arXiv:cond-mat/9406018

Pair fluctuation effects above $T_{\rm c}$

José Jesús Rodríguez-Núñez, Sandra Schafroth, R. Micnas, T. Schneider, Beck, H., M. H. Pedersen

Published Jun 3, 1994• 0 citations

Abstract

We explore the occurrence of pairing effects above $T_{\rm c}$ in the 2D attractive Hubbard model. The presence of pairs above $T_{\rm c}$ goes beyond the BCS approximation, in which pair formation and condensation occur at the same temperature. Using the fully self-conserving $T$-matrix formalism, which is valid above $T_{\rm c}$, we find that (1) the distribution function, $n(k)$, shows a 10\% change of the weight from ``below'' to ``above'' $k_{\rm F}$ with respect to the free case, and (2) the phase shift, $δϕ(ω,k)$, shows $Θ$-like behavior as a function of $ω$ for large momentum $k$ along the diagonal of the Brillouin zone. Our calculations have been carried out for an interaction of $U/t = - 4.0$ and a temperature of $T/t = 0.125$, where $t$ is the hopping matrix element between nearest neighbors. We conclude that for such an interaction value the Fermi surface is not a well-defined quantity.

DOI: 10.48550/arxiv.cond-mat/9406018

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Evidence for a Pseudogap Above the Critical Temperature in the Attractive Hubbard Model

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Temperature and Interaction Dependence of the Electron Spectrum in a 2D Attractive Hubbard Model

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The Pairing Interaction in the 2D Hubbard Model

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arXiv:cond-mat/9406018

Pair fluctuation effects above $T_{\rm c}$ Pair fluctuation effects above $T_{\rm c}$

José Jesús Rodríguez-Núñez, Sandra Schafroth, R. Micnas, T. Schneider, Beck, H., M. H. Pedersen

Published Jun 3, 1994• 0 citations

arXiv HTML PDF

Abstract

We explore the occurrence of pairing effects above

T_{\rm c}

in the 2D attractive Hubbard model. The presence of pairs above

T_{\rm c}

goes beyond the BCS approximation, in which pair formation and condensation occur at the same temperature. Using the fully self-conserving

T

-matrix formalism, which is valid above

T_{\rm c}

, we find that (1) the distribution function,

n(k)

, shows a 10% change of the weight from below'' to above''

k_{\rm F}

with respect to the free case, and (2) the phase shift,

δϕ(ω,k)

, shows

Θ

-like behavior as a function of

ω

for large momentum

k

along the diagonal of the Brillouin zone. Our calculations have been carried out for an interaction of

U/t = - 4.0

and a temperature of

T/t = 0.125

, where

t

is the hopping matrix element between nearest neighbors. We conclude that for such an interaction value the Fermi surface is not a well-defined quantity.

DOI: 10.48550/arxiv.cond-mat/9406018

Related papers

Evidence for a Pseudogap Above the Critical Temperature in the Attractive Hubbard Model

José Jesús Rodríguez-Núñez, Sandra Schafroth, R. Micnas +3 more

We explore the effect of fluctuations for $T \geq T_c$ in the 2-D negative Hubbard model within the framework of the selfconsistent T-matrix formalism, which goes beyond the BCS approximation and includes pair fluctuations. We enter the regime where correlations are important, namely, $U/t = -4.0$ , where $t$ denotes the hopping matrix element between nearest neighboors and $U$ is the strength of the on-site interaction. Our results include: 1) In comparison to the free case, the distribution function, $n({\bf k})$ , reveals considerable renormalization. This fact makes us conclude that the Fermi surface is blurred for correlated electron systems; 2) There is a pseudogap in the density of states, $N(ω)$ , around the chemical potential, $μ$ ; and 3) The real part of the T-matrix, at zero frequency and zero momentum, vs $T/t$ shows a divergence at a particular temperature. This last result shows that more approximate T-matrix calculations, extended to $T~=~0$ are meaningless.We explore the effect of fluctuations for $T \geq T_c$ in the 2-D negative Hubbard model within the framework of the selfconsistent T-matrix formalism, which goes beyond the BCS approximation and includes pair fluctuations. We enter the regime where correlations are important, namely, $U/t = -4.0$, where $t$ denotes the hopping matrix element between nearest neighboors and $U$ is the strength of the on-site interaction. Our results include: 1) In comparison to the free case, the distribution function, $n({\bf k})$, reveals considerable renormalization. This fact makes us conclude that the Fermi surface is blurred for correlated electron systems; 2) There is a pseudogap in the density of states, $N(ω)$, around the chemical potential, $μ$; and 3) The real part of the T-matrix, at zero frequency and zero momentum, vs $T/t$ shows a divergence at a particular temperature. This last result shows that more approximate T-matrix calculations, extended to $T~=~0$ are meaningless.

