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arXiv:cond-mat/9911002 · Physical review. B, Condensed matter

Exact duality and dual Monte Carlo simulation for the bosonic Hubbard model

F. Hébert, G. G. Batrouni, H. Mabilat

Published Apr 15, 2000• 3 citations
arXivHTMLPDF

Abstract

We derive the exact dual to the bosonic Hubbard model. The dual variables take the form of conserved current loops (local and global). Previously this has been done only for the very soft core model at very high density. No such approximations are made here. In particular, the dual of the hard core model is shown to have a very simple form, which is then used to construct an efficient Monte Carlo algorithm which is quite similar to the World Line algorithm but with some important differences. For example, with this algorithm we can measure easily the correlation function of the order parameter (Green function), a quantity which is extremely difficult to measure with the standard World Line algorithm. We demonstrate the algorithm for the one and two dimensional hardcore bosonic Hubbard models. We present results especially for the Green function and zero-mode filling fraction in the two dimensional hardcore model.

DOI
10.1103/physrevb.61.10725

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We derive the exact dual to the bosonic Hubbard model. The dual variables take the form of conserved current loops (local and global). Previously this has been done only for the very soft core model at very high density. No such approximations are made here. In particular, the dual of the hard core model is shown to have a very simple form, which is then used to construct an efficient Monte Carlo algorithm which is quite similar to the World Line algorithm but with some important differences. For example, with this algorithm we can measure easily the correlation function of the order parameter (Green function), a quantity which is extremely difficult to measure with the standard World Line algorithm. We demonstrate the algorithm for the one and two dimensional hardcore bosonic Hubbard models. We present results especially for the Green function and zero-mode filling fraction in the two dimensional hardcore model.
We report preliminary results of an investigation of the thermodynamic properties of the extended Hubbard model in one- dimension, calculated with the world-line Monte Carlo method described by Hirsch et al. With strictly continuous world-lines, we are able to measure the expectation of operators that conserve fermion number locally, such as the energy and (spatial) occupation number. By permitting the world-lines to be broken'' stochastically, we may also measure the expectation of operators that conserve fermion number only globally, such as the single-particle Green's function. For a 32 site lattice we present preliminary calculations of the average electron occupancy as a function of wavenumber when U = 4, V = 0 and {beta} = 16. For a half-filled band we find no indications of a Fermi surface. Slightly away from half-filling, we find Fermi-surface-like behavior similar to that found in other numerical investigations. 8 refs., 3 figs.
Recent developments have established exact bosonization and fermionization with a Z2\mathbb{Z}_2Z2​ symmetry as dualities through gauging. In this work, we apply fermionic gauging, which realizes generalized Jordan-Wigner transformations, to the Bose-Hubbard (BH) model with a global U(1)U(1)U(1) symmetry and derive an exact dual description in terms of fermionic composites, built from bosons and fermions. In 1D, this duality generalizes the exact mapping between the extended hard-core BH model and the spinless Fermi-Hubbard model to include soft-core bosons. At low energies, the mapping reduces to the well-known equivalence between the sine-Gordon model and the Thirring model. The oscillation wave vector of the fermionic composite correlation function in the gapless phase is fixed by their density, providing a novel manifestation of Luttinger's theorem. We verify the exact duality using density matrix renormalization group (DMRG) calculations and demonstrate that the gapless phase and the phase transition are governed by the compact boson conformal field theory. Our construction naturally extends to generic bosonic systems and higher dimensions, opening new avenues for studying Bose-Fermi mixtures in optical lattices and other strongly correlated quantum systems.
In this work we introduce the dual boson diagrammatic Monte Carlo technique for strongly interacting electronic systems. This method combines the strength of dynamical mean-filed theory for nonperturbative description of local correlations with the systematic account of nonlocal corrections in the dual boson theory by the diagrammatic Monte Carlo approach. It allows us to get a numerically exact solution of the dual boson theory at the two-particle local vertex level for the extended Hubbard model. We show that it can be efficiently applied to description of single-particle observables in a wide range of interaction strengths. We compare our exact results for the self-energy with the ladder dual boson approach and determine a physical regime, where the description of collective electronic effects requires more accurate consideration beyond the ladder approximation. Additionally, we find that the order-by-order analysis of the perturbative diagrammatic series for the single-particle Green's function allows to estimate the transition point to the charge density wave phase.
We introduce two quantum Monte Carlo algorithms and employ them to study the superfluid-insulator transition in a two-dimensional system of hard-core bosons. One of the algorithms is appropriate for zero temperature and is based upon the Green's function Monte Carlo method; the other is an extension of the finite-temperature world-line cluster method. In each case we find that the dynamical exponent is consistent with the theoretical prediction of z=2z\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2z=2 by Fisher and co-workers.
Two cluster algorithms, based on constructing and flipping loops, are presented for world-line quantum Monte Carlo simulations of fermions and are tested on the one-dimensional repulsive Hubbard model. We call these algorithms the loop-flip and loop-exchange algorithms. For these two algorithms and the standard world-line algorithm, we calculated the autocorrelation times for various physical quantities and found that the ordinary world-line algorithm, which uses only local moves, suffers from very long correlation times that makes not only the estimate of the error difficult but also the estimate of the average values themselves difficult. These difficulties are especially severe in the low-temperature, large-U regime. In contrast, we find that new algorithms, when used alone or in combinations with themselves and the standard algorithm, can have significantly smaller autocorrelation times, in some cases being smaller by three orders of magnitude. The new algorithms, which use nonlocal moves, are discussed from the point of view of a general prescription for developing cluster algorithms. The loop-flip algorithm is also shown to be ergodic and to belong to the grand canonical ensemble. Extensions to other models and higher dimensions are briefly discussed.
On the basis of a Feynman–Kac-type formula involving Poisson stochastic processes, a Monte Carlo algorithm has recently been introduced, which describes exactly the real- or imaginary-time evolution of many-body lattice quantum systems. We extend this algorithm to the exact simulation of time-dependent correlation functions. The techniques generally employed in Monte Carlo simulations to control fluctuations, namely reconfigurations and importance sampling, are adapted to the present algorithm and their validity is rigorously proved. We complete the analysis by several examples for the hard-core boson Hubbard model and for the Heisenberg model.