arXiv:cond-mat/0508645 · Physical Review B

Potential-energy-driven (BCS) to kinetic-energy-driven (BEC) pairing in the two-dimensional attractive Hubbard model: Cellular dynamical mean-field theory

Bumsoo Kyung, Antoine Georges, A.–M. S. Tremblay

Published Jul 5, 2006• 34 citations

Abstract

The BCS-BEC crossover within the two-dimensional attractive Hubbard model is studied by using the Cellular Dynamical Mean-Field Theory, both in the normal and superconducting ground states. Short-range spatial correlations incorporated in this theory remove the normal-state quasiparticle peak and the first-order transition found in the Dynamical Mean-Field Theory, rendering the normal state crossover smooth. For $U$ smaller than the bandwidth, pairing is driven by the potential energy, while in the opposite case it is driven by the kinetic energy, resembling a recent optical conductivity experiment in cuprates. Phase coherence leads to the appearance of a collective Bogoliubov mode in the density-density correlation function and to the sharpening of the spectral function.

DOI: 10.1103/physrevb.74.024501

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arXiv:cond-mat/0508645 · Physical Review B

Potential-energy-driven (BCS) to kinetic-energy-driven (BEC) pairing in the two-dimensional attractive Hubbard model: Cellular dynamical mean-field theory

Bumsoo Kyung, Antoine Georges, A.–M. S. Tremblay

Published Jul 5, 2006• 34 citations

arXiv HTML PDF

Abstract

U

smaller than the bandwidth, pairing is driven by the potential energy, while in the opposite case it is driven by the kinetic energy, resembling a recent optical conductivity experiment in cuprates. Phase coherence leads to the appearance of a collective Bogoliubov mode in the density-density correlation function and to the sharpening of the spectral function.

DOI: 10.1103/physrevb.74.024501

Related papers

BCS–BEC Crossover in the Two-Dimensional Attractive Hubbard Model: Variational Cluster Approach

Tatsuya Kaneko, Yukinori Ohta

We use the variational cluster approximation to study the superconducting ground state in the two-dimensional attractive Hubbard model, putting particular emphasis on the significance of quantum fluctuations of the system. We first show that the order parameter is suppressed in comparison with that obtained using the mean-field theory owing to the effects of spatial fluctuations in two-dimensional systems. We then show that the calculated Bogoliubov quasiparticle spectra and condensation amplitude clearly exhibit the character of Cooper pairs in momentum space and that the pair coherence length $ξ$ evaluated from the condensation amplitude demonstrates a smooth crossover in real space from a weakly paired BCS state ( $ξ\gg a$ ) to a BEC state of tightly bound pairs ( $ξ\ll a$ ), where $a$ is the lattice constant. The calculated kinetic and potential energies in the superconducting and normal ground states indicate that the superconducting state in the weak-coupling region is driven by the gain in potential energy, while that in the strongcoupling region is driven by the gain in kinetic energy.We use the variational cluster approximation to study the superconducting ground state in the two-dimensional attractive Hubbard model, putting particular emphasis on the significance of quantum fluctuations of the system. We first show that the order parameter is suppressed in comparison with that obtained using the mean-field theory owing to the effects of spatial fluctuations in two-dimensional systems. We then show that the calculated Bogoliubov quasiparticle spectra and condensation amplitude clearly exhibit the character of Cooper pairs in momentum space and that the pair coherence length $ξ$ evaluated from the condensation amplitude demonstrates a smooth crossover in real space from a weakly paired BCS state ($ξ\gg a$) to a BEC state of tightly bound pairs ($ξ\ll a$), where $a$ is the lattice constant. The calculated kinetic and potential energies in the superconducting and normal ground states indicate that the superconducting state in the weak-coupling region is driven by the gain in potential energy, while that in the strongcoupling region is driven by the gain in kinetic energy.

