We examine three different quantum Monte Carlo methods for studying systems of interacting particles. The determinantal quantum Monte Carlo method is compared to two different worldline simulations. The first worldline method consists of a simulation carried out in the real-space basis, while the second method is implemented using as basis the eigenstates of the Hamiltonian on blocks of the two-dimensional lattice. We look, in particular, at the Hubbard model on a 4\ifmmode\times\else\texttimes\fi{}4 lattice with periodic boundary conditions. The block method is superior to the real-space method in terms of the computational cost of the simulation, but shows a much worse negative sign problem. For larger values of U and away from half-filling it is found that the real-space method can provide results at lower temperatures than the determinantal method. We show that the sign problem in the block method can be slightly improved by an appropriate choice of basis.
We examine three different quantum Monte Carlo methods for studying systems of interacting particles. The determinantal quantum Monte Carlo method is compared to two different worldline simulations. The first worldline method consists of a simulation carried out in the real-space basis, while the second method is implemented using as basis the eigenstates of the Hamiltonian on blocks of the two-dimensional lattice. We look, in particular, at the Hubbard model on a 4\ifmmode\times\else\texttimes\fi{}4 lattice with periodic boundary conditions. The block method is superior to the real-space method in terms of the computational cost of the simulation, but shows a much worse negative sign problem. For larger values of U and away from half-filling it is found that the real-space method can provide results at lower temperatures than the determinantal method. We show that the sign problem in the block method can be slightly improved by an appropriate choice of basis.
Sign problem in quantum Monte Carlo (QMC) simulation appears to be an extremely hard yet interesting problem. In this article, we present a pedagogical overview on the origin of the sign problem in various quantum Monte Carlo simulation techniques, ranging from the world-line and stochastic series expansion Monte Carlo for boson and spin systems to the determinant and momentum-space quantum Monte Carlo for interacting fermions. We point out the basis dependency of the sign problem and summarize the progresses to cure, ease and even make use of the sign problem over the years, such as symmetry analysis of the underlying Hamiltonian, basis optimization in writting down the partition functions and many others. Moreover, we state that although traditional lore saying that in case of sign problem, the average sign in QMC simulation approaches zero exponentially fast with the space-time volume of the configurational space, there are recent breakthroughs showing this is not always the case and based on the properties of the partition function for finite size systems, one could distinguish when the average sign has the usual exponential scaling and when it is bestowed with an algebraic scaling at the low temperature limit. Fermionic QMC simulations with such algebraic sign problems have been successfully carried out for extended Hubbard-type and quantum Moiré lattice models.
Many strongly correlated quantum systems are difficult to study numerically because of the exponential growth of the Hilbert space combined with the failure of numerical methods in important parameter regimes. This includes the long-standing problems of high-temperature superconductivity and frustrated magnetism. A promising solution to this dilemma are quantum simulators, well-controlled (synthetic) quantum systems, emulating model Hamiltonians not only of condensed matter systems, where parameters can be tuned at will and model complexity can be added step by step. This dissertation presents numerical simulation studies that are motivated by quantum simulators. Using large-scale quantum Monte Carlo computer simulations and mean-field theory the semiquantitative ground state phase diagram of a Penning ion trap quantum simulator, realizing a two-dimensional model of quantum magnetism with long-range interactions, is obtained. We study also thermal and quantum phase transitions in one-dimensional transverse-field Ising models with long-range interactions, which exhibit a Kosterlitz-Thouless transition and universality classes with continuously varying critical exponents. The second part of this thesis is concerned with the Fermi-Hubbard model, which has been realized recently with ultracold atoms in optical lattices, both for repulsive and attractive interactions. Equipped with a quantum gas microscope, which allows single-site single-atom detection, these experimental setups can collect histograms of the particle number from repeated projective measurements. This dissertation presents a method to compute such full counting statistics for quadratic operators such as particle number or magnetization in the framework of determinantal quantum Monte Carlo. From an analysis of the full counting statistics of the particle number for different subsystem sizes the size of preformed pairs or Cooper pairs in the attractive Hubbard model is inferred. Furthermore, excellent agreement of the numerically computed probability distribution of the staggered magnetization with a recent quantum gas experiment demonstrates that current experiments are capable of resolving the differences between the Hubbard model and its limiting Heisenberg model. An extensive method section reviews the stochastic series expansion quantum Monte Carlo method for Ising models in a transverse field, which is free of the sign problem for arbitrary interactions, with special emphasis on the efficient simulation of long-range interactions. Also a self-contained introduction to the determinantal quantum Monte Carlo method is provided.
We tutorially review the determinantal Quantum Monte Carlo method for fermionic systems, using the Hubbard model as a case study. Starting with the basic ingredients of Monte Carlo simulations for classical systems, we introduce aspects such as importance sampling, sources of errors, and finite-size scaling analyses. We then set up the preliminary steps to prepare for the simulations, showing that they are actually carried out by sampling discrete Hubbard-Stratonovich auxiliary fields. In this process the Green's function emerges as a fundamental tool, since it is used in the updating process, and, at the same time, it is directly related to the quantities probing magnetic, charge, metallic, and superconducting behaviours. We also discuss the as yet unresolved ‘minus-sign problem', and two ways to stabilize the algorithm at low temperatures.
In this work we introduce a worldline-based fermion Monte Carlo algorithm for studying few body quantum mechanics of self-interacting fermions in the Hamiltonian lattice formulation. Our motivation to construct the method comes from our interest in studying renormalization of chiral nuclear effective-field theory with lattice regularization. In particular we wish to apply our method to compute the lattice spacing dependence of local lattice interactions as we take the continuum limit of the lattice theory. Our algorithm can compute matrix elements of the operator $exp(\ensuremath{-}\ensuremath{\beta}H)$, where H is the lattice Hamiltonian and $\ensuremath{\beta}$ is a free real parameter. These elements help us compute deep bound states that are well separated from scattering states even at values of $\ensuremath{\beta}$ which are not very large. Computing these bound-state energies accurately can help us study renormalization of the lattice theory. In addition to developing the algorithm, in this work we also introduce a finite-volume renormalization scheme for the lattice Hamiltonian of the leading pionless effective-field theory and show how it would work in the one- and two-body sectors.
We tutorially review the determinantal Quantum Monte Carlo method for fermionic systems, using the Hubbard model as a case study. Starting with the basic ingredients of Monte Carlo simulations for classical systems, we introduce aspects such as importance sampling, sources of errors, and finite-size scaling analyses. We then set up the preliminary steps to prepare for the simulations, showing that they are actually carried out by sampling discrete Hubbard-Stratonovich auxiliary fields. In this process the Green's function emerges as a fundamental tool, since it is used in the updating process, and, at the same time, it is directly related to the quantities probing magnetic, charge, metallic, and superconducting behaviours. We also discuss the as yet unresolved "minus-sign problem", and two ways to stabilize the algorithm at low temperatures.
We apply the Worldvolume Hybrid Monte Carlo (WV-HMC) method [arXiv:2012.08468] to the two-dimensional Hubbard model, which is known to suffer from a severe sign problem when the system is doped (away from half filling). We show that the method predicts physical observables with controlled statistical errors on an 8×8 lattice at temperature T/t=1/6.4≈0.156 and interaction strength U/t=8.0 (t is the hopping amplitude), for which the standard determinant quantum Monte Carlo fails.