Physics Procedia

Spin-gapped wave function as normal state in attractive Hubbard model

Shun Tamura, H. Yokoyama

Published Jan 1, 2012• 0 citations

Abstract

With the pseudogap in cuprate superconductors in mind, we study an attractive Hubbard model on the square lattice, in which the mechanism of superconductivity undergoes a crossover at |Uco|/t from a BCS type to a Bose-Einstein condensation type, as the interaction strength |U|/t increases. Correspondingly, for |U|>|Uco|, a normatate, realized for T>Tc, is considered to exhibit gapped behavior. We try to construct such a state and study its properties using a variational Monte Carlo method, which treats the local correlation accurately and can connect the weak-and strong-correlation limits continuously. We introduce an intersite correlation factor between antiparallel spins in nearest-neighbor sites, in addition to the onsite (Gutzwiller) projection. This factor brings about a first-order transition from a metal to a spin-gapped phase at |U=|Uc|=W (bandwidth). For |U|>|Uc|, a gap opens in the spin degree of freedom, whereas the charge sector remains gapless. In this phase, conductivity is caused by a hopping of doublons.

DOI: 10.1016/j.phpro.2012.03.414

Related papers

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

Shun Tamura, Hisatoshi Yokoyama

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

Crossover between BCS Superconductor and Doped Mott Insulator of d -Wave Pairing State in Two-Dimensional Hubbard Model

Hisatoshi Yokoyama, Masao Ogata, Yukio Tanaka +2 more

arXiv:1208.1102Journal of the Physical Society of JapanJan 15, 2013100 citations

Gutzwiller-Type Projected BCS Ground States for Attractive Hubbard Model in Infinite Dimensions

Y. Y. Suzuki, Shinichi Saito, Susumu Kurihara

Progress of Theoretical PhysicsNov 1, 19999 citations

Variational Monte Carlo Studies of Attractive Hubbard Model. I: Normal-State Properties

H. Yokoyama

arXiv:cond-mat/0205281Progress of Theoretical PhysicsJul 1, 200245 citations

BCS to Bose crossover in anisotropic superconductors

J. P. Wallington, James F. Annett

arXiv:cond-mat/9909043Physical review. B, Condensed matterJan 1, 200026 citations

Superconducting phase coherence and pairing gap in the three-dimensional attractive Hubbard model

T. K. Kopeć

Physical review. B, Condensed matterJan 1, 200211 citations

Physics Procedia

Spin-gapped wave function as normal state in attractive Hubbard model

Shun Tamura, H. Yokoyama

Published Jan 1, 2012• 0 citations

Publisher

Abstract

DOI: 10.1016/j.phpro.2012.03.414

Related papers

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

Crossover between BCS Superconductor and Doped Mott Insulator of d -Wave Pairing State in Two-Dimensional Hubbard Model

Hisatoshi Yokoyama, Masao Ogata, Yukio Tanaka +2 more

With high-Tc cuprates in mind, properties of correlated dx2-y2-wave superconducting (SC) and antiferromagnetic (AF) states are studied for the Hubbard (t-t'-U) model on square lattices, using a variational Monte Carlo method. We employ simple trial wave functions including only crucial parameters, such as a doublon-holon binding factor indispensable to describe correlated SC and normal states as doped Mott insulators. U/t, t'/t and δ(doping rate) dependence of relevant quantities are systematically calculated. As U/t increases, a sharp crossover of SC properties occurs at U_co/t \sim 10 from a conventional BCS type to a kinetic-energy-driven type for any t'/t. As δdecreases, U_co/t is smoothly connected to the Mott transition point at half filling. For U/t\lsim 5, steady superconductivity corresponding to the cuprates is not found, whereas the d-wave SC correlation function Pd^\infty rapidly increases for U/t\gsim 6 and becomes maximum at U=U_co. Comparing the δdependence of Pd^\infty with experimentally observed dome-shaped Tc and condensation energy, we find that the effective value of $U$ for the cuprates should be larger than the band width, for which the t-J model is valid. Analyzing the kinetic energy, we reveal that for U>U_co only doped holes (electrons) become charge carriers, which will make a small Fermi surface (hole pocket), but for UU_co is determined by two factors, i.e., the AF spin correlation, which creates singlet pairs (pseudogap), and the charge mobility dominated by Mott physics.With high-Tc cuprates in mind, properties of correlated dx2-y2-wave superconducting (SC) and antiferromagnetic (AF) states are studied for the Hubbard (t-t'-U) model on square lattices, using a variational Monte Carlo method. We employ simple trial wave functions including only crucial parameters, such as a doublon-holon binding factor indispensable to describe correlated SC and normal states as doped Mott insulators. U/t, t'/t and δ(doping rate) dependence of relevant quantities are systematically calculated. As U/t increases, a sharp crossover of SC properties occurs at U_co/t \sim 10 from a conventional BCS type to a kinetic-energy-driven type for any t'/t. As δdecreases, U_co/t is smoothly connected to the Mott transition point at half filling. For U/t\lsim 5, steady superconductivity corresponding to the cuprates is not found, whereas the d-wave SC correlation function Pd^\infty rapidly increases for U/t\gsim 6 and becomes maximum at U=U_co. Comparing the δdependence of Pd^\infty with experimentally observed dome-shaped Tc and condensation energy, we find that the effective value of $U$ for the cuprates should be larger than the band width, for which the t-J model is valid. Analyzing the kinetic energy, we reveal that for U>U_co only doped holes (electrons) become charge carriers, which will make a small Fermi surface (hole pocket), but for UU_co is determined by two factors, i.e., the AF spin correlation, which creates singlet pairs (pseudogap), and the charge mobility dominated by Mott physics.

