Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2505.04442 · Physical Review Research

Berry-phase-induced canonical structure of quantum fluctuations across the Mott-superfluid phase diagram

Idan S. Wallerstein, Eytan Grosfeld

Published May 28, 2026• 0 citations
arXivHTMLPDF

Abstract

We develop a unified field-theoretical description of the Bose-Hubbard model that captures low-energy dynamics across the insulator-to-superfluid transition. We show that commonly used cumulant expansions about the Mott ground state fail in the superfluid phase not only quantitatively but also fundamentally, because they impose an incorrect canonical structure. Expanding instead about the appropriate mean-field saddle point, we derive an effective action in which the dynamics of fluctuations are governed by a Berry-phase term, giving rise to an emergent magnetic field in the associated oscillator description. This field encodes particle-hole symmetry breaking in both the Mott-insulator phase and the superfluid, and vanishes at the point where a pure Goldstone mode emerges. The resulting effective hydrodynamic theory admits a speed of sound that approaches the Bogoliubov limit in the weakly interacting regime, demonstrating consistency across the entire phase diagram, from the Mott-insulator to the deep superfluid regime. While the total density remains continuous within mean-field theory, we find that quantum fluctuations generate a discontinuous contribution at the quantum phase transition, which vanishes at the tip of the Mott lobe.

DOI
10.1103/g274-2kj3

Related papers

Mott-insulator–to–superfluid transition in the Bose-Hubbard model: A strong-coupling approach

K. Sengupta, N. Dupuis

We present a strong-coupling expansion of the Bose-Hubbard model which describes both the superfluid and the Mott phases of ultracold bosonic atoms in an optical lattice. By performing two successive Hubbard-Stratonovich transformations of the intersite hopping term, we derive an effective action which provides a suitable starting point to study the strong-coupling limit of the Bose-Hubbard model. This action can be analyzed by taking into account Gaussian fluctuations about the mean-field approximation as in the Bogoliubov theory of the weakly interacting Bose gas. In the Mott phase, we reproduce results of previous mean-field theories and also calculate the momentum distribution function. In the superfluid phase, we find a gapless spectrum and compare our results with the Bogoliubov theory.

arXiv:cond-mat/0412204•Physical Review A•Mar 25, 2005•176 citations

Effective action approach to the p -band Mott insulator and superfluid transition

Xiaopeng Li, Erhai Zhao, W. Vincent Liu

Motivated by the recent experiment on $p$-orbital-band bosons in optical lattices, we study theoretically the quantum phases of Mott insulator and superfluidity in two dimensions. The system features a superfluid phase with transversely staggered orbital current at weak interaction and a Mott insulator phase with antiferro-orbital order at strong coupling and commensurate filling. We go beyond mean-field theory and derive from a microscopic model an effective action that is capable of describing both the $p$-band Mott insulating and superfluid phases in strong coupling. We further calculate the excitation spectra near the quantum critical point and find two gapless modes away from the tip of the Mott lobe but four gapless modes at the tip. Our effective theory reveals how the phase coherence peak builds up in the Mott regime when approaching the critical point. We also discuss the finite-temperature phase transition of $p$-band superfluidity.

arXiv:1104.1203•Physical Review A•Jun 20, 2011•22 citations

Effective action approach for quantum phase transitions in bosonic lattices

Barry Bradlyn, F. E. A. dos Santos, Axel Pelster

Based on standard field-theoretic considerations, we develop an effective action approach for investigating quantum phase transitions in lattice Bose systems at arbitrary temperature. We begin by adding to the Hamiltonian of interest a symmetry breaking source term. Using time-dependent perturbation theory, we then expand the grand-canonical free energy as a double power series in both the tunneling and the source term. From here, an order parameter field is introduced in the standard way and the underlying effective action is derived via a Legendre transformation. Determining the Ginzburg-Landau expansion to first order in the tunneling term, expressions for the Mott insulator-superfluid phase boundary, condensate density, average particle number, and compressibility are derived and analyzed in detail. Additionally, excitation spectra in the ordered phase are found by considering both longitudinal and transverse variations of the order parameter. Finally, these results are applied to the concrete case of the Bose-Hubbard Hamiltonian on a three-dimensional cubic lattice, and compared with the corresponding results from mean-field theory. Although both approaches yield the same Mott insulator--superfluid phase boundary to first order in the tunneling, the predictions of our effective action theory turn out to be superior to the mean-field results deeper into the superfluid phase.

