Bulletin of the American Physical Society
APS March Meeting 2014
Volume 59, Number 1
Monday–Friday, March 3–7, 2014; Denver, Colorado
Session S27: Quantum Many-Body Systems I |
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Sponsoring Units: DCOMP Chair: Robert DiStasio Jr., Princeton University Room: 501 |
Thursday, March 6, 2014 8:00AM - 8:12AM |
S27.00001: Path Integral Monte Carlo Methods for Fermions Ethan Ethan, Jonathan DuBois, David Ceperley In general, Quantum Monte Carlo methods suffer from a sign problem when simulating fermionic systems. This causes the efficiency of a simulation to decrease exponentially with the number of particles and inverse temperature. To circumvent this issue, a nodal constraint is often implemented, restricting the Monte Carlo procedure from sampling paths that cause the many-body density matrix to change sign. Unfortunately, this high-dimensional nodal surface is not a priori known unless the system is exactly solvable, resulting in uncontrolled errors. We will discuss two possible routes to extend the applicability of finite-temperatue path integral Monte Carlo. First we extend the regime where signful simulations are possible through a novel permutation sampling scheme. Afterwards, we discuss a method to variationally improve the nodal surface by minimizing a free energy during simulation. Applications of these methods will include both free and interacting electron gases, concluding with discussion concerning extension to inhomogeneous systems. [Preview Abstract] |
Thursday, March 6, 2014 8:12AM - 8:24AM |
S27.00002: Exact Dynamics via Poisson Process: a unifying Monte Carlo paradigm James Gubernatis A common computational task is solving a set of ordinary differential equations (o.d.e.'s). A little known theorem says that the solution of any set of o.d.e.'s is exactly solved by the expectation value over a set of arbitary Poisson processes of a particular function of the elements of the matrix that defines the o.d.e.'s. The theorem thus provides a new starting point to develop real and imaginary-time continous-time solvers for quantum Monte Carlo algorithms, and several simple observations enable various quantum Monte Carlo techniques and variance reduction methods to transfer to a new context. I will state the theorem, note a transformation to a very simple computational scheme, and illustrate the use of some techniques from the directed-loop algorithm in context of the wavefunction Monte Carlo method that is used to solve the Lindblad master equation for the dynamics of open quantum systems. I will end by noting that as the theorem does not depend on the source of the o.d.e.'s coming from quantum mechanics, it also enables the transfer of continuous-time methods from quantum Monte Carlo to the simulation of various classical equations of motion heretofore only solved deterministically. [Preview Abstract] |
Thursday, March 6, 2014 8:24AM - 8:36AM |
S27.00003: Algorithmic differentiation of diffusion Monte Carlo Tom Poole, Matthew Foulkes, James Spencer, Peter Haynes Algorithmic differentiation (AD) [1] is a programming technique for the efficient evaluation of the derivatives of a computed function. This approach proceeds via the application of the chain rule to the lines of source code that constitute the mathematical operation of a computer program, allowing access to the derivatives of functions that lack an algebraic representation. Another important element of the AD method is that the ``reverse mode'' of operation yields the derivative of a function output with respect to all inputs, simultaneously, in a small multiple of the computational cost of evaluating the underlying function in isolation. These features make this method particularly applicable to the diffusion Monte Carlo (DMC) algorithm where, despite a number of recent advances in the area, total energy derivatives have remained problematic. Here we present results illustrating accurate DMC energy derivatives with respect to both the input wave function parameters and the nuclear positions, with the former enabling DMC wave function optimization and the latter facilitating DMC molecular dynamics simulations.\\[4pt] [1] A. Griewank and A. Walther, Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, 2nd ed. (SIAM, Philadelphia IL, 2008). [Preview Abstract] |
Thursday, March 6, 2014 8:36AM - 8:48AM |