arXiv:1208.1102Journal of the Physical Society of JapanJan 15, 2013100 citations

Gutzwiller-Type Projected BCS Ground States for Attractive Hubbard Model in Infinite Dimensions

Y. Y. Suzuki, Shinichi Saito, Susumu Kurihara

In order to develop our theoretical understanding of the superconducting state of strongly correlated electron systems, we investigate the many-body ground state of the attractive Hubbard model in infinite dimensions using the Gutzwiller-type projected BCS variational (GBCS) wave function. We first calculate various physical quantities by numerically evaluating analytical equations for systems in the thermodynamic limit. We then show that the GBCS wave function has the lowest energy when the gap parameter has a mixture of s wave and extended s wave pairing symmetry and that the ground state energy of the GBCS is lower than that of either the BCS or Fermi liquid at any electron density and any strength of the attractive interactions in our model. We also show that with infinite strength interactions, each pair of electrons with opposite spins forms a tightly bound pair on a lattice site (BEC-like state). We find a smooth crossover from a BCS-like state with large, overlapping Cooper pairs (for weakly attractive interactions) to a BEC-like state of composite bosons formed out of the tightly bound pairs of fermions (for strongly attractive interactions).In order to develop our theoretical understanding of the superconducting state of strongly correlated electron systems, we investigate the many-body ground state of the attractive Hubbard model in infinite dimensions using the Gutzwiller-type projected BCS variational (GBCS) wave function. We first calculate various physical quantities by numerically evaluating analytical equations for systems in the thermodynamic limit. We then show that the GBCS wave function has the lowest energy when the gap parameter has a mixture of s wave and extended s wave pairing symmetry and that the ground state energy of the GBCS is lower than that of either the BCS or Fermi liquid at any electron density and any strength of the attractive interactions in our model. We also show that with infinite strength interactions, each pair of electrons with opposite spins forms a tightly bound pair on a lattice site (BEC-like state). We find a smooth crossover from a BCS-like state with large, overlapping Cooper pairs (for weakly attractive interactions) to a BEC-like state of composite bosons formed out of the tightly bound pairs of fermions (for strongly attractive interactions).

Progress of Theoretical PhysicsNov 1, 19999 citations

Variational Monte Carlo Studies of Attractive Hubbard Model. I: Normal-State Properties

