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Civil War Book Review

Study of Condensed Matter Systems with Monte Carlo Simulation on Heterogeneous Computing Systems

Sheng Feng

Published Apr 7, 2016• 0 citations
Publisher

Abstract

We study the Edwards-Anderson model on a simple cubic lattice with a finite constant external field. We employ an indicator composed of a ratio of susceptibilities at finite momenta, which was recently proposed to avoid the difficulties of a zero momentum quantity, for capturing the spin glass phase transition. Unfortunately, this new indicator is fairly noisy, so a large pool of samples at low temperature and small external field are needed to generate results with a sufficiently small statistical error for analysis. We thus implement the Monte Carlo method using graphics processing units to drastically speed up the simulation. We confirm previous findings that conventional indicators for the spin glass transition, including the Binder ratio and the correlation length do not show any indication of a transition for rather low temperatures. However, the ratio of spin glass susceptibilities does show crossing behavior, albeit a systematic analysis is beyond the reach of the present data. This reveals the difficulty with current numerical methods and computing capability in studying this problem. One of the fundamental challenges of theoretical condensed matter physics is the accurate solution of quantum impurity models. By taking expansion in the hybridization about an exactly solved local limit, one can formulate a quantum impurity solver. We implement the hybridization expansion quantum impurity solver on Intel Xeon Phi accelerators, and aim to apply this approach on the Dynamic Hubbard Models.

