arXiv:0804.2388

d-wave superconductivity in Hubbard model on the square lattice perturbed by weak 3D uniaxial anisotropy

J. M. P. Carmelo

Published Apr 15, 2008• 0 citations

Abstract

The Hubbard model on a square lattice is one of the most studied condensed-matter quantum problems.Here we find evidence that for intermediate $U/4t$ values and a hole-concentration range $x\in (x_c,x_*)$ the ground state of the Hubbard model on the square lattice perturbed by weak three-dimensional (3D) uniaxial anisotropy has long-range d-wave superconducting order. Here $t$ is the effective nearest-neighbor transfer integral and $U$ the effective on-site repulsion. The lower critical concentration $x_c$ involves the Ginzburg number Gi and is approximately given by $x_c\approx {\rm Gi}+x_0\approx 0.05$. Here $x_0

DOI: 10.48550/arxiv.0804.2388

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arXiv:0804.2388

d-wave superconductivity in Hubbard model on the square lattice perturbed by weak 3D uniaxial anisotropy

J. M. P. Carmelo

Published Apr 15, 2008• 0 citations

arXiv HTML PDF

Abstract

The Hubbard model on a square lattice is one of the most studied condensed-matter quantum problems.Here we find evidence that for intermediate

U/4t

values and a hole-concentration range

x\in (x_c,x_*)

the ground state of the Hubbard model on the square lattice perturbed by weak three-dimensional (3D) uniaxial anisotropy has long-range d-wave superconducting order. Here

t

is the effective nearest-neighbor transfer integral and

U

the effective on-site repulsion. The lower critical concentration

x_c

involves the Ginzburg number Gi and is approximately given by

x_c\approx {\rm Gi}+x_0\approx 0.05

. Here $x_0

DOI: 10.48550/arxiv.0804.2388

Related papers

d-wave superconductivity in the virtual-electron pair quantum liquid

J. M. P. Carmelo

We find evidence that for zero spin density $m=0$ , intermediate $U/4t$ values, and a range $x\in (x_c,x_*)$ of finite hole concentrations the ground state of the virtual-electron pair quantum liquid obtained from perturbing the square-lattice quantum liquid of Ref. \cite{companion2} by weak three-dimensional (3D) uniaxial anisotropy and intrinsic disorder has long-range d-wave superconducting order. Here $t$ is the effective nearest-neighbor transfer integral and $U$ the effective on-site repulsion. The long-range d-wave superconducting order emerges below a critical temperature $T_c$ for a hole concentration range centered at $x_{op}= (x_c+x_*)/2$ . It results from the effects of the residual interactions of the charge $c$ fermions and spin-neutral two-spinon $s1$ fermions of Ref. \cite{companion2}, as a by-product of the short-range spin correlations. Rather than the U(1) symmetry contained in the $η$ -spin SU(2) symmetry of the $SO(4) = [SU(2)\times SU(2)]/Z_2$ symmetry, the U(1) symmetry broken at $T_c$ is the $c$ fermion U(1) symmetry of Ref. \cite{bipartite}. It is contained in the extended global $SO(3)\times SO(3)\times U(1)=[SO(4)\times U(1)]/Z_2$ symmetry of the Hubbard model on the square lattice. Our preliminary results seem to indicate that combining the electronic correlations described by the square-lattice quantum liquid perturbed by 3D uniaxial anisotropy with the very weak effects of intrinsic disorder or superfluid-density anisotropy leads for the hole-concentration range $x\in (x_c,x_*)$ to a successful description of the universal properties of the hole-doped cuprate superconductors.We find evidence that for zero spin density $m=0$, intermediate $U/4t$ values, and a range $x\in (x_c,x_*)$ of finite hole concentrations the ground state of the virtual-electron pair quantum liquid obtained from perturbing the square-lattice quantum liquid of Ref. \cite{companion2} by weak three-dimensional (3D) uniaxial anisotropy and intrinsic disorder has long-range d-wave superconducting order. Here $t$ is the effective nearest-neighbor transfer integral and $U$ the effective on-site repulsion. The long-range d-wave superconducting order emerges below a critical temperature $T_c$ for a hole concentration range centered at $x_{op}= (x_c+x_*)/2$. It results from the effects of the residual interactions of the charge $c$ fermions and spin-neutral two-spinon $s1$ fermions of Ref. \cite{companion2}, as a by-product of the short-range spin correlations. Rather than the U(1) symmetry contained in the $η$-spin SU(2) symmetry of the $SO(4) = [SU(2)\times SU(2)]/Z_2$ symmetry, the U(1) symmetry broken at $T_c$ is the $c$ fermion U(1) symmetry of Ref. \cite{bipartite}. It is contained in the extended global $SO(3)\times SO(3)\times U(1)=[SO(4)\times U(1)]/Z_2$ symmetry of the Hubbard model on the square lattice. Our preliminary results seem to indicate that combining the electronic correlations described by the square-lattice quantum liquid perturbed by 3D uniaxial anisotropy with the very weak effects of intrinsic disorder or superfluid-density anisotropy leads for the hole-concentration range $x\in (x_c,x_*)$ to a successful description of the universal properties of the hole-doped cuprate superconductors.

