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Chinese Physics B

Auxiliary-field Monte Carlo method for frustrated spin systems

Ning Cai, Yuan 源 Gao 高, Wei Li, Yang Qi

Published Jan 1, 2025• 0 citations
Publisher

Abstract

Abstract We extend a semiclassical numerical method, bosonic auxiliary-field Monte Carlo, to quantum spin systems. This method breaks the lattice into clusters, solves each cluster precisely and couples them with classical auxiliary fields through classical Monte Carlo simulation. We test the method with antiferromagnetic spin models in one-dimensional chains, square lattices and triangular lattices, and obtain reasonable results at finite temperatures. This algorithm builds a bridge between classical Monte Carlo method and quantum methods. The algorithm can be improved with either progress in classical Monte Carlo sampling or the development of quantum solvers, and can also be further applied to systems with different lattices or interactions.

DOI
10.1088/1674-1056/ada758

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Abstract We extend a semiclassical numerical method, bosonic auxiliary-field Monte Carlo, to quantum spin systems. This method breaks the lattice into clusters, solves each cluster precisely and couples them with classical auxiliary fields through classical Monte Carlo simulation. We test the method with antiferromagnetic spin models in one-dimensional chains, square lattices and triangular lattices, and obtain reasonable results at finite temperatures. This algorithm builds a bridge between classical Monte Carlo method and quantum methods. The algorithm can be improved with either progress in classical Monte Carlo sampling or the development of quantum solvers, and can also be further applied to systems with different lattices or interactions.
We introduce a new numerical technique, the bosonic auxiliary-field Monte Carlo method, which allows us to calculate the thermal properties of large lattice-boson systems within a systematically improvable semiclassical approach, and which is virtually applicable to any bosonic model. Our method amounts to a decomposition of the lattice into clusters, and to an ansatz for the density matrix of the system in the form of a cluster-separable state-with nonentangled, yet classically correlated clusters. This approximation eliminates any sign problem, and can be systematically improved upon by using clusters of growing size. Extrapolation in the cluster size allows us to reproduce numerically exact results for the superfluid transition of hard-core bosons on the square lattice, and to provide a solid quantitative prediction for the superfluid and chiral transition of hardcore bosons on the frustrated triangular lattice.
We develop an efficient Monte Carlo algorithm, which accelerates slow Monte Carlo dynamics in quasi-one-dimensional Ising spin systems. The loop algorithm of the quantum Monte Carlo method is applied to the classical spin models with highly anisotropic exchange interactions. Both correlation time and real CPU time are reduced drastically. The algorithm is demonstrated in the layered triangular-lattice antiferromagnetic Ising model. We have obtained the relation between the transition temperature and the exchange interaction parameters, which modifies the result of the chain-mean-field theory.
Frustrated spin systems generically suffer from the negative sign problem inherent in Monte Carlo methods. Since the severity of this problem is formulation dependent, optimization strategies can be put forward. We introduce a phase pinning approach in the realm of the auxiliary field quantum Monte Carlo algorithm. If we can find an antiunitary operator that commutes with the one-body Hamiltonian coupled to the auxiliary field, then the phase of the action is pinned to 0 and $\ensuremath{\pi}$. For generalized Kitaev models, we can successfully apply this strategy and observe a remarkable improvement of the average sign. We use this method to study the thermodynamical and dynamical properties of the Kitaev-Heisenberg model down to temperatures corresponding to half of the exchange coupling constant. Our dynamical data reveal finite temperature properties of ordered and spin-liquid phases inherent in this model.
The sign problem poses a fundamental barrier to numerically exact quantum Monte Carlo simulations, severely hindering our understanding of quantum many-body physics across disciplines ranging from condensed matter to particle and nuclear physics. Geometrical frustration makes this barrier especially acute for interacting bosons. Here, we develop a bosonic auxiliary-field quantum Monte Carlo (BAFQMC) method and rigorously prove that time-reversal symmetry and reflection positivity each provide a general sufficient condition for sign-problem-free simulations. Crucially, under either criterion, BAFQMC evades the sign problem even on frustrated lattices where conventional world-line and stochastic-series-expansion methods fail. Benchmarking our approach on a two-flavor Bose-Hubbard model on the triangular lattice yields excellent quantitative agreement with finite-temperature exact diagonalization. By establishing unifying principles to solve the sign problem in BAFQMC, our work substantially expands the reach of unbiased, numerically exact simulations for frustrated bosonic matter.
Cold atoms in optical lattices offer unprecedented control over strongly correlatedmany-body states. For this reason they represent an excellent tool for the implementation ofa “quantum simulator”, which can be used to realize experimentally several Hamiltonians ofsystems of physical interest. In particular, they enable the engineering of artificial gaugefields, which gives access to the physics of frustrated magnetism. In this work, we study thethermodynamics of cold atoms both from a theoretical and a numerical point of view. Atpresent days, the most effective method used in this field is the quantum Monte Carlo. Butbecause of the so-called “sign problem” it can only be applied to a limited class of systems,which for example do not include frustrated systems. The interest of this thesis is to developof a new approximated method based on a Monte Carlo approach. The first part of this workis dedicated to theoretical considerations concerning the spatial structure of quantum andclassical correlations. These results permit to develop, in the second part, an approximationcalled quantum mean-field. This latter allows to propose, in the third part, a numericalmethod that we call “auxiliary-field Monte Carlo” and that we apply to some systems ofphysical interest, among which the frustrated triangular lattice.
Working within the stochastic series expansion framework, we introduce and characterize a plaquette-based quantum cluster algorithm for quantum Monte Carlo simulations of transverse field Ising models with frustrated Ising exchange interactions. As a demonstration of the capabilities of this algorithm, we show that a relatively small ferromagnetic next-nearest-neighbor coupling drives the transverse field Ising antiferromagnet on the triangular lattice from an antiferromagnetic three-sublattice ordered state at low temperature to a ferrimagnetic three-sublattice ordered state.