arXiv:cond-mat/9405071May 25, 19940 citations

Temperature and Interaction Dependence of the Electron Spectrum in a 2D Attractive Hubbard Model

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Journal of the Physical Society of JapanMar 15, 20040 citations

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A dynamic cluster quantum Monte Carlo approximation is used to study the effective pairing interaction of a 2D Hubbard model with a near neighbor hopping $t$ and an on-site Coulomb interaction $U$ . The effective pairing interaction is characterized in terms of the momentum and frequency dependence of the eigenfunction of the leading eigenvalue of the irreducible particle-particle vertex. The momentum dependence of this eigenfunction is found to vary as $(\cos k_x-\cos k_y)$ over most of the Brillouin zone and its frequency dependence is determined by the exchange energy $J$ . This implies that the effective pairing interaction is attractive for singlets formed between near-neighbor sites and retarded on a time scale set by $J^{-1}$ . The strength of the pairing interaction measured by the size of the d-wave eigenvalue peaks for $U$ of order the bandwidth $8t$ . It is found to increase as the system is underdoped.A dynamic cluster quantum Monte Carlo approximation is used to study the effective pairing interaction of a 2D Hubbard model with a near neighbor hopping $t$ and an on-site Coulomb interaction $U$ . The effective pairing interaction is characterized in terms of the momentum and frequency dependence of the eigenfunction of the leading eigenvalue of the irreducible particle-particle vertex. The momentum dependence of this eigenfunction is found to vary as $(\cos k_x-\cos k_y)$ over most of the Brillouin zone and its frequency dependence is determined by the exchange energy $J$. This implies that the effective pairing interaction is attractive for singlets formed between near-neighbor sites and retarded on a time scale set by $J^{-1}$. The strength of the pairing interaction measured by the size of the d-wave eigenvalue peaks for $U$ of order the bandwidth $8t$. It is found to increase as the system is underdoped.

arXiv:cond-mat/0606003May 31, 20060 citations

Effects of next-nearest neighbor hopping on the pairing and critical temperatures of the attractive Hubbard model on a square lattice