S27.00004: Advances in the application of diffusion Monte Carlo to solids L. Shulenburger, T.R. Mattsson The need for high fidelity electronic structure calculations has catalyzed an explosion in the development of new techniques. Improvements in DFT functionals, many body perturbation theory and dynamical mean field theory are starting to make significant headway towards reaching the accuracy required for a true predictive capability. One technique that is undergoing a resurgence is diffusion Monte Carlo (DMC). The early calculations with this method were of unquestionable accuracy (providing a valuable reference for DFT functionals) but were largely limited to model systems because of their high computational cost. Algorithmic advances and improvements in computer power have reached the point where this is no longer an insurmountable obstacle. In this talk I will present a broad study of DMC applied to condensed matter (arXiv:1310.1047). We have shown excellent agreement for the bulk modulus and lattice constant of solids exhibiting several different types of binding, including ionic, covalent and van der Waals. We will discuss both the opportunities for application of this method as well as opportunities for further theoretical improvements. [Preview Abstract] |
Thursday, March 6, 2014 8:48AM - 9:00AM |
S27.00005: Diffusion Monte Carlo calculations of solids using transcorrelated trial wave functions Yoshiyuki Yamamoto, Ryo Maezono, Masayuki Ochi, Shinji Tsuneyuki Diffusion Monte Carlo (DMC) method is an \textit{ab initio} wave-function theory that can treat correlated quantum systems to high accuracy within reasonable computational time, and enables us to calculate large systems such as solids. For electronic systems, DMC suffers from the fermion-sign problem, and in order to avoid it we have to use the fixed-node approximation. The amount of fixed-node error depends on the quality of the nodal structure of the trial wave function that we prepare in advance. A promising trial wave function is that of transcorrelated (TC) method, which is one of the wave-function theories. In this method, wave functions are approximated as the Slater-Jastrow form and the orbitals in the Slater determinant are relaxed by solving one-electron equations of similarity-transformed Hamiltonian. The nodal structure of the Slater-Jastrow wave function is determined by its determinantal part, so we can optimize the nodal structure of the Slater-Jastrow wave function by TC method. In this talk, we will present the fixed-node DMC energies of solids using TC trial wave functions and compare the energies with those using trial wave functions constructed from density functional theory and Hartree-Fock method. [Preview Abstract] |
Thursday, March 6, 2014 9:00AM - 9:12AM |
S27.00006: Construction of low energy effective Hamiltonians from Ab Initio Quantum Monte Carlo Hitesh Changlani, Lucas Wagner Solving the first principles quantum many-body Schroedinger equation can result in very high accuracy, but physical insight and generalization of the result can be hard to obtain. On the other hand, low energy effective model Hamiltonians often contain the essential physics of the problem, but may not provide sufficient accuracy needed to understand the properties of real materials. To connect the two approaches, we present a framework for obtaining low energy effective Hamiltonians from ab initio Quantum Monte Carlo calculations for molecular and extended systems. As a demonstration of the method, we focus on a few representative strongly correlated materials. We fit the Hamiltonian parameters to best reproduce the two body density matrix of the ground state obtained from ab initio calculations. We assess the accuracy of the resultant model, by comparing excited state properties to the original ab initio result. Such effective Hamiltonians are advantageous in reducing the computational complexity of the many-electron problem, and once generated, can be used for larger scale calculations using techniques designed for discrete systems. [Preview Abstract] |
Thursday, March 6, 2014 9:12AM - 9:24AM |
S27.00007: Flat histogram diagrammatic Monte Carlo method: Motion of a hole in a columnar antiferromagnet Nikolaos Diamantis, Efstratios Manousakis We will present a version of the diagrammatic Monte Carlo (Diag-MC) method in which we incorporate the flat histogram principle and we term the improved version ``Flat Histogram Diagrammatic Monte Carlo'' method. We demonstrate the superiority of the method over the standard Diag-MC in extracting the long-imaginary-time behavior of the Green's function, without incorporating any {\it a priori} knowledge about this function, by applying the technique to the polaron problem. We have also applied the technique to the motion of a hole inside a $J_1$-$J_2$ quantum antiferromagnet with columnar order. [Preview Abstract] |
Thursday, March 6, 2014 9:24AM - 9:36AM |