arXiv:1004.0923Apr 6, 20100 citations

A description of the Hubbard model on a square lattice consistent with its global $SO(3)\times SO(3)\times U(1)$ symmetryA description of the Hubbard model on a square lattice consistent with its global $SO(3)\times SO(3)\times U(1)$ symmetry

J. M. P. Carmelo, M. J. Sampaio

In this paper a description of the Hubbard model on the square lattice with nearest-neighbor transfer integral $t$ , on-site repulsion $U$ , and $N_a^2\gg 1$ sites consistent with its exact global $SO(3)\times SO(3)\times U(1)$ symmetry is constructed. Our studies profit from the interplay of that recently found global symmetry of the model on any bipartite lattice with the transformation laws under a suitable electron - rotated-electron unitary transformation of a well-defined set of operators and quantum objects. For $U/4t>0$ the occupancy configurations of these objects generate the energy eigenstates that span the one- and two-electron subspace. Such a subspace as defined in this paper contains nearly the whole spectral weight of the excitations generated by application onto the zero-spin-density ground state of one- and two-electron operators. Our description involves three basic objects: charge $c$ fermions, spin-1/2 spinons, and $η$ -spin-1/2 $η$ -spinons. Alike in chromodynamics the quarks have color but all quark-composite physical particles are color-neutral, the $η$ -spinon (and spinons) that are not invariant under that transformation have $η$ spin 1/2 (and spin 1/2) but are part of $η$ -spin-neutral (and spin-neutral) $2ν$ - $η$ -spinon (and $2ν$ -spinon) composite $ην$ fermions (and $sν$ fermions) where $ν=1,2,...$ is the number of $η$ -spinon (and spinon) pairs.The description introduced here is consistent with a Mott-Hubbard insulating ground state with antiferromagnetic long-range order for half filling at $x=0$ hole concentration and a ground state with short-range spin order for a well-defined range of finite hole concentrations $x>0$ .In this paper a description of the Hubbard model on the square lattice with nearest-neighbor transfer integral $t$, on-site repulsion $U$, and $N_a^2\gg 1$ sites consistent with its exact global $SO(3)\times SO(3)\times U(1)$ symmetry is constructed. Our studies profit from the interplay of that recently found global symmetry of the model on any bipartite lattice with the transformation laws under a suitable electron - rotated-electron unitary transformation of a well-defined set of operators and quantum objects. For $U/4t>0$ the occupancy configurations of these objects generate the energy eigenstates that span the one- and two-electron subspace. Such a subspace as defined in this paper contains nearly the whole spectral weight of the excitations generated by application onto the zero-spin-density ground state of one- and two-electron operators. Our description involves three basic objects: charge $c$ fermions, spin-1/2 spinons, and $η$-spin-1/2 $η$-spinons. Alike in chromodynamics the quarks have color but all quark-composite physical particles are color-neutral, the $η$-spinon (and spinons) that are not invariant under that transformation have $η$ spin 1/2 (and spin 1/2) but are part of $η$-spin-neutral (and spin-neutral) $2ν$-$η$-spinon (and $2ν$-spinon) composite $ην$ fermions (and $sν$ fermions) where $ν=1,2,...$ is the number of $η$-spinon (and spinon) pairs.The description introduced here is consistent with a Mott-Hubbard insulating ground state with antiferromagnetic long-range order for half filling at $x=0$ hole concentration and a ground state with short-range spin order for a well-defined range of finite hole concentrations $x>0$.

