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Spectral functions from Quantum Monte Carlo

Richard N. Silver

Published Jan 1, 1989• 0 citations
OpenAlex

Abstract

In his review, D. Scalapino identified two serious limitations on the application of Quantum Monte Carlo (QMC) methods to the models of interest in High {Tc} Superconductivity (HTS). One is the sign problem''. The other is the analytic continuation problem'', which is how to extract electron spectral functions from QMC calculations of the imaginary time Green's functions. Through-out this Symposium on HTS, the spectral functions have been the focus for the discussion of normal state properties including the applicability of band theory, Fermi liquid theory, marginal Fermi liquids, and novel non-perturbative states. 5 refs., 1 fig.

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In his review, D. Scalapino identified two serious limitations on the application of Quantum Monte Carlo (QMC) methods to the models of interest in High {Tc} Superconductivity (HTS). One is the sign problem''. The other is the analytic continuation problem'', which is how to extract electron spectral functions from QMC calculations of the imaginary time Green's functions. Through-out this Symposium on HTS, the spectral functions have been the focus for the discussion of normal state properties including the applicability of band theory, Fermi liquid theory, marginal Fermi liquids, and novel non-perturbative states. 5 refs., 1 fig.
We study cluster perturbation theory [Phys. Rev. Lett. 84, 522 (2000)] when auxiliary field quantum Monte Carlo method is used for solving the cluster hamiltonian. As a case study, we calculate the spectral functions of the Hubbard model in one and two dimensions and compare our results for the spectral functions to results obtained using exact diagonalization to solve the cluster hamiltonian. The main advantage of using quantum Monte Carlo results as a starting point is that the initial cluster size can be taken to be considerably larger and hence potentially capture more of the relevant physics. The drawback is that quantum Monte Carlo methods yield results at imaginary times with stochastic errors.
For important classes of many-fermion problems, quantum Monte Carlo (QMC) methods allow exact calculations of ground-state and finite-temperature properties without the sign problem. The list spans condensed matter, nuclear physics, and high-energy physics, including the half-filled repulsive Hubbard model, the spin-balanced atomic Fermi gas, and lattice quantum chromodynamics calculations at zero density with Wilson Fermions, and is growing rapidly as a number of problems have been discovered recently to be free of the sign problem. In these situations, QMC calculations are relied on to provide definitive answers. Their results are instrumental to our ability to understand and compute properties in fundamental models important to multiple subareas in quantum physics. It is shown, however, that the most commonly employed algorithms in such situations have an infinite variance problem. A diverging variance causes the estimated Monte Carlo statistical error bar to be incorrect, which can render the results of the calculation unreliable or meaningless. We discuss how to identify the infinite variance problem. An approach is then proposed to solve the problem. The solution does not require major modifications to standard algorithms, adding a "bridge link" to the imaginary-time path integral. The general idea is applicable to a variety of situations where the infinite variance problem may be present. Illustrative results are presented for the ground state of the Hubbard model at half-filling.
Recent years have seen a rise in the popularity of the physics of strongly interacting electronic systems, and in most of the more interesting cases the perturbative regime is not applicable. The continuous-time quantum Monte Carlo (CT-QMC) has already established its leading stochastic simulation method to address quantum impurity models that have gained central focus of DMFT. Lugiato-Lefever function approach, on the one hand, can provide a rigorous account of the electronic structure, excitation spectrum and phenomenon of correlation to most of the many-body systems. In the paper the connection between the CT-QMC technique and theory of Green functions are stated with a specification in the connection of the two, complementing each other in the solving of quantum many-body problems. We present the approach of CT-QMC that is numerically inexpensive but computes Green functions directly in the imaginary time that you then analytically continue to the real frequencies to obtain physical observables. Further, we discuss the ability to investigate quasiparticle behaviour, spectral density and transport phenomenon through self-energies acquired as Green functions in CT-QMC.
One of the fundamental questions about high-temperature cuprate superconductors is the size of the Fermi surface underlying the superconducting state. By analyzing the single-particle spectral function for the Fermi-Hubbard model as a function of repulsion UUU and chemical potential $\ensuremath{\mu}$, we find that the Fermi surface in the normal state undergoes a transition from a large Fermi surface matching the Luttinger volume as expected in a Fermi liquid, to a Fermi surface that encloses fewer electrons that we dub the ``Luttinger breaking'' phase, as the Mott insulator is approached. This transition into a non-Fermi-liquid phase that violates the Luttinger count occurs at a critical density in the absence of any other broken symmetry. We obtain the Fermi-surface contour from the spectral weight ${A}_{\mathbf{k}}(\ensuremath{\omega}=0)$ and from an analysis of the singularities of the Green's function ReGk(E=0)\mathrm{Re}\phantom{\rule{0.16em}{0ex}}{G}_{\mathbf{k}}(E=0)ReGk​(E=0), calculated using determinantal quantum Monte Carlo and analytic continuation methods. We discuss our numerical results in connection with experiments on Hall measurements, scanning tunneling spectroscopy, and angle-resolved photoemission spectroscopy.