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:1107.3626

BCS-BEC crossover in an optical lattice

Parag Ghosh

Published Jul 19, 2011• 0 citations
arXivHTMLPDF

Abstract

We model fermions with an attractive interaction in an optical lattice with a single-band Hubbard model away from half-filling with on-site attraction $U$ and nearest neighbor hopping $t$. Our goal is to understand the crossover from BCS superfluidity in the weak attraction limit to the BEC of molecules in the strong attraction limit, with particular emphasis on how this crossover in an optical lattice differs from the much better studied continuum problem. We use a large-$N$ theory with Sp(2N) symmetry to study the fluctuations beyond mean field theory. At T=0, we calculate across the crossover various observables, including chemical potential, gap, ground state energy, speed of sound and compressibility. The superfluid density $n_s$ is found to have non-trivial $U/t$ dependence in this lattice system. We show that the transition temperature $T_c$ scales with the energy gap in the weak coupling limit but crosses over to a $t^2/U$ scaling in the BEC limit, where phase fluctuations controlled by $n_s$ determine $T_c$. We also find, quite contrary to our expectations, that in the strong coupling limit, the large-$N$ theory gives qualitatively wrong trends for compressibility. A comparison with a simple Hartree shifted BCS theory, which takes into account both pairing and Hartree shifts, and correctly recovers the atomic limit and the right qualitative trend for compressibility, reveals that the large-$N$ theory on the lattice, although considers a larger number of diagrams, is in fact inferior to the simpler Hartree shifted BCS theory. The failure of the large-$N$ approach is explained by noting (i) the importance of Hartree shift in lattice problems, and (ii) inability of the large-$N$ approach to treat particle-particle and particle-hole channels at equal footing at the saddle point level.

DOI
10.48550/arxiv.1107.3626

Related papers

BCS-BEC crossover and effects of density fluctuations in a two-component Fermi gas described by the three-dimensional attractive Hubbard model

Hiromasa Tamaki, Yoji Ohashi, K. Miyake

We investigate the superfluid phase transition in a gas of Fermi atoms loaded on a three-dimensional optical lattice. When the lattice potential is strong, this system can be well described by an attractive Hubbard model. In this model, we calculate the superfluid phase transition temperature ${T}_{\mathrm{c}}$, including both superfluid and (spin and charge) density fluctuations within the self-consistent $t$-matrix theory and fluctuation exchange approximation, respectively. Since we treat these fluctuations in a consistent manner, our theory satisfies the required particle-hole symmetry over the entire BCS--Bose-Einstein-condensation (BEC) crossover region. We show that charge density fluctuations compete against superfluid fluctuations near the half-filling, leading to the suppression of ${T}_{\mathrm{c}}$. As a result, the maximum ${T}_{\mathrm{c}}$ is obtained away from the half-filling. Since the strong density fluctuations originate from the nesting property of the Fermi surface at the half-filling (which is absent in a uniform gas with no lattice potential), our results would be useful in considering lattice effects on strong-coupling superfluidity.

Physical Review A•Jun 18, 2008•19 citations

BCS-BEC crossover and effects of density fluctuations in a two-component Fermi gas loaded on an optical lattice

Hiromasa Tamaki, Yoji Ohashi, Miyake, K.

We investigate the superfluid phase transition in a gas of Fermi atoms loaded on a three-dimensional optical lattice. When the lattice potential is strong, this system can be well described by an attractive Hubbard model. In this model, we calculate the superfluid phase transition temperature Tc, including both superfluid and (spin and charge) density fluctuations within the self-consistent t-matrix theory and fluctuation exchange approximation, respectively. Since we treat these fluctuations in a consistent manner, our theory satisfies the required particle-hole symmetry over the entire BCS-BEC crossover region. We show that charge density fluctuations compete against superfluid fluctuations near the half-filling, leading to the suppression of Tc. As a result, the maximum Tc is obtained away the half-filling. Since the strong density fluctuations originate from the nesting property of the Fermi surface at the half filling (which is absent in a uniform gas with no lattice potential), our results would be useful in considering lattice effects on strong-coupling superfluidity.

