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arXiv:1606.00899 · Physical review. B./Physical review. B

Pairing of particle-hole symmetric composite fermions in half-filled Landau level

Zhiqiang Wang, Sudip Chakravarty

Published Oct 17, 2016• 8 citations
arXivHTMLPDF

Abstract

In a recent proposal of the half-filled Landau level, the composite fermions are taken to be Dirac particles and particle-hole symmetric. Cooper pairing of these composite fermions in different angular momentum channels, $\ensuremath{\ell}$, can give rise to different kinds of Pfaffian states. In addition to the well-known Moore-Read Pfaffian and anti-Pfaffian states, a new putative particle-hole symmetric Pfaffian state, corresponding to the $s$-wave pairing channel, was also proposed. However, the possible underlying pairing mechanism is not clear at all. In this work we provide a specific pairing mechanism for realizing some of these Pfaffian states. We show that there can be nonzero pairing in angular momentum channels $|\ensuremath{\ell}|\ensuremath{\ge}1$ depending on the magnitude of a coupling constant. There is a quantum phase transition from the Dirac composite Fermi-liquid state to Cooper pairing states in angular momentum channels $|\ensuremath{\ell}|\ensuremath{\ge}1$ as the coupling constant is tuned across its critical point value. Surprisingly the particle-hole symmetric $\ensuremath{\ell}=0$ channel pairing turns out to be impossible irrespective of the size of the coupling constant.