arXiv:0804.2358Apr 15, 20080 citations

Phase diagram of thet–U2Hamiltonian of the weak coupling Hubbard model

Takashi Yanagisawa

We determine the symmetry of Cooper pairs, on the basis of the perturbation theory in terms of the Coulomb interaction U , for the two-dimensional Hubbard model on the square lattice. The phase diagram is investigated in detail. The Hubbard model for small U is mapped on to an effective Hamiltonian with the attractive interaction using the canonical transformation: H eff = e S H e − S . The gap equation of the weak coupling formulation is solved without numerical ambiguity to determine the symmetry of Cooper pairs. The superconducting gap crucially depends on the position of the van Hove singularity. We show the phase diagram in the plane of the electron filling n e and the next nearest-neighbor transfer t '. The d-wave pairing is dominant for the square lattice in a wide range of n e and t '. The d-wave pairing is also stable for the square lattice with anisotropic t '. The three-band d–p model is also investigated, for which the d-wave pairing is stable in a wide range of n e and t pp (the transfer between neighboring oxygen atoms). In the weak coupling analysis, the second-neighbor transfer parameter - t ' could not be so large so that the optimum doping rate is in the range of 0.8 < n e < 0.85.We determine the symmetry of Cooper pairs, on the basis of the perturbation theory in terms of the Coulomb interaction U , for the two-dimensional Hubbard model on the square lattice. The phase diagram is investigated in detail. The Hubbard model for small U is mapped on to an effective Hamiltonian with the attractive interaction using the canonical transformation: H eff = e S H e − S . The gap equation of the weak coupling formulation is solved without numerical ambiguity to determine the symmetry of Cooper pairs. The superconducting gap crucially depends on the position of the van Hove singularity. We show the phase diagram in the plane of the electron filling n e and the next nearest-neighbor transfer t '. The d-wave pairing is dominant for the square lattice in a wide range of n e and t '. The d-wave pairing is also stable for the square lattice with anisotropic t '. The three-band d–p model is also investigated, for which the d-wave pairing is stable in a wide range of n e and t pp (the transfer between neighboring oxygen atoms). In the weak coupling analysis, the second-neighbor transfer parameter - t ' could not be so large so that the optimum doping rate is in the range of 0.8 < n e < 0.85.

arXiv:0803.1739New Journal of PhysicsFeb 12, 200831 citations

Quantum criticality and correlations in the cuprate superconductors

J. M. P. Carmelo

A description of the electronic correlations contained in the Hubbard model on the square-lattice perturbed by very weak three-dimensional uniaxial anisotropy in terms of the residual interactions of charge $c$ fermions and spin-neutral composite two-spinon $s1$ fermions is used to access further information on the origin of quantum critical behavior in the hole-doped cuprate superconductors. Excellent quantitative agreement with their anisotropic linear- $ω$ one-electron scattering rate and normal-state linear- $T$ resistivity is achieved. Our results provide strong evidence that the normal-state linear- $T$ resistivity is indeed a manifestation of low-temperature scale-invariant physics.A description of the electronic correlations contained in the Hubbard model on the square-lattice perturbed by very weak three-dimensional uniaxial anisotropy in terms of the residual interactions of charge $c$ fermions and spin-neutral composite two-spinon $s1$ fermions is used to access further information on the origin of quantum critical behavior in the hole-doped cuprate superconductors. Excellent quantitative agreement with their anisotropic linear-$ω$ one-electron scattering rate and normal-state linear-$T$ resistivity is achieved. Our results provide strong evidence that the normal-state linear-$T$ resistivity is indeed a manifestation of low-temperature scale-invariant physics.

arXiv:1004.1164Apr 7, 20100 citations

Fidelity study of the superconducting phase diagram in the two-dimensional single-band Hubbard model

