Physical review. B, Condensed matter

Disordereds-,d-, andp-wave superconductors: Exact results in infinite dimensions

Rostam Moradian, James F. Annett, Balazs L. Gyorffy

Published Aug 1, 2000• 13 citations

Abstract

The coherent potential approximation (CPA) has recently been applied to disordered s-wave, p-wave and d-wave superconductors. Here we show that for systems with hypercubic lattices and generalized local s-wave, d-wave, extended s-wave or p-wave superconductivity the exact average Green function is equal to the CPA Green function in the limit where the number of dimensions goes to infinity. As in the case of normal systems, the $\stackrel{\ensuremath{\rightarrow}}{d}\ensuremath{\infty}$ limit is regularized by demanding that the second moment of the quasiparticle density of states be finite. Surprisingly, this condition leads to the scalings ${t}_{\mathrm{ij}}{=t}^{*}/\sqrt{d}$ and ${\ensuremath{\Delta}}_{\mathrm{ij}}={\ensuremath{\Delta}}^{*}/\sqrt{d}$ for the hopping integral and the nonlocal pairing potential, respectively, but does not require scaling of the effective electron-electron interaction constant, ${U}_{\mathrm{ij}}$ even if i and j refer to different sites.

DOI: 10.1103/physrevb.62.3508

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Disordered two-dimensional superconductors: Roles of temperature and interaction strength

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Physical review. B, Condensed matter

Disordereds-,d-, andp-wave superconductors: Exact results in infinite dimensions

Rostam Moradian, James F. Annett, Balazs L. Gyorffy

Published Aug 1, 2000• 13 citations

Publisher

Abstract

\stackrel{\ensuremath{\rightarrow}}{d}\ensuremath{\infty}

limit is regularized by demanding that the second moment of the quasiparticle density of states be finite. Surprisingly, this condition leads to the scalings

{t}_{\mathrm{ij}}{=t}^{*}/\sqrt{d}

and

{\ensuremath{\Delta}}_{\mathrm{ij}}={\ensuremath{\Delta}}^{*}/\sqrt{d}

for the hopping integral and the nonlocal pairing potential, respectively, but does not require scaling of the effective electron-electron interaction constant,

{U}_{\mathrm{ij}}

even if i and j refer to different sites.

DOI: 10.1103/physrevb.62.3508

Related papers

Disordered two-dimensional superconductors: Roles of temperature and interaction strength

Felipe Mondaini, Thereza Paiva, Raimundo R. dos Santos +1 more

We have considered the half-filled disordered attractive Hubbard model on a square lattice, in which the on-site attraction is switched off on a fraction $f$ of sites, while keeping a finite $U$ on the remaining ones. Through quantum Monte Carlo simulations for several values of $f$ and $U$ and for system sizes ranging from $8\ifmmode\times\else\texttimes\fi{}8$ to $16\ifmmode\times\else\texttimes\fi{}16$, we have calculated the configurational averages of the equal-time pair structure factor ${P}_{s}$ and, for a more restricted set of variables, the helicity modulus ${\ensuremath{\rho}}_{s}$, as functions of temperature. Two finite-size scaling Ans\"atze for ${P}_{s}$ have been used: one for zero temperature and the other for finite temperatures. We have found that the system sustains superconductivity in the ground state up to a critical impurity concentration ${f}_{c}$ , which increases with $U$ , at least up to $U=4$ (in units of the hopping energy). Also, the normalized zero-temperature gap as a function of $f$ shows a maximum near $f\ensuremath{\sim}0.07$ for $2\ensuremath{\lesssim}U\ensuremath{\lesssim}6$. Analyses of the helicity modulus and of the pair structure factor led to the determination of the critical temperature as a function of $f$ for $U=3$ , 4, and 6: they also show maxima near $f\ensuremath{\sim}0.07$, with the highest ${T}_{c}$ increasing with $U$ in this range. We argue that, overall, the observed behavior results from both the breakdown of charge-density-wave-superconductivity degeneracy and the fact that free sites tend to ``push'' electrons toward attractive sites; the latter effect being more drastic at weak couplings.We have considered the half-filled disordered attractive Hubbard model on a square lattice, in which the on-site attraction is switched off on a fraction $f$ of sites, while keeping a finite $U$ on the remaining ones. Through quantum Monte Carlo simulations for several values of $f$ and $U$ and for system sizes ranging from $8\ifmmode\times\else\texttimes\fi{}8$ to $16\ifmmode\times\else\texttimes\fi{}16$, we have calculated the configurational averages of the equal-time pair structure factor ${P}_{s}$ and, for a more restricted set of variables, the helicity modulus ${\ensuremath{\rho}}_{s}$, as functions of temperature. Two finite-size scaling Ans\"atze for ${P}_{s}$ have been used: one for zero temperature and the other for finite temperatures. We have found that the system sustains superconductivity in the ground state up to a critical impurity concentration ${f}_{c}$, which increases with $U$, at least up to $U=4$ (in units of the hopping energy). Also, the normalized zero-temperature gap as a function of $f$ shows a maximum near $f\ensuremath{\sim}0.07$ for $2\ensuremath{\lesssim}U\ensuremath{\lesssim}6$. Analyses of the helicity modulus and of the pair structure factor led to the determination of the critical temperature as a function of $f$ for $U=3$, 4, and 6: they also show maxima near $f\ensuremath{\sim}0.07$, with the highest ${T}_{c}$ increasing with $U$ in this range. We argue that, overall, the observed behavior results from both the breakdown of charge-density-wave-superconductivity degeneracy and the fact that free sites tend to ``push'' electrons toward attractive sites; the latter effect being more drastic at weak couplings.