DOI
10.1103/physrevb.94.165138

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arXiv:2103.13958•Modern Physics Letters B•Jul 3, 2020•6 citations
In a recent proposal of the half-filled Landau level, the composite fermions are taken to be Dirac particles and particle-hole symmetric. Cooper pairing of these composite fermions in different angular momentum channels, \ensuremathℓ\ensuremath{\ell}\ensuremathℓ, can give rise to different kinds of Pfaffian states. In addition to the well-known Moore-Read Pfaffian and anti-Pfaffian states, a new putative particle-hole symmetric Pfaffian state, corresponding to the sss-wave pairing channel, was also proposed. However, the possible underlying pairing mechanism is not clear at all. In this work we provide a specific pairing mechanism for realizing some of these Pfaffian states. We show that there can be nonzero pairing in angular momentum channels ∣\ensuremathℓ∣\ensuremath≥1|\ensuremath{\ell}|\ensuremath{\ge}1∣\ensuremathℓ∣\ensuremath≥1 depending on the magnitude of a coupling constant. There is a quantum phase transition from the Dirac composite Fermi-liquid state to Cooper pairing states in angular momentum channels ∣\ensuremathℓ∣\ensuremath≥1|\ensuremath{\ell}|\ensuremath{\ge}1∣\ensuremathℓ∣\ensuremath≥1 as the coupling constant is tuned across its critical point value. Surprisingly the particle-hole symmetric \ensuremathℓ=0\ensuremath{\ell}=0\ensuremathℓ=0 channel pairing turns out to be impossible irrespective of the size of the coupling constant.
We propose a particle-hole symmetric theory of the Fermi-liquid ground state of a half-filled Landau level. This theory should be applicable for a Dirac fermion in the magnetic field at charge neutrality, as well as for the v = 1/2 quantum Hall ground state of nonrelativistic fermions in the limit of negligible inter-Landau-level mixing. We argue that when particle-hole symmetry is exact, the composite fermion is a massless Dirac fermion, characterized by a Berry phase of π around the Fermi circle. We write down a tentative effective field theory of such a fermion and discuss the discrete symmetries, in particular, C P. The Dirac composite fermions interact through a gauge, but non-Chern-Simons, interaction. The particle-hole conjugate pair of Jain-sequence states at filling factors n/(2n + 1) and (n + 1)/(2n + 1), which in the conventional composite fermion picture corresponds to integer quantum Hall states with different filling factors, n and n + 1, is now mapped to the same half-integer filling factor n + 1/2 of the Dirac composite fermion. The Pfaffian and anti-Pfaffian states are interpreted as d -wave Bardeen-Cooper-Schrieffer paired states of the Dirac fermion with orbital angular momentum of opposite signs, while s -wave pairing would give rise to a particle-hole symmetric non-Abelian gapped phase. When particle-hole symmetry is not exact, the Dirac fermion has a C P -breaking mass. The conventional fermionic Chern-Simons theory is shown to emerge in the nonrelativistic limit of the massive theory.
Recently, a Dirac (particle-hole symmetric) description of composite fermions in the half-filled Landau level (LL) was proposed [D. T. Son, Phys. Rev. X 5, 031027 (2015)], and we study its possible consequences on BCS (Cooper) pairing of composite fermions (CFs). One of the main consequences is the existence of anisotropic states in single-layer and bilayer systems, which was previously suggested in Jeong and Park [J. S. Jeong and K. Park, Phys. Rev. B 91, 195119 (2015)]. We argue that in the half-filled LL in the single-layer case the gapped states may sustain anisotropy, because isotropic pairings may coexist with anisotropic ones. Furthermore, anisotropic pairings with the addition of a particle-hole symmetry-breaking mass term may evolve into rotationally symmetric states, i.e., Pfaffian states of Halperin-Lee-Read (HLR) ordinary CFs. On the basis of the Dirac formalism, we argue that in the quantum Hall bilayer at total filling factor 1, with decreasing distance between the layers, weak pairing of ppp-wave paired CFs is gradually transformed from Dirac to ordinary, HLR-like, with a concomitant decrease in the CF number. Global characterization of low-energy spectra based on the Dirac CFs agrees well with previous calculations performed by exact diagonalization on a torus. Finally, we discuss features of the Dirac formalism when applied in this context.
Here, the authors investigate the possibility that particle-hole symmetry in the lowest Landau level at half filling arises as an emergent low-energy symmetry. Specifically, they start with the traditional approach to the half-filled Landau level: an effective field theory known as the composite Fermi liquid pioneered by Halperin, Lee, and Read. In this theory, composite fermions arise when flux is attached to electrons in a half-filled Landau level. However, an equivalent starting point is one where one starts with a filled Landau level, depletes electrons (or equivalently, adds holes) to the point of half filling. One then attaches flux to the holes, and the resulting composite hole description is another viable description of the half-filled Landau level. The authors construct a particle-hole symmetric theory by studying a system consisting of alternating quasi-one-dimensional strips of composite Fermi liquid (CFL) and composite hole liquid (CHL), both of which break particle-hole symmetry. Gauge invariance requires the existence of gapless edge modes at the interface between the CFL and CHL. Out of these modes, they construct a neutral Dirac fermion coupled to an emergent U(1) gauge field. Thus, at distances long compared to the strip width, they demonstrate that the system is described by a Dirac fermion coupled to an emergent gauge field, with an anti-unitary particle-hole symmetry, precisely the form conjectured recently by Son to be the low-energy description of a particle-hole symmetric half-filled Landau level.
