We start this series of lectures on quantum Monte Carlo methods with the simplest model for a strongly correlated system, namely an antiferromagnetic Heisenberg S‐1/2 chain. We will review methods for its simulation starting with the world‐line algorithm and then introducing the loop‐algorithm with global updates. We discuss next a model for doped antiferromagnets, with emphasis on the strong correlation limit that is central for high temperature superconductors and related materials, namely the so‐called t‐J model. An exact canonical transformation for this model that leads to a formulation with separated charge and spin degrees of freedom will be discussed. Results for a single hole in antiferromagnetic chains and planes will be summarized, giving an initial picture of charge dynamics in a quantum antiferromagnet. An additional algorithmic element, namely the determinantal method, will be introduced next, in order to deal with a finite number of charge degrees of freedom. Merging the loop‐ and the determinantal methods leads to a new one, the hybrid‐loop algorithm, that combines the efficiency of the loop‐algorithm for spins with the one of the determinantal method for fermions. Finally we discuss a first application of the hybrid‐loop algorithm, where the spectral function of the t‐J model in one dimension with finite doping is determined. There we can see how charge spin separation shows up in the one‐particle spectral function, and a detailed description can be achieved by comparison with an analitically soluble model, namely the t‐J model with 1/r2 interaction. The results show that in addition to spinons and holons expected in one‐dimensional metals, antiholons with charge Q = 2e, spin S = 0, and twice the mass of the holons are necessary to describe the inverse photoemission spectra at finite doping.
We start this series of lectures on quantum Monte Carlo methods with the simplest model for a strongly correlated system, namely an antiferromagnetic Heisenberg S‐1/2 chain. We will review methods for its simulation starting with the world‐line algorithm and then introducing the loop‐algorithm with global updates. We discuss next a model for doped antiferromagnets, with emphasis on the strong correlation limit that is central for high temperature superconductors and related materials, namely the so‐called t‐J model. An exact canonical transformation for this model that leads to a formulation with separated charge and spin degrees of freedom will be discussed. Results for a single hole in antiferromagnetic chains and planes will be summarized, giving an initial picture of charge dynamics in a quantum antiferromagnet. An additional algorithmic element, namely the determinantal method, will be introduced next, in order to deal with a finite number of charge degrees of freedom. Merging the loop‐ and the determinantal methods leads to a new one, the hybrid‐loop algorithm, that combines the efficiency of the loop‐algorithm for spins with the one of the determinantal method for fermions. Finally we discuss a first application of the hybrid‐loop algorithm, where the spectral function of the t‐J model in one dimension with finite doping is determined. There we can see how charge spin separation shows up in the one‐particle spectral function, and a detailed description can be achieved by comparison with an analitically soluble model, namely the t‐J model with 1/r2 interaction. The results show that in addition to spinons and holons expected in one‐dimensional metals, antiholons with charge Q = 2e, spin S = 0, and twice the mass of the holons are necessary to describe the inverse photoemission spectra at finite doping.
In this work, we present results for the dynamics of the doped one-dimensional t-J model for very large systems and in a wide parameter range. These results are obtained by quantum Monte Carlo simulation based on a newly developed algorithm, the hybrid-loop Monte Carlo. Within this algorithm the loop-algorithm is used to update the spin degrees of freedom, assuring optimal efficiency, whereas the charge degrees of freedom are evolved exactly for a given spin background in the frame of the determinantal algorithm, so that both static and dynamical observables can be measured. The study of the one-particle spectral function in the whole Luttinger-liquid regime of the phase diagram shows a splitting of the spectral weight into two branches on the inverse photoemission side, in contradiction with what is expected for an ordinary metal. This feature can be understood if we assume that the elementary excitation content of the nearest-neighbor t-J model and their dispersion is the same as the one analytically obtained at J/t=2 from a t-J model with both hopping and exchange interaction decaying as 1/r². In this case the elementary excitations are spinons with charge Q=0 and spin S=1/2, holons with charge Q=-e and spin S=0, and antiholon with charge Q=2e and spin S=0. Since the antiholon has twice the mass of the holon and charge with opposite sign, holon and antiholon are not charge conjugate but two independent elementary excitations. The agreement between the analytic results derived from our assumptions and the Monte Carlo data for a wide range of parameters and for different values of doping strongly indicate, that antiholons are generic excitations in the nearest-neighbor t-J model. Since the quantum numbers of the antiholons are the same as those of for a Cooper pair, the question arises whether these new excitations are related to superconductivity, that is indeed one of the possible phases in the one-dimensional nearest-neighbor t-J model. A further study of the system in the phase-separated region shows that the spectral function is very discontinuous in the photoemission side. This feature reminds an interference scenario generated by superposition of a continuous dispersion with a regular pattern. The nature of a phase-separated regime in a simulation with periodic boundary condition can be the origin of the regular pattern.
These lecture notes introduce quantum spin systems and several computational methods for studying their ground‐state and finite‐temperature properties. Symmetry‐breaking and critical phenomena are first discussed in the simpler setting of Monte Carlo studies of classical spin systems, to illustrate finite‐size scaling at continuous and first‐order phase transitions. Exact diagonalization and quantum Monte Carlo (stochastic series expansion) algorithms and their computer implementations are then discussed in detail. Applications of the methods are illustrated by results for some of the most essential models in quantum magnetism, such as the S = 1/2 Heisenberg antiferromagnet in one and two dimensions, as well as extended models useful for studying quantum phase transitions between antiferromagnetic and magnetically disordered states.
A review of the loop algorithm , its generalizations, and its relation to some other Monte Carlo techniques is given. The loop algorithm is a quantum Monte Carlo procedure that employs non-local changes of worldline configurations, determined by local stochastic decisions. It is based on a formulation of quantum models of any dimension in an extended ensemble of worldlines and graphs, and is related to Swendsen-Wang algorithms. It can be represented directly on an operator level, both with a continuous imaginary time path integral and with the stochastic series expansion. It overcomes many of the difficulties of traditional worldline simulations. Autocorrelations are reduced by orders of magnitude. Grand-canonical ensembles, off-diagonal operators, and variance reduced estimators are accessible. In some cases, infinite systems can be simulated. For a restricted class of models, the fermion sign problem can be overcome. Transverse magnetic fields are handled efficiently, in contrast to strong diagonal fields. The method has been applied successfully to a variety of models for spin and charge degrees of freedom, including Heisenberg and XYZ spin models, hard-core bosons, Hubbard and t - J -models. Owing to the improved efficiency, precise calculations of asymptotic behaviour and of quantum critical exponents have been possible.
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.
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 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.