Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1505.04865 · Physical Review B

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

Motoharu Kitatani, Naoto Tsuji, Hideo Aoki

Published Aug 3, 2015• 58 citations
arXivHTMLPDF

Abstract

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 ${T}_{c}$.

DOI
10.1103/physrevb.92.085104

Related papers

High-Tc superconductivity studied by diagrammatic extensions of the dynamical mean field theory

基治 北谷

While it is over 30 years since the high-T c superconductor was discovered, the cuprates have attracted great interests in many scientific fields.Understanding high-T c is one of the most challenging issues in the condensed matter theory.The mechanism of the superconductivity in these materials is still not fully understood, and continues to be discussed intensely.Specifically, the two-dimensional repulsive Hubbard model, which is a representative model for the cuprates, still has many questions, and many analytical and numerical methods have been proposed for studying the superconductivity, magnetism and other instabilities.In the present thesis, we propose to combine the non-perturbative dynamical mean field theory (DMFT) with the perturbative diagrammatic expansion techniques to treat the local correlation effects ("Mott physics") and momentum dependent pairing interaction (that produces the d-wave pairing superconductivity) to explore the superconductivity and various instabilities in the Hubbard model.First, we formulate the combination of the fluctuation exchange (FLEX) approximation and DMFT in a self-consistent manner in terms of the Luttinger-Ward functional for superconducting phases.The resultant phase diagram exhibits a clear T c dome structure against the band filling as observed experimentally, while in the ordinary FLEX, T c almost monotonically increases toward the half-filling.We trace the origin of the T c -dome back to a combined effect of the filling dependence of the local vertex correction and the filling dependence of the pairing interaction.As another effect inherent in the strongly correlated systems, we study a phase transition for the Pomeranchuk instability, where the four-fold rotational symmetry in the electronic structure on the tetragonal lattice is spontaneously broken into two-fold due to an interaction effect.We find that the Pomeranchuk instability is remarkably more sensitive to the Fermi surface warping than the superconducting T c dome.An interesting question is the effect of the distorted Fermi surface due to the Pomeranchuk instability on the superconductivity, and we have revealed that the gap function is slightly changed from ordinary d-wave pairing to (d+s)-wave.Furthermore, we find that the Fermi surface distortion can enhance the superconducting T c in frustrated systems.i Finally, we consider the extension of the DMFT vertex to study the vertex correction effect on nonlocal self-energy and the pairing interaction.For this purpose, we employ the DΓA (dynamical vertex approximation) formalism, which is extended here to accommodate superconducting phases.For the Hubbard model we obtain the T c -dome due to a combined effect as in the FLEX+DMFT case, but the transition temperature is reduced by the vertex correction.We also find that the pairing interaction is strongly screened by the vertex correction effect.ii

•Mar 23, 2017•0 citations

Interplay of Pomeranchuk instability and superconductivity in the two-dimensional repulsive Hubbard model

Motoharu Kitatani, Naoto Tsuji, Hideo Aoki

Interplay of Pomeranchuk instability (spontaneous symmetry breaking of the Fermi surface) and $d$-wave superconductivity is studied for the repulsive Hubbard model on the square lattice with the dynamical mean-field theory combined with the fluctuation exchange approximation (FLEX+DMFT). We show that the fourfold symmetric Fermi surface becomes unstable against a spontaneous distortion into twofold near the van Hove filling, where the symmetry of superconductivity coexisting with the Pomeranchuk-distorted Fermi surface is modified from the $d$-wave pairing to the $(d+s)$ wave. By systematically shifting the position of van Hove filling with varied second- and third-neighbor hoppings, we find that the transition temperature ${T}_{\mathrm{c}}^{\mathrm{PI}}$ for the Pomeranchuk instability is more sensitively affected by the position of van Hove filling than the superconducting ${T}_{\mathrm{c}}^{\mathrm{SC}}$. This implies that the filling region for strong Pomeranchuk instability and that for the ${T}_{\mathrm{c}}^{\mathrm{SC}}$ dome can be separated, and that Pomeranchuk instability can appear even if the peak of ${T}_{\mathrm{c}}^{\mathrm{PI}}$ is lower than the peak of ${T}_{\mathrm{c}}^{\mathrm{SC}}$. An interesting finding is that the Fermi surface distortion can enhance the superconducting ${T}_{\mathrm{c}}^{\mathrm{SC}}$ in the overdoped regime, which is explained with a perturbational picture for small distortions.