arXiv:0809.0706•Physical Review A•Jan 15, 2009•34 citations

Spectral weight redistribution in strongly correlated bosons in optical lattices

C. Menotti, Nandini Trivedi

We calculate the single-particle spectral function for the one-band Bose-Hubbard model within the random-phase approximation (RPA). In the strongly correlated superfluid, in addition to the gapless phonon excitations, we find extra gapped modes, which become particularly relevant near the superfluid-Mott quantum phase transition (QPT). The strength in one of the gapped modes, a precursor of the Mott phase, grows as the QPT is approached and evolves into a hole (particle) excitation in the Mott insulator depending on whether the chemical potential $\ensuremath{\mu}$ is above (below) the tip of the lobe. The sound velocity $c$ of the Goldstone modes remains finite when the transition is approached at constant density; otherwise, it vanishes at the transition. It agrees well with Bogoliubov theory except close to the transition. We also calculate the spatial correlations for bosons in an inhomogeneous trapping potential creating alternating shells of Mott insulator and superfluid. Finally, we discuss the capability of the RPA to correctly account for quantum fluctuations in the vicinity of the QPT.

arXiv:0801.4672•Physical Review B•Jun 25, 2008•81 citations

Berezinskii-Kosterlitz-Thouless transition and BCS-Bose crossover in the two-dimensional attractive Hubbard model

N. Dupuis

We study the two-dimensional attractive Hubbard model using the mapping onto the half-filled repulsive Hubbard model in a uniform magnetic field coupled to the fermion spins. The low-energy effective action for charge and pairing fluctuations is obtained in the hydrodynamic regime. We recover the action of a Bose superfluid where half the fermion density is identified as the conjugate variable of the phase of the superconducting order parameter. By integrating out charge fluctuations, we obtain a phase-only action. In the zero-temperature superconducting state, this action describes a collective phase mode smoothly evolving from the Anderson-Bogoliubov mode at weak coupling to the Bogoliubov mode of a Bose superfluid at strong coupling. At finite temperature, the phase-only action can be used to extract an effective $XY$ model and thus obtain the Berezinskii-Kosterlitz-Thouless (BKT) phase transition temperature. We also identify a renormalized classical regime of superconducting fluctuations above the BKT phase transition, and a regime of incoherent pairs at higher temperature. Special care is devoted to the nearly half-filled case where the symmetry of the order parameter is enlarged to SO(3) due to strong $\mathbf{q}=(\ensuremath{\pi},\ensuremath{\pi})$ charge fluctuations. The low-energy effective action is then an SO(3) nonlinear $\ensuremath{\sigma}$ model with a (symmetry breaking) magnetic field proportional to the doping. In the strong-coupling limit, the attractive Hubbard model can be mapped onto the Heisenberg model in a magnetic field, which reduces to the quantum $XY$ model (except for a weak magnetic field, i.e., in the low-density limit of the attractive model). In the low-density limit, the Heisenberg model allows one to recover the action of a Bose superfluid, including the ${(\mathbf{\ensuremath{\nabla}}\ensuremath{\rho})}^{2}$ term (with $\ensuremath{\rho}$ the density), and in turn the Gross-Pitaevskii equation.