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Grzegorz Litak A

arXiv:cond-mat/0106315Jun 16, 20010 citations

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V. Janiš

I review the quantum theory of the electron moving in a random environment. First, the quantum mechanics of individual particles scattered on a random potential is discussed. The quantum-mechanical description is extended to many-body systems by using many-body Green functions. The many-body approach is used to derive the coherent-potential approximation and to show how it fits into the dynamical mean-field theory. The generating functional of the coherent-potential approximation is obtained in an analytic form from the limit to infinite dimensions of the general many-body description of non-interacting electrons in random lattices. The analytic generating functional of the mean-field description of random systems is extended to the Falicov-Kimball model with thermally equilibrated scattering lattice potential. The many-body Green functions are then used to calculate transport properties. The electrical conductivity of the coherent-potential approximation is derived from the two-particle Green function calculated in infinite spatial dimensions. Finally, a perturbation theory for the vertex corrections to the mean-field conductivity is introduced. In particular, it is shown how to make the expansion beyond the local approximations conserving.I review the quantum theory of the electron moving in a random environment. First, the quantum mechanics of individual particles scattered on a random potential is discussed. The quantum-mechanical description is extended to many-body systems by using many-body Green functions. The many-body approach is used to derive the coherent-potential approximation and to show how it fits into the dynamical mean-field theory. The generating functional of the coherent-potential approximation is obtained in an analytic form from the limit to infinite dimensions of the general many-body description of non-interacting electrons in random lattices. The analytic generating functional of the mean-field description of random systems is extended to the Falicov-Kimball model with thermally equilibrated scattering lattice potential. The many-body Green functions are then used to calculate transport properties. The electrical conductivity of the coherent-potential approximation is derived from the two-particle Green function calculated in infinite spatial dimensions. Finally, a perturbation theory for the vertex corrections to the mean-field conductivity is introduced. In particular, it is shown how to make the expansion beyond the local approximations conserving.

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Grzegorz Litak A

Chinese Journal of PhysicsFeb 1, 20060 citations

Superconductivity in disordered materials

Fabio Siringo

— The real-space single-particle Green’s function formalism is extended to the attractive disordered Hubbard Hamiltonian by matrix Nambu notation, thus providing a general approach to the superconductive instability of a disordered system in quantum mean-field approximation. Configurational averages may be performed by a recent method, already employed for the repulsive model, thus allowing for spatial fluctuations of the local order parameter. The method is shown to be a natural way of investigating the validity limit of the Anderson theorem. An exact analytical solution is provided for a Bethe lattice with no diagonal disorder and infinitely large connectivity.— The real-space single-particle Green’s function formalism is extended to the attractive disordered Hubbard Hamiltonian by matrix Nambu notation, thus providing a general approach to the superconductive instability of a disordered system in quantum mean-field approximation. Configurational averages may be performed by a recent method, already employed for the repulsive model, thus allowing for spatial fluctuations of the local order parameter. The method is shown to be a natural way of investigating the validity limit of the Anderson theorem. An exact analytical solution is provided for a Bethe lattice with no diagonal disorder and infinitely large connectivity.

