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.
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, which is much smaller than the inverse healing length
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 and
kh are of the order of the inverse lattice spacing.
We study the phase diagram of the boson Hubbard model at
T=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=1 and
d=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.
We propose an implementation of the nonperturbative renormalization group (NPRG) which applies to lattice models. Contrary to the usual NPRG approach where the initial condition of the RG flow is the mean-field solution, the lattice NPRG uses the (local) limit of decoupled sites as the (initial) reference system. In the long-distance limit, it is equivalent to the usual NPRG formulation and therefore yields identical results for the critical properties. We discuss both a lattice field theory defined on a d-dimensional hypercubic lattice and classical spin models. The simplest approximation, the local potential approximation, is sufficient to obtain the critical temperature and the magnetization of the three-dimensional Ising, XY, and Heisenberg models to an accuracy on the order of 1%. We show how the local potential approximation can be improved to include a nonzero anomalous dimension η and discuss the Berezinskii-Kosterlitz-Thouless transition of the two-dimensional XY model on a square lattice.
We present a lattice nonperturbative renormalization group (NPRG) approach to quantum
XY spin models by using a mapping onto hardcore bosons. The NPRG takes as initial condition of the renormalization group flow the (local) limit of decoupled sites, allowing us to take into account the hardcore constraint exactly. The initial condition of the flow is equivalent to the large
S classical results of the corresponding spin system. Furthermore, the hardcore constraint is conserved along the RG flow, and we can describe both local and long-distance fluctuations in a nontrivial way. We discuss a simple approximation scheme, and solve the corresponding flow equations. We compute both the zero-temperature thermodynamics and the finite temperature phase diagram on the square and cubic lattices. The NPRG allows us to recover the correct critical physics at finite temperature in two and three dimensions. The results compare well with numerical simulations.
We study the thermodynamics near the generic (density-driven) superfluid--Mott-insulator transition in the three-dimensional Bose-Hubbard model using the nonperturbative renormalization-group approach. At low energy, the physics is controlled by the Gaussian fixed point and becomes universal. Thermodynamic quantities can then be expressed in terms of the universal scaling functions of the dilute Bose gas universality class while the microscopic physics enters only via two nonuniversal parameters, namely, the effective mass
m∗ and the ``scattering length''
a∗ of the elementary excitations at the quantum critical point between the superfluid and Mott-insulating phases. A notable exception is the condensate density in the superfluid phase which is proportional to the quasiparticle weight
Zqp of the elementary excitations. The universal regime is defined by ${m}^{*}{a}^{*}{}^{2}T\ensuremath{\ll}1$ and ${m}^{*}{a}^{*}{}^{2}|\ensuremath{\delta}\ensuremath{\mu}|\ensuremath{\ll}1$ or, equivalently, $|\overline{n}\ensuremath{-}{\overline{n}}_{c}|{a}^{*}{}^{3}\ensuremath{\ll}1$, where $\ensuremath{\delta}\ensuremath{\mu}=\ensuremath{\mu}\ensuremath{-}{\ensuremath{\mu}}_{c}$ is the chemical potential shift from the quantum critical point $(\ensuremath{\mu}={\ensuremath{\mu}}_{c},\phantom{\rule{0.28em}{0ex}}T=0)$ and $\overline{n}\ensuremath{-}{\overline{n}}_{c}$ the doping with respect to the commensurate density
nc of the
T=0 Mott insulator. We compute
Zqp,
m∗, and
a∗ and find that they vary strongly with both the ratio
t/U between hopping amplitude and onsite repulsion and the value of the (commensurate) density
nc. Finally, we discuss the experimental observation of universality and the measurement of
Zqp,
m∗, and
a∗ in a cold-atomic gas in an optical lattice.