Rodrigo A. Fontenele, Natanael C. Costa, Thereza Paiva +1 more

The attractive Hubbard model plays a paradigmatic role in the study of superconductivity (superfluidity) and has become directly realizable in ultracold atom experiments on optical lattices. However, the critical temperatures, $T_c$ 's, remain lower than the lowest temperatures currently achievable in experiments. Here, we explore a possible route to enhance $T_c$ by introducing an additional next-nearest-neighbor (NNN) hopping, $t^\prime$ , in a two-dimensional square lattice. We perform sign-problem-free determinant quantum Monte Carlo simulations to compute response functions such as pairing correlation functions, superfluid density, and uniform spin susceptibility. Our results show that a judicious choice of $t^\prime$ can increase $Tc$ by up to $50\%$ compared to the case with only nearest-neighbor hopping. In contrast, the preformed pairs temperature scale, named pairing temperature, $T_p$ , decreases with increasing $|t^{\prime}/t|$ , which should represent a reduction of the pseudogap region, favoring a more BCS-like behavior at intermediate coupling. We further analyze the interacting density of states to characterize the transition from a pseudogap regime to a fully gapped superconducting state. These findings suggest that NNN hopping could be a viable route to increase $T_c$ to values closer to experimentally accessible temperature scales.The attractive Hubbard model plays a paradigmatic role in the study of superconductivity (superfluidity) and has become directly realizable in ultracold atom experiments on optical lattices. However, the critical temperatures, $T_c$'s, remain lower than the lowest temperatures currently achievable in experiments. Here, we explore a possible route to enhance $T_c$ by introducing an additional next-nearest-neighbor (NNN) hopping, $t^\prime$, in a two-dimensional square lattice. We perform sign-problem-free determinant quantum Monte Carlo simulations to compute response functions such as pairing correlation functions, superfluid density, and uniform spin susceptibility. Our results show that a judicious choice of $t^\prime$ can increase $Tc$ by up to $50\%$ compared to the case with only nearest-neighbor hopping. In contrast, the preformed pairs temperature scale, named pairing temperature, $T_p$, decreases with increasing $|t^{\prime}/t|$, which should represent a reduction of the pseudogap region, favoring a more BCS-like behavior at intermediate coupling. We further analyze the interacting density of states to characterize the transition from a pseudogap regime to a fully gapped superconducting state. These findings suggest that NNN hopping could be a viable route to increase $T_c$ to values closer to experimentally accessible temperature scales.

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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.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:2201.02156Jan 6, 20221 citation

Pairing order and condensate fraction across the BCS-BEC crossover in the two-dimensional attractive Hubbard model

Hao Du, Qian Sun, Zhuotao Xie +1 more

We report a systematic study of the superconducting ground state in the two-dimensional attractive Hubbard model, employing the {\it numerically exact} auxiliary-field quantum Monte Carlo method. We focus primarily on the pairing order and condensate fraction across the BCS-BEC crossover. Our numerical calculations cover a broad and physically relevant parameter regime, with the interaction strength $U/t=2\sim 12$ (where $t$ is the nearest-neighbor hopping amplitude) and fermion density $n=0.25\sim 1.0$ . We systematically characterize the dependence of the pairing order and condensate fraction on both the interaction strength and fermion density, and perform finite-size extrapolations to obtain their thermodynamic-limit values. As complementary quantities, we also provide high-precision numerical results for the double occupancy and total energy. Furthermore, we investigate the effects of next-nearest-neighbor hopping on the superconducting properties of the model. Our results offer a comprehensive set of unbiased numerical data for the attractive Hubbard model, establishing quantitative benchmarks for understanding superconductivity across the BCS-BEC crossover and for validating future theoretical and numerical approaches.We report a systematic study of the superconducting ground state in the two-dimensional attractive Hubbard model, employing the {\it numerically exact} auxiliary-field quantum Monte Carlo method. We focus primarily on the pairing order and condensate fraction across the BCS-BEC crossover. Our numerical calculations cover a broad and physically relevant parameter regime, with the interaction strength $U/t=2\sim 12$ (where $t$ is the nearest-neighbor hopping amplitude) and fermion density $n=0.25\sim 1.0$. We systematically characterize the dependence of the pairing order and condensate fraction on both the interaction strength and fermion density, and perform finite-size extrapolations to obtain their thermodynamic-limit values. As complementary quantities, we also provide high-precision numerical results for the double occupancy and total energy. Furthermore, we investigate the effects of next-nearest-neighbor hopping on the superconducting properties of the model. Our results offer a comprehensive set of unbiased numerical data for the attractive Hubbard model, establishing quantitative benchmarks for understanding superconductivity across the BCS-BEC crossover and for validating future theoretical and numerical approaches.

arXiv:2610.06319Oct 5, 20260 citations

Pair fluctuation effects above

T_{\rm c}

We explore the occurrence of pairing effects above

T_{\rm c}

in the 2D attractive Hubbard model. The presence of pairs above

T_{\rm c}

goes beyond the BCS approximation, in which pair formation and condensation occur at the same temperature. Using the fully self-conserving

