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Lettere al nuovo cimento della societa italiana di fisica/Lettere al nuovo cimento

Calculation of the two-body correlation function for Fermion systems

Stefano Fantoni, Sergio Rosati

Published Jul 1, 1974• 73 citations
Publisher
DOI
10.1007/bf02784779

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Numerical studies of low-dimensional quantum many-body systems.

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Exact solutions of many-body problems hold a very special and significant place in condensed matter physics, helping us to understand the behaviors of interacting particles. In this thesis, we focus our attention on study the exact solutions of two interesting many-body problems in stronglycorrelated \n \nsystems. One is to calculate the exact correlation functions of \n \n1D chains of spinless fermions and hardcore bosons. The other is to calculate the single-particle dispersion relation of Hubbard model of spinless fermions on 2D square lattices.

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We construct a quantum Monte Carlo algorithm for interacting fermions using the two-body density as the fundamental quantity. The central idea is mapping the interacting fermionic system onto an auxiliary system of interacting bosons. The correction term is approximated using correlated wave functions for the interacting system, resulting in an effective potential that represents the nodal surface. We calculate the properties of 3He and find good agreement with experiment and with other theoretical work. In particular, our results for the total energy agree well with other calculations where the same approximations were implemented but the standard quantum Monte Carlo algorithm was used.
In the framework of exactly solvable quantum many-body systems we study models of interacting spin one-half Fermions in one dimension. The first part deals with systems of spin one-half Fermions which interact via repulsive contact interaction. A reformulation of the Bethe-Ansatz solvable many-body wave function is presented. This simplifies considerably the calculations for the highly imbalanced case, where very few particles of one species (minority Fermions) are present. For the other particle species (majority Fermions) the thermodynamic limit is taken. We assume the majority Fermions to be in the ground state such that their non-interacting momentum distribution is a Fermi-sea. Upon this we consider excitations where the particles of the minority species may occupy an arbitrary state within the Fermi-sea. In the case of only a single minority Fermion, the many-body wave function can be expressed as a determinant. This allows us to derive exact thermodynamic expressions for several expectation values as well as for the density-density correlation function. Moreover it is possible to find closed expressions for the single particle Green's function. All of the above mentioned quantities show a non-trivial dependence on the minority particle's momentum. In particular the Green's function in the Tonks-Girardeau regime of hardcore interaction is shown to undergo a transition from the one of impenetrable Bosons to that of free Fermions as the extra particle's momentum varies from the core to the edge of the Fermi-sea. This transition becomes manifest in an algebraic asymptotic decay of the Green's function. If two minority Fermions are present, the many-body wave function turns out to be more complicated. Nevertheless it is possible to derive exact expressions for the two and the three particle density-density correlation functions. Furthermore we calculate the system's total energy and based on that, identify terms which have a natural interpretation as effective interaction energy for the two minority Fermions in the presence of the Fermi-sea. The second part is devoted to the study of one-dimensional systems consisting of two fermionic particle species with different masses. We show that for specific kinds of interaction potentials and for certain relations between the masses and the coupling constants, the particle creation and annihilation operators of such a system can be constructed exactly.
Exact solutions of many-body problems hold a very special and significant place in condensed matter physics, helping us to understand the behaviors of interacting particles. In this thesis, we focus our attention on study the exact solutions of two interesting many-body problems in stronglycorrelated \n \nsystems. One is to calculate the exact correlation functions of \n \n1D chains of spinless fermions and hardcore bosons. The other is to calculate the single-particle dispersion relation of Hubbard model of spinless fermions on 2D square lattices.