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arXiv:1609.05759 · Physical review. B./Physical review. B

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

Motoharu Kitatani, Naoto Tsuji, Hideo Aoki

Published Feb 3, 2017• 24 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.95.075109

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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.
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
Divergencies appearing in perturbation expansions of interacting many-body systems can often be removed by expanding around a suitably chosen renormalized (instead of the noninteracting) Hamiltonian. We describe such a renormalized perturbation expansion for interacting Fermi systems, which treats Fermi surface shifts and superconductivity with an arbitrary gap function via additive counterterms. The expansion is formulated explicitly for the Hubbard model to second order in the interaction. Numerical solutions of the self-consistency condition determining the Fermi surface and the gap function are calculated for the two-dimensional case. For the repulsive Hubbard model close to half-filling we find a superconducting state with d-wave symmetry, as expected. For Fermi levels close to the van Hove singularity a Pomeranchuk instability leads to Fermi surfaces with a broken square lattice symmetry, whose topology can be closed or open. For the attractive Hubbard model the second-order calculation yields an s-wave superconductivity with a weakly momentum dependent gap, whose size is reduced compared to the mean-field result.
We analyze a mean-field model of electrons on a square lattice with two types of interaction: forward scattering favoring a ddd-wave Pomeranchuk instability and a BCS pairing interaction driving ddd-wave superconductivity. By tuning the interaction parameters a rich variety of phase diagrams is obtained. If the BCS interaction is not too strong, Fermi surface symmetry breaking is stabilized around van Hove filling, and coexists with superconductivity at low temperatures. For pure forward scattering Fermi surface symmetry breaking occurs typically via a first order transition at low temperatures. The presence of superconductivity reduces the first order character of this transition and, if strong enough, can turn it into a continuous one. This gives rise to a quantum critical point within the superconducting phase. The superconducting gap tends to suppress Fermi surface symmetry breaking. For a relatively strong BCS interaction, Fermi surface symmetry breaking can be limited to intermediate temperatures, or can be suppressed completely by pairing.
The deformation of the Fermi surface induced by Coulomb interactions is investigated in the $t\ensuremath{-}{t}^{\ensuremath{'}}$-Hubbard model. The interplay of the local U and extended V interactions is analyzed. It is found that the interactions V enhance small anisotropies producing deformations of the Fermi surface that break the point group symmetry of the square lattice at the Van Hove filling. This Pomeranchuk instability competes with magnetic instabilities and is suppressed at a critical value U(V).U(V).U(V). The interaction V effectively renormalizes the ${t}^{\ensuremath{'}}$ parameter to smaller values, which favors nesting. It also induces changes in the topology of the Fermi surface, which can go from holelike to electronlike. This may explain the results of recent angle-resolved photoemission spectroscopy experiments.
We present a stability analysis of the two-dimensional $t\ensuremath{-}{t}^{\ensuremath{'}}$ Hubbard model for various values of the next-nearest-neighbor hopping ${t}^{\ensuremath{'}},$ and electron concentrations close to the Van Hove filling by means of the flow equation method. For ${t}^{\ensuremath{'}}>~\ensuremath{-}t/3$ a ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave Pomeranchuk instability dominates (apart from antiferromagnetism at small ${t}^{\ensuremath{'}}).$ At ${t}^{\ensuremath{'}}<\ensuremath{-}t/3,$ the leading instabilities are a g-wave Pomeranchuk instability and p-wave particle-hole instability in the triplet channel at temperatures T<0.15t,T<0.15t,T<0.15t, and an s∗{s}^{*}s∗-magnetic phase for T>0.15t;T>0.15t;T>0.15t; upon increasing the electron concentration, the triplet analog of the flux phase occurs at low temperatures. Other weaker instabilities are also found.
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​.