H. Yokoyama

Normal states of the attractive Hubbard model, mainly in two dimensions, are studied in the context of a transition from a Fermi liquid to an insulating or gapped state. A series of variational Monte Carlo calculations with better statistics was carried out to estimate accurately the expectation values of several many-body wave functions. Although a relatively clear crossover is observed, even for the plain Gutzwiller wave function, the states in both regimes are metallic. Meanwhile, a substantial metal-insulator transition takes place at ∣U∣∼W (band width) for an improved wave function, into which an intersite correlation is introduced by taking account of virtual states in the second-order perturbation in the infinite ∣U∣ limit. The critical value compares favorably with recent results of the dynamical-mean-field approximation. In contrast, a conventional Jastrow-type wave function scarcely improves the normal state. In addition, the issue of the Brinkman-Rice metal-insulator transition is reconsidered with much larger systems.Normal states of the attractive Hubbard model, mainly in two dimensions, are studied in the context of a transition from a Fermi liquid to an insulating or gapped state. A series of variational Monte Carlo calculations with better statistics was carried out to estimate accurately the expectation values of several many-body wave functions. Although a relatively clear crossover is observed, even for the plain Gutzwiller wave function, the states in both regimes are metallic. Meanwhile, a substantial metal-insulator transition takes place at ∣U∣∼W (band width) for an improved wave function, into which an intersite correlation is introduced by taking account of virtual states in the second-order perturbation in the infinite ∣U∣ limit. The critical value compares favorably with recent results of the dynamical-mean-field approximation. In contrast, a conventional Jastrow-type wave function scarcely improves the normal state. In addition, the issue of the Brinkman-Rice metal-insulator transition is reconsidered with much larger systems.

arXiv:cond-mat/0205281Progress of Theoretical PhysicsJul 1, 200245 citations

BCS to Bose crossover in anisotropic superconductors

J. P. Wallington, James F. Annett

In this work we use functional integral techniques to examine the nearest-neighbor attractive Hubbard model on a quasi-two-dimensional lattice. It is a simple phenomenological model for the high- ${T}_{c}$ cuprates that allows both extended (nonlocal) s- and d-wave singlet superconductivity as well as mixed symmetry states. The Hartree-Gor'kov mean-field theory of the model has a finite-temperature phase diagram which shows a transition from pure s-wave to pure d-wave superconductivity, via a mixed symmetry $s+id$ state, as a function of doping. Including Gaussian fluctuations we examine the crossover from weak-coupling BCS superconductivity to the strong-coupling Bose-Einstein condensation of composite s- or d-wave bosons and comment on the origin and symmetry of the pseudogap.In this work we use functional integral techniques to examine the nearest-neighbor attractive Hubbard model on a quasi-two-dimensional lattice. It is a simple phenomenological model for the high-${T}_{c}$ cuprates that allows both extended (nonlocal) s- and d-wave singlet superconductivity as well as mixed symmetry states. The Hartree-Gor'kov mean-field theory of the model has a finite-temperature phase diagram which shows a transition from pure s-wave to pure d-wave superconductivity, via a mixed symmetry $s+id$ state, as a function of doping. Including Gaussian fluctuations we examine the crossover from weak-coupling BCS superconductivity to the strong-coupling Bose-Einstein condensation of composite s- or d-wave bosons and comment on the origin and symmetry of the pseudogap.

arXiv:cond-mat/9909043Physical review. B, Condensed matterJan 1, 200026 citations

Superconducting phase coherence and pairing gap in the three-dimensional attractive Hubbard model

