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Journal of Mathematical Physics

Boson approximants for lattice Fermi systems

Piotr Garbaczewski

Published Aug 1, 1985• 5 citations
Publisher

Abstract

We demonstrate that Bose systems can be used to simulate properties of their Fermi partners in thermal bath, provided the spectral property HB =PHBP +(1−P)HB(1−P), HF =PHBP can be proved in the state space of the Bose system. The approximation accuracy can be made arbitrarily good by varying the (free) coupling parameter λ∈(0,∞), and permits studying of fermionic partition or correlation functions in a finite volume by means of the standard techniques (Bose:path integral, Monte Carlo, etc).

DOI
10.1063/1.526876

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We demonstrate that Bose systems can be used to simulate properties of their Fermi partners in thermal bath, provided the spectral property HB =PHBP +(1−P)HB(1−P), HF =PHBP can be proved in the state space of the Bose system. The approximation accuracy can be made arbitrarily good by varying the (free) coupling parameter λ∈(0,∞), and permits studying of fermionic partition or correlation functions in a finite volume by means of the standard techniques (Bose:path integral, Monte Carlo, etc).
The author establishes relationships between Bose, Fermi and spin-1/2 systems on lattices of arbitrary dimension and arbitrary but finite size which are defined to live in the (common) state space of the boson system. The underlying systems display an essential affinity by being described by precisely the same form of Hamiltonians, which are the polynomial functions of canonical operator variables: boson, fermion or spin-1/2 variables, respectively. For large enough, densely populated finite volume lattices it entails the computation of partition and correlation functions for lattice fermions in terms of bosons only. Albeit an approximation, its accuracy increases as the continuum limit is approached.
Exploratory simulations of Bose-Fermi mixtures on the three-dimensional optical lattice at finite temperature are performed by adopting the lattice quantum chromodynamics technique. We analyze the bosonic superfluid transition and its dependence on the strength of the boson-fermion coupling. The particle densities and the pair occupancies are also studied to understand the effect of the boson-fermion coupling to the microscopic properties of the system. The effect of the induced fermion-fermion interaction by the boson density fluctuation is clearly seen.
An exact spectral solution of the Hubbard problem on the tetrahedron is given. It is demonstrated that it provides at the same time a partial spectral solution for the Bose version of this (Fermi) model. A remarkable relationship of the Hamiltonians HB=HF+(1−P)HB(1−P) is established, where P is a projection in the state space of the interacting Bose system. This partial observation is then extended to the spatial lattice composed of the four-site Hubbard cells, which interact due to the superimposed perturbation of the hopping type, affecting all sites of the lattice; H=∑(i j) ∑σ=± bi j c*iσ cjσ+U ∑i ni+ni− being the corresponding Hamiltonian. It is thus demonstrated that the joint Bose–Fermi spectral problem (previously established in 1+1 dimensions) makes sense in 1+3 dimensions as well.
Two-component Fermi gases model the behavior of many systems in different fields of physics, and one of their interesting features is that they condense into superfluids at low temperatures. Ultracold atoms experiments represent one realization of such Fermi gases, and their great flexibility sparked active research into their phase structure. In the literature, there are studies of the phase structure using functional methods, mean-field approximations, and other approaches. In the present work, we aim to perform ab-initio Monte-Carlo simulations of the system to probe their phase structure for inhomogeneous phases in the presence of spin imbalance. Such simulations generally require a bosonization of the theoretical description of the system that rewrites the theory in terms of an auxiliary bosonic field. For this auxiliary field, many possible choices achieve this, and previous studies often use a field that corresponds to a density of fermions. In the present work, we develop a novel approach to this problem by bosonizing the system in terms of the so-called pairing field which corresponds to the superfluid order parameter. Even in the absence of mass- or spin-imbalance, this approach exhibits a sign problem that presents a problem to many Monte-Carlo methods. To circumvent this sign problem, we base our simulation on the Complex Langevin method. The central question of this thesis is if a simulation of a pairing-field-based formalism using the Complex-Langevin method is suitable to study the phase structure of two-component Fermi gases in the presence of spin imbalance. To this end, we develop a lattice theory based on the pairing field by discretizing the continuous Hamiltonian and performing a rigorous derivation of the path integral from there. We pay special attention to the derivation of lattice derivative operators and rescale the theory to be dimensionless. Beyond that, we develop an efficient notation for lattice theories that makes their handling arguably easier than that of continuum theories. With the obtained theory, we derive the Langevin equation for the system together with the expressions we need to sample observables and develop a numerical simulation on that basis. We use the simulation to study 0+1-dimensional systems as a proof-of-concept. Specifically, we calculate density equations of state and two-point functions for a range of dimensionless couplings and compare them to exact solutions. Additionally, we compare our obtained results for the two-point functions to general discussions of the analytic properties of correlation functions to gain additional insight into the qualitative and numerical behavior of our simulation. We found that our results for both density equations of state and two-point functions are in excellent agreement with the exact solutions. Going forward, we plan to study systems in d>0 spatial dimensions, beginning with simulations of 1+1-dimensional systems. In d=3 dimensions the pairing field's correspondence to the superfluid order parameter may allow us to efficiently study the spontaneous breaking of the U(1) symmetry of the system. On that basis, we can probe the phase diagram of the system for inhomogeneous phases at finite spin imbalance in future studies.
This work deals with the development and application of novel quantum Monte Carlo methods to simulate fermion-boson models. Our developments are based on the path-integral formalism, where the bosonic degrees of freedom are integrated out exactly to obtain a retarded fermionic interaction. We give an overview of three methods that can be used to simulate retarded interactions. In particular, we develop a novel quantum Monte Carlo method with global directed-loop updates that solves the autocorrelation problem of previous approaches and scales linearly with system size. We demonstrate its efficiency for the Peierls transition in the Holstein model and discuss extensions to other fermion-boson models as well as spin-boson models. Furthermore, we show how with the help of generating functionals bosonic observables can be recovered directly from the Monte Carlo configurations. This includes estimators for the boson propagator, the fidelity susceptibility, and the specific heat of the Holstein model. The algorithmic developments of this work allow us to study the specific heat of the spinless Holstein model covering its entire parameter range. Its key features are explained from the single-particle spectral functions of electrons and phonons. In the adiabatic limit, the spectral properties are calculated exactly as a function of temperature using a classical Monte Carlo method and compared to results for the Su-Schrieffer-Heeger model.