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arXiv:1106.5585 · Physical Review B

Nonperturbative renormalization group approach to strongly correlated lattice bosons

Adam Rançon, N. Dupuis

Published Nov 21, 2011• 61 citations
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Abstract

We present a nonperturbative renormalization-group approach to the Bose-Hubbard model. By taking as initial condition of the renormalization-group flow the (local) limit of decoupled sites, we take into account both local and long-distance fluctuations in a nontrivial way. This approach yields a phase diagram in very good quantitative agreement with quantum Monte Carlo simulations, and reproduces the two universality classes of the superfluid-Mott-insulator transition. The critical behavior near the multicritical points, where the transition takes place at constant density, agrees with the original predictions of Fisher et al. [Phys. Rev. B 40, 546 (1989)] based on simple scaling arguments. At a generic transition point, the critical behavior is mean-field like with logarithmic corrections in two dimensions. In the weakly correlated superfluid phase (far away from the Mott insulating phase), the renormalization-group flow is controlled by the Bogoliubov fixed point down to a characteristic (Ginzburg) momentum scale ${k}_{G}$, which is much smaller than the inverse healing length ${k}_{h}$. In the vicinity of the multicritical points, when the density is commensurate, we identify a sharp crossover from a weakly to a strongly correlated superfluid phase where the condensate density and the superfluid stiffness are strongly suppressed and both ${k}_{G}$ and ${k}_{h}$ are of the order of the inverse lattice spacing.

DOI
10.1103/physrevb.84.174513

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We present a nonperturbative renormalization-group approach to the Bose-Hubbard model. By taking as initial condition of the renormalization-group flow the (local) limit of decoupled sites, we take into account both local and long-distance fluctuations in a nontrivial way. This approach yields a phase diagram in very good quantitative agreement with quantum Monte Carlo simulations, and reproduces the two universality classes of the superfluid-Mott-insulator transition. The critical behavior near the multicritical points, where the transition takes place at constant density, agrees with the original predictions of Fisher et al. [Phys. Rev. B 40, 546 (1989)] based on simple scaling arguments. At a generic transition point, the critical behavior is mean-field like with logarithmic corrections in two dimensions. In the weakly correlated superfluid phase (far away from the Mott insulating phase), the renormalization-group flow is controlled by the Bogoliubov fixed point down to a characteristic (Ginzburg) momentum scale kG{k}_{G}kG​, which is much smaller than the inverse healing length kh{k}_{h}kh​. In the vicinity of the multicritical points, when the density is commensurate, we identify a sharp crossover from a weakly to a strongly correlated superfluid phase where the condensate density and the superfluid stiffness are strongly suppressed and both kG{k}_{G}kG​ and kh{k}_{h}kh​ are of the order of the inverse lattice spacing.
We present a nonperturbative renormalization group (RG) approach to the Bose-Hubbard model. By taking as initial condition of the RG flow the (local) limit of decoupled sites, we take into account both local and long-distance fluctuations in a nontrivial way. This approach yields a phase diagram in very good quantitative agreement with the quantum Monte Carlo results and reproduces the two universality classes of the superfluid--Mott-insulator transition with a good estimate of the critical exponents. Furthermore, it reveals the crucial role of the ``Ginzburg length'' as a crossover length between a weakly and a strongly correlated superfluid phase.
We present a non-perturbative renormalization-group approach to the Bose-Hubbard model. By taking as initial condition of the RG flow the (local) limit of decoupled sites, we take into account both local and long-distance fluctuations in a nontrivial way. This approach yields a phase diagram in very good quantitative agreement with the quantum Monte Carlo results and reproduces the two universality classes of the superfluid--Mott-insulator transition with a good estimate of the critical exponents. Furthermore, it reveals the crucial role of the Ginzburg as a crossover length between a weakly- and a strongly-correlated superfluid phase.
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We present a strong-coupling expansion of the Bose-Hubbard model based on a mean-field treatment of the hopping term, while onsite fluctuations are taken into account exactly. This random phase approximation describes the universal features of the generic Mott-insulator--superfluid transition (induced by a density change) and the superfluid state near the phase transition. The critical quasiparticles at the quantum critical point have a quadratic dispersion with an effective mass m∗{m}^{*}m∗ and their mutual interaction is described by an effective sss-wave scattering length a∗{a}^{*}a∗. The singular part of the pressure takes the same form as in a dilute Bose gas, provided we replace the boson mass mmm and the scattering length in vacuum aaa by m∗{m}^{*}m∗ and a∗{a}^{*}a∗, and the density nnn by the excess density $|n\ensuremath{-}{n}_{\mathrm{MI}}|$ of particles (or holes) with respect to the Mott insulator. We define a universal two-body contact Cuniv{C}_{\mathrm{univ}}Cuniv​ that controls the high-momentum tail $\ensuremath{\sim}1/{|\mathbf{k}|}^{4}$ of the singular part nksing{n}_{\mathbf{k}}^{\mathrm{sing}}nksing​ of the momentum distribution. We also apply the strong-coupling RPA to a lattice model of hard-core bosons and find that the high-momentum distribution is controlled by a universal contact, in complete agreement with the Bose-Hubbard model. Finally, we discuss a continuum model of bosons in an optical lattice and define two additional two-body contacts: a short-distance universal contact Cunivsd{C}_{\mathrm{univ}}^{\mathrm{sd}}Cunivsd​ which controls the high-momentum tail of nksing{n}_{\mathbf{k}}^{\mathrm{sing}}nksing​ at scales larger than the inverse lattice spacing, and a full contact CCC, which controls the high-momentum tail of the full-momentum distribution nk{n}_{\mathbf{k}}nk​.
We study the phase diagram of the boson Hubbard model at T=0T=0T=0 without disorder in a one-site and a two-site mean-field approximations. The two-site mean-field significantly improves the results compared to ordinary mean-field solutions and allows to use a simple scheme of mean-field renormalization group (MFRG). Some results obtained with MFRG are even better including the critical exponents for the density-driven transition between the Mott insulator and superfluid phases. However, when we evaluate critical exponents for transitions through the tip of the insulating lobes (special transitions without density change), we get similar exponents indicating that the MFRG apparently does not distinguish between both type of transitions and places them in the same universality class. The results for the first insulating lobe in d=1d=1d=1 and d=2d=2d=2 are in good agreement with a perturbative method and with the more reliable quantum Monte Carlo results. Our MFRG method provides analytical results for any lobe in any dimension and can be systematically improved if we consider larger clusters.
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