arXiv:1308.0664Journal of the Physical Society of JapanJan 21, 20149 citations

Energetic balance of the superconducting transition across the BCS—Bose Einstein crossover in the attractive Hubbard model

A. Toschi, Massimo Capone, Claudio Castellani

We investigate by means of Dynamical mean-field theory the crossover from BCS superconductivity to Bose-Einstein (BE) condensation of preformed pairs which occurs in the attractive Hubbard model by increasing the attraction strength. Fixing the density to a generic value $n=0.75$ , we follow the evolution of the two energy scales underlying the superconducting phenomenon, the gap ${\ensuremath{\Delta}}_{0}$, and the superfluid stiffness ${D}_{S}$ , which controls the phase coherence. The BCS--BE crossover is clearly mirrored in a change in the hierarchy of these two scales, the smallest of the two controlling the critical temperature. In the whole intermediate-to-strong-coupling region ${T}_{c}$ scales with ${D}_{S}$ , while ${T}_{C}$ is proportional to ${\ensuremath{\Delta}}_{0}$ only in the BCS regime. This evolution as a function of the interaction qualitatively resembles what happens in the cuprates when the doping is decreased towards the Mott insulator. This continuous change reflects also in the energetic balance at the superconducting transition. While, as it is well known, superconductivity is stabilized by a potential energy gain in the BCS regime, the strong-coupling superconductivity is made stable by a reduction of kinetic energy. Interestingly the intermediate-coupling region, where the maximum ${T}_{c}$ is achieved, behaves similarly to the strong-coupling regime, and its gain in kinetic energy is the largest as a function of the coupling. Since the integral of the optical conductivity is proportional to the kinetic energy, the above finding implies that the attractive Hubbard model can account qualitatively for the anomalous behavior of optical spectra around ${T}_{c}$ , where an increase of spectral weight is observed in under and optimally doped cuprates, while the overdoped samples have a more standard behavior. This qualitative agreement is lost in the normal phase, specifically at strong coupling, calling for the inclusion of strong correlation effects in the theoretical description.We investigate by means of Dynamical mean-field theory the crossover from BCS superconductivity to Bose-Einstein (BE) condensation of preformed pairs which occurs in the attractive Hubbard model by increasing the attraction strength. Fixing the density to a generic value $n=0.75$, we follow the evolution of the two energy scales underlying the superconducting phenomenon, the gap ${\ensuremath{\Delta}}_{0}$, and the superfluid stiffness ${D}_{S}$, which controls the phase coherence. The BCS--BE crossover is clearly mirrored in a change in the hierarchy of these two scales, the smallest of the two controlling the critical temperature. In the whole intermediate-to-strong-coupling region ${T}_{c}$ scales with ${D}_{S}$, while ${T}_{C}$ is proportional to ${\ensuremath{\Delta}}_{0}$ only in the BCS regime. This evolution as a function of the interaction qualitatively resembles what happens in the cuprates when the doping is decreased towards the Mott insulator. This continuous change reflects also in the energetic balance at the superconducting transition. While, as it is well known, superconductivity is stabilized by a potential energy gain in the BCS regime, the strong-coupling superconductivity is made stable by a reduction of kinetic energy. Interestingly the intermediate-coupling region, where the maximum ${T}_{c}$ is achieved, behaves similarly to the strong-coupling regime, and its gain in kinetic energy is the largest as a function of the coupling. Since the integral of the optical conductivity is proportional to the kinetic energy, the above finding implies that the attractive Hubbard model can account qualitatively for the anomalous behavior of optical spectra around ${T}_{c}$, where an increase of spectral weight is observed in under and optimally doped cuprates, while the overdoped samples have a more standard behavior. This qualitative agreement is lost in the normal phase, specifically at strong coupling, calling for the inclusion of strong correlation effects in the theoretical description.

