We present a systematic stability analysis for the two-dimensional Hubbard model, which is based on a new renormalization group method for interacting Fermi systems. The flow of effective interactions and susceptibilities confirms the expected existence of a d-wave pairing instability driven by antiferromagnetic spin fluctuations. More unexpectedly, we find that strong forward scattering interactions develop which may lead to a Pomeranchuk instability breaking the tetragonal symmetry of the Fermi surface.
We present a systematic stability analysis for the two-dimensional Hubbard model, which is based on a new renormalization group method for interacting Fermi systems. The flow of effective interactions and susceptibilities confirms the expected existence of a d-wave pairing instability driven by antiferromagnetic spin fluctuations. More unexpectedly, we find that strong forward scattering interactions develop which may lead to a Pomeranchuk instability breaking the tetragonal symmetry of the Fermi surface.
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 TcPI for the Pomeranchuk instability is more sensitively affected by the position of van Hove filling than the superconducting TcSC. This implies that the filling region for strong Pomeranchuk instability and that for the TcSC dome can be separated, and that Pomeranchuk instability can appear even if the peak of TcPI is lower than the peak of TcSC. An interesting finding is that the Fermi surface distortion can enhance the superconducting TcSC in the overdoped regime, which is explained with a perturbational picture for small distortions.
The repulsive two-dimensional Hubbard model is analysed within a functional renormalisation group (RG) approach. The flow equation for the effective interactions is evaluated on 1-loop level. The effective interactions diverge at a finite energy scale which is exponentially small for small bare interactions. To analyse the nature of the instabilities signalled by the diverging interactions, we compute the flow of the superconducting susceptibilities for various pairing symmetries and also charge and spin density susceptibilities. Depending on the choice of the model parameters (hopping amplitudes, interaction strength and band-filling) we find spin density wave instabilities or d-wave superconductivity as leading instability close to half-filling. This calculation establishes the existence of d-wave superconductivity in the Hubbard model beyond doubt.
In connection with the symmetry-breaking phenomena found in cuprate superconductors, the instability toward x-y and (x + y)-(xy) anisotropy (Pomeranchuk instability) is studied for a strongly correlated Hubbard model (U/t = 12) on a square lattice with the next-nearest-neighbor transfer t ′ , using a variational Monte Carlo method.By analyzing a normal (paramagnetic), d x 2 -y 2 -wave superconducting, and antiferromagnetic states for t ′ /t = 0, ±0.3 and 0 ≤ δ ≤ 0.3 (δ: doping rate), we argue that the possibility of Pomeranchuk instability is limited to the normal state with t ′ /t = 0 and 0.05 ≲ δ ≲ 0.20 in the x-y anisotropy.Band-renormalization effects (Fermi-surface changes), which are introduced to each state, play a crucial role for the argument.
We analyze a mean-field model of electrons on a square lattice with two types of interaction: forward scattering favoring a d-wave Pomeranchuk instability and a BCS pairing interaction driving d-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.
We describe a Wick ordered functional renormalization group method for interacting Fermi systems, where the complete flow from the bare action of the microscopic model to the effective low-energy action is obtained from a differential flow equation. We apply this renormalization group approach to a prototypical two-dimensional lattice electron system, the Hubbard model on a square lattice. The flow equation for the effective interactions is evaluated numerically on 1-loop level. The effective interactions diverge at a finite energy scale which is exponentially small for small bare interactions. To analyze the nature of the instabilities signalled by the diverging interactions we compute the flow of the singlet superconducting susceptibilities for various pairing symmetries and also charge and spin density susceptibilities. Depending on the choice of the model parameters (hopping amplitudes, interaction strength and band-filling) we find antiferromagnetic order or d-wave superconductivity as leading symmetry breaking instability.
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