Chunjing Jia, Brian Moritz, Cheng-Chien Chen +2 more

Extensive numerical studies have demonstrated that the two-dimensional single-band Hubbard model contains much of the key physics in cuprate high-temperature superconductors. However, there is no definitive proof that the Hubbard model truly possesses a superconducting ground state or, if it does, of how it depends on model parameters. To answer these longstanding questions, we study an extension of the Hubbard model including an infinite-range $d$ -wave pair field term, which precipitates a superconducting state in the $d$ -wave channel. Using exact diagonalization on 16-site square clusters, we study the evolution of the ground state as a function of the strength of the pairing term. This is achieved by monitoring the fidelity metric of the ground state, as well as determining the ratio between the two largest eigenvalues of the $d$ -wave pair/spin/charge-density matrices. The calculations show a $d$ -wave superconducting ground state in doped clusters bracketed by a strong antiferromagnetic state at half filling controlled by the Coulomb repulsion $U$ and a weak short-range checkerboard charge ordered state at larger hole doping controlled by the next-nearest-neighbor hopping ${t}^{\ensuremath{'}}$. We also demonstrate that negative ${t}^{\ensuremath{'}}$ plays an important role in facilitating $d$ -wave superconductivity.Extensive numerical studies have demonstrated that the two-dimensional single-band Hubbard model contains much of the key physics in cuprate high-temperature superconductors. However, there is no definitive proof that the Hubbard model truly possesses a superconducting ground state or, if it does, of how it depends on model parameters. To answer these longstanding questions, we study an extension of the Hubbard model including an infinite-range $d$-wave pair field term, which precipitates a superconducting state in the $d$-wave channel. Using exact diagonalization on 16-site square clusters, we study the evolution of the ground state as a function of the strength of the pairing term. This is achieved by monitoring the fidelity metric of the ground state, as well as determining the ratio between the two largest eigenvalues of the $d$-wave pair/spin/charge-density matrices. The calculations show a $d$-wave superconducting ground state in doped clusters bracketed by a strong antiferromagnetic state at half filling controlled by the Coulomb repulsion $U$ and a weak short-range checkerboard charge ordered state at larger hole doping controlled by the next-nearest-neighbor hopping ${t}^{\ensuremath{'}}$. We also demonstrate that negative ${t}^{\ensuremath{'}}$ plays an important role in facilitating $d$-wave superconductivity.

arXiv:1012.4013Physical Review BSep 9, 201124 citations

One-electron scattering rate and normal-state linear- $T$ resistivity of the cupratesOne-electron scattering rate and normal-state linear-$T$ resistivity of the cuprates

J. M. P. Carmelo

Here we use a description of the electronic correlations contained in the Hubbard model on the square-lattice perturbed by very weak three-dimensional uniaxial anisotropy in terms of the residual interactions of charge $c$ fermions and spin-neutral composite two-spinon $s1$ fermions. Excellent quantitative agreement with the anisotropic linear- $ω$ one-electron scattering rate and normal-state linear- $T$ resistivity observed in experiments on hole-doped cuprates with critical concentrations $x_c\approx 0.05$ and $x_*\approx 0.27$ is achieved. Our results provide strong evidence that the normal-state linear- $T$ resistivity is a manifestation of low-temperature scale-invariant physics.Here we use a description of the electronic correlations contained in the Hubbard model on the square-lattice perturbed by very weak three-dimensional uniaxial anisotropy in terms of the residual interactions of charge $c$ fermions and spin-neutral composite two-spinon $s1$ fermions. Excellent quantitative agreement with the anisotropic linear-$ω$ one-electron scattering rate and normal-state linear-$T$ resistivity observed in experiments on hole-doped cuprates with critical concentrations $x_c\approx 0.05$ and $x_*\approx 0.27$ is achieved. Our results provide strong evidence that the normal-state linear-$T$ resistivity is a manifestation of low-temperature scale-invariant physics.

arXiv:0804.2393Apr 15, 20080 citations

The Hubbard model on a square lattice is one of the most studied condensed-matter quantum problems.Here we find evidence that for intermediate

U/4t

values and a hole-concentration range

x\in (x_c,x_*)

the ground state of the Hubbard model on the square lattice perturbed by weak three-dimensional (3D) uniaxial anisotropy has long-range d-wave superconducting order. Here

t

is the effective nearest-neighbor transfer integral and

U

the effective on-site repulsion. The lower critical concentration

x_c

involves the Ginzburg number Gi and is approximately given by

x_c\approx {\rm Gi}+x_0\approx 0.05

. Here $x_0

We find evidence that for zero spin density

m=0

, intermediate

U/4t

values, and a range

x\in (x_c,x_*)

of finite hole concentrations the ground state of the virtual-electron pair quantum liquid obtained from perturbing the square-lattice quantum liquid of Ref. \cite{companion2} by weak three-dimensional (3D) uniaxial anisotropy and intrinsic disorder has long-range d-wave superconducting order. Here

t

is the effective nearest-neighbor transfer integral and

U

the effective on-site repulsion. The long-range d-wave superconducting order emerges below a critical temperature