DOI
10.31390/gradschool_dissertations.3741

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•Jan 1, 2017•0 citations
We study the Edwards-Anderson model on a simple cubic lattice with a finite constant external field. We employ an indicator composed of a ratio of susceptibilities at finite momenta, which was recently proposed to avoid the difficulties of a zero momentum quantity, for capturing the spin glass phase transition. Unfortunately, this new indicator is fairly noisy, so a large pool of samples at low temperature and small external field are needed to generate results with a sufficiently small statistical error for analysis. We thus implement the Monte Carlo method using graphics processing units to drastically speed up the simulation. We confirm previous findings that conventional indicators for the spin glass transition, including the Binder ratio and the correlation length do not show any indication of a transition for rather low temperatures. However, the ratio of spin glass susceptibilities does show crossing behavior, albeit a systematic analysis is beyond the reach of the present data. This reveals the difficulty with current numerical methods and computing capability in studying this problem. One of the fundamental challenges of theoretical condensed matter physics is the accurate solution of quantum impurity models. By taking expansion in the hybridization about an exactly solved local limit, one can formulate a quantum impurity solver. We implement the hybridization expansion quantum impurity solver on Intel Xeon Phi accelerators, and aim to apply this approach on the Dynamic Hubbard Models.
A spin-glass transition has been investigated for a long time but we have not reached a conclusion yet due to difficulties in the simulation studies. They are slow dynamics, strong finite-size effects, and sample-to-sample dependencies. We found that a size of the spin-glass order reaches a lattice boundary within a very short Monte Carlo step. A competition between the spin-glass order and a boundary condition causes these difficulties. Once the boundary effect was removed, physical quantities exhibited quite normal behaviors. They became self-averaging in a limit of large replica numbers. These findings suggest that the nonequilibrium relaxation method is a good choice for solving the difficulties if a lattice size and a replica number are set sufficiently large. A dynamic scaling analysis on nonequilibrium relaxation functions gave a result that the spin-glass transition and the chiral-glass transition occurs at the same temperature in the Heisenberg model in three dimensions. The estimated critical exponent ν agrees with the experimental result.
We study the nonequilibrium aging behavior of the $\ifmmode\pm\else\textpm\fi{}J$ Edwards-Anderson model in three dimensions for samples of size up to N=1283N={128}^{3}N=1283 and for up to 108{10}^{8}108 Monte Carlo sweeps. In particular we are interested in the change of the aging when crossing from the spin-glass phase to the ferromagnetic phase. The necessary long simulation times are reached by employing a CUDA-based GPU implementation, which allows for single-spin flip times as small as 8 ps. We measure typical spin-glass correlation functions in space and time to determine the growing length scale and extract the constituting exponents. We observe a clear signature of the disorder-driven equilibrium transition in the nonequilibrium behavior.
The study of the low temperature phase of spin glass models by means of Monte Carlo simulations is a challenging task, because of the very slow dynamics and the severe finite-size effects they show. By exploiting at the best the capabilities of standard modern CPUs (especially the streaming single instruction, multiple data extensions), we have been able to simulate the four-dimensional Edwards-Anderson model with Gaussian couplings up to sizes L=70 and for times long enough to accurately measure the asymptotic behavior. By quenching systems of different sizes to the critical temperature and to temperatures in the whole low temperature phase, we have been able to identify the regime where finite-size effects are negligible: ξ(t)≲L/7. Our estimates for the dynamical exponent (z≃1/T) and for the replicon exponent (α≃1.0 and T independent), that controls the decay of the spatial correlation in the zero overlap sector, are consistent with the replica symmetry breaking theory, but the latter differs from the theoretically conjectured value.
We present, in this dissertation, a numerical simulation method to study interacting fermion systems. The general simulation procedures are discussed in connection with a description of a Quantum Monte Carlo simulation algorithm for interacting electrons on lattices. The algorithm presented here has been used to simulate interacting electrons on lattices, and it makes possible the study of substantially larger systems than can be studied by other numerical methods. As long as certain limits of applicability are respected, model Hamiltonians of interacting electrons can be studied without resort to uncontrolled approximations. The method then provides nearly exact solutions to model Hamiltonians of many-body systems, in the sense that the degree of error can be controlled. We also discuss some results obtained from simulations of the extended Hubbard model and and the periodic Anderson model. For the extended Hubbard model in two dimensions, different regions are identified where electron correlation produces antiferromagnetism, charge-density-waves, and singlet pairing superconducting behaviors. We also find regions where transitions from one type of correlation to another occur. For the symmetric periodic Anderson model, we observe the formation of local spin moments at high temperatures and their quenching at low temperatures, as in the single impurity Anderson model. In addition, we find antiferromagnetic interaction between f local moments at low temperatures which are not present in a single impurity problem.
Spin glasses have been the subject of intense study and considerable controversy for decades, and the low-temperature phase of short-range spin glasses is still poorly understood. Our main goal is to improve our understanding in this area and find an answer to the following question: Are there only a single pair or a countable infinity of pure states in the low temperature phase of the EA spin glass? To that aim we first start by introducing spin glasses and provide a brief history of their research, then proceed to describe our method of simulation, the parallel tempering Monte Carlo algorithm. Next, we present the results of a large-scale numerical study of the equilibrium three-dimensional Edwards-Anderson Ising spin glass with Gaussian disorder. In order to understand how the parallel tempering algorithm works, we measure various static, as well as dynamical quantities, such as the autocorrelation times and round-trip times for the parallel tempering Monte Carlo method. We examine the correlation between static and dynamic observables for _ 5000 disorder realizations and up to 1000 spins down to temperatures at 20% of the critical temperature, and our results show that autocorrelation times are directly correlated with the roughness of the free-energy landscape. In the following chapters, the three- and four-dimensional Edwards-Anderson and mean-field Sherrington-Kirkpatrick Ising spin glasses are studied again via large-scale Monte Carlo simulations at low temperatures, deep within the spinglass phase. Performing a careful statistical analysis of several thousand independent disorder realizations and using an observable that detects peaks in the overlap distribution, we show that the Sherrington-Kirkpatrick and Edwards-Anderson models have a distinctly different low-temperature behavior. We arrive to the following conclusion: The structure of the spin-glass overlap distribution for the Edwards-Anderson model suggests that its low-temperature phase has only a single pair of pure states. Finally we present results for several new observables, along with a few preliminary studies and suggestions for future research.
This work is motivated by the fact that the investigation of non-equilibrium phenomena in strongly correlated electron systems has developed into one of the most active and exciting branches of condensed matter physics as it provides rich new insights that could not be obtained from the study of equilibrium situations. However, a theoretical description of those phenomena is missing. Therefore, in this thesis, we develop a numerical method that can be used to study two minimal models -- the Hubbard model and the Anderson impurity model with general parameter range and time dependence. We begin by introducing the theoretical framework and the general features of the Hubbard model. We then describe the dynamical mean field theory (DMFT), which was first invented by Georges in 1992. It provides a feasible way to approach strongly correlated electron systems and reduces the complexity of the calculations via a mapping of lattice models onto quantum impurity models subject to a self-consistency condition. We employ the non-equilibrium extension of DMFT and map the Hubbard model to the single impurity Anderson model (SIAM). Since the fundamental component of the DMFT method is a solver of the single impurity Anderson model, we continue with a description of the formalism to study the real-time dynamics of the impurity model staring at its thermal equilibrium state. We utilize the non-equilibrium strong-coupling perturbation theory and derive semi-analytical approximation methods such as the non-crossing approximation (NCA) and the one-crossing approximation (OCA). We then use the Quantum Monte-Carlo method (QMC) as a numerically exact method and present proper measurements of local observables, current and Green's functions. We perform simulations of the current after a quantum quench from equilibrium by rapidly applying a bias voltage in a wide range of initial temperatures. The current exhibits short equilibrium times and saturates upon the decrease of temperature at all times, indicating Kondo behavior both in the transient regime and in the steady state. However, this bare QMC solver suffers from a dynamical sign problem for long time propagations. To overcome the limitations of this bare treatment, we introduce the ``Inchworm algorithm'', based on iteratively reusing the information obtained in previous steps to extend the propagation to longer times and stabilize the calculations. We show that this algorithm greatly reduces the required order for each simulation and re-scales the exponential challenge to quadratic in time. We introduce a method to compute Green's functions, spectral functions, and currents for inchworm Monte Carlo and show how systematic error assessments in real time can be obtained. We illustrate the capabilities of the algorithm with a study of the behavior of quantum impurities after an instantaneous voltage quench from a thermal equilibrium state. We conclude with the applications of the unbiased inchworm impurity solver to DMFT calculations. We employ the methods for a study of the one-band paramagnetic Hubbard model on the Bethe lattice in equilibrium, where the DMFT approximation becomes exact. We begin with a brief introduction of the Mott metal insulator phase diagram. We present the results of both real time Green's functions and spectral functions from our nonequilibrium calculations. We observe the metal-insulator crossover as the on-site interaction is increased and the formation of a quasi-particle peak as the temperature is lowered. We also illustrate the convergence of our algorithms in different aspects.