We examine the validity of vector-mean-field theory (VMFT) for statistical transmutation on large lattices with a high density of particles per site (1/2 and 1/4). We take as a difficult test case the representation of hard-core bosons as fermions plus attached flux tubes. We use a variational Monte Carlo method to test the variational properties of the mean-field ground-state wave function against the predictions of the VMFT. We find a discrepancy of order 1 in the thermodynamic limit. This leads us to postulate that a better formulation on a lattice may be that of a renormalized vector-mean-field theory. We show that the renormalization coefficients can be understood by an analysis of the phase fluctuations (whose magnitude we estimate) of the long-range gauge interaction. These phase fluctuations are of order \ensuremath{\pi} on the lattice (thus leading to a breakdown of VMFT on the lattice) while they vanish in a continuum formulation. We give a detailed discussion of the qualitative differences of VMFT on the lattice versus the continuum. In particular, we examine the effect of having lines of zeros (lattice) versus points of zeros (continuum) for the nodes of the many-body wave function. In addition, a remarkable variational theorem is discovered for the ground-state wave function of the VMFT.
We examine the validity of vector-mean-field theory (VMFT) for statistical transmutation on large lattices with a high density of particles per site (1/2 and 1/4). We take as a difficult test case the representation of hard-core bosons as fermions plus attached flux tubes. We use a variational Monte Carlo method to test the variational properties of the mean-field ground-state wave function against the predictions of the VMFT. We find a discrepancy of order 1 in the thermodynamic limit. This leads us to postulate that a better formulation on a lattice may be that of a renormalized vector-mean-field theory. We show that the renormalization coefficients can be understood by an analysis of the phase fluctuations (whose magnitude we estimate) of the long-range gauge interaction. These phase fluctuations are of order \ensuremath{\pi} on the lattice (thus leading to a breakdown of VMFT on the lattice) while they vanish in a continuum formulation. We give a detailed discussion of the qualitative differences of VMFT on the lattice versus the continuum. In particular, we examine the effect of having lines of zeros (lattice) versus points of zeros (continuum) for the nodes of the many-body wave function. In addition, a remarkable variational theorem is discovered for the ground-state wave function of the VMFT.
We study wave functions for generalized flux phases by the variational Monte Carlo method. Flux tubes are attached to the holes, which produce additional phase factors besides those due to the motion of electrons. The phase factors cancel each other when the electrons and holes interchange position at some special electron densities and, as a result, there are no orbital currents in these states. We investigate especially the one-third filled case and find a slightly lower kinetic energy than the original commensurate flux phase. We also calculate the chirality to confirm that time-reversal symmetry is also broken in these states. The analogy to the quantum-Hall-effect wave functions leads us to study modified forms of off-diagonal long-range order in the pair-correlation function. The existence of this long-range order is shown explicitly by variational Monte Carlo calculations and renormalized mean-field approximations. The flux phases do not have the usual off-diagonal long-range order but do have a modified form similar to that in the quantum Hall effect. The relationships to the slave-boson mean-field theories and anyon systems are also discussed.
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.
Motivated by recent experimental progress in the context of ultracold multicolor fermionic atoms in optical lattices, we have investigated the properties of the SU(N) Heisenberg chain with totally antisymmetric irreducible representations, the effective model of Mott phases with m<N particles per site. These models have been studied for arbitrary N and m with non-Abelian bosonization [I. Affleck, Nucl. Phys. B 265, 409 (1986); I. Affleck, Nucl. Phys. B 305, 582 (1988)], leading to predictions about the nature of the ground state (gapped or critical) in most but not all cases. Using exact diagonalization and variational Monte Carlo based on Gutzwiller projected fermionic wave functions, we have been able to verify these predictions for a representative number of cases with $N\ensuremath{\le}10$ and $m\ensuremath{\le}N/2$, and we have shown that the opening of a gap is associated to a spontaneous dimerization or trimerization depending on the value of m and N. We have also investigated the marginal cases where Abelian bosonization did not lead to any prediction. In these cases, variational Monte Carlo predicts that the ground state is critical with exponents consistent with conformal field theory.
Lattice gauge theories coupled to fermionic matter account for many interesting phenomena in both high-energy physics and condensed-matter physics. Certain regimes, e.g., at finite fermion density, are difficult to simulate with traditional Monte Carlo algorithms due to the so-called sign problem. We present a variational, sign-problem-free Monte Carlo method for lattice gauge theories with continuous gauge groups and apply it to (2+1)-dimensional compact QED with dynamical fermions at finite density. The variational ansatz is formulated in the full gauge-field basis, i.e., without having to resort to truncation schemes for the U(1) gauge-field Hilbert space. The ansatz consists of two parts: first, a pure gauge part based on Jastrow-type ansatz states (which can be connected to certain neural-network ansatz states) and, second, a fermionic part based on gauge-field-dependent fermionic Gaussian states. These are designed in such a way that the gauge-field integral over all fermionic Gaussian states is gauge-invariant and at the same time still efficiently tractable. To ensure the validity of the method we benchmark the pure gauge part of the ansatz against another variational method and the full ansatz against an existing Monte Carlo simulation where the sign problem is absent. Moreover, in limiting cases where the exact ground state is known we show that our ansatz is able to capture this behavior. Finally, we study a sign-problem affected regime by probing density-induced phase transitions.
A method is introduced to renormalize the zero-range interaction for use in mean-field and many-body theory, starting from two-body calculations. The density-renormalized $\ensuremath{\delta}$-function interaction is then applied using mean-field theory to a two-component fermion gas and compared with diffusion Monte Carlo simulations and conventional mean-field calculations. In the unitarity limit, the equation of state exhibits the expected behavior $\ensuremath{\mu}\ensuremath{\propto}{\ensuremath{\rho}}^{2∕3}$, with a parameter $\ensuremath{\beta}=\ensuremath{-}0.492$, which is consistent with recent experiments.