Constrained-path Monte Carlo Method Research Articles

Pairing correlation of the kagome-lattice Hubbard model with the nearest-neighbor interaction

Abstract A recently discovered family of kagome lattice materials, AV3Sb5 (A = K,Rb,Cs), has attracted great interest, especially in the debate over their dominant superconducting pairing symmetry. To explore this issue, we study the superconducting pairing behavior within the kagome-lattice Hubbard model through the constrained path Monte Carlo method. It is found that doping around the Dirac point generates a dominant next-nearest-neighbor-d pairing symmetry driven by on-site Coulomb interaction U. However, when considering the nearest-neighbor interaction V, it may induce nearest-neighbor-p pairing to become the preferred pairing symmetry. Our results provide useful information to identify the dominant superconducting pairing symmetry in the AV3Sb5 family.

Benchmark study of the two-dimensional Hubbard model with auxiliary-field quantum Monte Carlo method

Ground state properties of the Hubbard model on a two-dimensional square lattice are studied by the auxiliary-field quantum Monte Carlo method. Accurate results for energy, double occupancy, effective hopping, magnetization, and momentum distribution are calculated for interaction strengths of U/t from 2 to 8, for a range of densities including half-filling and n = 0.3, 0.5, 0.6, 0.75, and 0.875. At half-filling, the results are numerically exact. Away from half-filling, the constrained path Monte Carlo method is employed to control the sign problem. Our results are obtained with several advances in the computational algorithm, which are described in detail. We discuss the advantages of generalized Hartree-Fock trial wave functions and its connection to pairing wave functions, as well as the interplay with different forms of Hubbard-Stratonovich decompositions. We study the use of different twist angle sets when applying the twist averaged boundary conditions. We propose the use of quasi-random sequences, which improves the convergence to the thermodynamic limit over pseudo-random and other sequences. With it and a careful finite size scaling analysis, we are able to obtain accurate values of ground state properties in the thermodynamic limit. Detailed results for finite-sized systems up to 16 \times 16 are also provided for benchmark purposes.

CPMC-Lab: A Matlab package for Constrained Path Monte Carlo calculations

We describe CPMC-Lab, a Matlab program for the constrained-path and phaseless auxiliary-field Monte Carlo methods. These methods have allowed applications ranging from the study of strongly correlated models, such as the Hubbard model, to ab initio calculations in molecules and solids. The present package implements the full ground-state constrained-path Monte Carlo (CPMC) method in Matlab with a graphical interface, using the Hubbard model as an example. The package can perform calculations in finite supercells in any dimensions, under periodic or twist boundary conditions. Importance sampling and all other algorithmic details of a total energy calculation are included and illustrated. This open-source tool allows users to experiment with various model and run parameters and visualize the results. It provides a direct and interactive environment to learn the method and study the code with minimal overhead for setup. Furthermore, the package can be easily generalized for auxiliary-field quantum Monte Carlo (AFQMC) calculations in many other models for correlated electron systems, and can serve as a template for developing a production code for AFQMC total energy calculations in real materials. Several illustrative studies are carried out in one- and two-dimensional lattices on total energy, kinetic energy, potential energy, and charge- and spin-gaps. Program summaryProgram title: CPMC-LabCatalogue identifier: AEUD_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEUD_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 2850No. of bytes in distributed program, including test data, etc.: 24838Distribution format: tar.gzProgramming language:Matlab.Computer: The non-interactive scripts can be executed on any computer capable of running Matlab with all Matlab versions. The GUI requires Matlab R2010b (version 7.11) and above. Operating system: Windows, Mac OS X, Linux.RAM: Variable.Classification: 7.3.External routines:MatlabNature of problem:Obtaining ground state energy of a repulsive Hubbard model in a supercell in any number of dimensions.Solution method:In the Constrained Path Monte Carlo (CPMC) method, the ground state of a many-fermion system is projected from an initial trial wave function by a branching random walk in an overcomplete basis of Slater determinants. Constraining the determinants according to a trial wave function |ΨT〉 removes the exponential decay of the signal-to-noise ratio characteristic of the sign problem. The method is exact if |ΨT〉 is exact.Unusual features:Direct and interactive environment with a Graphical User Interface for beginners to learn and study the Constrained Path Monte Carlo method with minimal overhead for setup.Running time:The sample provided takes a few seconds to run, the batch sample a few minutes.