arXiv:1609.05759•Physical review. B./Physical review. B•Feb 3, 2017•24 citations

SO(4) FLEX+DMFT formalism with SU(2)$\otimes$SU(2)-symmetric impurity solver for superconductivity in the repulsive Hubbard model

Sharareh Sayyad, Naoto Tsuji, Massimo Capone +1 more

Here we have developed a FLEX+DMFT formalism, where the symmetry properties of the system are incorporated by constructing a SO(4) generalization of the conventional fluctuation-exchange approximation (FLEX) coupled self-consistently to the dynamical mean-field theory (DMFT). Along with this line, we emphasize that the SO(4) symmetry is the lowest group-symmetry that enables us to investigate superconductivity and antiferromagnetism on an equal footing. We have imposed this by decomposing the electron operator into auxiliary fermionic and slave-boson constituents that respect SU(2)$_{\rm spin}\otimes$SU(2)$_{η{\rm spin}}$. This is used not in a mean-field treatment as in the usual slave-boson formalisms, but instead in the DMFT impurity solver with an SU(2)$_{\rm spin}\otimes$SU(2)$_{η{\rm spin}}$ hybridization function to incorporate the FLEX-generated bath information into DMFT iterations. While there have been attempts such as the doublon-less SU(2) slave-boson formalism, the present "full-SU(2)" slave-boson formalism is expected to provide a new platform for addressing the underlying physics for various quantum orders, which compete with each other and can coexist.

arXiv:1903.05800•Mar 14, 2019•0 citations

On the dangers of partial diagrammatic summations: Benchmarks for the two-dimensional Hubbard model in the weak-coupling regime

Jan Gukelberger, Li Huang, Philipp Werner

We study the two-dimensional Hubbard model in the weak-coupling regime and compare the self-energy obtained from various approximate diagrammatic schemes to the result of diagrammatic Monte Carlo simulations, which sum up all weak-coupling diagrams up to a given order. While dynamical mean-field theory provides a good approximation for the local part of the self-energy, including its frequency dependence, the partial summation of bubble and/or ladder diagrams typically yields worse results than second-order perturbation theory. Even widely used self-consistent schemes such as $GW$ or the fluctuation-exchange approximation (FLEX) are found to be unreliable. Combining the dynamical mean-field self-energy with the nonlocal component of $GW$ in $GW + \mathrm{DMFT}$ yields improved results for the local self-energy and nonlocal self-energies of the correct order of magnitude, but here, too, a more reliable scheme is obtained by restricting the nonlocal contribution to the second-order diagram. FLEX + DMFT is found to give accurate results in the low-density regime, but even worse results than FLEX near half-filling.

arXiv:1501.04960•Physical Review B•Jun 9, 2015•74 citations

Screening and nonlocal correlations in the extended Hubbard model from self-consistent combined G W and dynamical mean field theory

Thomas Ayral, Silke Biermann, Philipp Werner

We describe a recent implementation of the combined $GW$ and dynamical mean field method ($GW$+DMFT) for the two-dimensional Hubbard model with onsite and nearest-neighbor repulsion. We clarify the relation of the $GW$+DMFT scheme to alternative approaches in the literature, and discuss the corresponding approximations to the free-energy functional of the model. Furthermore, we describe a numerically exact technique for the solution of the $GW$+DMFT equations, namely, the hybridization expansion continuous-time algorithm for impurity models with retarded interactions. We compute the low-temperature phase diagram of the half-filled extended Hubbard model, addressing the metal-insulator transition at small intersite interactions and the transition to a charge-ordered state for stronger intersite repulsions. $GW$+DMFT introduces a nontrivial momentum dependence into the many-body self-energy and polarization. We find that the charge fluctuations included in the present approach have a larger impact on the latter than on the former. Finally, within the $GW$+DMFT framework, as in extended DMFT, the intersite repulsion translates into a frequency dependence of the local effective interaction. We analyze this dependence and show how it affects the local spectral function.

arXiv:1210.2712•Physical Review B•Mar 29, 2013•168 citations

Spectral Properties of Correlated Materials: Local Vertex and Nonlocal Two-Particle Correlations from Combined G W and Dynamical Mean Field Theory