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Superconductivity in the cuprates exhibits many unusual features. We study the two-dimensional Hubbard model with plaquette dynamical mean-field theory to address these unusual features and relate them to other normal-state phenomena, such as the pseudogap. Previous studies with this method found that upon doping the Mott insulator at low temperature a pseudogap phase appears. The low-temperature transition between that phase and the correlated metal at higher doping is first-order. A series of crossovers emerge along the Widom line extension of that first-order transition in the supercritical region. Here we show that the highly asymmetric dome of the dynamical mean-field superconducting transition temperature Tc(d), the maximum of the condensation energy as a function of doping, the correlation between maximum Tc(D) and normal-state scattering rate, the change from potential-energy driven to kinetic-energy driven pairing mechanisms can all be understood as remnants of the normal state first-order transition and its associated crossovers that also act as an organizing principle for the superconducting state.Superconductivity in the cuprates exhibits many unusual features. We study the two-dimensional Hubbard model with plaquette dynamical mean-field theory to address these unusual features and relate them to other normal-state phenomena, such as the pseudogap. Previous studies with this method found that upon doping the Mott insulator at low temperature a pseudogap phase appears. The low-temperature transition between that phase and the correlated metal at higher doping is first-order. A series of crossovers emerge along the Widom line extension of that first-order transition in the supercritical region. Here we show that the highly asymmetric dome of the dynamical mean-field superconducting transition temperature Tc(d), the maximum of the condensation energy as a function of doping, the correlation between maximum Tc(D) and normal-state scattering rate, the change from potential-energy driven to kinetic-energy driven pairing mechanisms can all be understood as remnants of the normal state first-order transition and its associated crossovers that also act as an organizing principle for the superconducting state.

arXiv:1602.02278Scientific ReportsMar 11, 201672 citations

BCS-BEC crossover at T=0 : A dynamical mean-field theory approach

Arti Garg, H. R. Krishnamurthy, Mohit Randeria

We study the $T=0$ crossover from the BCS superconductivity to Bose-Einstein condensation in the attractive Hubbard Model within dynamical mean field theory (DMFT) in order to examine the validity of Hartree-Fock-Bogoliubov (HFB) mean field theory, usually used to describe this crossover, and to explore the physics beyond it. Quantum fluctuations are incorporated using iterated perturbation theory as the DMFT impurity solver. We find that these fluctuations lead to large quantitative effects in the intermediate coupling regime, leading to a reduction of both the superconducting order parameter and the energy gap relative to the HFB results. A qualitative change is found in the single-electron spectral function, which now shows an incoherent spectral weight for energies larger than three times the gap, in addition to the usual Bogoliubov quasiparticle peaks.We study the $T=0$ crossover from the BCS superconductivity to Bose-Einstein condensation in the attractive Hubbard Model within dynamical mean field theory (DMFT) in order to examine the validity of Hartree-Fock-Bogoliubov (HFB) mean field theory, usually used to describe this crossover, and to explore the physics beyond it. Quantum fluctuations are incorporated using iterated perturbation theory as the DMFT impurity solver. We find that these fluctuations lead to large quantitative effects in the intermediate coupling regime, leading to a reduction of both the superconducting order parameter and the energy gap relative to the HFB results. A qualitative change is found in the single-electron spectral function, which now shows an incoherent spectral weight for energies larger than three times the gap, in addition to the usual Bogoliubov quasiparticle peaks.

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Variational Monte Carlo Study of Spin-Gapped Normal State and BCS–BEC Crossover in Two-Dimensional Attractive Hubbard Model