Dec 1, 19970 citations

Impurity States and the Absence of Quasiparticle Localization in Disorderedd-Wave Superconductors

Alexander V. Balatsky, M. I. Salkola

The absence of localization of impurity-induced low-energy quasiparticle states in a 2D d-wave superconductor is shown for any amount of disorder in the limit of unitary scatterers. This result follows from the fact that a unitary impurity produces a marginally bound state at zero energy which decays as a power law along the nodes of the d-wave energy gap. Consequently, for finite density of impurities, the impurity-induced states are coupled by long-range overlaps yielding extended quasiparticle states below a characteristic energy scale ${\ensuremath{\omega}}_{c}$. Simple scaling arguments suggest that ${\ensuremath{\omega}}_{c}\ensuremath{\propto}{e}^{{\ensuremath{-}c/n}_{\mathrm{imp}}}$, with ${n}_{\mathrm{imp}}$ the impurity density and c a positive constant.The absence of localization of impurity-induced low-energy quasiparticle states in a 2D d-wave superconductor is shown for any amount of disorder in the limit of unitary scatterers. This result follows from the fact that a unitary impurity produces a marginally bound state at zero energy which decays as a power law along the nodes of the d-wave energy gap. Consequently, for finite density of impurities, the impurity-induced states are coupled by long-range overlaps yielding extended quasiparticle states below a characteristic energy scale ${\ensuremath{\omega}}_{c}$. Simple scaling arguments suggest that ${\ensuremath{\omega}}_{c}\ensuremath{\propto}{e}^{{\ensuremath{-}c/n}_{\mathrm{imp}}}$, with ${n}_{\mathrm{imp}}$ the impurity density and c a positive constant.

arXiv:cond-mat/9602034Physical Review LettersMar 25, 199679 citations

\stackrel{\ensuremath{\rightarrow}}{d}\ensuremath{\infty}

limit is regularized by demanding that the second moment of the quasiparticle density of states be finite. Surprisingly, this condition leads to the scalings

{t}_{\mathrm{ij}}{=t}^{*}/\sqrt{d}

and

{\ensuremath{\Delta}}_{\mathrm{ij}}={\ensuremath{\Delta}}^{*}/\sqrt{d}

for the hopping integral and the nonlocal pairing potential, respectively, but does not require scaling of the effective electron-electron interaction constant,

{U}_{\mathrm{ij}}

even if i and j refer to different sites.

We have considered the half-filled disordered attractive Hubbard model on a square lattice, in which the on-site attraction is switched off on a fraction

f

of sites, while keeping a finite

U

on the remaining ones. Through quantum Monte Carlo simulations for several values of

f

and

U

and for system sizes ranging from $8\ifmmode\times\else\texttimes\fi{}8$ to $16\ifmmode\times\else\texttimes\fi{}16$, we have calculated the configurational averages of the equal-time pair structure factor

{P}_{s}

and, for a more restricted set of variables, the helicity modulus ${\ensuremath{\rho}}_{s}$, as functions of temperature. Two finite-size scaling Ans\"atze for

{P}_{s}

have been used: one for zero temperature and the other for finite temperatures. We have found that the system sustains superconductivity in the ground state up to a critical impurity concentration

{f}_{c}

, which increases with

U

, at least up to

U=4

(in units of the hopping energy). Also, the normalized zero-temperature gap as a function of

f

shows a maximum near $f\ensuremath{\sim}0.07$ for $2\ensuremath{\lesssim}U\ensuremath{\lesssim}6$. Analyses of the helicity modulus and of the pair structure factor led to the determination of the critical temperature as a function of

f

for

U=3

, 4, and 6: they also show maxima near $f\ensuremath{\sim}0.07$, with the highest

{T}_{c}

increasing with

U

in this range. We argue that, overall, the observed behavior results from both the breakdown of charge-density-wave-superconductivity degeneracy and the fact that free sites tend to ``push'' electrons toward attractive sites; the latter effect being more drastic at weak couplings.

{n}_{\mathrm{imp}}

the impurity density and c a positive constant.