T

-matrix formalism, which is valid above

T_{\rm c}

, we find that (1) the distribution function,

n(k)

, shows a 10% change of the weight from below'' to above''

k_{\rm F}

with respect to the free case, and (2) the phase shift,

δϕ(ω,k)

, shows

Θ

-like behavior as a function of

ω

for large momentum

k

along the diagonal of the Brillouin zone. Our calculations have been carried out for an interaction of

U/t = - 4.0

and a temperature of

T/t = 0.125

, where

t

is the hopping matrix element between nearest neighbors. We conclude that for such an interaction value the Fermi surface is not a well-defined quantity.

We explore the effect of fluctuations for

T \geq T_c

in the 2-D negative Hubbard model within the framework of the selfconsistent T-matrix formalism, which goes beyond the BCS approximation and includes pair fluctuations. We enter the regime where correlations are important, namely,

U/t = -4.0

, where

t

denotes the hopping matrix element between nearest neighboors and

U

is the strength of the on-site interaction. Our results include: 1) In comparison to the free case, the distribution function,

n({\bf k})

, reveals considerable renormalization. This fact makes us conclude that the Fermi surface is blurred for correlated electron systems; 2) There is a pseudogap in the density of states,

N(ω)

, around the chemical potential,

μ

; and 3) The real part of the T-matrix, at zero frequency and zero momentum, vs

T/t

shows a divergence at a particular temperature. This last result shows that more approximate T-matrix calculations, extended to

T~=~0

are meaningless.

A dynamic cluster quantum Monte Carlo approximation is used to study the effective pairing interaction of a 2D Hubbard model with a near neighbor hopping

t

and an on-site Coulomb interaction

U

. The effective pairing interaction is characterized in terms of the momentum and frequency dependence of the eigenfunction of the leading eigenvalue of the irreducible particle-particle vertex. The momentum dependence of this eigenfunction is found to vary as

(\cos k_x-\cos k_y)

over most of the Brillouin zone and its frequency dependence is determined by the exchange energy

J

. This implies that the effective pairing interaction is attractive for singlets formed between near-neighbor sites and retarded on a time scale set by

J^{-1}

. The strength of the pairing interaction measured by the size of the d-wave eigenvalue peaks for

U

of order the bandwidth

8t

. It is found to increase as the system is underdoped.

T_c

's, remain lower than the lowest temperatures currently achievable in experiments. Here, we explore a possible route to enhance

T_c

by introducing an additional next-nearest-neighbor (NNN) hopping,

t^\prime

, in a two-dimensional square lattice. We perform sign-problem-free determinant quantum Monte Carlo simulations to compute response functions such as pairing correlation functions, superfluid density, and uniform spin susceptibility. Our results show that a judicious choice of

t^\prime

can increase

Tc

by up to

50\%

compared to the case with only nearest-neighbor hopping. In contrast, the preformed pairs temperature scale, named pairing temperature,

T_p

, decreases with increasing

|t^{\prime}/t|

, which should represent a reduction of the pseudogap region, favoring a more BCS-like behavior at intermediate coupling. We further analyze the interacting density of states to characterize the transition from a pseudogap regime to a fully gapped superconducting state. These findings suggest that NNN hopping could be a viable route to increase

T_c

to values closer to experimentally accessible temperature scales.

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.

U/t=2\sim 12

(where

t

is the nearest-neighbor hopping amplitude) and fermion density

n=0.25\sim 1.0

. We systematically characterize the dependence of the pairing order and condensate fraction on both the interaction strength and fermion density, and perform finite-size extrapolations to obtain their thermodynamic-limit values. As complementary quantities, we also provide high-precision numerical results for the double occupancy and total energy. Furthermore, we investigate the effects of next-nearest-neighbor hopping on the superconducting properties of the model. Our results offer a comprehensive set of unbiased numerical data for the attractive Hubbard model, establishing quantitative benchmarks for understanding superconductivity across the BCS-BEC crossover and for validating future theoretical and numerical approaches.