The authors demonstrate that the two-body contact theory, which links thermodynamics to the tail of the momentum distribution, can be applied to bosons on a lattice near the superfluid--Mott-insulator transition. This is possible because an excess of particles (or holes) compared to the Mott insulator behaves like a dilute gas.

arXiv:2501.14884Physical Review ASep 11, 20251 citation

Strong-coupling random-phase-approximation theory of a Bose gas near the superfluid–Mott-insulator transition: Universal thermodynamics and two-body contact

N. Dupuis, Moksh Bhateja, Adam Rançon

arXiv:2501.14887Physical Review ASep 11, 20252 citations

Bulletin of the American Physical Society

Universal thermodynamics of the superfluid to Mott insulator transition

N. Dupuis, Adam Rançon

Published Mar 7, 2014• 0 citations

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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 ${Z}_{\mathrm{qp}}$ 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 ${\overline{n}}_{c}$ of the $T=0$ Mott insulator. We compute ${Z}_{\mathrm{qp}}$ , ${m}^{*}$ , 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 ${\overline{n}}_{c}$ . Finally, we discuss the experimental observation of universality and the measurement of ${Z}_{\mathrm{qp}}$ , ${m}^{*}$ , and ${a}^{*}$ in a cold-atomic 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 ${m}^{*}$ 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 ${Z}_{\mathrm{qp}}$ 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 ${\overline{n}}_{c}$ of the $T=0$ Mott insulator. We compute ${Z}_{\mathrm{qp}}$, ${m}^{*}$, 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 ${\overline{n}}_{c}$. Finally, we discuss the experimental observation of universality and the measurement of ${Z}_{\mathrm{qp}}$, ${m}^{*}$, and ${a}^{*}$ in a cold-atomic gas in an optical lattice.

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arXiv:2501.14884Physical Review ASep 11, 20251 citation

Strong-coupling random-phase-approximation theory of a Bose gas near the superfluid–Mott-insulator transition: Universal thermodynamics and two-body contact

N. Dupuis, Moksh Bhateja, Adam Rançon

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 ${m}^{*}$ and their mutual interaction is described by an effective $s$ -wave scattering length ${a}^{*}$ . The singular part of the pressure takes the same form as in a dilute Bose gas, provided we replace the boson mass $m$ and the scattering length in vacuum $a$ by ${m}^{*}$ and ${a}^{*}$ , and the density $n$ 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 ${C}_{\mathrm{univ}}$ that controls the high-momentum tail $\ensuremath{\sim}1/{|\mathbf{k}|}^{4}$ of the singular part ${n}_{\mathbf{k}}^{\mathrm{sing}}$ 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 ${C}_{\mathrm{univ}}^{\mathrm{sd}}$ which controls the high-momentum tail of ${n}_{\mathbf{k}}^{\mathrm{sing}}$ at scales larger than the inverse lattice spacing, and a full contact $C$ , which controls the high-momentum tail of the full-momentum distribution ${n}_{\mathbf{k}}$ .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 ${m}^{*}$ and their mutual interaction is described by an effective $s$-wave scattering length ${a}^{*}$. The singular part of the pressure takes the same form as in a dilute Bose gas, provided we replace the boson mass $m$ and the scattering length in vacuum $a$ by ${m}^{*}$ and ${a}^{*}$, and the density $n$ 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 ${C}_{\mathrm{univ}}$ that controls the high-momentum tail $\ensuremath{\sim}1/{|\mathbf{k}|}^{4}$ of the singular part ${n}_{\mathbf{k}}^{\mathrm{sing}}$ 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 ${C}_{\mathrm{univ}}^{\mathrm{sd}}$ which controls the high-momentum tail of ${n}_{\mathbf{k}}^{\mathrm{sing}}$ at scales larger than the inverse lattice spacing, and a full contact $C$, which controls the high-momentum tail of the full-momentum distribution ${n}_{\mathbf{k}}$.

arXiv:2501.14887Physical Review ASep 11, 20252 citations

{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

{Z}_{\mathrm{qp}}

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

{\overline{n}}_{c}

of the

T=0

Mott insulator. We compute

{Z}_{\mathrm{qp}}

{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

{\overline{n}}_{c}

. Finally, we discuss the experimental observation of universality and the measurement of

{Z}_{\mathrm{qp}}

{m}^{*}

, and

{a}^{*}

in a cold-atomic gas in an optical lattice.

{m}^{*}

and their mutual interaction is described by an effective

s

-wave scattering length

{a}^{*}

. The singular part of the pressure takes the same form as in a dilute Bose gas, provided we replace the boson mass

m

and the scattering length in vacuum

a

{m}^{*}

and

{a}^{*}

, and the density

n

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

{C}_{\mathrm{univ}}

that controls the high-momentum tail $\ensuremath{\sim}1/{|\mathbf{k}|}^{4}$ of the singular part

{n}_{\mathbf{k}}^{\mathrm{sing}}

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

{C}_{\mathrm{univ}}^{\mathrm{sd}}

which controls the high-momentum tail of

{n}_{\mathbf{k}}^{\mathrm{sing}}

at scales larger than the inverse lattice spacing, and a full contact

C

, which controls the high-momentum tail of the full-momentum distribution

{n}_{\mathbf{k}}