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arXiv:2309.05166 · Physical review. B./Physical review. B

Quantum Monte Carlo algorithm for Bose-Hubbard models on arbitrary graphs

E. Akaturk, Itay Hen

Published Apr 25, 2024• 6 citations
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Abstract

We propose a quantum Monte Carlo algorithm capable of simulating the Bose-Hubbard model on arbitrary graphs, obviating the need for devising lattice-specific updates for different input graphs. We show that with our method, which is based on the recently introduced permutation matrix representation quantum Monte Carlo [Gupta, Albash, and Hen, J. Stat. Mech. (2020) 073105], the problem of adapting the simulation to a given geometry amounts to generating a cycle basis for the graph on which the model is defined, a procedure that can be carried out efficiently and in an automated manner. To showcase the versatility of our approach, we provide simulation results for Bose-Hubbard models defined on two-dimensional lattices as well as on a number of random graphs.

DOI
10.1103/physrevb.109.134519

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We propose a quantum Monte Carlo algorithm capable of simulating the Bose-Hubbard model on arbitrary graphs, obviating the need for devising lattice-specific updates for different input graphs. We show that with our method, which is based on the recently introduced permutation matrix representation quantum Monte Carlo [Gupta, Albash, and Hen, J. Stat. Mech. (2020) 073105], the problem of adapting the simulation to a given geometry amounts to generating a cycle basis for the graph on which the model is defined, a procedure that can be carried out efficiently and in an automated manner. To showcase the versatility of our approach, we provide simulation results for Bose-Hubbard models defined on two-dimensional lattices as well as on a number of random graphs.
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We present a universal parameter-free quantum Monte Carlo (QMC) algorithm designed to simulate arbitrary spin- 1/2 Hamiltonians. To ensure the convergence of the Markov chain to equilibrium for every conceivable case, we devise a clear and simple automated protocol that produces QMC updates that are provably ergodic and satisfy detailed balance. We demonstrate the applicability and versatility of our method by considering several illustrative examples, including the simulation of the XY model on a triangular lattice, the toric code, and random k -local Hamiltonians. Published by the American Physical Society 2024
Abstract We present a quantum Monte Carlo algorithm for the simulation of general quantum and classical many-body models within a single unifying framework. The algorithm builds on a power series expansion of the quantum partition function in its off-diagonal terms and is both parameter-free and Trotter error-free. In our approach, the quantum dimension consists of products of elements of a permutation group. As such, it allows for the study of a very wide variety of models on an equal footing. To demonstrate the utility of our technique, we use it to clarify the emergence of the sign problem in the simulations of non-stoquastic physical models. We showcase the flexibility of our algorithm and the advantages it offers over existing state-of-the-art by simulating transverse-field Ising model Hamiltonians and comparing the performance of our technique against that of the stochastic series expansion algorithm. We also study a transverse-field Ising model augmented with randomly chosen two-body transverse-field interactions.
We propose algorithms of the path-ingetral-based quantum Monte Carlo simulation, which is otherwise prohibitively slow. While the basic idea is the loop-cluster update, there are some important 'tricks' that are vital to make the simulation practical. In the present paper, we show two such techniques and their successful applications to the two-dimensional SU( N ) Heisenberg model and the three-dimensional Bose Hubbard model. In the former, we obtain a new type of the valence-bond-solid state for the two-boson representation, while in the latter we equillibrate a system of which the size is comparable to a typical experiment of optical lattices.
We extend a Quantum Monte Carlo (QMC) algorithm to the Hubbard model containing bond-dependent off-diagonal hoppings, which describes the manganites and other similar systems, such as systems with spin-flip hoppings. To show correctness of this extension, we detect two simple examples of one-dimensional chains. The first one is half-filled and the hopping matrixes between arbitrary nearest neighboring pairs are uniform, i.e., ti,i+1=ti+1,i+2{t}_{i,i+1}{=t}_{i+1,i+2}ti,i+1​=ti+1,i+2​ (i=1,2,3,…).(i=1,2,3,\dots{}).(i=1,2,3,…). The other one is quarter-filled and the hopping matrixes alternate every four lattice constants with ${t}_{i,i+1}{=t}_{i+1,i+2}\ensuremath{\ne}{t}_{i+2,i+3}{=t}_{i+3,i+4}$ (i=1,5,9,…).(i=1,5,9,\dots{}).(i=1,5,9,…). Results of the two examples are reasonable.
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.