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arXiv:1001.1552 · Physical Review B

Describing systems of interacting fermions by boson models: Mapping in arbitrary dimension and applications

K. B. Efetov, C. Pépin, Hannes Meier

Published Dec 14, 2010• 11 citations
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Abstract

We develop a method that allows us to map models of interacting fermions onto bosonic models describing collective excitations in an arbitrary dimension. We present arguments that this mapping may become exact in the thermodynamic limit in the presence of a bath. The boson models can be written either in the form of a model of noninteracting bosons in a fluctuating auxiliary Hubbard-Stratonovich field or in the form of a superfield theory of interacting bosons. We show how one can study the latter version within the perturbation theory. Using a diagrammatic technique developed for the boson model we compare the first two orders of the perturbation theory with the corresponding results for the original fermion model and find a perfect agreement. As concerns the former representation, we suggest a scheme that may be suitable for Monte Carlo simulations. We demonstrate both analytically and numerically that the partition function of the boson model in the auxiliary Hubbard-Stratonovich field is, in contrast to the fermion model, always positive and thus the boson model can be suitable for the Monte Carlo routine. At the same time, our arguments showing the equivalence of the boson and fermion models in the thermodynamic limit are not completely rigorous and it is still to be investigated numerically how accurate the boson model can approximate the original fermion one.

DOI
10.1103/physrevb.82.235120

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We develop a method that allows us to map models of interacting fermions onto bosonic models describing collective excitations in an arbitrary dimension. We present arguments that this mapping may become exact in the thermodynamic limit in the presence of a bath. The boson models can be written either in the form of a model of noninteracting bosons in a fluctuating auxiliary Hubbard-Stratonovich field or in the form of a superfield theory of interacting bosons. We show how one can study the latter version within the perturbation theory. Using a diagrammatic technique developed for the boson model we compare the first two orders of the perturbation theory with the corresponding results for the original fermion model and find a perfect agreement. As concerns the former representation, we suggest a scheme that may be suitable for Monte Carlo simulations. We demonstrate both analytically and numerically that the partition function of the boson model in the auxiliary Hubbard-Stratonovich field is, in contrast to the fermion model, always positive and thus the boson model can be suitable for the Monte Carlo routine. At the same time, our arguments showing the equivalence of the boson and fermion models in the thermodynamic limit are not completely rigorous and it is still to be investigated numerically how accurate the boson model can approximate the original fermion one.
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In this work we study the recently developed parametrized partition function formulation and show how we can infer the thermodynamic properties of fermions based on numerical simulation of bosons and distinguishable particles at various temperatures. In particular, we show that in the three dimensional space defined by energy, temperature and the parameter characterizing parametrized partition function, we can map the energies of bosons and distinguishable particles to fermionic energies through constant-energy contours. We apply this idea to both noninteracting and interacting Fermi systems and show it is possible to infer the fermionic energies at all temperatures, thus providing a practical and efficient approach to obtain thermodynamic properties of large fermion systems with numerical simulation. As an example, we present energies for up to 50 noninteracting fermions and up to 20 interacting fermions at all temperatures and show good agreement with the analytical result for noninteracting case.
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