T. K. Kopeć

We consider a lattice version of the ``negative- $U''$ Hubbard model in three dimensions, to study the crossover from a superconducting phase transition (dominated by phase fluctuation) to a regime where the amplitude of the superconducting order parameter controls the critical temperature. Starting from a fermionic Hamiltonian, we give a microscopic derivation of the effective action in terms of the relevant physical fields: the modulus and phase of the superconducting order parameter. Furthermore, by employing to the resulting coarse--grained quantum $\mathrm{XY}$ model the phase fluctuation algebra between number and phase operators (given by the Euclidean group ${E}_{2}),$ we map the ``phase-only'' action onto a solvable quantum spherical model. We establish the self--consistent theory (involving both fermionic and bosonic degrees of freedom), and calculate the superconducting phase coherence transition temperature ${T}_{c}$ as a function of the coupling strength $|U|,$ exhibiting a maximum where the behavior crosses over from BCS to Bose-Einstein (BE) condensation. We examine a pseudogap in the normal state which emerges naturally as a precursor of superconductivity at the second characteristic temperature ${T}_{g}$ due to a state with bound pairs but without long--range phase coherence. Furthermore, we demonstrate that the reduced gaps of the model $2{\ensuremath{\Delta}}_{0}{/k}_{B}{T}_{g}\ensuremath{\sim}4$ is almost independent on the pairing strength $|U|/t,$ whereas $2{\ensuremath{\Delta}}_{0}{/k}_{B}{T}_{c}$ increases with increasing $|U|$ ---in striking analogy with the phenomenology of the high--temperature cuprate superconductors when interpreted in terms of the BCS-BC crossover scenario.We consider a lattice version of the ``negative-$U''$ Hubbard model in three dimensions, to study the crossover from a superconducting phase transition (dominated by phase fluctuation) to a regime where the amplitude of the superconducting order parameter controls the critical temperature. Starting from a fermionic Hamiltonian, we give a microscopic derivation of the effective action in terms of the relevant physical fields: the modulus and phase of the superconducting order parameter. Furthermore, by employing to the resulting coarse--grained quantum $\mathrm{XY}$ model the phase fluctuation algebra between number and phase operators (given by the Euclidean group ${E}_{2}),$ we map the ``phase-only'' action onto a solvable quantum spherical model. We establish the self--consistent theory (involving both fermionic and bosonic degrees of freedom), and calculate the superconducting phase coherence transition temperature ${T}_{c}$ as a function of the coupling strength $|U|,$ exhibiting a maximum where the behavior crosses over from BCS to Bose-Einstein (BE) condensation. We examine a pseudogap in the normal state which emerges naturally as a precursor of superconductivity at the second characteristic temperature ${T}_{g}$ due to a state with bound pairs but without long--range phase coherence. Furthermore, we demonstrate that the reduced gaps of the model $2{\ensuremath{\Delta}}_{0}{/k}_{B}{T}_{g}\ensuremath{\sim}4$ is almost independent on the pairing strength $|U|/t,$ whereas $2{\ensuremath{\Delta}}_{0}{/k}_{B}{T}_{c}$ increases with increasing $|U|$---in striking analogy with the phenomenology of the high--temperature cuprate superconductors when interpreted in terms of the BCS-BC crossover scenario.

Physical review. B, Condensed matterJan 1, 200211 citations

|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.

U

for the cuprates should be larger than the band width, for which the t-J model is valid. Analyzing the kinetic energy, we reveal that for U>U_co only doped holes (electrons) become charge carriers, which will make a small Fermi surface (hole pocket), but for UU_co is determined by two factors, i.e., the AF spin correlation, which creates singlet pairs (pseudogap), and the charge mobility dominated by Mott physics.

In this work we use functional integral techniques to examine the nearest-neighbor attractive Hubbard model on a quasi-two-dimensional lattice. It is a simple phenomenological model for the high-

{T}_{c}

cuprates that allows both extended (nonlocal) s- and d-wave singlet superconductivity as well as mixed symmetry states. The Hartree-Gor'kov mean-field theory of the model has a finite-temperature phase diagram which shows a transition from pure s-wave to pure d-wave superconductivity, via a mixed symmetry

s+id

state, as a function of doping. Including Gaussian fluctuations we examine the crossover from weak-coupling BCS superconductivity to the strong-coupling Bose-Einstein condensation of composite s- or d-wave bosons and comment on the origin and symmetry of the pseudogap.

We consider a lattice version of the ``negative-

U''

Hubbard model in three dimensions, to study the crossover from a superconducting phase transition (dominated by phase fluctuation) to a regime where the amplitude of the superconducting order parameter controls the critical temperature. Starting from a fermionic Hamiltonian, we give a microscopic derivation of the effective action in terms of the relevant physical fields: the modulus and phase of the superconducting order parameter. Furthermore, by employing to the resulting coarse--grained quantum

\mathrm{XY}

model the phase fluctuation algebra between number and phase operators (given by the Euclidean group

{E}_{2}),

we map the ``phase-only'' action onto a solvable quantum spherical model. We establish the self--consistent theory (involving both fermionic and bosonic degrees of freedom), and calculate the superconducting phase coherence transition temperature

{T}_{c}

as a function of the coupling strength

|U|,

exhibiting a maximum where the behavior crosses over from BCS to Bose-Einstein (BE) condensation. We examine a pseudogap in the normal state which emerges naturally as a precursor of superconductivity at the second characteristic temperature

{T}_{g}

due to a state with bound pairs but without long--range phase coherence. Furthermore, we demonstrate that the reduced gaps of the model $2{\ensuremath{\Delta}}_{0}{/k}_{B}{T}_{g}\ensuremath{\sim}4$ is almost independent on the pairing strength

|U|/t,

whereas $2{\ensuremath{\Delta}}_{0}{/k}_{B}{T}_{c}$ increases with increasing

|U|

---in striking analogy with the phenomenology of the high--temperature cuprate superconductors when interpreted in terms of the BCS-BC crossover scenario.