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arXiv:1107.1314 · Physical Review A

Quantum criticality of a Bose gas in an optical lattice near the Mott transition

Adam Rançon, N. Dupuis

Published Jan 4, 2012• 13 citations
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Abstract

We derive the equation of state of bosons in an optical lattice in the framework of the Bose-Hubbard model. Near the density-driven Mott transition, the expression of the pressure $P(\ensuremath{\mu},T)$ versus chemical potential and temperature is similar to that of a dilute Bose gas but with renormalized mass ${m}^{*}$ and scattering length ${a}^{*}$. Here ${m}^{*}$ is the mass of the elementary excitations at the quantum critical point governing the transition from the superfluid phase to the Mott-insulating phase, while ${a}^{*}$ is related to their effective interaction at low energy. We use a nonperturbative renormalization-group approach to compute these parameters as a function of the ratio $t/U$ between hopping amplitude and on-site repulsion.

DOI
10.1103/physreva.85.011602

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We derive the equation of state of bosons in an optical lattice in the framework of the Bose-Hubbard model. Near the density-driven Mott transition, the expression of the pressure P(\ensuremathμ,T)P(\ensuremath{\mu},T)P(\ensuremathμ,T) versus chemical potential and temperature is similar to that of a dilute Bose gas but with renormalized mass m∗{m}^{*}m∗ and scattering length a∗{a}^{*}a∗. Here m∗{m}^{*}m∗ is the mass of the elementary excitations at the quantum critical point governing the transition from the superfluid phase to the Mott-insulating phase, while a∗{a}^{*}a∗ is related to their effective interaction at low energy. We use a nonperturbative renormalization-group approach to compute these parameters as a function of the ratio t/Ut/Ut/U between hopping amplitude and on-site repulsion.
We derive the equation of state of a two-dimensional Bose gas in an optical lattice in the framework of the Bose-Hubbard model. We focus on the vicinity of the multicritical points where the quantum phase transition between the Mott insulator and the superfluid phase occurs at fixed density and belongs to the three-dimensional XY model universality class. Using a nonperturbative renormalization-group approach, we compute the pressure P(μ,T)P(μ,T)P(μ,T) as a function of chemical potential and temperature. Our results compare favorably with a calculation based on the quantum O(2) model -- we find the same universal scaling function -- and allow us to determine the region of the phase diagram in the vicinity of a quantum multicritical point where the equation of state is universal. We also discuss the possible experimental observation of quantum XY criticality in a ultracold gas in an optical lattice.
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∗{m}^{*}m∗ and the ``scattering length'' a∗{a}^{*}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{Z}_{\mathrm{qp}}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 n‾c{\overline{n}}_{c}nc​ of the T=0T=0T=0 Mott insulator. We compute Zqp{Z}_{\mathrm{qp}}Zqp​, m∗{m}^{*}m∗, and a∗{a}^{*}a∗ and find that they vary strongly with both the ratio t/Ut/Ut/U between hopping amplitude and onsite repulsion and the value of the (commensurate) density n‾c{\overline{n}}_{c}nc​. Finally, we discuss the experimental observation of universality and the measurement of Zqp{Z}_{\mathrm{qp}}Zqp​, m∗{m}^{*}m∗, and a∗{a}^{*}a∗ in a cold-atomic gas in an optical lattice.
We study the quantum phase transitions and identify a tricritical point between a normal Bose superfluid, a superfluid that breaks additional Z(2) Ising symmetry, and a Mott insulator in a recent shaken optical lattice experiment. We show that near the transition between normal and Z(2) symmetry breaking superfluids, bosons can condense into a momentum state with high or even locally maximum kinetic energies due to the interaction effect. We present a general low-energy effective field theory that treats both the superfluid transition and the Ising transition in a uniform framework. Using the perturbative renormalization group method, we find that the critical behavior of the quantum phase transition belongs to a universality class different from that of a dilute Bose gas.
Abstract The physics of bosonic atoms in optical lattices is described. Starting from the basic experimental tools available to create such systems, the chapter introduces a simple model (known as the Bose—Hubbard model) for such a system. A mean-field type picture is developed to determine the basic character of the superfluid and Mott-insulating phases, and the physical conditions under which each should arise. A number of experimental methods are outlined to detect the properties of the gas. The fact that two thermodynamic ground states exist at zero temperature with different types of order implies the existence of a quantum phase transition. The behavior of the gas near this transition is described using an entirely different framework—quantum criticality—formed around the concepts of scaling symmetry and the renormalization group.
The Bose-Hubbard Hamiltonian of spin-2 cold bosons with repulsive interaction in an optical lattice is proposed. After neglecting the hopping term, the site-independent Hamiltonian and its energy eigenvalues and eigenstates are obtained. We consider the hopping term as a perturbation to do the calculations in second order and draw the phase diagrams for different cases. The phase diagrams show that there is a phase transition from Mott insulator with integer number bosons to superfluid when the ratio c0/t{c}_{0}/tc0​/t (c0{(c}_{0}(c0​ is the spin-independent on-site interaction and t is the hopping matrix element between adjacent lattice sites) is decreased to a critical value and that there is a different phase boundary between superfluid and Mott-insulator phase for different Zeeman level component in some ground states. We find that the position of phase boundary for different Zeeman level component is related to its average population in the Mott ground state.
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​.