arXiv:0803.2084•Mar 14, 2008•1 citation

Fermions with attractive interactions on optical lattices and implications for correlated systems

Chih-Chun Chien, Qijin Chen, Kathryn Levin

In this paper we address the behavior of the superfluid transition temperature ${T}_{c}$ in the attractive Hubbard model. The attractive Hubbard model can be regarded as the generalization of the BCS to Bose-Einstein condensation (BEC) crossover to a lattice and, as such, may have implications for future optical lattice studies. Nevertheless, the BEC limit of the Hubbard model is very different from that of the better-studied Fermi gases, owing to the strong intersite repulsion between pairs. Here we address systematically the effects of pairing fluctuations for all filling fractions over the entire range of attractive interaction strength. A central conclusion of our work is that in a lattice, around half filling, the smooth evolution from the BCS to the BEC limits is interrupted: ${T}_{c}$ vanishes when the system approaches the bosonic regime with increasing interaction strength. We suggest that this interruption of crossover may signal a quantum critical transition to another form of superfluid not continuously connected to a BCS-like phase. A simple variational ansatz for an alternate ground state in this more strongly coupled superfluid is presented. A generalization of the ($s$-wave) Hubbard model to $d$-wave pairing allows us to address issues of relevance to high-${T}_{c}$ superconductivity. The phase diagram (representing the pairing or pseudogap onset temperature ${T}^{*}$ and ${T}_{c}$) shows that here too one observes a vanishing of ${T}_{c}$ when ${T}^{*}$ becomes sufficiently large. Given this predicted breakdown of the crossover, and given the striking similarity to features of the cuprate phase diagram, we suggest that future experiments on ultracold fermions in optical lattices should not be exclusively limited to the repulsive Hubbard model, but should address the attractive model in order to elucidate features of high-temperature superconductivity.

arXiv:0808.1900•Physical Review A•Oct 23, 2008•19 citations

Topological superfluids and the BEC-BCS crossover in the attractive Haldane-Hubbard model

Yi‐Cai Zhang, Zhihao Xu, Shizhong Zhang

Motivated by the recent realization of the Haldane model in a shaking optical lattice, we investigate the effects of attractive interaction and the BEC-BCS crossover in this model at and away from half-filling. We show that, contrary to the usual $s$-wave BEC-BCS crossover in the lattice, a topological superfluid with Chern number $C=2$ appears in an extended region of the phase space for intermediate strength of the attractive interaction on the interaction-density plane. When inversion symmetry is broken, a gapless weak topological state is realized. We also investigate the fluctuations in these superfluid phases and show that the Anderson-Bogoliubov mode is quadratic due to time-reversal symmetry breaking and the existence of an undamped Leggett mode in the strong-coupling limit. Near the topological phase transition, the damping of the Leggett mode reaches its maximum.

