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Journal of Physics Conference Series

The Hubbard-Stratonovich Transformation and Exchange Boson Contributions

Nguyen Van Hoa, Nguyen Tri Lan

Published Sep 23, 2014• 1 citation
Publisher

Abstract

The Hubbard - Stratonovich transformation of many - body theory is reconsidered in various versions. Learning from its failures in coping with competing channels of collective phenomena, a combined Hubbard - Stratonovich transformation is proposed to overcome these difficulties. That yields a variety of fluctuating bosonic quantum fields describing the competition of different phases in the system under consideration. As a consequence of results, the validity of the Hubbard - Stratonovich transformation and its relation to a given order parameter are discussed.

DOI
10.1088/1742-6596/537/1/012004

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The Hubbard - Stratonovich transformation of many - body theory is reconsidered in various versions. Learning from its failures in coping with competing channels of collective phenomena, a combined Hubbard - Stratonovich transformation is proposed to overcome these difficulties. That yields a variety of fluctuating bosonic quantum fields describing the competition of different phases in the system under consideration. As a consequence of results, the validity of the Hubbard - Stratonovich transformation and its relation to a given order parameter are discussed.
Through the development of many-body methodology and algorithms, it has become possible to describe quantum systems composed of a large number of particles with great accuracy. Essential to all these methods is the application of auxiliary fields via the Hubbard-Stratonovich transformation. This transformation effectively reduces two-body interactions to interactions of one particle with the auxiliary field, thereby improving the computational scaling of the respective algorithms. The relevance of collective phenomena and interactions grows with the number of particles. For many theories, e.g. Chiral Perturbation Theory, the inclusion of three-body forces has become essential in order to further increase the accuracy on the many-body level. In this proceeding, the an-alytical framework for establishing a Hubbard-Stratonovich-like transformation, which allows for the systematic and controlled inclusion of contact three-and more-body inter-actions, is presented.
\nThrough the development of many-body methodology and algorithms, it has become possible to describe quantum systems composed of a large number of particles with great accuracy. Essential to all these methods is the application of auxiliary fields via the Hubbard-Stratonovich transformation. This transformation effectively reduces two-body interactions to interactions of one particle with the auxiliary field, thereby improving the computational scaling of the respective algorithms. The relevance of collective phenomena and interactions grows with the number of particles. For many theories, e.g. Chiral Perturbation Theory, the inclusion of three-body forces has become essential in order to further increase the accuracy on the many-body level. In this proceeding, the an-alytical framework for establishing a Hubbard-Stratonovich-like transformation, which allows for the systematic and controlled inclusion of contact three-and more-body inter-actions, is presented.\n
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.
We analyze a variety of Gaussian integrals with the aim of revisiting the derivation of the Hubbard–Stratonovich transformation as given in standard graduate-level textbooks and provide an overview of its applications. We pinpoint problematic steps in the usual discussions and propose careful derivations of the Hubbard–Stratonovich identity pertinent to a variety of situations relevant to statistical physics and quantum field theory. These derivations are based on direct use of either a resolution identity or a series expansion. A few homework problems for students are suggested.
Despite its apparent simplicity, the two-dimensional Hubbard model for locally interacting fermions on a square lattice is widely considered as a promising approach for the understanding of Cooper pair formation in the quasi two-dimensional high-Tc cuprate materials. In the present work this model is investigated by means of the functional renormalization group, based on an exact flow equation for the effective average action. In addition to the fermionic degrees of freedom of the Hubbard Hamiltonian, bosonic fields are introduced which correspond to the different possible collective orders of the system, for example magnetism and superconductivity. The interactions between bosons and fermions are determined by means of the method of “rebosonization” (or “flowing bosonization”), which can be described as a continuous, scale-dependent Hubbard-Stratonovich transformation. This method allows an efficient parameterization of the momentumdependent effective two-particle interaction between fermions (four-point vertex), and it makes it possible to follow the flow of the running couplings into the regimes exhibiting spontaneous symmetry breaking, where bosonic fluctuations determine the types of order which are present on large length scales. Numerical results for the phase diagram are presented, which include the mutual influence of different, competing types of order.