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Journal of Experimental and Theoretical Physics Letters

Fermi Surface Topology in the Case of Spontaneously Broken Rotational Symmetry

С. С. Панкратов, M. Baldo, MIKHAIL V. ZVEREV

Published Dec 1, 2017• 2 citations
Publisher

Abstract

The relation between the broken rotational symmetry of a system and the topology of its Fermi surface is studied for the two-dimensional system with the quasiparticle interaction f ( p , p' ) having a sharp peak at | p − p' | = q 0 . It is shown that, in the case of attraction and q 0 = 2 p F the first instability manifesting itself with the growth of the interaction strength is the Pomeranchuk instability. This instability appearing in the L = 2 channel gives rise to a second order phase transition to a nematic phase. The Monte Carlo calculations demonstrate that this transition is followed by a sequence of the first and second order phase transitions corresponding to the changes in the symmetry and topology of the Fermi surface. In the case of repulsion and small values of q 0 , the first transition is a topological transition to a state with the spontaneously broken rotational symmetry, namely, corresponding to the nucleation of L ≃ π( p F / q 0 − 1) small hole pockets at the distance p F − q 0 from the center and the deformation of the outer Fermi surface with the characteristic multipole number equal to L . At q 0 → 0, when the model under study transforms to the two-dimensional Nozières model, the multipole number characterizing the spontaneous deformation is L → ∞, whereas the infinitely folded Fermi curve acquires the Hausdorff dimension D = 2 which corresponds to the state with the fermion condensate.

DOI
10.1134/s0021364017230096

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The relation between the broken rotational symmetry of a system and the topology of its Fermi surface is studied for the two-dimensional system with the quasiparticle interaction f ( p , p' ) having a sharp peak at | p − p' | = q 0 . It is shown that, in the case of attraction and q 0 = 2 p F the first instability manifesting itself with the growth of the interaction strength is the Pomeranchuk instability. This instability appearing in the L = 2 channel gives rise to a second order phase transition to a nematic phase. The Monte Carlo calculations demonstrate that this transition is followed by a sequence of the first and second order phase transitions corresponding to the changes in the symmetry and topology of the Fermi surface. In the case of repulsion and small values of q 0 , the first transition is a topological transition to a state with the spontaneously broken rotational symmetry, namely, corresponding to the nucleation of L ≃ π( p F / q 0 − 1) small hole pockets at the distance p F − q 0 from the center and the deformation of the outer Fermi surface with the characteristic multipole number equal to L . At q 0 → 0, when the model under study transforms to the two-dimensional Nozières model, the multipole number characterizing the spontaneous deformation is L → ∞, whereas the infinitely folded Fermi curve acquires the Hausdorff dimension D = 2 which corresponds to the state with the fermion condensate.
We address at the mean field level the emergence of a Pomeranchuk instability in a uniform Fermi liquid with central particle-particle interactions. We find that Pomeranchuk instabilities with all symmetries except l=1l=1l=1 can take place if the interaction is repulsive and has a finite range r0{r}_{0}r0​ of the order of the interparticle distance. We demonstrate this by solving the mean field equations analytically for an explicit model interaction, as well as numerical results for more realistic potentials. We find in addition to the Pomeranchuk instability other, subtler phase transitions in which the Fermi surface changes topology without rotational symmetry breaking. We argue that such interaction-driven topological transitions may be as generic to such systems as the Pomeranchuk instability.
Strong interaction between fermions may lead to a novel quantum ground state where the rotational symmetry is spontaneously broken while the translational symmetry is still preserved. Such a 'nematic' Fermi liquid has a spontaneously deformed Fermi surface and supports a gapless Goldstone mode associated with the broken rotational symmetry. We consider a nematic Fermi liquid of fermions carrying two internal quantum numbers that, for example, may represent the layer degree of freedom of a bilayer system. It is found that the effective interaction mediated by a collective mode gives rise to a pairing instability of the underlying fermions under certain conditions, leading to the conclusion that the nematic order could be useful for superconductivity.
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 discuss shape (Pomeranchuk) instabilities of the Fermi surface of a two-dimensional Fermi system using bosonization. We consider in detail the quantum critical behavior of the transition of a two-dimensional Fermi fluid to a nematic state which breaks spontaneously the rotational invariance of the Fermi liquid. We show that higher dimensional bosonization reproduces the quantum critical behavior expected from the Hertz-Millis analysis, and verify that this theory has dynamic critical exponent z=3z=3z=3. Going beyond this framework, we study the behavior of the fermion degrees of freedom directly, and show that at quantum criticality as well as in the quantum nematic phase (except along a set of measure zero of symmetry-dictated directions) the quasiparticles of the normal Fermi liquid are generally wiped out. Instead, they exhibit short-ranged spatial correlations that decay faster than any power law, with the law $\ensuremath{\mid}x{\ensuremath{\mid}}^{\ensuremath{-}1}\mathrm{exp}(\ensuremath{-}\mathrm{const}\phantom{\rule{0.3em}{0ex}}\ensuremath{\mid}x{\ensuremath{\mid}}^{1∕3})$ and we verify explicitly the vanishing of the fermion residue utilizing this expression. In contrast, the fermion autocorrelation function has the behavior $\ensuremath{\mid}t{\ensuremath{\mid}}^{\ensuremath{-}1}\mathrm{exp}(\ensuremath{-}\mathrm{const}\phantom{\rule{0.3em}{0ex}}\ensuremath{\mid}t{\ensuremath{\mid}}^{\ensuremath{-}2∕3})$. In this regime we also find that, at low frequency, the single-particle fermion density of states behaves as ${N}^{*}(\ensuremath{\omega})={N}^{*}(0)+B{\ensuremath{\omega}}^{2∕3}\mathrm{ln}\ensuremath{\omega}+\ensuremath{\cdots}$, where N∗(0){N}^{*}(0)N∗(0) is larger than the free Fermi value, N(0)N(0)N(0), and BBB is a constant. These results confirm the non-Fermi liquid nature of both the quantum critical theory and of the nematic phase.
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.
Interactions in Fermi systems can induce a "Pomeranchuk instability" leading to orientational symmetry breaking, that is, nematic order. In a metallic system close to such an instability the Fermi surface is easily deformed by anisotropic perturbations, and exhibits enhanced collective fluctuations. We discuss electrons on a square lattice near a Pomeranchuk instability with d-wave symmetry. The strong response of such a system to a small orthorhombic perturbation can explain naturally the large in-plane anisotropy of electronic and magnetic properties observed in detwinned YBCO crystals. Fluctuations in a quantum critical regime near the instability provide a mechanism for non-Fermi liquid behavior. They lead to a singular forward scattering interaction, which destroys fermionic quasi-particles on the whole Fermi surface except at "cold spots" on the Brillouin zone diagonal. The decay rate for DC transport is linear in temperature except at the cold spots, where conventional Fermi liquid behavior survives. In clean systems this gives rise to a T 3/2 temperature dependence of the DC resistivity, as observed in some overdoped cuprates. In the presence of disorder, the resistivity is linear in T at low temperatures.