We study Cooper pairing in the Dirac composite fermion (CF) system. The presence of the mass term in the Dirac CF description (which may simulate Landau level mixing), i.e., breaking of particle-hole (PH) symmetry in this system, is a necessary condition for the existence of a PH Pfaffian-like topological state. In the scope of the random-phase approximation (RPA) and hydrodynamic approach, we find some signatures of pairing at finite frequencies. Motivated by this insight, we extend our analysis to the case of a different but still Dirac quasiparticle (CF) representation, appropriate in the presence of a mass term, and discuss the likelihood of PH Pfaffian and Pfaffian pairings in general. On the basis of gauge field effects, we find for a small Dirac mass, an anti-Pfaffian or Pfaffian instability depending on the sign of mass, while for large mass (Landau level mixing), irrespective of its sign, we find a PH Pfaffian-like instability.
We begin by scrutinizing a recent proposal that presents an alternate description of the half-filled Landau level in terms of massless Dirac fermions. In Chapter \\ref{chap:HFLL}, we examine the possibility of pairing of these Dirac fermions by numerically solving the coupled Eliashberg equations unlike a related previous calculation (Wang and Chakravarty, 2016). In addition, vertex corrections are calculated to be zero from the Ward identity. We find that pairing is possible in non-zero angular momentum channels; only differences are minor numerical shifts. As before, the pairing leads to the gapped Pfaffian and anti-Pfaffian states. However, in our approximation scheme, pairing is not possible in the putative particle–hole symmetric state for l=0l = 0l=0 angular momentum. The specific heat at low temperatures of a system of massless Dirac fermions interacting with a transverse gauge field, expected to be relevant for the half-filled Landau level, is calculated. Using the Luttinger formula, it is found to be proptoTlnT\\propto T \\ln TproptoTlnT in the leading low temperature limit, due to the exchange of transverse gauge bosons. The result agrees with the corresponding one in the nonrelativistic composite fermion theory of Halperin, Lee and Read of the half-filled Landau level.The rest of the thesis concerns the cuprate high-TcT_cTc​ superconductors (``cuprates''). Conventional wisdom says that beyond the superconducting ``dome'' in cuprates, the material behaves as a Fermi liquid. However, this picture does not help explain the disappearance of the superconducting order parameter, and there are some anomalous measurements that cannot be explained by a Fermi liquid phase. It was proposed by Kopp, Ghosal, and Chakravarty that there is a \\emph{ferromagnetic} phase at zero temperature beyond the superconducting dome, and that fluctuations of the ferromagnetic order parameter compete with the superconducting order parameter and work to suppress the superconducting transition temperature \\cite{Kopp:2007fn}. In Chapter \\ref{chap:ferro} we summarize the experimental evidence for ferromagnetic fluctuations in the overdoped cuprates, and present several calculations supporting the existence of a ferromagnetic ground state in the 2D single-band Hubbard model, which model is thought to provide an adequate description of the cuprate superconductors.Another region of interest in the cuprate phase diagram is the pseudogap phase. It is unclear which of a host of competing order parameters is responsible for the behavior in this phase, such as the recent observation of an anomalous thermal Hall conductance in the cuprate La2−x{}_{2-x}2−x​Srx{}_xx​CuO4{}_44​ \\cite{Grissonnanche:2019gx}. One promising candidate is the ddd-density wave state. In Chapter \\ref{chap:DMI}, we investigate the effect that density wave states have on the localized spins of a square lattice. We derive the effective Dzyaloshinskii-Moriya (DM) interaction from first principles and study its effects on both ferromagnetic and antiferromagnetic backgrounds. We find that topologically nontrivial density wave states can induce stable DM interactions among the localized spins of the lattice when an external magnetic field is present. Furthermore, these density wave-induced DM vectors point along the external magnetic field's direction--implying that they break time-reversal and spin rotation symmetries in the same manner. Due to these symmetry considerations alone we find that the underlying magnon excitations cannot induce any thermal Hall effect. Utilizing a Holstein-Primakoff substitution about a mean-field ground state expansion we calculate the topological density wave corrections to magnetic ground state energy, spin canting angles, and the dispersion of the magnons for both the ferromagnetic and antiferromagnetic cases.
In this review, the physics of Pfaffian paired states, in the context of fractional quantum Hall effect, is discussed using field-theoretical approaches. The Pfaffian states are prime examples of topological ([Formula: see text]-wave) Cooper pairing and are characterized by non-Abelian statistics of their quasiparticles. Here we focus on conditions for their realization and competition among them at half-integer filling factors. Using the Dirac composite fermion description, in the presence of a mass term, we study the influence of Landau level mixing in selecting a particular Pfaffian state. While Pfaffian and anti-Pfaffian are selected when Landau level mixing is not strong, and can be taken into account perturbatively, the particle–hole (PH) Pfaffian state requires non-perturbative inclusion of at least two Landau levels. Our findings, for small Landau level mixing, are in accordance with numerical investigations in the literature, and call for a non-perturbative approach in the search for PH Pfaffian correlations. We demonstrated that a method based on the Chern–Simons field-theoretical approach can be used to generate characteristic interaction pseudo-potentials for Pfaffian paired states.