arXiv:1511.03833•Physical Review A•Apr 28, 2017•15 citations

Two-dimensional attractive Hubbard model and the BCS-BEC crossover

Rodrigo A. Fontenele, Natanael C. Costa, Raimundo R. dos Santos +1 more

Recent experiments with ultracold fermionic atoms in optical lattices have provided a tuneable and clean realization of the attractive Hubbard model (AHM). In view of this, several physical properties may be thoroughly studied across the crossover between weak [Bardeen-Cooper-Schrieffer, (BCS)] and strong [Bose-Einstein condensation (BEC)] couplings. Here we report on extensive determinant Quantum Monte Carlo (DQMC) studies of the AHM on a square lattice from which several different quantities have been calculated and should be useful as a road map to experiments. We have obtained a detailed phase diagram for the critical superconducting temperature ${T}_{c}$ in terms of the band filling $\ensuremath{\langle}n\ensuremath{\rangle}$ and interaction strength $U$ from which we pinpoint a somewhat wide region $|U|/t\ensuremath{\approx}5\ifmmode\pm\else\textpm\fi{}1$ ($t$ is the hopping amplitude) and $\ensuremath{\langle}n\ensuremath{\rangle}\ensuremath{\approx}0.79\ifmmode\pm\else\textpm\fi{}0.09$ leading to a maximum ${T}_{c}\ensuremath{\approx}0.16t$. Two additional temperature scales, namely, pairing ${T}_{p}$ and degeneracy ${T}_{d}$ have been highlighted: The former sets the scale for pair formation (believed to be closely related to the scale for the gap of spin excitations in cuprates), whereas the latter sets the scale for dominant quantum effects. Our DQMC data for the distribution of doubly occupied sites for the momentum distribution function, and for the quasiparticle weight show distinctive features on both sides of the BCS-BEC crossover, being also suggestive of an underlying crossover between Fermi- and non-Fermi liquid behaviors.

Physical review. B./Physical review. B•May 2, 2022•34 citations

The 2D attractive Hubbard model and the BCS-BEC crossover

Rodrigo A. Fontenele, Natanael C. Costa, Raimundo R. dos Santos +1 more

Recent experiments with ultracold fermionic atoms in optical lattices have provided a tuneable and clean realization of the attractive Hubbard model (AHM). In view of this, several physical properties may be thoroughly studied across the crossover between weak (Bardeen-Cooper-Schrieffer, BCS) and strong (Bose-Einstein condensation, BEC) couplings. Here we report on extensive determinant Quantum Monte Carlo (DQMC) studies of the AHM on a square lattice, from which several different quantities have been calculated and should be useful as a roadmap to experiments. We have obtained a detailed phase diagram for the critical superconducting temperature, $T_c$, in terms of the band filling, $\ave{n}$, and interaction strength, $U$, from which we pinpoint a somewhat wide region $|U|/t \approx 5 \pm 1$ ($t$ is the hopping amplitude) and $\ave{n} \approx 0.79 \pm 0.09$ leading to a maximum $T_c \approx 0.16 t$. Two additional temperature scales, namely pairing, $T_p$, and degeneracy, $T_d$, have been highlighted: the former sets the scale for pair formation (believed to be closely related to the scale for the gap of spin excitations in cuprates), while the latter sets the scale for dominant quantum effects. Our DQMC data for the distribution of doubly occupied sites, for the momentum distribution function, and for the quasiparticle weight show distinctive features on both sides of the BCS-BEC crossover, being also suggestive of an underlying crossover between Fermi- and non-Fermi liquid behaviors.

