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Journal of Experimental and Theoretical Physics Letters

Mott-insulator-superfluid-liquid transition in a one-dimensional bosonic Hubbard model: Quantum Monte Carlo method

V. А. Kashurnikov, Andrey V. Krasavin, Boris V. Svistunov

Published Jul 1, 1996• 40 citations
Publisher

Abstract

The values of the insulator gap Δ in one-dimensional systems of interacting bosons described by the Hubbard Hamiltonian are calculated at low temperatures by the quantum world-line Monte Carlo algorithm. The dependence of Δ on the size of the system, the temperature, and the parameters of the model is investigated. It is shown that a chain with N a =50 sites is already sufficient to estimate the thermodynamic value of the critical quantity ( t/U ) c for which a transition from the insulator into the superfluid state occurs in a commensurate system. To within the computational error, this value, ( t/U ) c =0.300±0.005, agrees with the value ( t/U ) c =0.304±0.002 obtained previously by the combined “exact diagonalization + renormalization-group analysis” method. The characteristic Kosterlitz-Thouless behavior of the insulator gap is demonstrated near the critical region: Δ∼exp[− b (1− t/t c ) −1/2 ].

DOI
10.1134/1.567139

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arXiv:cond-mat/0401256•Physical Review B•Oct 6, 2004•17 citations
The values of the insulator gap Δ in one-dimensional systems of interacting bosons described by the Hubbard Hamiltonian are calculated at low temperatures by the quantum world-line Monte Carlo algorithm. The dependence of Δ on the size of the system, the temperature, and the parameters of the model is investigated. It is shown that a chain with N a =50 sites is already sufficient to estimate the thermodynamic value of the critical quantity ( t/U ) c for which a transition from the insulator into the superfluid state occurs in a commensurate system. To within the computational error, this value, ( t/U ) c =0.300±0.005, agrees with the value ( t/U ) c =0.304±0.002 obtained previously by the combined “exact diagonalization + renormalization-group analysis” method. The characteristic Kosterlitz-Thouless behavior of the insulator gap is demonstrated near the critical region: Δ∼exp[− b (1− t/t c ) −1/2 ].
We present numerical simulations on a lattice model of bosons in two dimensions using a path integral quantum Monte Carlo algorithm. At finite temperature we observe the Kosterlitz-Thouless transition and by a finite-size analysis determine the transition temperature. In the limit of zero temperature we find two distinct phases: at a special commensurate density (number of bosons integer multiple of the number of sites), the system is superfluid for small values of the interaction, but the superfluidity vanishes at a critical value of the interaction and the system becomes a Mott insulator. For an incommensurate density, the system remains superfluid at all values of the repulsive interaction between bosons. The superfluid density, however, vanishes as a commensurate density is approached for sufficiently large interaction. We estimate the critical exponents for these transitions.
We use the density-matrix renormalization group to study the one-dimensional bosonic Hubbard model, with and without disorder. We obtain the gaps in all phases, certain correlation functions, and the superfluid density. A finite-size-scaling analysis enables us to obtain an accurate phase diagram and the $\ensuremath{\beta}$ function at the superfluid--Mott insulator transition and the Kosterlitz-Thouless essential singularity in the pure case. We also obtain coupling constants used in effective field theories for this system.