arXiv:cond-mat/0311374•Physical Review B•Oct 13, 2004•25 citations

Role of Quantum Fluctuations in Phase Transitions of Ultra-Cold Bosonic Systems

Dr. Narendra Kumar Ravi Nath Choudhry

Quantum fluctuations play a pivotal role in determining the ground state properties and phase transitions of ultra-cold bosonic systems. This study presents a comprehensive theoretical investigation of fluctuation effects beyond mean-field theory in dilute Bose gases, examining their impact on the superfluid–Mott insulator transition, quantum droplet formation, and critical phenomena. Beginning with the Gross–Pitaevskii mean-field framework , we systematically incorporate quantum corrections through Bogoliubov theory, yielding the Lee–Huang–Yang (LHY) energy correction scaling as . The quantum depletion quantifies the fraction of atoms driven from the condensate by zero-point fluctuations. For the Bose–Hubbard model , quantum Monte Carlo simulations establish the superfluid–Mott insulator critical point at , shifted 15% from the mean-field prediction. Critical exponents and confirm 3D XY universality. Quantum droplets in binary mixtures are stabilized by the repulsive LHY correction against mean-field collapse, with equilibrium densities matching experimental observations in K mixtures. Finite-size scaling analysis and the Ginzburg criterion delineate regimes where fluctuation effects dominate. All theoretical predictions are validated against precision measurements in ultracold atomic gases.

