmannian nature of the variables involved in the path integral expression of the grand canonical partition function for self--interacting fermionic models to show, in one-space dimension, a general relation among the terms of it expansion in the high temperature limit and a combination of co-factors of a suitable matrix with commuting entries. As an application, we apply this framework to calculate the exact coefficients, up to order β^3, of the expansion of the grand canonical partition function for the Hubbard model in d=(1+1) in the high temperature limit. The results are valid for any set of parameters that characterize the model.
The numerical transfer matrix for the partition function of discrete lattice models is generalized to allow the calculation of the density of states \ensuremath{\Omega}(E), and the restricted density of states \ensuremath{\Omega}(E,M). Given \ensuremath{\Omega}(E,M) the partition function is expressed as a polynomial in the variables x=${\mathit{e}}^{\mathrm{\ensuremath{\beta}}\mathit{h}}$ and y=${\mathit{e}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\beta}}}$. These algorithms are illustrated with calculations for the Ising model on finite square lattices. The zeros of the partition function are examined in both the complex x and y planes. Finite size scaling analysis of the zeros leads to very accurate estimates for the critical temperature and critical exponents.
Variational calculations play an important role in quantum mechanics, particularly in determining the ground-state energy of physical systems. Didactic examples in which one can see how the exact energy and wave function are approached by using a family of trial functions are not very numerous. An example was given in a recent book [M. Moshinsky, The Harmonic Oscillator in Modern Physics: From Atoms to Quarks (Gordon and Breach, New York, 1969)] which discusses the ground state of the hydrogen atom using as trial wave function a linear combination of harmonic-oscillator states. In the present paper we carry out a similar analysis for the ground state of a particle of mass m in a three-dimensional square-well potential. We discuss not only the approach to the exact energy when we vary the frequency and the number of oscillator states but also analyze the overlap of the exact and variational wave functions and compare the exact and approximate form factors.