Thomas Ayral, Philipp Werner, Silke Biermann

We present a fully self-consistent combined GW and dynamical mean field (DMFT) study of the extended two-dimensional Hubbard model. The inclusion of the local dynamical vertex stemming from the DMFT self-energy and polarization is shown to cure the known problems of self-consistent GW. We calculate momentum-resolved spectral functions, two-particle polarizations, and electron-loss spectra, as well as the effective dynamical interaction induced by nonlocal screening. The momentum-dependence introduced by GW into the extended DMFT description leads to a narrowing of the quasiparticle width and more pronounced Hubbard bands in the metallic regime as one approaches the charge-ordering transition. It further affects the shape of collective modes, giving rise to dispersive plasmon-like long-wavelength and stripe modes.

arXiv:1205.5553•Physical Review Letters•Nov 28, 2012•94 citations
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 ddd-wave superconductivity in the two-dimensional repulsive Hubbard model. The critical temperature (Tc)({T}_{c})(Tc​) 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 Tc{T}_{c}Tc​-dome structure is qualitatively reproduced due to the same vertex correction effect, but a crucial difference from FLEX+DMFT is that Tc{T}_{c}Tc​ 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 Tc{T}_{c}Tc​.
While it is over 30 years since the high-T c superconductor was discovered, the cuprates have attracted great interests in many scientific fields.Understanding high-T c is one of the most challenging issues in the condensed matter theory.The mechanism of the superconductivity in these materials is still not fully understood, and continues to be discussed intensely.Specifically, the two-dimensional repulsive Hubbard model, which is a representative model for the cuprates, still has many questions, and many analytical and numerical methods have been proposed for studying the superconductivity, magnetism and other instabilities.In the present thesis, we propose to combine the non-perturbative dynamical mean field theory (DMFT) with the perturbative diagrammatic expansion techniques to treat the local correlation effects ("Mott physics") and momentum dependent pairing interaction (that produces the d-wave pairing superconductivity) to explore the superconductivity and various instabilities in the Hubbard model.First, we formulate the combination of the fluctuation exchange (FLEX) approximation and DMFT in a self-consistent manner in terms of the Luttinger-Ward functional for superconducting phases.The resultant phase diagram exhibits a clear T c dome structure against the band filling as observed experimentally, while in the ordinary FLEX, T c almost monotonically increases toward the half-filling.We trace the origin of the T c -dome back to a combined effect of the filling dependence of the local vertex correction and the filling dependence of the pairing interaction.As another effect inherent in the strongly correlated systems, we study a phase transition for the Pomeranchuk instability, where the four-fold rotational symmetry in the electronic structure on the tetragonal lattice is spontaneously broken into two-fold due to an interaction effect.We find that the Pomeranchuk instability is remarkably more sensitive to the Fermi surface warping than the superconducting T c dome.An interesting question is the effect of the distorted Fermi surface due to the Pomeranchuk instability on the superconductivity, and we have revealed that the gap function is slightly changed from ordinary d-wave pairing to (d+s)-wave.Furthermore, we find that the Fermi surface distortion can enhance the superconducting T c in frustrated systems.i Finally, we consider the extension of the DMFT vertex to study the vertex correction effect on nonlocal self-energy and the pairing interaction.For this purpose, we employ the DΓA (dynamical vertex approximation) formalism, which is extended here to accommodate superconducting phases.For the Hubbard model we obtain the T c -dome due to a combined effect as in the FLEX+DMFT case, but the transition temperature is reduced by the vertex correction.We also find that the pairing interaction is strongly screened by the vertex correction effect.ii
Interplay of Pomeranchuk instability (spontaneous symmetry breaking of the Fermi surface) and ddd-wave superconductivity is studied for the repulsive Hubbard model on the square lattice with the dynamical mean-field theory combined with the fluctuation exchange approximation (FLEX+DMFT). We show that the fourfold symmetric Fermi surface becomes unstable against a spontaneous distortion into twofold near the van Hove filling, where the symmetry of superconductivity coexisting with the Pomeranchuk-distorted Fermi surface is modified from the ddd-wave pairing to the (d+s)(d+s)(d+s) wave. By systematically shifting the position of van Hove filling with varied second- and third-neighbor hoppings, we find that the transition temperature TcPI{T}_{\mathrm{c}}^{\mathrm{PI}}TcPI​ for the Pomeranchuk instability is more sensitively affected by the position of van Hove filling than the superconducting TcSC{T}_{\mathrm{c}}^{\mathrm{SC}}TcSC​. This implies that the filling region for strong Pomeranchuk instability and that for the TcSC{T}_{\mathrm{c}}^{\mathrm{SC}}TcSC​ dome can be separated, and that Pomeranchuk instability can appear even if the peak of TcPI{T}_{\mathrm{c}}^{\mathrm{PI}}TcPI​ is lower than the peak of TcSC{T}_{\mathrm{c}}^{\mathrm{SC}}TcSC​. An interesting finding is that the Fermi surface distortion can enhance the superconducting TcSC{T}_{\mathrm{c}}^{\mathrm{SC}}TcSC​ in the overdoped regime, which is explained with a perturbational picture for small distortions.
