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arXiv:1209.5638

Dynamical Structure Factors in Quantum Many-Body Systems from Quantum Monte Carlo Calculations

Alessandro Roggero, Francesco Pederiva, Giuseppina Orlandini

Published Sep 25, 2012• 0 citations
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Abstract

An ab-initio method for determining the dynamical structure function of an interacting many--body quantum system has been devised by combining a generalized integral transform method with Quantum Monte Carlo methods. As a first application, the coherent and, separately, the incoherent excitation spectrum of bulk atomic 4He has been computed, both in the low and intermediate momentum range. The peculiar form of the kernel in the integral transform of the dynamical structure function allows to predict, without using any model, both position and width of the collective excitations in the maxon--roton region, as well as the second collective peak. A prediction of the dispersion of the single--particle modes described by the incoherent part is also presented.

DOI
10.48550/arxiv.1209.5638

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An ab-initio method for determining the dynamical structure function of an interacting many--body quantum system has been devised by combining a generalized integral transform method with Quantum Monte Carlo methods. As a first application, the coherent and, separately, the incoherent excitation spectrum of bulk atomic 4He has been computed, both in the low and intermediate momentum range. The peculiar form of the kernel in the integral transform of the dynamical structure function allows to predict, without using any model, both position and width of the collective excitations in the maxon--roton region, as well as the second collective peak. A prediction of the dispersion of the single--particle modes described by the incoherent part is also presented.
An ab initio method for determining the dynamical structure function of an interacting many-body quantum system has been devised by combining a generalized integral transform method with quantum Monte Carlo (QMC) methods. A kernel has been found that (i) lets the transform be calculable with QMC methods and (ii) is a representation of the $\ensuremath{\delta}$ function, allowing an inversion of the transform with a much higher predictive power than the inverse Laplace transform. As a first application, the excitation spectrum of bulk atomic 4{}^{4}4He has been computed, both in the low and intermediate momentum ranges. The peculiar form of the kernel allows us to predict, without using any model, both positions and widths of the collective excitations in the maxon-roton region, as well as the existence of the second collective peak. A prediction of the dispersion of the single-particle modes described by the incoherent part is also presented.
We review an ab-initio method for calculating the dynamical structure function of an interacting many-body quantum system. The method consists in coupling a generalized integral transform approach with imaginary time Quantum Monte Carlo calculations. The strength of the method has been tested on the excitation spectrum of bulk atomic 4 He. The peculiar form of the kernel as a representation of the delta-function has allowed to minimize the ill-posedness of the integral transform inversion. In fact it has been possible to obtain, at a considerable degree of reliability, both position and width of the collective excitations in the maxon-roton region, as well as the second collective peak. What we stress here is the ability of such a 4-function-like kernel, for which one can control position and width, to maintain in the transformed space the characteristics of the collective structures. The application to the coherent and incoherent density excitation spectrum of liquid 4 He is discussed.
We investigate the dynamic structure factor of a system of Bose particles at zero temperature using quantum Monte Carlo methods. Interactions are modeled using a hard-sphere potential of size aaa and simulations are performed for values of the gas parameter na3n{a}^{3}na3 ranging from the dilute regime up to densities nnn where the thermodynamically stable phase is a solid. With increasing density, we observe a crossover of the dispersion of elementary excitations from a Bogoliubov-type spectrum to a phonon-maxon-roton curve and the emergence of a broad multiphonon contribution accompanying the single-quasiparticle peak. In particular, for na3=0.2138n{a}^{3}=0.2138na3=0.2138, which corresponds to superfluid 4{}^{4}4He at equilibrium density, the extracted spectrum turns out to be in good agreement with the experimental energy-momentum dispersion relation in the roton region and for higher momenta. The behavior of the spectral function at the same density in the stable solid and metastable gas phase above the freezing point is also discussed.
Many-body physics poses one of the greatest challenges to science in the 21st century. Still more daunting is the problem of accurately calculating the properties of quantum many-body systems in the strongly correlated regime. Cold atomic gases provide an excellent test ground, for both experimentalists and theorists, to study the exotic and sometimes counterintuitive behavior of quantum many-body problems. Of particular interest is the appearance of collective excitations in these systems, such as the famous Goldstone mode and the elusive Higgs mode. It is particularly important to assess the robustness of theoretical and computational techniques to study such excitations. We build on the unprecedented opportunity provided by the fact that, in some cases, exact numerical predictions can be obtained through quantum Monte Carlo. We use these predictions to assess the accuracy of the Random Phase Approximation, which is widely considered to be a method of choice for the study of the collective excitations in a cold atomic Fermi gas modeled with a Fermi–Hubbard Hamiltonian. We found good agreement between the two methodologies for the dynamic properties, particularly for the position of the Goldstone mode. We also explored the possibility of using a renormalized, effective potential in place of the physical potential. We determined that using a renormalized potential is likely too simplistic a method for improving the accuracy of generalized Random Phase Approximation and that a more sophisticated approach is needed.
We obtain ab-initio estimations of the dynamic structure factor, S ( q ,ω), of Bose gases at zero temperature. More precisely, we use the Genetic Inversion via Falsification of Theories (GIFT) algorithm to perform analytic continuations of imaginary time correlation functions computed via an exact Path Integral projector method. Using the hard-sphere potential to model the two-body interactions between the atoms, we compute S ( q ,ω) changing the gas parameter from the dilute regime ( na 3 = 10 −4 ) up to the density corresponding to superfluid 4 He at equilibrium ( na 3 = 0.2138). With increasing density, we observe the emergence of a broad multiphonon contribution accompanying the quasiparticle peak and a crossover of the dispersion of elementary excitations from a Bogoliubov-like spectrum to a phonon-maxon- roton curve. Apart from the low wave vector region, for na 3 = 0.2138 the energy-momentum dispersion relation and the static density response function, χ( q ), turns out to be in good agreement with the superfluid 4 He experimental data at equilibrium density.
We introduce an exact framework to compute the positive frequency moments ${M}^{(\ensuremath{\alpha})}(\mathbf{q})=\ensuremath{\langle}{\ensuremath{\omega}}^{\ensuremath{\alpha}}\ensuremath{\rangle}$ of different dynamic properties from imaginary-time quantum Monte Carlo data. As a practical example, we obtain the first five moments of the dynamic structure factor $S(\mathbf{q},\ensuremath{\omega})$ of the uniform electron gas at the electronic Fermi temperature based on ab initio path integral Monte Carlo simulations. We find excellent agreement with known sum rules for $\ensuremath{\alpha}=1,3$, and present results for $\ensuremath{\alpha}=2,4,5$. Our idea can be straightforwardly generalized to other dynamic properties such as the single-particle spectral function $A(\mathbf{q},\ensuremath{\omega})$, and will be useful for a number of applications, including the study of ultracold atoms, exotic warm dense matter, and condensed matter systems.