We report results of quantum Monte Carlo simulations of the Bose-Hubbard model in three dimensions. Critical parameters for the superfluid-to-Mott-insulator transition are determined with significantly higher accuracy than has been done in the past. In particular, the position of the critical point at filling factor n=1n=1n=1 is found to be at (U∕t)c=29.34(2){(U∕t)}_{\mathrm{c}}=29.34(2)(U∕t)c​=29.34(2), and the insulating gap $\ensuremath{\Delta}$ is measured with accuracy of a few percent of the hopping amplitude ttt. We obtain the effective mass of particle and hole excitations in the insulating state---with explicit demonstration of the emerging particle-hole symmetry and relativistic dispersion law at the transition tip---along with the sound velocity in the strongly correlated superfluid phase. These parameters are the necessary ingredients to perform analytic estimates of the low temperature $(T⪡\ensuremath{\Delta})$ thermodynamics in macroscopic samples. We present accurate thermodynamic curves, including these for specific heat and entropy, for typical insulating (U∕t=40)(U∕t=40)(U∕t=40) and superfluid (t∕U=0.0385)(t∕U=0.0385)(t∕U=0.0385) phases. Our data can serve as a basis for accurate experimental thermometry, and a guide for appropriate initial conditions if one attempts to use interacting bosons in quantum information processing.
We study ground-state properties of the one-dimensional truncated (no more than two particles on a site) bosonic Hubbard model in both repulsive and attractive regimes by exact diagonalization and exact world-line Monte Carlo simulation. In the commensurate case (one particle per site) we demonstrate that the point of Mott-insulator--superfluid transition, ${(U/t)}_{c}=0.50\ifmmode\pm\else\textpm\fi{}0.05$, is remarkably far from that of the full model. In the attractive region we observe the phase transition from a one-particle superfluid to a two-particle one. The paring gap demonstrates a linear behavior in the vicinity of the critical point. The critical state features marginal response to the gauge phase. We argue that the two-particle superfluid is a macroscopic analog of a peculiar phase observed earlier in a spin-1 model with axial anisotropy.
We use the exact-diagonalization results for a one-dimensional finite-size boson Hubbard-like Hamiltonians to initialize the renormalization-group equations in the vicinity of the superfluid-insulator transition and demonstrate that this provides a rather accurate method of extrapolation of the finite-size results to larger system sizes, and in particular, pinpointing the critical parameters of the Hamiltonians. With our approach we reproduce with the accuracy \ensuremath{\sim}2.5% the known analytical result for the transition point in the half-filled system of hard-core bosons, obtain with a controllable accuracy the critical ratio (t/U)c{)}_{\mathit{c}})c​=0.304\ifmmode\pm\else\textpm\fi{}0.002 for the boson Hubbard model, and find out that, in contrast to the general belief, the reduced Hamiltonian (with the constraint that the site occupation numbers be less than 3) is far from being a good approximation to the full model. \textcopyright{} 1996 The American Physical Society.
We examine the critical behavior near the metal-insulator transition (MIT) in the one-dimensional extended Hubbard model with the on-site and the nearest-neighbor interactions UUU and VVV at quarter filling using a combined method of the numerical diagonalization and the renormalization group (RG). The Luttinger-liquid parameter ${K}_{\ensuremath{\rho}}$ is calculated with the exact diagonalization for finite size systems and is substituted into the RG equation as an initial condition to obtain ${K}_{\ensuremath{\rho}}$ in the infinite size system. This approach also yields the charge gap $\ensuremath{\Delta}$ in the insulating state near the MIT. The results agree very well with the available exact results for $U=\ensuremath{\infty}$ even in the critical regime of the MIT where the characteristic energy becomes exponentially small and the usual finite size scaling is not applicable. When the system approaches the MIT critical point $V\ensuremath{\rightarrow}{V}_{c}$ for a fixed UUU, ${K}_{\ensuremath{\rho}}$ and $\ensuremath{\Delta}$ behave as ${\ensuremath{\mid}\mathrm{l}\mathrm{n}\ensuremath{\Delta}\ensuremath{\mid}}^{\ensuremath{-}2}={c}_{\ensuremath{\Delta}}(V∕{V}_{c}\ensuremath{-}1)$ and ${({K}_{\ensuremath{\rho}}\ensuremath{-}1∕4)}^{2}={c}_{K}(1\ensuremath{-}V∕{V}_{c})$, where the critical value Vc{V}_{c}Vc​ and the coefficients ${c}_{\ensuremath{\Delta}}$ and cK{c}_{K}cK​ are functions of UUU. These critical properties, which are known to be exact for $U=\ensuremath{\infty}$, are observed also for finite UUU case. We also observe the same critical behavior in the limit of the MIT critical point $U\ensuremath{\rightarrow}{U}_{c}$ when UUU is varied for a fixed VVV.