S27.00008: Approximations beyond the initiator approach for ameliorating the sign problem Adam Holmes, Cyrus Umrigar, Bryan Clark Full CI Quantum Monte Carlo [1-2] is a computationally expensive method that projects out the ground state in a given basis without resorting to the fixed-node approximation. Instead, it has an initiator bias, which disappears as one approaches the infinite walker limit. While the semistochastic improvement on the method increases the efficiency by about three orders of magnitude [3], converging the ground state energy with respect to the walker population can still require prohibitively many walkers, not only as the number of electrons is increased, but even as the size of the basis is increased for a fixed number of electrons. We therefore investigate other approaches to ameliorating the sign problem, e.g., fixed-node and partial-node approximations [4], and compare the tradeoffs between accuracy and efficiency. [1] George H. Booth, Alex JW Thom and Ali Alavi, J. Chem. Phys. {\bf 131}, 054106 (2009). [2] Cleland, Deidre, George H. Booth and Ali Alavi, J. Chem. Phys. {\bf 132}, 041103 (2010). [3] F. R. Petruzielo, A. A. Holmes, Hitesh J. Changlani, M. P. Nightingale and C. J. Umrigar, Phys. Rev. Lett. {\bf 109}, 30201 (2012). [4] M. Kolodrubetz and B. K. Clark. Phys. Rev. B {\bf 86}, 075109 (2012). [Preview Abstract] |
Thursday, March 6, 2014 9:36AM - 9:48AM |
S27.00009: Spins as variables in electronic structure quantum Monte Carlo calculations Lubos Mitas, Minyi Zhu, Shi Guo Current electronic structure quantum Monte Carlo (QMC) methods keep particle spins static in configurations that correspond to spin-space symmetries of calculated states. Here we present a generalization of the QMC approaches for treating fermionic spin degrees of freedom as variables. The developed method possesses two key properties that make it suitable for high accuracy calculations of real systems. First, the spinors entering the trial function are kept intact during the imaginary time evolution. Second, the approach has the zero variance property pointwise for arbitrary configurations of spatial and appropriately chosen spin coordinates. The spin coordinates are overcomplete and therefore can be smoothly evolved in the imaginary time propagation. The method is illustrated on molecules and atomic excitations of heavy elements with spin-orbit interactions and on 2D electron gas with the Rashba interaction. The performance of the method is similar to the commonly used static spin calculations in several aspects such as achieved accuracy and energy fluctuations. The corresponding wave functions have lower symmetries and therefore can exhibit potentially stronger multi-reference character as is observed in some cases. [Preview Abstract] |
Thursday, March 6, 2014 9:48AM - 10:00AM |
S27.00010: Monte Carlo simulations of two-dimensional fermion models with string bond tensor-network states Jeong-Pil Song, R.T. Clay We present computational results using the string-bond tensor network ansatz for Fermionic lattice models in two dimensions. We use quantum Monte Carlo to calculate ground state quantities combined with stochastic optimization to optimize the matrix elements of matrix-product state strings. We apply the approach to a two-dimensional spinless fermion model with nearest-neighbor Coulomb repulsion under periodic boundary conditions. We test the numerical accuracy and convergence with matrix size D of the method with comparisons with the free fermion system, exact diagonalization results, and density matrix renormalization group results. The phase boundary between low entangled charge ordered and highly entangled metallic phases can be determined using finite size scaling of charge structure factor in the thermodynamic limit. Since this stochastic approach does not suffer from a fermion sign problem, it can handle frustrations and be applied to the Hubbard models with periodic boundaries in two dimensions. [Preview Abstract] |
Thursday, March 6, 2014 10:00AM - 10:12AM |
S27.00011: Symmetry-projected Hartree-Fock wave functions in quantum Monte Carlo calculations Hao Shi, Carlos Hoyos, Rayner Rodriguez, Gustavo Scuseria, Shiwei Zhang Symmetry-projected Hartree-Fock wave functions provide an ansatz which accounts for static correlations while preserving symmetry. We implement such wave functions in constrained path (CP) auxiliary-field quantum Monte Carlo (AFQMC) calculations as the trial wave function. Unlike usual multi-determinant trial wave functions obtained from a configuration interaction picture, the computational cost of this class of trial wave functions can be made to scale as a low power with system size. A systematic test is carried out in the two-dimensional Hubbard model on the accuracy of the approach. It is found that the accuracy of the calculated ground-state energy increases as more symmetries are restored, while the statistical variance decreases. We find that wave functions with space group and spin symmetry significantly reduce the CP systematic error in AFQMC compared to simple Hartree-Fock trial wave functions. Essentially all the correlation energy is recovered by the AFQMC when the fully symmetry-projected trial wave function is used. Correlation functions are accurately predicted. [Preview Abstract] |
