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arXiv:1808.09953 · Physical review. B./Physical review. B

Universal collective modes in two-dimensional chiral superfluids

Wei-Han Hsiao

Published Sep 6, 2019• 6 citations
arXivHTMLPDF

Abstract

In this work, we utilize semiclassical kinetic equations to investigate the order parameter collective modes of a class of two-dimensional superfluids. Extending the known results for $p$-wave superfluids, we show that for any chiral ground state of angular momentum $L\ensuremath{\ge}1$, there exists a subgap mode with a mass $\sqrt{2}\phantom{\rule{0.16em}{0ex}}\mathrm{\ensuremath{\Delta}}$ in the BCS limit, where $\mathrm{\ensuremath{\Delta}}$ is the magnitude of the ground-state gap. We determine the most significant Landau parameter that contributes to the mass renormalization and show explicitly that the renormalized modes become massless at the Pomeranchuk instability of the fermion vacuum. Particularly for $L=1$, we propose a continuous field theory to include the Fermi liquid effect in a quadrupolar channel and produce the same result under consistent approximations. They provide potential diagnostics for distinguishing two-dimensional chiral ground states of different angular momenta with the order parameter collective modes and reveal another low-energy degree of freedom near the nematic transition.

DOI
10.1103/physrevb.100.094510

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In this work, we utilize semiclassical kinetic equations to investigate the order parameter collective modes of a class of two-dimensional superfluids. Extending the known results for ppp-wave superfluids, we show that for any chiral ground state of angular momentum L\ensuremath≥1L\ensuremath{\ge}1L\ensuremath≥1, there exists a subgap mode with a mass 2\ensuremathΔ\sqrt{2}\phantom{\rule{0.16em}{0ex}}\mathrm{\ensuremath{\Delta}}2​\ensuremathΔ in the BCS limit, where \ensuremathΔ\mathrm{\ensuremath{\Delta}}\ensuremathΔ is the magnitude of the ground-state gap. We determine the most significant Landau parameter that contributes to the mass renormalization and show explicitly that the renormalized modes become massless at the Pomeranchuk instability of the fermion vacuum. Particularly for L=1L=1L=1, we propose a continuous field theory to include the Fermi liquid effect in a quadrupolar channel and produce the same result under consistent approximations. They provide potential diagnostics for distinguishing two-dimensional chiral ground states of different angular momenta with the order parameter collective modes and reveal another low-energy degree of freedom near the nematic transition.
Motivated by the search for quantum liquid crystal phases in a gas of ultracold atoms and molecules, we study the density-wave and nematic instabilities of dipolar fermions on the two-dimensional square lattice (in the $x\text{\ensuremath{-}}y$ plane) with dipoles pointing to the zzz direction. We determine the phase diagram using two complementary methods, the Hartree-Fock mean-field theory and the linear-response analysis of compressibility. Both give consistent results. In addition to the staggered $(\ensuremath{\pi},\ensuremath{\pi})$ density wave, over a finite range of densities and hopping parameters, the ground state of the system first becomes nematic and then smectic, when the dipolar interaction strength is increased. Both phases are characterized by the same broken fourfold (C4)({C}_{4})(C4​) rotational symmetry. The difference is that the nematic phase has a closed Fermi surface but the smectic does not. The transition from the nematic to the smectic phase is associated with a jump in the nematic order parameter. This jump is closely related to the Van Hove singularities. We derive the kinetic equation for collective excitations in the normal isotropic phase and find that the zero sound mode is strongly Landau damped and thus is not a well-defined excitation. Experimental implications of our results are discussed.
