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arXiv:0712.2493 · Physical Review E

Evaluation of the permutational structure of quantum gases at finite temperature

Ryan Springall, Manolo C. Per, Ian K. Snook

Published Mar 30, 2007• 1 citation
arXivHTMLPDF

Abstract

Proposed is an alternative method for permutational sampling in quantum gases using the path integral formulation of statistical mechanics. It is shown that in principle we are able to use two operators which enable us to construct a Markov chain through a graph of the irreducible representation of the symmetric group. As an illustration of this method, a test calculation of four particles in a harmonic trap is performed.

DOI
10.1103/physreve.75.031124

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Proposed is an alternative method for permutational sampling in quantum gases using the path integral formulation of statistical mechanics. It is shown that in principle we are able to use two operators which enable us to construct a Markov chain through a graph of the irreducible representation of the symmetric group. As an illustration of this method, a test calculation of four particles in a harmonic trap is performed.
This thesis is composed of two parts. In part one we report results of two quantum Monte Carlo methods - variational Monte Carlo and diffusion Monte Carlo - on the potential energy curve of the helium dimer, the prototypical van der Waals system. In contrast to previous quantum Monte Carlo calculations on this system, we have employed trial wave functions of the Slater-Jastrow form and used the fixed node approximation for the fermion nodal surface. We find both methods to be in excellent agreement with the best theoretical results at short range. In addition, the diffusion Monte Carlo results give very good agreement across the whole potential energy curve, while the Slater-Jastrow wave function fails to bind the dimer at all. In part two we switch to investigations of many-body systems at finite temperature. We use the path integral representation of statistical mechanics to investigate the symmetry properties of the canonical ensemble partition function and find a representation in terms of irreducible representations of the symmetric group. We use this as a foundation to propose a novel technique for the stochastic sampling of the bosonic partition function for quantum gases. It is shown that in principle we are able to use two operators which enable us to construct a Markov chain through a graph of the irreducible representation of the symmetric group. As an illustration of this method, a test calculation of four particles in a harmonic trap is performed.
A rigorous expression for the partition function of N identical fermions with spin 1/2 is obtained in a form suitable for a path-integral Monte Carlo (PIMC) simulation of a many-electron system at a finite temperature. A sum over permutations is reduced to a sum over classes with readily determinable factors. A distribution over projections and values of the total spin is also obtained, enabling PIMC calculations of spin-dependent quantities. An MC algorithm is proposed which accounts simultaneously for all types of trajectory linkages. A test calculation, four electrons in a spherical cavity with a radius of 1 nm at a temperature of 300--2000 K, is performed.
Quantum-mechanical concepts can be formulated in constructive finite terms without loss of their empirical content if we replace a general unitary group by a unitary representation of a finite group. Any linear representation of a finite group can be realized as a subrepresentation of a permutation representation. Thus, quantum-mechanical problems can be expressed in terms of permutation groups. This approach allows us to clarify the meaning of a number of physical concepts. Combining methods of computational group theory with Monte Carlo simulation we study a model based on representations of permutation groups.
A compact representation is obtained for the quantum statistical sum of indistinguishable nonrelativistic spin-1/2 fermions in the form of Feynman path integrals which can be used as the basis to develop a fundamentally exact method of computer modeling for systems of strongly interacting electrons at nonzero temperature. A basis of symmetrized wave functions is constructed using Young symmetry operators. An exact permutation symmetrization procedure leads to an avalanche-like multiplication in the number of diagrams of linked Feynman integrals of the order of N !. The partition function can be simplified without introducing any approximations and this is performed numerically by computer by direct sorting of diagrams. The control tables obtained, containing combinatorial weights of diagrams, direct the Markov random walk process in virtual trajectory space which is achieved numerically by computer. The equilibrium characteristics of the quantum system are calculated by averaging. This approach is an expansion of the Monte Carlo-Metropolis method to systems of quantum indistinguishable particles with spin. Demonstration numerical calculations using this method were made for the simplest exchange systems, for a hydrogen molecule, a Be + ion, and a Li atom. The ground state of the hydrogen molecule is reproduced with a statistical error of 0.2%. Exchange-correlation effects lead to nontrivial structural changes in the thermally excited electron shells of ions in a state of strong plasma compression.
This chapter follows on from the previous chapter in quantum statistical mechanics but specialising on systems with identical particles. Using Gibbs prescritpion on generic states from Chapter 3, the occupation number representation is introduced. Constraints imposed on statistics by irreducible representations of the permutation group are discussed. These group-theoretic considerations are used to justify the use of Gentile’s parastatistics. Fermions and bosons are introduced as special cases of Gentile’s statistics, corresponding to the trivial representation of the permutation group for bosons, and the sign representation of the permutation group for fermions. Basic applications to fermions and bosons is given, including the Fermi–Dirac and Bose–Einstein statistics. A detailed expose of why photons are said to have zero chemical potential is also proposed.