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Register a new userStatistical and computational intelligence approach to analytic continuation in Quantum Monte Carlo
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Added by
Gianluca Bertaina Dr
Publication date
2016
fields
Physics
and research's language is
English
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The term analytic continuation emerges in many branches of Mathematics, Physics, and, more generally, applied Science. Generally speaking, in many situations, given some amount of information that could arise from experimental or numerical measurements, one is interested in extending the domain of such information, to infer the values of some variables which are central for the study of a given problem. For example, focusing on Condensed Matter Physics, state-of-the-art methodologies to study strongly correlated quantum physical systems are able to yield accurate estimations of dynamical correlations in imaginary time. Those functions have to be extended to the whole complex plane, via analytic continuation, in order to infer real-time properties of those physical systems. In this Review, we will present the Genetic Inversion via Falsification of Theories method, which allowed us to compute dynamical properties of strongly interacting quantum many--body systems with very high accuracy. Even though the method arose in the realm of Condensed Matter Physics, it provides a very general framework to face analytic continuation problems that could emerge in several areas of applied Science. Here we provide a pedagogical review that elucidates the approach we have developed.
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The Quantum Monte Carlo (QMC) method can yield the imaginary-time dependence of a correlation function $C(tau)$ of an operator $hat O$. The analytic continuation to real-time proceeds by means of a numerical inversion of these data to find the response function or spectral density $A(omega)$ corresponding to $hat O$. Such a technique is very sensitive to the statistical errors in $C(tau)$ especially for large values of $tau$, when we are interested in the low-energy excitations. In this paper, we find that if we use the flat histogram technique in the QMC method, in such a way to make the {it histogram of} $C(tau)$ flat, the results of the analytic continuation for low-energy excitations improve using the same amount of computational time. To demonstrate the idea we select an exactly soluble version of the single-hole motion in the $t-J$ model and the diagrammatic Monte Carlo technique.
Quantum Monte Carlo simulations, while being efficient for bosons, suffer from the negative sign problem when applied to fermions - causing an exponential increase of the computing time with the number of particles. A polynomial time solution to the sign problem is highly desired since it would provide an unbiased and numerically exact method to simulate correlated quantum systems. Here we show, that such a solution is almost certainly unattainable by proving that the sign problem is NP-hard, implying that a generic solution of the sign problem would also solve all problems in the complexity class NP (nondeterministic polynomial) in polynomial time.
We explore the extended Koopmans theorem (EKT) within the phaseless auxiliary-field quantum Monte Carlo (AFQMC) method. The EKT allows for the direct calculation of electron addition and removal spectral functions using reduced density matrices of the $N$-particle system, and avoids the need for analytic continuation. The lowest level of EKT with AFQMC, called EKT1-AFQMC, is benchmarked using small molecules, 14-electron and 54-electron uniform electron gas supercells, and diamond at the $Gamma$-point. Via comparison with numerically exact results (when possible) and coupled-cluster methods, we find that EKT1-AFQMC can reproduce the qualitative features of spectral functions for Koopmans-like charge excitations with errors in peak locations of less than 0.25 eV in a finite basis. We also note the numerical difficulties that arise in the EKT1-AFQMC eigenvalue problem, especially when back-propagated quantities are very noisy. We show how a systematic higher order EKT approach can correct errors in EKT1-based theories with respect to the satellite region of the spectral function. Our work will be of use for the study of low-energy charge excitations and spectral functions in correlated molecules and solids where AFQMC can be reliably performed.
The fidelity susceptibility is a general purpose probe of phase transitions. With its origin in quantum information and in the differential geometry perspective of quantum states, the fidelity susceptibility can indicate the presence of a phase transition without prior knowledge of the local order parameter, as well as reveal the universal properties of a critical point. The wide applicability of the fidelity susceptibility to quantum many-body systems is, however, hindered by the limited computational tools to evaluate it. We present a generic, efficient, and elegant approach to compute the fidelity susceptibility of correlated fermions, bosons, and quantum spin systems in a broad range of quantum Monte Carlo methods. It can be applied both to the ground-state and non-zero temperature cases. The Monte Carlo estimator has a simple yet universal form, which can be efficiently evaluated in simulations. We demonstrate the power of this approach with applications to the Bose-Hubbard model, the spin-$1/2$ XXZ model, and use it to examine the hypothetical intermediate spin-liquid phase in the Hubbard model on the honeycomb lattice.
The inverse problem for a disordered system involves determining the interparticle interaction parameters consistent with a given set of experimental data. Recently, Rutledge has shown (Phys. Rev. E63, 021111 (2001)) that such problems can be generally expressed in terms of a grand canonical ensemble of polydisperse particles. Within this framework, one identifies a polydisperse attribute (`pseudo-species) $sigma$ corresponding to some appropriate generalized coordinate of the system to hand. Associated with this attribute is a composition distribution $barrho(sigma)$ measuring the number of particles of each species. Its form is controlled by a conjugate chemical potential distribution $mu(sigma)$ which plays the role of the requisite interparticle interaction potential. Simulation approaches to the inverse problem involve determining the form of $mu(sigma)$ for which $barrho(sigma)$ matches the available experimental data. The difficulty in doing so is that $mu(sigma)$ is (in general) an unknown {em functional} of $barrho(sigma)$ and must therefore be found by iteration. At high particle densities and for high degrees of polydispersity, strong cross coupling between $mu(sigma)$ and $barrho(sigma)$ renders this process computationally problematic and laborious. Here we describe an efficient and robust {em non-equilibrium} simulation scheme for finding the equilibrium form of $mu[barrho(sigma)]$. The utility of the method is demonstrated by calculating the chemical potential distribution conjugate to a specific log-normal distribution of particle sizes in a polydisperse fluid.
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