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.
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 analyze a two-dimensional Pomeranchuk-nematic instability, trigger by the Landau parameter F2<0, in the presence of a small magnetic field. Using Landau Fermi-liquid theory in the isotropic phase, we analyze the collective modes near the quantum critical point ${F}_{2}=\ensuremath{-}1,{\ensuremath{\omega}}_{c}=0$ (where ${\ensuremath{\omega}}_{c}$ is the cyclotron frequency). We focus on the effects of parity symmetry breaking on the Fermi-surface deformation. We show that the linear response approximation of the Landau-Silin equation is not sufficient to study the critical regime and it is necessary to compute corrections at least of order ${\ensuremath{\omega}}_{c}^{2}$. Identifying the slowest oscillation mode in the disordered phase, we compute the phase diagram for the isotropic/nematic phase transition in terms of F2 and ${\ensuremath{\omega}}_{c}$.
We analyze the behavior of an itinerant two-dimensional Fermi system near a charge nematic (n=2) Pomeranchuk instability in terms of the Landau Fermi-liquid (FL) theory. A key object of our study is the fully renormalized vertex function ${\ensuremath{\Gamma}}^{\ensuremath{\Omega}}$, related to the Landau interaction function. We derive ${\ensuremath{\Gamma}}^{\ensuremath{\Omega}}$ for a model case of the long-range interaction in the nematic channel. Already within the random-phase approximation (RPA), the vertex is singular near the instability. The full vertex, obtained by resumming the ladder series composed of the RPA vertices, differs from the RPA result by a multiplicative renormalization factor ${Z}_{\ensuremath{\Gamma}}$, related to the single-particle residue Z and effective-mass renormalization ${m}^{\ensuremath{\ast}}/m$. We employ the Pitaevski-Landau identities, which express the derivatives of the self-energy in terms of ${\ensuremath{\Gamma}}^{\ensuremath{\Omega}}$, to obtain and solve a set of coupled nonlinear equations for ${Z}_{\ensuremath{\Gamma}}$, Z, and ${m}^{\ensuremath{\ast}}/m$. We show that near the transition the system enters a critical FL regime, where ${Z}_{\ensuremath{\Gamma}}\ensuremath{\sim}Z\ensuremath{\propto}{(1+{g}_{c,2})}^{1/2}$ and ${m}^{\ensuremath{\ast}}/m\ensuremath{\approx}1/Z$, where gc,2 is the n=2 charge Landau component which approaches $\ensuremath{-}1$ at the instability. We construct the Landau function of the critical FL and show that all but gc,2 Landau components diverge at the critical point. We also show that in the critical regime the one-loop result for the self-energy $\ensuremath{\Sigma}(K)\ensuremath{\propto}\ensuremath{\int}dPG(P)D(K\ensuremath{-}P)$ is asymptotically exact if one identifies the effective interaction D with the RPA form of ${\ensuremath{\Gamma}}^{\ensuremath{\Omega}}$.
We present a theory of the isotropic-nematic quantum phase transition in the composite Fermi liquid arising in half-filled Landau levels. We show that the quantum phase transition between the isotropic and the nematic phase is triggered by an attractive quadrupolar interaction between electrons, as in the case of conventional Fermi liquids. We derive the theory of the nematic state and of the phase transition. This theory is based on the flux attachment procedure, which maps an electron liquid in half-filled Landau levels into the composite Fermi liquid close to a nematic transition. We show that the local fluctuations of the nematic order parameters act as an effective dynamical metric interplaying with the underlying Chern-Simons gauge fields associated with the flux attachment. Both the fluctuations of the Chern-Simons gauge field and the nematic order parameter can destroy the composite fermion quasiparticles and drive the system into a non-Fermi liquid state. The effective-field theory for the isotropic-nematic phase transition is shown to have z=3 dynamical exponent due to the Landau damping of the dense Fermi system. We show that there is a Berry-phase-type term that governs the effective dynamics of the nematic order parameter fluctuations, which can be interpreted as a nonuniversal ``Hall viscosity'' of the dynamical metric. We also show that the effective-field theory of this compressible fluid has a Wen-Zee-type term. Both terms originate from the time-reversal breaking fluctuation of the Chern-Simons gauge fields. We present a perturbative (one-loop) computation of the Hall viscosity and also show that this term is also obtained by a Ward identity. We show that the topological excitation of the nematic fluid, the disclination, carries an electric charge. We show that a resonance observed in radio-frequency conductivity experiments can be interpreted as a Goldstone nematic mode gapped by lattice effects.