SO(4) FLEX+DMFT formalism with SU(2)⊗\otimes⊗SU(2)-symmetric impurity solver for superconductivity in the repulsive Hubbard model
Here we have developed a FLEX+DMFT formalism, where the symmetry properties of the system are incorporated by constructing a SO(4) generalization of the conventional fluctuation-exchange approximation (FLEX) coupled self-consistently to the dynamical mean-field theory (DMFT). Along with this line, we emphasize that the SO(4) symmetry is the lowest group-symmetry that enables us to investigate superconductivity and antiferromagnetism on an equal footing. We have imposed this by decomposing the electron operator into auxiliary fermionic and slave-boson constituents that respect SU(2)spin⊗_{\rm spin}\otimesspin​⊗SU(2)ηspin_{η{\rm spin}}ηspin​. This is used not in a mean-field treatment as in the usual slave-boson formalisms, but instead in the DMFT impurity solver with an SU(2)spin⊗_{\rm spin}\otimesspin​⊗SU(2)ηspin_{η{\rm spin}}ηspin​ hybridization function to incorporate the FLEX-generated bath information into DMFT iterations. While there have been attempts such as the doublon-less SU(2) slave-boson formalism, the present "full-SU(2)" slave-boson formalism is expected to provide a new platform for addressing the underlying physics for various quantum orders, which compete with each other and can coexist.
We study the two-dimensional Hubbard model in the weak-coupling regime and compare the self-energy obtained from various approximate diagrammatic schemes to the result of diagrammatic Monte Carlo simulations, which sum up all weak-coupling diagrams up to a given order. While dynamical mean-field theory provides a good approximation for the local part of the self-energy, including its frequency dependence, the partial summation of bubble and/or ladder diagrams typically yields worse results than second-order perturbation theory. Even widely used self-consistent schemes such as GWGWGW or the fluctuation-exchange approximation (FLEX) are found to be unreliable. Combining the dynamical mean-field self-energy with the nonlocal component of GWGWGW in GW+DMFTGW + \mathrm{DMFT}GW+DMFT yields improved results for the local self-energy and nonlocal self-energies of the correct order of magnitude, but here, too, a more reliable scheme is obtained by restricting the nonlocal contribution to the second-order diagram. FLEX + DMFT is found to give accurate results in the low-density regime, but even worse results than FLEX near half-filling.
We describe a recent implementation of the combined GWGWGW and dynamical mean field method (GWGWGW+DMFT) for the two-dimensional Hubbard model with onsite and nearest-neighbor repulsion. We clarify the relation of the GWGWGW+DMFT scheme to alternative approaches in the literature, and discuss the corresponding approximations to the free-energy functional of the model. Furthermore, we describe a numerically exact technique for the solution of the GWGWGW+DMFT equations, namely, the hybridization expansion continuous-time algorithm for impurity models with retarded interactions. We compute the low-temperature phase diagram of the half-filled extended Hubbard model, addressing the metal-insulator transition at small intersite interactions and the transition to a charge-ordered state for stronger intersite repulsions. GWGWGW+DMFT introduces a nontrivial momentum dependence into the many-body self-energy and polarization. We find that the charge fluctuations included in the present approach have a larger impact on the latter than on the former. Finally, within the GWGWGW+DMFT framework, as in extended DMFT, the intersite repulsion translates into a frequency dependence of the local effective interaction. We analyze this dependence and show how it affects the local spectral function.
We present a fully self-consistent combined GW and dynamical mean field (DMFT) study of the extended two-dimensional Hubbard model. The inclusion of the local dynamical vertex stemming from the DMFT self-energy and polarization is shown to cure the known problems of self-consistent GW. We calculate momentum-resolved spectral functions, two-particle polarizations, and electron-loss spectra, as well as the effective dynamical interaction induced by nonlocal screening. The momentum-dependence introduced by GW into the extended DMFT description leads to a narrowing of the quasiparticle width and more pronounced Hubbard bands in the metallic regime as one approaches the charge-ordering transition. It further affects the shape of collective modes, giving rise to dispersive plasmon-like long-wavelength and stripe modes.