Thursday, March 6, 2014 10:12AM - 10:24AM |
S27.00012: Downfolding calculations in solids by auxiliary-field quantum Monte Carlo Fengjie Ma, Wirawan Purwanto, Shiwei Zhang, Henry Krakauer We present a recent development in {\it ab initio} auxiliary-field quantum Monte Carlo (AFQMC) \footnote{S.~\ Zhang and H.~\ Krakauer, Phys. Rev. Lett. {\bf 90}, 136401 (2003)} calculations of solid systems using downfolded Hamiltonians. For a given system, the many-body downfolded Hamiltonian is expressed with respect to a truncated basis set of Kohn-Sham orbitals, which are obtained from a high-quality density-functional calculation. This approach allows many-body calculations to treat a much simpler Hamiltonian while retaining material-specific properties. Typical size of the basis set is more than an order of magnitude smaller than the original (the number of plane-waves), leading to large savings in AFQMC computation. The Hamiltonians are systematically improvable and allow one to dial, in principle, between the simplest model and the full Hamiltonian. As a by-product of this approach, pseudopotential errors can essentially be eliminated \footnote{W.~\ Purwanto, S.~\ Zhang, H.~\ Krakauer, J. Chem. Theory Comput. {\bf 9}, 4825 (2013)}. The method is demonstrated by calculating the lattice constant and bulk modulus of solids, including classic semiconductors (Si and diamond), an ionic insulator (NaCl), and metallic systems (Na and Al). [Preview Abstract] |
Thursday, March 6, 2014 10:24AM - 10:36AM |
S27.00013: Possibility of Deconfined Criticality in SU($N$) Heisenberg Models at Small $N$ Kenji Harada, Takafumi Suzuki, Tsuyoshi Okubo, Haruhiko Matsuo, Jie Lou, Hiroshi Watanabe, Synge Todo, Naoki Kawashima To examine the validity of the scenario of the deconfined critical phenomena[1], we carry out quantum Monte Carlo simulation for the SU($N$) generalization of the Heisenberg model with four-body and six-body interactions[2]. The quantum phase transition between the SU($N$) N\'eel and valence-bond solid phases is characterized for $N=2,3,$ and $4$ on the square and honeycomb lattices. While finite-size scaling analysis works well up to the maximum lattice size ($L=256$) and indicates the continuous nature of the phase transition, a clear systematic change towards the first-order transition is observed in the estimates of the critical exponent $y \equiv 1/\nu$ as the system size increases. We discuss the details of finite-size scaling analysis. [1] T. Senthil, A. Vishwanath, L. Balentz, S. Sachdev and M.P.A. Fisher, Science 303 (2004). [2] K. Harada, T. Suzuki, T. Okubo, H. Matsuo, J. Lou, H. Watanabe, S. Todo, and N. Kawashima, arXiv:1307.0501. [Preview Abstract] |
Thursday, March 6, 2014 10:36AM - 10:48AM |
S27.00014: Phase diagram of easy-plane deformations of SU(N) magnets Jonathan Demidio, Ribhu K. Kaul We consider Hamiltonians of SU($N$) quantum magnets with easy-plane deformations, leaving a U(1) rotation symmetry about each of the $N-1$ diagonal generators and a discrete $Z_{N}$ symmetry. For $N=2$ our model reduces to the XY model and can hence be considered as a larger-$N$ generalization of this well-studied model. We present numerical data from quantum Monte Carlo simulations which allows us to map the phase diagram of these models as a function of $N$, including both two-spin and four-spin interactions. [Preview Abstract] |
Thursday, March 6, 2014 10:48AM - 11:00AM |
S27.00015: Quantum Monte Carlo Gap Analysis of Neel-VBS Phase Transition Hidemaro Suwa, Anders Sandvik We have developed a generalized moment method for calculating excitation gaps in finite-temperature and ground-state projector quantum Monte Carlo simulations. We show analytically that this estimator is unbiased in the low-temperature or long-projection-length limit. Not only the first gap but also the second gap for each quantum number can be calculated without any systematic errors. As a demonstration, we have applied this approach to the Neel--valence-bond-solid transition of the two-dimensional J-Q spin model. The transition point was successfully obtained from the singlet-triplet level crossing. Interestingly, the size-scaling of the crossing point is the same as in the one-dimensional case ($1/L^2$, $L$ being the system length). [Preview Abstract] |
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