Magnetic and pairing properties of a two-orbital model for the pnictide superconductors: a quantum Monte Carlo study

Using the constrained-path Monte Carlo method, a two-orbital model for the pnictide superconductors is studied at half filling and in both the electron- and hole-doped cases. At half filling, a stable (π, 0)/(0, π) magnetic order is explicitly observed and the system tends to be in an orthomagnetic order rather than the striped antiferromagnetic order on increasing the Coulomb repulsion U. In the electron-doped case, the (π, 0)/(0, π) magnetic order is enhanced upon doping and suppressed eventually and a s± pairing state dominates all the possible nearest-neighbour-bond pairings. Whereas in the hole-doped case, the magnetic order is straightforwardly suppressed and two nearly degenerate A1g and B1g intraband pairings become the dominant ones.

Ground-state pairing correlations in the S4 symmetric microscopic model for iron-based superconductors

We present the ground-state pairing correlations in the S4 symmetric microscopic model for iron-based superconductors, computed with the constrained-path Monte Carlo method. For various electron fillings and interaction strengths, we find that the sxy pairing dominates over other pairing correlations and is positive when the pair separation exceeds several lattice constants, both for iron-pnictides and iron-chalcogenides. These ground-state properties, especially the long-range part pairing correlations reconfirm the previous finite-temperature results published in Phys. Rev. Lett., 110 (2013) 107002. We further our study by including the nearest-neighbour interaction V and it is found that the sxy pairing correlation is slightly suppressed by increasing V.

The pairing correlations in the phased t – U Hubbard model

In this paper we have studied the phased t–U Hubbard model. Using the constrained path Monte Carlo method, we investigated the effects of phase factor on pairing correlation functions in the ground state and we found that the long-range correlations are dependent on the choice of phase factor and for some special values of phase factor there exist long-range pairing correlations only in the strong coupling region.

Quantum Monte Carlo study of pairing correlations in two-dimensional extended one-band and three-band Hubbard models

Using the recently developed constrained-path Monte Carlo method, we have completed a series of ground state computations of pairing correlation functions in two-dimensional extended one-band and three-band Hubbard models. We found that d x 2− y 2 -wave pairing correlations dominated s-wave correlations but that these correlations were suppressed by increasing the strength of the Hubbard repulsion U and the lattice size and that this picture was not significantly altered by adding a next nearest neighbor hopping t ′ and a nearest neighbor Coulomb attraction V to the one band model or by adding a nearest-neighbor O–O hopping and Cu–O repulsion to the three-band model. The trends suggest an unlikeliness that these models superconduct.

Ferromagnetism in the two-dimensional periodic Anderson model

Using the constrained-path Monte Carlo method, we studied the magnetic properties of the two-dimensional periodic Anderson model for electron fillings between 1/4 and 1/2. We also derived two effective low energy theories to assist in interpreting the numerical results. For 1/4 filling we found that the system can be a Mott or a charge transfer insulator, depending on the relative values of the Coulomb interaction and the charge transfer gap between the two non-interacting bands. The insulator may be a paramagnet or antiferromagnet. We concentrated on the effect of electron doping on these insulating phases. Upon doping we obtained a partially saturated ferromagnetic phase for low concentrations of conduction electrons. If the system were a charge transfer insulator, we would find that the ferromagnetism is induced by the well-known RKKY interaction. However, we found a novel correlated hopping mechanism inducing the ferromagnetism in the region where the non-doped system is a Mott insulator. Our regions of ferromagnetism spanned a much smaller doping range than suggested by recent slave boson and dynamical mean field theory calculations, but they were consistent with that obtained by density matrix renormalization group calculations of the one-dimensional periodic Anderson model.