We study the orbital angular momentum (OAM) Lz{L}_{z}Lz​ in two-dimensional chiral $({p}_{x}+i{p}_{y}{)}^{\ensuremath{\nu}}$-wave superfluids (SFs) of NNN fermions on a disk at zero temperature, in terms of spectral asymmetry and spectral flow. It is shown that ${L}_{z}=\ensuremath{\nu}N/2$ for any integer $\ensuremath{\nu}$, in the Bose-Einstein condensation regime. In contrast, in the BCS limit, while the OAM is Lz=N/2{L}_{z}=N/2Lz​=N/2 for the p+ipp+ipp+ip-wave SF, for chiral SFs with $\ensuremath{\nu}\ensuremath{\ge}2$, the OAM is remarkably suppressed as ${L}_{z}=N\ifmmode\times\else\texttimes\fi{}O({\mathrm{\ensuremath{\Delta}}}_{0}/{ϵ}_{F})\ensuremath{\ll}N$, where ${\mathrm{\ensuremath{\Delta}}}_{0}$ is the gap amplitude and ϵF{ϵ}_{F}ϵF​ is the Fermi energy. We demonstrate that the difference between the p+ipp+ipp+ip-wave SF and the other chiral SFs in the BCS regimes originates from the nature of edge modes and related depairing effects.
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{F}_{2}<0F2​<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{F}_{2}F2​ and ${\ensuremath{\omega}}_{c}$.
The spectra of fermionic excitations, pairing correlations, and edge currents confined near the boundary of a chiral ppp-wave superfluid are calculated to leading order in $\ensuremath{\hbar}/{p}_{f}\ensuremath{\xi}$. Results for the energy- and momentum-resolved spectral functions, including the spectral current density, of a chiral ppp-wave superfluid near a confining boundary are reported. The spectral functions reveal the subtle role of the chiral edge states in relation to the edge current and the angular momentum of a chiral ppp-wave superfluid, including the rapid suppression of Lz(T){L}_{z}(T)Lz​(T) for $0\ensuremath{\lesssim}T\ensuremath{\ll}{T}_{c}$ in the fully gapped two-dimensional chiral superfluid. The edge current and ground-state angular momentum are shown to be sensitive to boundary conditions, and as a consequence the topology and geometry of the confining boundaries. For perfect specular boundaries, the edge current accounts for the ground-state angular momentum, ${L}_{z}=(N/2)\ensuremath{\hbar}$, of a cylindrical disk of a chiral superfluid with N/2N/2N/2 fermion pairs. Nonspecular scattering can dramatically suppress the edge current. In the limit of perfect retroreflection, the edge states form a flat band of zero modes that are nonchiral and generate no edge current. For a chiral superfluid film confined in a cylindrical toroidal geometry, the ground-state angular momentum is, in general, nonextensive, and can have a value ranging from ${L}_{z}>(N/2)\ensuremath{\hbar}$ to ${L}_{z}<\ensuremath{-}(N/2)\ensuremath{\hbar}$ depending on the ratio of the inner and outer radii and the degree of backscattering on the inner and outer surfaces. Nonextensive scaling of Lz{L}_{z}Lz​, and the reversal of the ground-state angular momentum for a toroidal geometry, would provide a signature of broken time-reversal symmetry of the ground state of superfluid 3{}^{3}3He-AAA, as well as direct observation of chiral edge currents.
Abstract Using the Landau kinetic equation to study the non-equilibrium behavior of interacting Fermi systems is one of the crowning achievements of Landau’s Fermi liquid theory. While thorough study of transport modes has been done for standard three-dimensional Fermi liquids, an equally in-depth analysis for two dimensional Fermi liquids is lacking. In applying the Landau kinetic equation (LKE) to a two-dimensional Fermi liquid, we obtain unconventional behavior of the zero sound mode c 0 . As a function of the usual dimensionless parameter s = ω / q v F , we find two peculiar results: first, for | s | > 1 we see the propagation of an undamped mode for weakly interacting systems. This differs from the three dimensional case where an undamped mode only propagates for repulsive interactions and the mode experiences Landau damping for any arbitrary attractive interaction. Second, we find that regardless of interaction strength, a propagating mode is forbidden for | s | < 1. This is profoundly different from the three-dimensional case where a mode can propagate, albeit damped. In addition, we present a revised Pomeranchuk instability condition for a two-dimensional Fermi liquid as well as equations of motion for the fluid that follow directly from the LKE. In two dimensions, we find a constant minimum for all Landau parameters for ℓ ⩾ 1 which differs from the three dimensional case. Finally we discuss the effect of a Coulomb interaction on the system resulting in the plasmon frequency ω p exhibiting a crossover to the zero sound mode.