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Journal of Mathematical Physics

A contribution to the many-fermion problem

Daniel Charles Mattis

Published Aug 16, 2012• 2 citations
Publisher

Abstract

In this work we reexamine the many-fermion problem in arbitrary dimensions. It is shown that in two dimensions or higher, the Hamiltonian of interacting fermions can be separated into individual nonintersecting sectors labeled by the wave-vector $\vec q$q⃗. Within each sector the Hamiltonian maps onto a generalized version of the one-dimensional Luttinger model that resembles a boson string. These are chain-like quadratic forms in boson operators that are readily diagonalized in the absence of “exchange” corrections. Moreover, in a simple example involving SU(2) fermions, that of the Hubbard model, we show that it can be possible also to incorporate exchange terms and express them entirely within an enlarged set of string variables.

DOI
10.1063/1.4742966

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We describe briefly a new method of approach to the interacting fermion and boson systems. Namely, we determine the explicit form of the single-particle wave functions {wi_{i}i​(r)} appearing in the microscopic parameters of models in the second-quantization representation. The method is illustrated on the examples of H2_{2}2​ molecule and He atom, the Hubbard chain, and a ring of N ≤\leq≤ 10 atoms.
We develop and describe new approaches to the problem of interacting fermions in spatial dimensions greater than 1. These approaches are based on generalizations of powerful tools previously applied to problems in one spatial dimension. We begin with a review of one-dimensional interacting fermions. We then introduce a simplified model in two spatial dimensions to study the role that spin and perfect nesting play in destabilizing fermion liquids. The complicated functional renormalization-group equations of the full problem are made tractable in our model by replacing the continuum of points that make up the closed Fermi line with four Fermi points. Despite this drastic approximation, the model exhibits physically reasonable behavior both at half-filling (where instabilities occur) and away from half-filling (where a Luttinger liquid arises). Next we implement the bosonization of higher-dimensional Fermi surfaces introduced by Luther and advocated most recently by Haldane. Bosonization incorporates the phase-space and small-angle scattering processes neglected in our model (but does not, as yet, address questions of stability). The charge sector is equivalent to an exactly solvable Gaussian quantum field theory; the spin sector, however, must be solved semiclassically. Using the Luther-Haldane approach we recover the collective mode equation of Fermi-liquid theory and in three dimensions reproduce the T3{\mathit{T}}^{3}T3ln(T) contribution to the specific heat due to small-angle scattering processes. We conclude with a discussion of our results and some speculation about future possibilities.
We derive the exact form of the bosonized Hamiltonian for a many-body fermion system in one spatial dimension with arbitrary dispersion relations, using the droplet bosonization method. For a single-particle Hamiltonian polynomial in the momentum, the bosonized Hamiltonian is a polynomial of one degree higher in the bosonic ``boundary'' field and includes subleading lower-order and derivative terms. This generalizes the known results for massless relativistic and nonrelativistic fermions (quadratic and cubic bosonic Hamiltonians, respectively). We also consider two-body interactions and demonstrate that they lead to interesting collective behavior and phase transitions in the Fermi sea.
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.