Pairing, charge, and spin correlations in the three-band Hubbard model

Using the constrained path Monte Carlo method, we simulated the two-dimensional, three-band Hubbard model to study pairing, charge, and spin correlations as a function of electron and hole doping and the Coulomb repulsion ${V}_{\mathrm{pd}}$ between charges on neighboring Cu and O lattice sites. As a function of distance, both the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave and extended s-wave pairing correlations decayed quickly. In the charge-transfer regime, increasing ${V}_{\mathrm{pd}}$ decreased the long-range part of the correlation functions in both channels, while in the mixed-valent regime, it increased the long-range part of the s-wave behavior but decreased that of the d-wave behavior. Still the d-wave behavior dominated. At a given doping, increasing ${V}_{\mathrm{pd}}$ increased the spin-spin correlations in the charge-transfer regime but decreased them in the mixed-valent regime. Also, increasing ${V}_{\mathrm{pd}}$ suppressed the charge-charge correlations between neighboring Cu and O sites. Electron and hole doping away from half-filling was accompanied by a rapid suppression of antiferromagnetic correlations. Our results suggest that adding a repulsive ${V}_{\mathrm{pd}}$ to the model does not enhance its tendency toward superconductivity even though the behavior of the spin structure factor is consistent with the properties of some high-temperature superconducting materials.

Random walk beyond Hartree–Fock

We give a brief discussion of the recently developed Constrained-Path Monte Carlo method. This method is a quantum Monte Carlo technique that eliminates the fermion sign problem plaguing simulations of systems of interacting electrons. The elimination is accomplished by trading an exact procedure for an approximate one that has been demonstrated to give very accurate estimates of energies and many-body correlation functions. We also give a short review of its applications, a discussion of several strategies for parallelizing it, and some speculation of its future extensions.

Search for superconductivity in the two-dimensional Hubbard model using the Constrained Path Monte Carlo

The search for a superconducting state in the two-dimensional one-band Hubbard model (HM) has been intense in the last few years. Monte Carlo methods, haunted by the infamous fermion sign problem, have been unable to study large enough systems and low enough temperatures. The Constrained Path Monte Carlo method (CPMC) evades the sign problem by introducing an approximate constraining state. We introduce a new formulation of the CPMC that allows the implementation of a whole family of generalized mean-field states as constraint, and calculate superconducting pairing correlation functions in the HM for lattices up to 12×12. We compare the results obtained using a BCS and a free-electron constraining wave-function. We find that the d-wave correlations are insensitive to the constraining state used.

Issues and observations on applications of the constrained-path Monte Carlo method to many-fermion systems

We report several important observations that underscore the distinctions between the constrained-path Monte Carlo method and the continuum and lattice versions of the fixed-node method. The main distinctions stem from the differences in the state space in which the random walk occurs and in the manner in which the random walkers are constrained. One consequence is that in the constrained-path method the so-called mixed estimator for the energy is not an upper bound to the exact energy, as previously claimed. Several ways of producing an energy upper bound are given, and relevant methodological aspects are illustrated with simple examples.

Constrained path Monte Carlo method for fermion ground states

We describe and discuss a recently proposed quantum Monte Carlo algorithm to compute the ground-state properties of various systems of interacting fermions. In this method, the ground state is projected from an initial wave function by a branching random walk in an over-complete basis of Slater determinants. By constraining the determinants according to a trial wave function $|\psi_T\rangle$, we remove the exponential decay of signal-to-noise ratio characteristic of the sign problem. The method is variational and is exact if $|\psi_T\rangle$ is exact. We illustrate the method by describing in detail its implementation for the two-dimensional one-band Hubbard model. We show results for lattice sizes up to $16\times 16$ and for various electron fillings and interaction strengths. Besides highly accurate estimates of the ground-state energy, we find that the method also yields reliable estimates of other ground-state observables, such as superconducting pairing correlation functions. We conclude by discussing possible extensions of the algorithm.