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arXiv:cond-mat/0204037 · Journal of Physics Condensed Matter

Pairing instability in a nematic Fermi liquid

Yong Baek Kim, Hae‐Young Kee

Published Apr 24, 2004• 26 citations
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Abstract

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.

DOI
10.1088/0953-8984/16/18/015

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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.
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.
Abstract We explore the interplay between nematicity (spontaneous breaking of the sixfold rotational symmetry), superconductivity, and non-Fermi liquid behavior in partially flat-band (PFB) models on the triangular lattice. A key result is that the nematicity (Pomeranchuk instability), which is driven by many-body effect and stronger in flat-band systems, enhances superconducting transition temperature in a systematic manner on the T c dome. There, a plausible s x 2 + y 2 − d x 2 − y 2 − d x y -wave symmetry, in place of the conventional d x 2 − y 2 -wave, governs the nematicity-enhanced pairing with a sharp rise in the T c dome on the filling axis. When the sixfold symmetry is spontaneously broken, the pairing interaction is shown to become stronger with more compact pairs in real space than when the symmetry is enforced. These are accompanied by a non-Fermi character of electrons in the PFBs with many-body interactions.
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.
We develop a microscopic theory of the electronic nematic phase proximate to an isotropic Fermi liquid in both two and three dimensions. Explicit expressions are obtained for the small amplitude collective excitations in the ordered state; remarkably, the nematic Goldstone mode (the director wave) is overdamped except along special directions dictated by symmetry. At the quantum critical point we find a dynamical exponent of z=3,z=3,z=3, implying stability of the Gaussian fixed point. The leading perturbative effect of the overdamped Goldstone modes leads to a breakdown of Fermi-liquid theory in the nematic phase and to strongly angle-dependent electronic self energies around the Fermi surface. Other metallic liquid-crystal phases, e.g., a quantum hexatic, behave analogously.
Nematicity in quantum Hall systems has been experimentally well established at excited Landau levels. The mechanism of the symmetry breaking, however, is still unknown. Pomeranchuk instability of Fermi liquid parameter F_{ℓ}≤-1 in the angular momentum ℓ=2 channel has been argued to be the relevant mechanism, yet there are no definitive theoretical proofs. Here we calculate, using the variational Monte Carlo technique, Fermi liquid parameters F_{ℓ} of the composite fermion Fermi liquid with a finite layer width. We consider F_{ℓ} in different Landau levels n=0, 1, 2 as a function of layer width parameter η. We find that unlike the lowest Landau level, which shows no sign of Pomeranchuk instability, higher Landau levels show nematic instability below critical values of η. Furthermore, the critical value η_{c} is higher for the n=2 Landau level, which is consistent with observation of nematic order in ambient conditions only in the n=2 Landau levels. The picture emerging from our work is that approaching the true 2D limit brings half-filled higher Landau-level systems to the brink of nematic Pomeranchuk instability.
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.