Shun Tamura, Hisatoshi Yokoyama

We study properties of normal, superconducting (SC) and CDW states for an attractive Hubbard model on the square lattice, using a variational Monte Carlo method. In trial wave functions, we introduce an interspinon binding factor, indispensable to induce a spin-gap transition in the normal state, in addition to the onsite attractive and intersite repulsive factors. It is found that, in the normal state, as the interaction strength $|U|/t$ increases, a first-order spin-gap transition arises at $|U_{\rm c}|\sim W$ ( $W$ : band width) from a Fermi liquid to a spin-gapped state, which is conductive through hopping of doublons. In the SC state, we confirm by analysis of various quantities that the mechanism of superconductivity undergoes a smooth crossover at around $|U_{\ma{co}}|\sim |U_{\rm c}|$ from a BCS type to a Bose-Einstein condensation (BEC) type, as $|U|/t$ increases. For $|U||U_{\ma{co}}|$, quantities such as the energy gain in the SC transition and superfluid stiffness, which is related to the cost of phase coherence, behave as $\sim t^2/|U|\propto T_{\rm c}$ , as expected in a bosonic scheme. In this regime, the SC transition is induced by a gain in kinetic energy, in contrast with the BCS theory. We refer to the relevance to the pseudogap in cuprate superconductors.We study properties of normal, superconducting (SC) and CDW states for an attractive Hubbard model on the square lattice, using a variational Monte Carlo method. In trial wave functions, we introduce an interspinon binding factor, indispensable to induce a spin-gap transition in the normal state, in addition to the onsite attractive and intersite repulsive factors. It is found that, in the normal state, as the interaction strength $|U|/t$ increases, a first-order spin-gap transition arises at $|U_{\rm c}|\sim W$ ($W$: band width) from a Fermi liquid to a spin-gapped state, which is conductive through hopping of doublons. In the SC state, we confirm by analysis of various quantities that the mechanism of superconductivity undergoes a smooth crossover at around $|U_{\ma{co}}|\sim |U_{\rm c}|$ from a BCS type to a Bose-Einstein condensation (BEC) type, as $|U|/t$ increases. For $|U||U_{\ma{co}}|$, quantities such as the energy gain in the SC transition and superfluid stiffness, which is related to the cost of phase coherence, behave as $\sim t^2/|U|\propto T_{\rm c}$, as expected in a bosonic scheme. In this regime, the SC transition is induced by a gain in kinetic energy, in contrast with the BCS theory. We refer to the relevance to the pseudogap in cuprate superconductors.

arXiv:1203.1719Journal of the Physical Society of JapanJun 4, 20129 citations

From BCS-like superconductivity to condensation of local pairs: A numerical study of the attractive Hubbard model

Johannes Maria Singer, M. H. Pedersen, T. Schneider +2 more

We investigate the two-dimensional attractive Hubbard model with quantum Monte Carlo techniques to reveal the crossover from a BCS-type superconductivity in the weak-coupling regime to a superconductivity properly described by a Bose-Einstein condensation (BEC) of local, preformed pairs. The crossover from BCS to BEC is particularly well exposed in the temperature dependence of both the spin susceptibility and the double occupancy, as well as by the appearance of a pseudogap in the density of states far above ${\mathit{T}}_{\mathit{c}}$ . These features are also mirrored in the shape of the specific-heat peak around ${\mathit{T}}_{\mathit{c}}$ , the separation of the temperature regimes where pair formation and their condensation occur, and in the transfer of spectral weight from the single-particle excitation branch to a pair band in the normal state. \textcopyright{} 1996 The American Physical Society.We investigate the two-dimensional attractive Hubbard model with quantum Monte Carlo techniques to reveal the crossover from a BCS-type superconductivity in the weak-coupling regime to a superconductivity properly described by a Bose-Einstein condensation (BEC) of local, preformed pairs. The crossover from BCS to BEC is particularly well exposed in the temperature dependence of both the spin susceptibility and the double occupancy, as well as by the appearance of a pseudogap in the density of states far above ${\mathit{T}}_{\mathit{c}}$. These features are also mirrored in the shape of the specific-heat peak around ${\mathit{T}}_{\mathit{c}}$, the separation of the temperature regimes where pair formation and their condensation occur, and in the transfer of spectral weight from the single-particle excitation branch to a pair band in the normal state. \textcopyright{} 1996 The American Physical Society.