arXiv:2201.02156•Jan 6, 2022•1 citation
We model fermions with an attractive interaction in an optical lattice with a single-band Hubbard model away from half-filling with on-site attraction UUU and nearest neighbor hopping ttt. Our goal is to understand the crossover from BCS superfluidity in the weak attraction limit to the BEC of molecules in the strong attraction limit, with particular emphasis on how this crossover in an optical lattice differs from the much better studied continuum problem. We use a large-NNN theory with Sp(2N) symmetry to study the fluctuations beyond mean field theory. At T=0, we calculate across the crossover various observables, including chemical potential, gap, ground state energy, speed of sound and compressibility. The superfluid density nsn_sns​ is found to have non-trivial U/tU/tU/t dependence in this lattice system. We show that the transition temperature TcT_cTc​ scales with the energy gap in the weak coupling limit but crosses over to a t2/Ut^2/Ut2/U scaling in the BEC limit, where phase fluctuations controlled by nsn_sns​ determine TcT_cTc​. We also find, quite contrary to our expectations, that in the strong coupling limit, the large-NNN theory gives qualitatively wrong trends for compressibility. A comparison with a simple Hartree shifted BCS theory, which takes into account both pairing and Hartree shifts, and correctly recovers the atomic limit and the right qualitative trend for compressibility, reveals that the large-NNN theory on the lattice, although considers a larger number of diagrams, is in fact inferior to the simpler Hartree shifted BCS theory. The failure of the large-NNN approach is explained by noting (i) the importance of Hartree shift in lattice problems, and (ii) inability of the large-NNN approach to treat particle-particle and particle-hole channels at equal footing at the saddle point level.
We investigate the superfluid phase transition in a gas of Fermi atoms loaded on a three-dimensional optical lattice. When the lattice potential is strong, this system can be well described by an attractive Hubbard model. In this model, we calculate the superfluid phase transition temperature Tc{T}_{\mathrm{c}}Tc​, including both superfluid and (spin and charge) density fluctuations within the self-consistent ttt-matrix theory and fluctuation exchange approximation, respectively. Since we treat these fluctuations in a consistent manner, our theory satisfies the required particle-hole symmetry over the entire BCS--Bose-Einstein-condensation (BEC) crossover region. We show that charge density fluctuations compete against superfluid fluctuations near the half-filling, leading to the suppression of Tc{T}_{\mathrm{c}}Tc​. As a result, the maximum Tc{T}_{\mathrm{c}}Tc​ is obtained away from the half-filling. Since the strong density fluctuations originate from the nesting property of the Fermi surface at the half-filling (which is absent in a uniform gas with no lattice potential), our results would be useful in considering lattice effects on strong-coupling superfluidity.
We investigate the superfluid phase transition in a gas of Fermi atoms loaded on a three-dimensional optical lattice. When the lattice potential is strong, this system can be well described by an attractive Hubbard model. In this model, we calculate the superfluid phase transition temperature Tc, including both superfluid and (spin and charge) density fluctuations within the self-consistent t-matrix theory and fluctuation exchange approximation, respectively. Since we treat these fluctuations in a consistent manner, our theory satisfies the required particle-hole symmetry over the entire BCS-BEC crossover region. We show that charge density fluctuations compete against superfluid fluctuations near the half-filling, leading to the suppression of Tc. As a result, the maximum Tc is obtained away the half-filling. Since the strong density fluctuations originate from the nesting property of the Fermi surface at the half filling (which is absent in a uniform gas with no lattice potential), our results would be useful in considering lattice effects on strong-coupling superfluidity.
In this paper we address the behavior of the superfluid transition temperature Tc{T}_{c}Tc​ in the attractive Hubbard model. The attractive Hubbard model can be regarded as the generalization of the BCS to Bose-Einstein condensation (BEC) crossover to a lattice and, as such, may have implications for future optical lattice studies. Nevertheless, the BEC limit of the Hubbard model is very different from that of the better-studied Fermi gases, owing to the strong intersite repulsion between pairs. Here we address systematically the effects of pairing fluctuations for all filling fractions over the entire range of attractive interaction strength. A central conclusion of our work is that in a lattice, around half filling, the smooth evolution from the BCS to the BEC limits is interrupted: Tc{T}_{c}Tc​ vanishes when the system approaches the bosonic regime with increasing interaction strength. We suggest that this interruption of