Physical Review B

Modeling superconducting states of arbitrary pairing symmetry with the fluctuation exchange approximation applied to Hubbard-like models

J. J. Deisz, T. Slife

Published Sep 21, 2009• 3 citations

Abstract

We present an algorithm for microscopic modeling of superconducting states of arbitrary pairing symmetry via an implementation of self-consistent perturbation theory, the fluctuation exchange approximation (FLEX), applied to Hubbard-like models. Like the original formulation of FLEX, all ring diagrams that are generated with the bare interaction vertex are included in the generating functional for the self-energy. Thus, this algorithm is suitable for FLEX-based Hubbard model calculations in both the attractive and repulsive cases.

DOI: 10.1103/physrevb.80.094516

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Physical Review B

Modeling superconducting states of arbitrary pairing symmetry with the fluctuation exchange approximation applied to Hubbard-like models

J. J. Deisz, T. Slife

Published Sep 21, 2009• 3 citations

Publisher

Abstract

DOI: 10.1103/physrevb.80.094516

Related papers

FLEX+DMFT approach to the d -wave superconducting phase diagram of the two-dimensional Hubbard model

Motoharu Kitatani, Naoto Tsuji, Hideo Aoki

The dynamical mean-field theory (DMFT) combined with the fluctuation exchange (FLEX) method, namely, FLEX+DMFT, is an approach for correlated electron systems to incorporate both local and nonlocal long-range correlations in a self-consistent manner. We formulate FLEX+DMFT in a systematic way starting from a Luttinger-Ward functional, and apply it to study the $d$ -wave superconductivity in the two-dimensional repulsive Hubbard model. The critical temperature $({T}_{c})$ curve obtained in the FLEX+DMFT exhibits a dome structure as a function of the filling, which has not been clearly observed in the FLEX approach alone. We trace back the origin of the dome to the local vertex correction from DMFT that renders a filling dependence in the FLEX self-energy. We compare the results with those of GW+DMFT, where the ${T}_{c}$ -dome structure is qualitatively reproduced due to the same vertex correction effect, but a crucial difference from FLEX+DMFT is that ${T}_{c}$ is always estimated below the N\'eel temperature in GW+DMFT. The single-particle spectral function obtained with FLEX+DMFT exhibits a double-peak structure as a precursor of the Hubbard bands at temperatures above ${T}_{c}$ .The dynamical mean-field theory (DMFT) combined with the fluctuation exchange (FLEX) method, namely, FLEX+DMFT, is an approach for correlated electron systems to incorporate both local and nonlocal long-range correlations in a self-consistent manner. We formulate FLEX+DMFT in a systematic way starting from a Luttinger-Ward functional, and apply it to study the $d$-wave superconductivity in the two-dimensional repulsive Hubbard model. The critical temperature $({T}_{c})$ curve obtained in the FLEX+DMFT exhibits a dome structure as a function of the filling, which has not been clearly observed in the FLEX approach alone. We trace back the origin of the dome to the local vertex correction from DMFT that renders a filling dependence in the FLEX self-energy. We compare the results with those of GW+DMFT, where the ${T}_{c}$-dome structure is qualitatively reproduced due to the same vertex correction effect, but a crucial difference from FLEX+DMFT is that ${T}_{c}$ is always estimated below the N\'eel temperature in GW+DMFT. The single-particle spectral function obtained with FLEX+DMFT exhibits a double-peak structure as a precursor of the Hubbard bands at temperatures above ${T}_{c}$.

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José Jesús Rodríguez-Núñez, Sandra Schafroth

We implement the numerical method of summing Green function diagrams on the Matsubara frequency axis for the fluctuation exchange (FLEX) approximation. Our method has previously been applied to the attractive Hubbard model for low density. Here we apply our numerical algorithm to the Hubbard model close to half filling (ρ =0.40), and for T/t = 0.03, in order to study the dynamics of one- and two-particle Green functions. For the values of the chosen parameters we see the formation of three branches which we associate with the two-peak structure in the imaginary part of the self-energy. From the imaginary part of the self-energy we conclude that our system is a Fermi liquid (for the temperature investigated here), since Im Σ( k , ω) ≈ w2 around the chemical potential. We have compared our fully self-consistent FLEX solutions with a lower order approximation where the internal Green functions are approximated by free Green functions. These two approches, i.e., the fully self-consistent and the non-self-consistent ones give different results for the parameters considered here. However, they have similar global results for small densities.We implement the numerical method of summing Green function diagrams on the Matsubara frequency axis for the fluctuation exchange (FLEX) approximation. Our method has previously been applied to the attractive Hubbard model for low density. Here we apply our numerical algorithm to the Hubbard model close to half filling (ρ =0.40), and for T/t = 0.03, in order to study the dynamics of one- and two-particle Green functions. For the values of the chosen parameters we see the formation of three branches which we associate with the two-peak structure in the imaginary part of the self-energy. From the imaginary part of the self-energy we conclude that our system is a Fermi liquid (for the temperature investigated here), since Im Σ( k , ω) ≈ w2 around the chemical potential. We have compared our fully self-consistent FLEX solutions with a lower order approximation where the internal Green functions are approximated by free Green functions. These two approches, i.e., the fully self-consistent and the non-self-consistent ones give different results for the parameters considered here. However, they have similar global results for small densities.

arXiv:cond-mat/9707305International Journal of Modern Physics COct 1, 19971 citation

Numerical renormalization-group approach to fluctuation exchange in the presence of electron-phonon coupling: Pairing in the Holstein-Hubbard model