Physical review. B, Condensed matterJul 1, 199697 citations

U

ξ

evaluated from the condensation amplitude demonstrates a smooth crossover in real space from a weakly paired BCS state (

ξ\gg a

) to a BEC state of tightly bound pairs (

ξ\ll a

), where

a

is the lattice constant. The calculated kinetic and potential energies in the superconducting and normal ground states indicate that the superconducting state in the weak-coupling region is driven by the gain in potential energy, while that in the strongcoupling region is driven by the gain in kinetic energy.

n=0.75

, we follow the evolution of the two energy scales underlying the superconducting phenomenon, the gap ${\ensuremath{\Delta}}_{0}$, and the superfluid stiffness

{D}_{S}

, which controls the phase coherence. The BCS--BE crossover is clearly mirrored in a change in the hierarchy of these two scales, the smallest of the two controlling the critical temperature. In the whole intermediate-to-strong-coupling region

{T}_{c}

scales with

{D}_{S}

, while

{T}_{C}

is proportional to ${\ensuremath{\Delta}}_{0}$ only in the BCS regime. This evolution as a function of the interaction qualitatively resembles what happens in the cuprates when the doping is decreased towards the Mott insulator. This continuous change reflects also in the energetic balance at the superconducting transition. While, as it is well known, superconductivity is stabilized by a potential energy gain in the BCS regime, the strong-coupling superconductivity is made stable by a reduction of kinetic energy. Interestingly the intermediate-coupling region, where the maximum

{T}_{c}

is achieved, behaves similarly to the strong-coupling regime, and its gain in kinetic energy is the largest as a function of the coupling. Since the integral of the optical conductivity is proportional to the kinetic energy, the above finding implies that the attractive Hubbard model can account qualitatively for the anomalous behavior of optical spectra around

{T}_{c}

, where an increase of spectral weight is observed in under and optimally doped cuprates, while the overdoped samples have a more standard behavior. This qualitative agreement is lost in the normal phase, specifically at strong coupling, calling for the inclusion of strong correlation effects in the theoretical description.

We study the

T=0

crossover from the BCS superconductivity to Bose-Einstein condensation in the attractive Hubbard Model within dynamical mean field theory (DMFT) in order to examine the validity of Hartree-Fock-Bogoliubov (HFB) mean field theory, usually used to describe this crossover, and to explore the physics beyond it. Quantum fluctuations are incorporated using iterated perturbation theory as the DMFT impurity solver. We find that these fluctuations lead to large quantitative effects in the intermediate coupling regime, leading to a reduction of both the superconducting order parameter and the energy gap relative to the HFB results. A qualitative change is found in the single-electron spectral function, which now shows an incoherent spectral weight for energies larger than three times the gap, in addition to the usual Bogoliubov quasiparticle peaks.

|U|/t

increases, a first-order spin-gap transition arises at

|U_{\rm c}|\sim W

(

W

: band width) from a Fermi liquid to a spin-gapped state, which is conductive through hopping of doublons. In the SC state, we confirm by analysis of various quantities that the mechanism of superconductivity undergoes a smooth crossover at around $|U_{\ma{co}}|\sim |U_{\rm c}|$ from a BCS type to a Bose-Einstein condensation (BEC) type, as

|U|/t

increases. For $|U||U_{\ma{co}}|$, quantities such as the energy gain in the SC transition and superfluid stiffness, which is related to the cost of phase coherence, behave as

\sim t^2/|U|\propto T_{\rm c}

, as expected in a bosonic scheme. In this regime, the SC transition is induced by a gain in kinetic energy, in contrast with the BCS theory. We refer to the relevance to the pseudogap in cuprate superconductors.

{\mathit{T}}_{\mathit{c}}

. These features are also mirrored in the shape of the specific-heat peak around

{\mathit{T}}_{\mathit{c}}

, the separation of the temperature regimes where pair formation and their condensation occur, and in the transfer of spectral weight from the single-particle excitation branch to a pair band in the normal state. \textcopyright{} 1996 The American Physical Society.