T_c

for a hole concentration range centered at

x_{op}= (x_c+x_*)/2

. It results from the effects of the residual interactions of the charge

c

fermions and spin-neutral two-spinon

s1

fermions of Ref. \cite{companion2}, as a by-product of the short-range spin correlations. Rather than the U(1) symmetry contained in the

η

-spin SU(2) symmetry of the

SO(4) = [SU(2)\times SU(2)]/Z_2

symmetry, the U(1) symmetry broken at

T_c

is the

c

fermion U(1) symmetry of Ref. \cite{bipartite}. It is contained in the extended global

SO(3)\times SO(3)\times U(1)=[SO(4)\times U(1)]/Z_2

symmetry of the Hubbard model on the square lattice. Our preliminary results seem to indicate that combining the electronic correlations described by the square-lattice quantum liquid perturbed by 3D uniaxial anisotropy with the very weak effects of intrinsic disorder or superfluid-density anisotropy leads for the hole-concentration range

x\in (x_c,x_*)

to a successful description of the universal properties of the hole-doped cuprate superconductors.

A description of the Hubbard model on a square lattice consistent with its global

SO(3)\times SO(3)\times U(1)

symmetry

In this paper a description of the Hubbard model on the square lattice with nearest-neighbor transfer integral

t

, on-site repulsion

U

, and

N_a^2\gg 1

sites consistent with its exact global

SO(3)\times SO(3)\times U(1)

symmetry is constructed. Our studies profit from the interplay of that recently found global symmetry of the model on any bipartite lattice with the transformation laws under a suitable electron - rotated-electron unitary transformation of a well-defined set of operators and quantum objects. For

U/4t>0

the occupancy configurations of these objects generate the energy eigenstates that span the one- and two-electron subspace. Such a subspace as defined in this paper contains nearly the whole spectral weight of the excitations generated by application onto the zero-spin-density ground state of one- and two-electron operators. Our description involves three basic objects: charge

c

fermions, spin-1/2 spinons, and

η

-spin-1/2

η

-spinons. Alike in chromodynamics the quarks have color but all quark-composite physical particles are color-neutral, the

η

-spinon (and spinons) that are not invariant under that transformation have

η

spin 1/2 (and spin 1/2) but are part of

η

-spin-neutral (and spin-neutral)

2ν

η

-spinon (and

2ν

-spinon) composite

ην

fermions (and

sν

fermions) where

ν=1,2,...

is the number of

η

-spinon (and spinon) pairs.The description introduced here is consistent with a Mott-Hubbard insulating ground state with antiferromagnetic long-range order for half filling at

x=0

hole concentration and a ground state with short-range spin order for a well-defined range of finite hole concentrations

x>0

c

fermions and spin-neutral composite two-spinon

s1

fermions is used to access further information on the origin of quantum critical behavior in the hole-doped cuprate superconductors. Excellent quantitative agreement with their anisotropic linear-

ω

one-electron scattering rate and normal-state linear-

T

resistivity is achieved. Our results provide strong evidence that the normal-state linear-

T

resistivity is indeed a manifestation of low-temperature scale-invariant physics.

d

-wave pair field term, which precipitates a superconducting state in the

d

-wave channel. Using exact diagonalization on 16-site square clusters, we study the evolution of the ground state as a function of the strength of the pairing term. This is achieved by monitoring the fidelity metric of the ground state, as well as determining the ratio between the two largest eigenvalues of the

d

-wave pair/spin/charge-density matrices. The calculations show a

d

-wave superconducting ground state in doped clusters bracketed by a strong antiferromagnetic state at half filling controlled by the Coulomb repulsion

U

and a weak short-range checkerboard charge ordered state at larger hole doping controlled by the next-nearest-neighbor hopping ${t}^{\ensuremath{'}}$. We also demonstrate that negative ${t}^{\ensuremath{'}}$ plays an important role in facilitating

d

-wave superconductivity.

One-electron scattering rate and normal-state linear-

T

resistivity of the cuprates

c

fermions and spin-neutral composite two-spinon

s1

fermions. Excellent quantitative agreement with the anisotropic linear-

ω

one-electron scattering rate and normal-state linear-

T

resistivity observed in experiments on hole-doped cuprates with critical concentrations

x_c\approx 0.05

and

x_*\approx 0.27

is achieved. Our results provide strong evidence that the normal-state linear-

T

resistivity is a manifestation of low-temperature scale-invariant physics.