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arXiv:cond-mat/0502238 · Physical Review B

Mean-field theory for symmetry-breaking Fermi surface deformations on a square lattice

Hiroyuki Yamase, Vadim Oganesyan, Walter Metzner

Published Jul 14, 2005• 126 citations
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Abstract

We analyze a mean-field model of electrons with pure forward scattering interactions on a square lattice which exhibits spontaneous Fermi surface symmetry breaking with a $d$-wave order parameter, the surface expands along the ${k}_{x}$ axis and shrinks along the ${k}_{y}$ axis (or vice versa). The symmetry-broken phase is stabilized below a dome-shaped transition line ${T}_{c}(\ensuremath{\mu})$, with a maximal ${T}_{c}$ near van Hove filling. The phase transition is usually first order at the edges of the transition line, and always second order around its center. The $d$-wave compressibility of the Fermi surface is however strongly enhanced even near the first order transition down to zero temperature. In the weak coupling limit the phase diagram is fully determined by a single nonuniversal energy scale, and hence dimensionless ratios of different characteristic quantities are universal. Adding a uniform repulsion to the forward scattering interaction, the two tricritical points at the ends of the second order transition line are shifted to lower temperatures. For a particularly favorable choice of hopping and interaction parameters one of the first order edges is replaced completely by a second order transition line, leading to a quantum critical point.

DOI
10.1103/physrevb.72.035114

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We analyze a mean-field model of electrons with pure forward scattering interactions on a square lattice which exhibits spontaneous Fermi surface symmetry breaking with a ddd-wave order parameter, the surface expands along the kx{k}_{x}kx​ axis and shrinks along the ky{k}_{y}ky​ axis (or vice versa). The symmetry-broken phase is stabilized below a dome-shaped transition line Tc(\ensuremathμ){T}_{c}(\ensuremath{\mu})Tc​(\ensuremathμ), with a maximal Tc{T}_{c}Tc​ near van Hove filling. The phase transition is usually first order at the edges of the transition line, and always second order around its center. The ddd-wave compressibility of the Fermi surface is however strongly enhanced even near the first order transition down to zero temperature. In the weak coupling limit the phase diagram is fully determined by a single nonuniversal energy scale, and hence dimensionless ratios of different characteristic quantities are universal. Adding a uniform repulsion to the forward scattering interaction, the two tricritical points at the ends of the second order transition line are shifted to lower temperatures. For a particularly favorable choice of hopping and interaction parameters one of the first order edges is replaced completely by a second order transition line, leading to a quantum critical point.
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.
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.
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.
A metallic electron system near an orientational symmetry breaking Pomeranchuk instability is characterized by a ``soft'' Fermi surface with enhanced collective fluctuations. We analyze fluctuation effects in a two-dimensional electron system on a square lattice in the vicinity of a Pomeranchuk instability with ddd-wave symmetry, using a phenomenological model which includes interactions with a small momentum transfer only. We compute the dynamical density correlations with a ddd-wave form factor for small momenta and frequencies, the dynamical effective interaction due to a fluctuation exchange, and the electron self-energy. At the quantum critical point the density correlations and the dynamical forward scattering interaction diverge with a dynamical exponent z=3z=3z=3. The singular forward scattering leads to large self-energy corrections, which destroy Fermi liquid behavior over the whole Fermi surface except near the Brillouin zone diagonal. The decay rate of single-particle excitations, which is related to the width of the peaks in the spectral function, exceeds the excitation energy in the low-energy limit. The dispersion of maxima in the spectra flattens strongly near those portions of the Fermi surface which are remote from the zone diagonal. The contribution from classical fluctuations to the self-energy spoils $(\ensuremath{\omega}∕T)$ scaling in the quantum critical regime.
We apply the self-consistent renormalized perturbation theory to the Hubbard model on the square lattice at finite temperatures to study the evolution of the Fermi surface (FS) as a function of temperature and doping. Previously, a nematic phase for the same model has been reported to appear at weak coupling near a Lifshitz transition from closed to open FS at zero temperature where the self-consistent renormalized perturbation theory was shown to be sensitive to small deformations of the FS. We find that the competition with the superconducting order leads to a maximal nematic order appearing at nonzero temperature. We explicitly observe the two competing phases near the onset of nematic instability, and by comparing the grand canonical potentials, we find that the transitions are first order. We explain the origin of the interaction-driven spontaneous symmetry breaking to a nematic phase in a system with several symmetry-related Van Hove points and discuss the required conditions.
Spinless fermions on the honeycomb lattice with repulsive nearest-neighbor interactions are known to harbour a quantum critical point at half-filling, with critical behavior in the Gross-Neveu (chiral Ising) universality class. The critical interaction strength separates a weak-coupling semimetallic regime from a commensurate charge-density-wave phase. The phase diagram of this basic model of correlated fermions on the honeycomb lattice beyond half-filling is, however, less well established. Here, we perform an analysis of its many-body instabilities using the functional renormalization group method with a basic Fermi surface patching scheme, which allows us to treat instabilities in competing channels on equal footing also away from half-filling. Between half-filling and the Van Hove filling, the free Fermi surface is holelike and we again find a charge-density wave instability to be dominant at large interactions. Moreover, its characteristics are those of the half-filled case. Directly at the Van Hove filling, the nesting property of the free Fermi surface stabilizes a dimerized bond-order phase. At lower filling, the free Fermi surface becomes electronlike and a superconducting instability with fff-wave symmetry is found to emerge from the interplay of intra-unit-cell repulsion and collective fluctuations in the proximity to the charge-density wave instability. We estimate the extent of the various phases and extract the corresponding order parameters from the effective low-energy Hamiltonians.