International journal of engineering science & humanities.•Jun 20, 2020•0 citations
We develop a unified field-theoretical description of the Bose-Hubbard model that captures low-energy dynamics across the insulator-to-superfluid transition. We show that commonly used cumulant expansions about the Mott ground state fail in the superfluid phase not only quantitatively but also fundamentally, because they impose an incorrect canonical structure. Expanding instead about the appropriate mean-field saddle point, we derive an effective action in which the dynamics of fluctuations are governed by a Berry-phase term, giving rise to an emergent magnetic field in the associated oscillator description. This field encodes particle-hole symmetry breaking in both the Mott-insulator phase and the superfluid, and vanishes at the point where a pure Goldstone mode emerges. The resulting effective hydrodynamic theory admits a speed of sound that approaches the Bogoliubov limit in the weakly interacting regime, demonstrating consistency across the entire phase diagram, from the Mott-insulator to the deep superfluid regime. While the total density remains continuous within mean-field theory, we find that quantum fluctuations generate a discontinuous contribution at the quantum phase transition, which vanishes at the tip of the Mott lobe.
We present a strong-coupling expansion of the Bose-Hubbard model which describes both the superfluid and the Mott phases of ultracold bosonic atoms in an optical lattice. By performing two successive Hubbard-Stratonovich transformations of the intersite hopping term, we derive an effective action which provides a suitable starting point to study the strong-coupling limit of the Bose-Hubbard model. This action can be analyzed by taking into account Gaussian fluctuations about the mean-field approximation as in the Bogoliubov theory of the weakly interacting Bose gas. In the Mott phase, we reproduce results of previous mean-field theories and also calculate the momentum distribution function. In the superfluid phase, we find a gapless spectrum and compare our results with the Bogoliubov theory.
Motivated by the recent experiment on ppp-orbital-band bosons in optical lattices, we study theoretically the quantum phases of Mott insulator and superfluidity in two dimensions. The system features a superfluid phase with transversely staggered orbital current at weak interaction and a Mott insulator phase with antiferro-orbital order at strong coupling and commensurate filling. We go beyond mean-field theory and derive from a microscopic model an effective action that is capable of describing both the ppp-band Mott insulating and superfluid phases in strong coupling. We further calculate the excitation spectra near the quantum critical point and find two gapless modes away from the tip of the Mott lobe but four gapless modes at the tip. Our effective theory reveals how the phase coherence peak builds up in the Mott regime when approaching the critical point. We also discuss the finite-temperature phase transition of ppp-band superfluidity.
Based on standard field-theoretic considerations, we develop an effective action approach for investigating quantum phase transitions in lattice Bose systems at arbitrary temperature. We begin by adding to the Hamiltonian of interest a symmetry breaking source term. Using time-dependent perturbation theory, we then expand the grand-canonical free energy as a double power series in both the tunneling and the source term. From here, an order parameter field is introduced in the standard way and the underlying effective action is derived via a Legendre transformation. Determining the Ginzburg-Landau expansion to first order in the tunneling term, expressions for the Mott insulator-superfluid phase boundary, condensate density, average particle number, and compressibility are derived and analyzed in detail. Additionally, excitation spectra in the ordered phase are found by considering both longitudinal and transverse variations of the order parameter. Finally, these results are applied to the concrete case of the Bose-Hubbard Hamiltonian on a three-dimensional cubic lattice, and compared with the corresponding results from mean-field theory. Although both approaches yield the same Mott insulator--superfluid phase boundary to first order in the tunneling, the predictions of our effective action theory turn out to be superior to the mean-field results deeper into the superfluid phase.
We calculate the single-particle spectral function for the one-band Bose-Hubbard model within the random-phase approximation (RPA). In the strongly correlated superfluid, in addition to the gapless phonon excitations, we find extra gapped modes, which become particularly relevant near the superfluid-Mott quantum phase transition (QPT). The strength in one of the gapped modes, a precursor of the Mott phase, grows as the QPT is approached and evolves into a hole (particle) excitation in the Mott insulator depending on whether the chemical potential $\ensuremath{\mu}$ is above (below) the tip of the lobe. The sound velocity ccc of the Goldstone modes remains finite when the transition is approached at constant density; otherwise, it vanishes at the transition. It agrees well with Bogoliubov theory except close to the transition. We also calculate the spatial correlations for bosons in an inhomogeneous trapping potential creating alternating shells of Mott insulator and superfluid. Finally, we discuss the capability of the RPA to correctly account for quantum fluctuations in the vicinity of the QPT.
We study the two-dimensional attractive Hubbard model using the mapping onto the half-filled repulsive Hubbard model in a uniform magnetic field coupled to the fermion spins. The low-energy effective action for charge and pairing fluctuations is obtained in the hydrodynamic regime. We recover the action of a Bose superfluid where half the fermion density is identified as the conjugate variable of the phase of the superconducting order parameter. By integrating out charge fluctuations, we obtain a phase-only action. In the zero-temperature superconducting state, this action describes a collective phase mode smoothly evolving from the Anderson-Bogoliubov mode at weak coupling to the Bogoliubov mode of a Bose superfluid at strong coupling. At finite temperature, the phase-only action can be used to extract an effective XYXYXY model and thus obtain the Berezinskii-Kosterlitz-Thouless (BKT) phase transition temperature. We also identify a renormalized classical regime of superconducting fluctuations above the BKT phase transition, and a regime of incoherent pairs at higher temperature. Special care is devoted to the nearly half-filled case where the symmetry of the order parameter is enlarged to SO(3) due to strong $\mathbf{q}=(\ensuremath{\pi},\ensuremath{\pi})$ charge fluctuations. The low-energy effective action is then an SO(3) nonlinear $\ensuremath{\sigma}$ model with a (symmetry breaking) magnetic field proportional to the doping. In the strong-coupling limit, the attractive Hubbard model can be mapped onto the Heisenberg model in a magnetic field, which reduces to the quantum XYXYXY model (except for a weak magnetic field, i.e., in the low-density limit of the attractive model). In the low-density limit, the Heisenberg model allows one to recover the action of a Bose superfluid, including the ${(\mathbf{\ensuremath{\nabla}}\ensuremath{\rho})}^{2}$ term (with $\ensuremath{\rho}$ the density), and in turn the Gross-Pitaevskii equation.
Quantum fluctuations play a pivotal role in determining the ground state properties and phase transitions of ultra-cold bosonic systems. This study presents a comprehensive theoretical investigation of fluctuation effects beyond mean-field theory in dilute Bose gases, examining their impact on the superfluid–Mott insulator transition, quantum droplet formation, and critical phenomena. Beginning with the Gross–Pitaevskii mean-field framework , we systematically incorporate quantum corrections through Bogoliubov theory, yielding the Lee–Huang–Yang (LHY) energy correction scaling as . The quantum depletion quantifies the fraction of atoms driven from the condensate by zero-point fluctuations. For the Bose–Hubbard model , quantum Monte Carlo simulations establish the superfluid–Mott insulator critical point at , shifted 15% from the mean-field prediction. Critical exponents and confirm 3D XY universality. Quantum droplets in binary mixtures are stabilized by the repulsive LHY correction against mean-field collapse, with equilibrium densities matching experimental observations in K mixtures. Finite-size scaling analysis and the Ginzburg criterion delineate regimes where fluctuation effects dominate. All theoretical predictions are validated against precision measurements in ultracold atomic gases.