C.-H. Pao, H.-B. Schüttler

The fluctuation exchange (FLEX) approximation is applied to study the Holstein-Hubbard model. Due to the retarded nature of the phonon-mediated electron-electron interaction, neither fast Fourier transform nor previously developed numerical renormalization-group (NRG) methods for Hubbard-type purely electronic models are applicable, while brute force solutions are limited by the demands on computational time and storage which increase rapidly at low temperature T. Here, we describe a NRG technique to solve the FLEX equations efficiently. Several orders of magnitude of CPU time and storage can be saved at low T $(\ensuremath{\sim}80\mathrm{K})$. To test our approach, we compare our NRG results to brute force calculations on small lattices at elevated temperatures. Both s-wave and d-wave superconducting phase diagrams are then obtained by applying the NRG approach at low T. The isotope effect for s-wave pairing is BCS-like in a realistic phonon frequency range, but vanishes at unphysically large phonon frequency $(\ensuremath{\sim}\mathrm{b}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{w}\mathrm{i}\mathrm{d}\mathrm{t}\mathrm{h})$. For d-wave pairing, the isotope exponent is negative and small compared to the typical observed values in nonoptimally doped cuprates.The fluctuation exchange (FLEX) approximation is applied to study the Holstein-Hubbard model. Due to the retarded nature of the phonon-mediated electron-electron interaction, neither fast Fourier transform nor previously developed numerical renormalization-group (NRG) methods for Hubbard-type purely electronic models are applicable, while brute force solutions are limited by the demands on computational time and storage which increase rapidly at low temperature T. Here, we describe a NRG technique to solve the FLEX equations efficiently. Several orders of magnitude of CPU time and storage can be saved at low T $(\ensuremath{\sim}80\mathrm{K})$. To test our approach, we compare our NRG results to brute force calculations on small lattices at elevated temperatures. Both s-wave and d-wave superconducting phase diagrams are then obtained by applying the NRG approach at low T. The isotope effect for s-wave pairing is BCS-like in a realistic phonon frequency range, but vanishes at unphysically large phonon frequency $(\ensuremath{\sim}\mathrm{b}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{w}\mathrm{i}\mathrm{d}\mathrm{t}\mathrm{h})$. For d-wave pairing, the isotope exponent is negative and small compared to the typical observed values in nonoptimally doped cuprates.

arXiv:cond-mat/9809321Physical review. B, Condensed matterJul 1, 19995 citations

Phase diagram for the attractive Hubbard model in two dimensions in a conserving approximation

J. J. Deisz, D. W. Hess, Joseph W. Serene

We calculate the superconducting transition temperature as a function of interaction strength and density for a two-dimensional attractive Hubbard model in the fluctuation exchange approximation (FEA). Significant corrections to mean-field theory are apparent, including a maximum in ${T}_{c}$ as a function of U and a fluctuation-broadened specific heat peak. At $n=1,$ competing pair fluctuations and charge-density fluctuations fail to drive ${T}_{c}$ to zero. This failure of the FEA reflects an inadequate account of the coupling among fluctuations. Because ${T}_{c}$ is determined from self-consistent calculations of thermodynamic potentials, the calculations are extremely sensitive to numerical errors, requiring accuracy up to 1 part in ${10}^{6}.$ We outline a procedure for systematically reducing numerical errors associated with the finite-frequency cutoff in numerical many-body calculations such as ours.We calculate the superconducting transition temperature as a function of interaction strength and density for a two-dimensional attractive Hubbard model in the fluctuation exchange approximation (FEA). Significant corrections to mean-field theory are apparent, including a maximum in ${T}_{c}$ as a function of U and a fluctuation-broadened specific heat peak. At $n=1,$ competing pair fluctuations and charge-density fluctuations fail to drive ${T}_{c}$ to zero. This failure of the FEA reflects an inadequate account of the coupling among fluctuations. Because ${T}_{c}$ is determined from self-consistent calculations of thermodynamic potentials, the calculations are extremely sensitive to numerical errors, requiring accuracy up to 1 part in ${10}^{6}.$ We outline a procedure for systematically reducing numerical errors associated with the finite-frequency cutoff in numerical many-body calculations such as ours.

Physical review. B, Condensed matterJul 25, 200218 citations

d

-wave superconductivity in the two-dimensional repulsive Hubbard model. The critical temperature

({T}_{c})

curve obtained in the FLEX+DMFT exhibits a dome structure as a function of the filling, which has not been clearly observed in the FLEX approach alone. We trace back the origin of the dome to the local vertex correction from DMFT that renders a filling dependence in the FLEX self-energy. We compare the results with those of GW+DMFT, where the

{T}_{c}

-dome structure is qualitatively reproduced due to the same vertex correction effect, but a crucial difference from FLEX+DMFT is that

{T}_{c}

is always estimated below the N\'eel temperature in GW+DMFT. The single-particle spectral function obtained with FLEX+DMFT exhibits a double-peak structure as a precursor of the Hubbard bands at temperatures above

{T}_{c}

{T}_{c}

as a function of U and a fluctuation-broadened specific heat peak. At

n=1,

competing pair fluctuations and charge-density fluctuations fail to drive

{T}_{c}

to zero. This failure of the FEA reflects an inadequate account of the coupling among fluctuations. Because

{T}_{c}

is determined from self-consistent calculations of thermodynamic potentials, the calculations are extremely sensitive to numerical errors, requiring accuracy up to 1 part in

{10}^{6}.

We outline a procedure for systematically reducing numerical errors associated with the finite-frequency cutoff in numerical many-body calculations such as ours.