scholarly journals Efficient calculation of imaginary-time-displaced correlation functions in the projector auxiliary-field quantum Monte Carlo algorithm

2001 ◽  
Vol 63 (7) ◽  
Author(s):  
M. Feldbacher ◽  
F. F. Assaad
Keyword(s):  
Monte Carlo ◽  
Correlation Functions ◽  
Quantum Monte Carlo ◽  
Monte Carlo Algorithm ◽  
Auxiliary Field ◽  
Imaginary Time ◽  
Efficient Calculation

scholarly journals AUXILIARY FIELDS AND THE SIGN PROBLEM

1995 ◽  
Vol 06 (03) ◽  
pp. 427-465 ◽  
Author(s):  
J.H. SAMSON
Keyword(s):  
Monte Carlo ◽  
Correlation Functions ◽  
Quantum Monte Carlo ◽  
Heisenberg Model ◽  
Functional Integral ◽  
Auxiliary Field ◽  
Operator Product ◽  
Phase Problem ◽  
Stochastic Field ◽  
Sign Problem

The auxiliary-field quantum Monte Carlo method is reviewed. The Hubbard-Stratonovich transformation converts an interacting Hamiltonian into a non-interacting Hamiltonian in a time-dependent stochastic field, allowing calculation of the resulting functional integral by Monte Carlo methods. The method is presented in a sufficiently general form to be applicable to any Hamiltonian with one- and two-body terms, with special reference to the Heisenberg model and one- and many-band Hubbard models. Many physical correlation functions can be related to correlation functions of the auxiliary field; general results are given here. Issues relating to the choice of auxiliary fields are addressed; operator product identities change the relative dimensionalities of the attractive and repulsive parts of the interaction. Frequently the integrand is not positive-definite, rendering numerical evaluation unstable. If the auxiliary field violates time-reversal invariance, the integrand is complex and this sign problem becomes a phase problem. The origin of this sign or phase is examined from a number of geometrical and other viewpoints and illustrated by simple examples: the phase problem by the spin (1/2) Heisenberg model, and the sign problem by the attractive SU(N) Hubbard model on a triangular molecule with negative hopping integrals. In the latter case, widely studied in the Jahn-Teller literature, the sign is due neither to fermions nor spin, but to frustration. This system is used to illustrate a number of suggested interpretations of the sign problem.

NEGATIVE WEIGHTS IN QUANTUM MONTE CARLO SIMULATIONS AT FINITE-TEMPERATURES USING THE AUXILIARY FIELD METHOD

1994 ◽  
Vol 05 (03) ◽  
pp. 599-613 ◽  
Author(s):  
J.E. GUBERNATIS ◽  
X.Y. ZHANG
Keyword(s):  
Monte Carlo ◽  
Quantum Monte Carlo ◽  
Topological Defects ◽  
Field Method ◽  
Monte Carlo Algorithm ◽  
Auxiliary Field ◽  
Sign Problem ◽  
Time Patterns ◽  
Ill Conditioned Matrices ◽  
Quantum Monte Carlo Simulations

We study the conditions under which negative weights (the sign problem) can exist in the finite-temperature, auxiliary field, quantum Monte Carlo algorithm of Blankenbecler, Scalapino, and Sugar. We specifically consider whether the sign problem arises from round-off error resulting from operations involving very ill-conditioned matrices or from topological defects in the auxiliary fields mirroring the space-time patterns of the physical fields. While we demonstrate these situations can generate negative weights, the results of our numerical tests suggest that these factors are most likely not the dominant sources of the problem. We also argue that the negative weights should not be considered as just a fermion problem. If it exists for the fermion problem, it will also exist for an analogous boson problem.

scholarly journals Imaginary time correlations and the phaseless auxiliary field quantum Monte Carlo

2014 ◽  
Vol 140 (2) ◽  
pp. 024107 ◽  
Author(s):  
M. Motta ◽  
D. E. Galli ◽  
S. Moroni ◽  
E. Vitali
Keyword(s):  
Monte Carlo ◽  
Quantum Monte Carlo ◽  
Auxiliary Field ◽  
Imaginary Time ◽  
Time Correlations

scholarly journals The ALF (Algorithms for Lattice Fermions) project release 1.0. Documentation for the auxiliary field quantum Monte Carlo code

2017 ◽  
Vol 3 (2) ◽  
Author(s):  
Martin Bercx ◽  
Florian Goth ◽  
Johannes Stephan Hofmann ◽  
Fakher Assaad
Keyword(s):  
Monte Carlo ◽  
Hubbard Model ◽  
Quantum Monte Carlo ◽  
Kondo Lattice ◽  
Monte Carlo Algorithm ◽  
Auxiliary Field ◽  
Honeycomb Lattice ◽  
Monte Carlo Code ◽  
Single Body ◽  
Lattice Fermions

The Algorithms for Lattice Fermions package provides a general code for the finite temperature auxiliary field quantum Monte Carlo algorithm. The code is engineered to be able to simulate any model that can be written in terms of sums of single-body operators, of squares of single-body operators and single-body operators coupled to an Ising field with given dynamics. We provide predefined types that allow the user to specify the model, the Bravais lattice as well as equal time and time displaced observables. The code supports an MPI implementation. Examples such as the Hubbard model on the honeycomb lattice and the Hubbard model on the square lattice coupled to a transverse Ising field are provided and discussed in the documentation. We furthermore discuss how to use the package to implement the Kondo lattice model and the SU(N)SU(N)-Hubbard-Heisenberg model. One can download the code from our Git instance at and sign in to file issues.

scholarly journals Quantum adiabatic algorithms, small gaps, and different paths

2011 ◽  
Vol 11 (3&4) ◽  
pp. 181-214
Author(s):  
Edward Farhi ◽  
Jeffrey Goldstone ◽  
David Gosset ◽  
Sam Gutmann ◽  
Harvey B. Meyer ◽  
...  
Keyword(s):  
Monte Carlo ◽  
Excited State ◽  
Ground State ◽  
Quantum Monte Carlo ◽  
Monte Carlo Algorithm ◽  
Imaginary Time ◽  
Monte Carlo Data ◽  
First Excited State ◽  
Time Quantum

We construct a set of instances of 3SAT which are not solved efficiently using the simplest quantum adiabatic algorithm. These instances are obtained by picking random clauses all consistent with two disparate planted solutions and then penalizing one of them with a single additional clause. We argue that by randomly modifying the beginning Hamiltonian, one obtains (with substantial probability) an adiabatic path that removes this difficulty. This suggests that the quantum adiabatic algorithm should in general be run on each instance with many different random paths leading to the problem Hamiltonian. We do not know whether this trick will help for a random instance of 3SAT (as opposed to an instance from the particular set we consider), especially if the instance has an exponential number of disparate assignments that violate few clauses. We use a continuous imaginary time Quantum Monte Carlo algorithm in a novel way to numerically investigate the ground state as well as the first excited state of our system. Our arguments are supplemented by Quantum Monte Carlo data from simulations with up to 150 spins.

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