A variational Monte Carlo method for bosonic lattice models is introduced. The method is based on the Baeriswyl projected wave function. The Baeriswyl wave function consists of a kinetic energy based projection applied to the wave function at infinite interaction, and is related to the shadow wave function already used in the study of continuous models of bosons. The wave function at infinite interaction, and the projector, are represented in coordinate space, leading to an expression for expectation values which can be evaluated via Monte Carlo sampling. We calculate the phase diagram and other properties of the bosonic Hubbard model. The calculated phase diagram is in excellent agreement with known quantum Monte Carlo results. We also analyze correlation functions.
A variational Monte Carlo method for bosonic lattice models is introduced. The method is based on the Baeriswyl projected wave function. The Baeriswyl wave function consists of a kinetic energy based projection applied to the wave function at infinite interaction, and is related to the shadow wave function already used in the study of continuous models of bosons. The wave function at infinite interaction, and the projector, are represented in coordinate space, leading to an expression for expectation values which can be evaluated via Monte Carlo sampling. We calculate the phase diagram and other properties of the bosonic Hubbard model. The calculated phase diagram is in excellent agreement with known quantum Monte Carlo results. We also analyze correlation functions.
We calculate the superfluid weight and the polarization amplitude for the one-dimensional bosonic Hubbard model with focus on the strong-coupling regime via variational, exact diagonalization, and strong coupling calculations. Our variational approach is based on the Baeriswyl wave function, implemented via Monte Carlo sampling. We derive the superfluid weight appropriately in a variational setting. We emphasize the importance of implementing the Peierls phase in position space and to allow for many-body interference effects, rather than implementing the Peierls phase as single particle momentum shifts. At integer filling, the Baeriswyl wave function gives zero superfluid response at any coupling. At half filling our variational superfluid weight is in reasonable agreement with exact diagonalization results. We also calculate the polarization amplitude, the variance of the total position, and the associated size scaling exponent, which corroborate that this variational approach produces an insulating state at integer filling. Our Baeriswyl based variational method is applicable to significantly larger system sizes than exact diagonalization or quantum Monte Carlo.
Quantum Averages and Statistical Samplings In this chapter, we discuss the general framework in which the variational Monte Carlo methods are defined and few important implementations for interacting systems of bosons and fermions on the lattice. The main advantage of considering this approach relies on the variational principle that has been shown in section 1.4: the energy of a given quantum state is always bounded from below by the exact ground-state one, giving us the route to obtain the best possible solution to the problem. In most cases, a suitable parametrization of the variational wave function allows us to consider a wide range of different quantum phases (e.g., metals, superconductors, and insulators). By performing an optimization of the parameters, we can reach the lowest-energy state, which is expected to capture the correct ground-state behavior. Therefore, variational wave functions represent a flexible and valuable approach to get important insights into the low-energy properties of models that cannot be solved by exact methods. By contrast, the main limitation of this approach is the fact that it is based on a given Ansatz , which may contain a relevant bias that cannot be removed within the chosen parametrization. We also mention that variational wave functions can be easily defined and treated for a wide class of models, irrespective of the range of interactions and the dimension of the local (i.e., single-site) Hilbert space, which can be even infinite and does not need the use of an uncontrolled cutoff to work with a finite-dimensional space. In this respect, variational Monte Carlo is better than other methods, like densitymatrix renormalization group or tensor-network methods in which the complexity dramatically increases with both the range of the interaction and the dimension of the local Hilbert space (White, 1992; Schollwöck, 2005, 2011). Let us start by describing the general framework in which variational Monte Carlo methods are defined. First of all, we fix a complete basis set in the Hilbert space, in which (for simplicity) the states are taken to be orthogonal and normalized such that: Then, any quantum state can be written as: In turn, the expectation value of an operator over a given variational wave function takes the following form: The main problem in evaluating the expectation value is that the number of configurations in the sum is exponentially large with the number of particles.
We study a model of strongly interacting lattice bosons with a Gutzwiller-type wave function that contains only on-site correlations. The variational energy and the condensate fraction associated with the variational wave function are exactly evaluated for both finite and infinite systems and compared with exact quantum Monte Carlo results in two dimensions. This ansatz for the wave function gives the correct qualitative picture of the phase diagram of this system; at commensurate densities, this system enters a Mott-insulator phase for large values of the interaction.
We propose a Monte Carlo method, which is a hybrid method of the quantum Monte Carlo method and variational Monte Carlo theory, to study the Hubbard model. The theory is based on the off-diagonal and the Gutzwiller type correlation factors which are taken into account by a Monte Carlo algorithm. In the 4x4 system our method is able to reproduce the exact results obtained by the diagonalization. An application is given to investigate the half-filled band case of two-dimensional square lattice. The energy is favorably compared with quantum Monte Carlo data.
The fixed-node Monte Carlo method is extended to lattice fermion models. This method replaces the problem of finding the ground state of a many-fermion system by an effective eigenvalue problem of finding the lowest-energy wave function in a given region of the configuration space. It has previously only been applied to fermions moving in continuous space. The discreteness of the configuration space causes the algorithm to differ from that of the continuum. The method is tested against a known limiting case where exact results are available for comparison. Good agreement is found. Using the fixed-node Monte Carlo method, we study the domain-wall phase in the ground state of the two-dimensional Hubbard model. The existence of a domain-wall phase which has domains of antiferromagnetic phases separated by walls of holes is recently suggested by inhomogeneous Hartree-Fock (HF) and variational Monte Carlo (VMC) calculations. Large improvements of the energies are found. The domain walls are broader than those obtained by the HF and VMC calculations.
Abstract This study extended the fixed-node QMC method to a lattice model for hopping fermions using Green’s function techniques. In general, the modifications of the procedures for continuous systems to adapt them to a lattice system were minor, and there appeared to be no surprises. Applications were made to the two-dimensional Hubbard model with different on-site interactions, lattice dimensions, boundary conditions, node specifications, and trial wavefunctions for importance sampling. Results were compared with those of Hartree-Fock and variational QMC calculations. For the HF calculations with zero interaction, the exact solutions are known, and the Green’s function QMC results agreed with those within statistical error. Trial wavefunctions specifying the nodes were obtained directly as those of HF determinantal solutions for zero interaction, or as variationally optimized determinantal functions with correlation terms. The procedure for fixed nodes was to eliminate walkers crossing the boundaries of a nodal region, a procedure not quite correct, but probably a reasonable approximation. The results in all cases were consistent with the HF and variational QMC calculations, giving lower energies than either for nonzero interaction energies. The existence of a domain-wall phase suggested by the HF and variational QMC results was confirmed by the fixed-node Green’s function results. It was noted that the fixed-node method would be applicable to systems other than the Hubbard model.