crossover may signal a quantum critical transition to another form of superfluid not continuously connected to a BCS-like phase. A simple variational ansatz for an alternate ground state in this more strongly coupled superfluid is presented. A generalization of the (sss-wave) Hubbard model to ddd-wave pairing allows us to address issues of relevance to high-Tc{T}_{c}Tc​ superconductivity. The phase diagram (representing the pairing or pseudogap onset temperature T∗{T}^{*}T∗ and Tc{T}_{c}Tc​) shows that here too one observes a vanishing of Tc{T}_{c}Tc​ when T∗{T}^{*}T∗ becomes sufficiently large. Given this predicted breakdown of the crossover, and given the striking similarity to features of the cuprate phase diagram, we suggest that future experiments on ultracold fermions in optical lattices should not be exclusively limited to the repulsive Hubbard model, but should address the attractive model in order to elucidate features of high-temperature superconductivity.
Motivated by the recent realization of the Haldane model in a shaking optical lattice, we investigate the effects of attractive interaction and the BEC-BCS crossover in this model at and away from half-filling. We show that, contrary to the usual sss-wave BEC-BCS crossover in the lattice, a topological superfluid with Chern number C=2C=2C=2 appears in an extended region of the phase space for intermediate strength of the attractive interaction on the interaction-density plane. When inversion symmetry is broken, a gapless weak topological state is realized. We also investigate the fluctuations in these superfluid phases and show that the Anderson-Bogoliubov mode is quadratic due to time-reversal symmetry breaking and the existence of an undamped Leggett mode in the strong-coupling limit. Near the topological phase transition, the damping of the Leggett mode reaches its maximum.
Recent experiments with ultracold fermionic atoms in optical lattices have provided a tuneable and clean realization of the attractive Hubbard model (AHM). In view of this, several physical properties may be thoroughly studied across the crossover between weak [Bardeen-Cooper-Schrieffer, (BCS)] and strong [Bose-Einstein condensation (BEC)] couplings. Here we report on extensive determinant Quantum Monte Carlo (DQMC) studies of the AHM on a square lattice from which several different quantities have been calculated and should be useful as a road map to experiments. We have obtained a detailed phase diagram for the critical superconducting temperature Tc{T}_{c}Tc​ in terms of the band filling $\ensuremath{\langle}n\ensuremath{\rangle}$ and interaction strength UUU from which we pinpoint a somewhat wide region $|U|/t\ensuremath{\approx}5\ifmmode\pm\else\textpm\fi{}1$ (ttt is the hopping amplitude) and $\ensuremath{\langle}n\ensuremath{\rangle}\ensuremath{\approx}0.79\ifmmode\pm\else\textpm\fi{}0.09$ leading to a maximum ${T}_{c}\ensuremath{\approx}0.16t$. Two additional temperature scales, namely, pairing Tp{T}_{p}Tp​ and degeneracy Td{T}_{d}Td​ have been highlighted: The former sets the scale for pair formation (believed to be closely related to the scale for the gap of spin excitations in cuprates), whereas the latter sets the scale for dominant quantum effects. Our DQMC data for the distribution of doubly occupied sites for the momentum distribution function, and for the quasiparticle weight show distinctive features on both sides of the BCS-BEC crossover, being also suggestive of an underlying crossover between Fermi- and non-Fermi liquid behaviors.
Recent experiments with ultracold fermionic atoms in optical lattices have provided a tuneable and clean realization of the attractive Hubbard model (AHM). In view of this, several physical properties may be thoroughly studied across the crossover between weak (Bardeen-Cooper-Schrieffer, BCS) and strong (Bose-Einstein condensation, BEC) couplings. Here we report on extensive determinant Quantum Monte Carlo (DQMC) studies of the AHM on a square lattice, from which several different quantities have been calculated and should be useful as a roadmap to experiments. We have obtained a detailed phase diagram for the critical superconducting temperature, TcT_cTc​, in terms of the band filling, $\ave{n}$, and interaction strength, UUU, from which we pinpoint a somewhat wide region ∣U∣/t≈5±1|U|/t \approx 5 \pm 1∣U∣/t≈5±1 (ttt is the hopping amplitude) and $\ave{n} \approx 0.79 \pm 0.09$ leading to a maximum Tc≈0.16tT_c \approx 0.16 tTc​≈0.16t. Two additional temperature scales, namely pairing, TpT_pTp​, and degeneracy, TdT_dTd​, have been highlighted: the former sets the scale for pair formation (believed to be closely related to the scale for the gap of spin excitations in cuprates), while the latter sets the scale for dominant quantum effects. Our DQMC data for the distribution of doubly occupied sites, for the momentum distribution function, and for the quasiparticle weight show distinctive features on both sides of the BCS-BEC crossover, being also suggestive of an underlying crossover between Fermi- and non-Fermi liquid behaviors.