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 present a theory of spontaneous Fermi surface deformations for half-filled Landau levels (filling factors of the form ν=2n+1/2). We assume the half-filled level to be in a compressible, Fermi liquid state with a circular Fermi surface. The Landau level projection is incorporated via a modified effective electron-electron interaction and the resulting band structure is described within the Hartree-Fock approximation. We regulate the infrared divergences in the theory and probe the intrinsic tendency of the Fermi surface to deform through Pomeranchuk instabilities. We find that the corresponding susceptibility never diverges, though the system is asymptotically unstable in the n→∞ limit.
"This thesis is devoted to the investigation of the quantum liquid crystal phases in strongly correlated electronic systems. Such phases are characterized by their partially broken spatial symmetries and are observed in various strongly correlated systems as being summarized in Chapter 1. In quantum liquid crystal phases formed by fermions, behaviors beyond Landau's theory of the Fermi liquids appear naturally, signed by the absence of well-defined quasiparticle excitations. Although quantum liquid crystal phases often involve strong-coupling physics, the associated symmetry analysis can often be carried out also in a relatively weak coupling regime and this is the main approach taken in this thesis. Two different quantum liquid crystal phases, the nematic state and the smectic state, are studied in Chapters 2 and 3 respectively, by investigating various instabilities of a Landau Fermi liquid. A nematic phase is a state that breaks the rotational symmetry spontaneously but preserves the translational symmetries. It can be reached by the Pomeranchuk instability of a Landau Fermi liquid. The Pomeranchuk instability in the charge (spin-singlet) sector of a Fermi liquid was studied previously within a random phase approximation and by a high-dimensional bosonization method. In Chapter 2, we concentrate on the Pomeranchuk instability in the spin-triplet sector, and two different classes of ordered phases are discovered, dubbed the a and ,8-phases by analogy to the superfluid 3He-A and B-phases. The Fermi surfaces in the a-phases exhibit spontaneous anisotropic distortions, while in the ,8-phases the Fermi surfaces remain circular or spherical with topologically non-trivial spin configurations in momentum space. The behaviors of the low energy bosonic excitations are studied both in the order phases (a and .B) and at the quantum critical point separating the ordered phases from the Fermi liquid phase. In Chapter 3, the smectic phase and the quantum phase transition between a metallic nematic state and a metallic smectic phase are investigated using the random phase approximation, by considering the formation of a unidirectional charge density wave in an electronic nematic state. The associated quantum critical point is investigated within the Hertz-Millis approach. Non-Fermi liquid behaviors are discovered both at the quantum phase transition and in the smectic ordered phase. The theory we studied here provides a simple description for the so-called ""fluctuating stripe"" phase in cuprates. Similar to its classical counterpart, the phase transition between the nematic and smectic phases are described by a ""gauge-like"" theory as required by the rotational invariance in close analogy with the McMillan-deGennes theory of the smectic-nematic phase transition in classical liquid crystals. However, unlike the classical McMillandeGennes theory, in which the coupling between the ""gauge-like"" nematic fluctuations and the ""matter-like"" smectic fields is relevant, in the sense of renormalization group, this coupling is irrelevant at the quantum phase transition. This property enables the study of low energy fluctuations and the fate of fermions near this quantum critical point. In Chapter 4, quantum liquid crystal phases with a broken time-reversal symmetry are investigated. We start by studying the general scenario of timereversal symmetry breaking phases in the absence of magnetic ordering in systems with long-lived low-energy fermionic quasiparticle excitations in two dimensions. Using a Berry phase approach, we classified possible time-reversal symmetry breaking phases by the accompanying spatial symmetry breaking patterns. Two of the simplest situations are referred to as the type I state, which breaks simultaneously the space-inversion symmetry and the time-reversal symmetry, and the type I I state, in which the chiral symmetry is broken in addition to the breaking of the time-reversal symmetry. Although the starting point is general and has little relation with the quantum liquid crystal phases, both type I and type II states defined above are realized by (generalized) Pomeranchuk instabilities. In fact , a type I I state is usually a nematic state which also breaks the time-reversal symmetry. Interestingly, the type II states also shows spontaneously generated Hall effect even in the absence of a magnetic field, which is known as the spontaneous anomalous Hall effect. In Chapter 5, we investigate the stability of a quadratic band-crossing point in two-dimensional fermionic systems that provide an explicit example, within well controlled approximations, for the nematic ordering discussed in Chapter 2 and spontaneous anomalous Hall effect discussed in Chapter 4. At the non-interacting level, we show that a quadratic band-crossing point exists and is topologically stable when the Berry flux of this crossing point is ±27r and the point symmetry group at this point has either fourfold or sixfold rotational symmetries. Strikingly, we find that this would-be topologically stable quadratic band-crossing is marginally unstable against short-range interactions: a quadratic band-crossing point in two dimensions is unstable against shortrange repulsion, no matter how weak! Four ordered phases are found in this system, the quantum anomalous Hall state, the quantum spin Hall phase, the nematic phase, and the nematic-spin-nematic phase."