Neutral Fermions at Ν = 1/2 Typeset Using Revt E X

We present a theory to describe the low energy physics of a two-dimensional electron gas at filling factor ν = 1/2. In this theory the elementary excitations are dipole-like fermionic particles each made up of charge ±1/2 quasiparticles. These neutral fermions obey a constrained dynamics, do not respect time reversal symmetry, and do not form a conventional Fermi liquid. Sufficiently strong disorder breaks up the dipoles and causes the system to realize the critical state of the ν = 0→1 plateau transition. We argue that the present description is consistent with the particle-hole symmetry in the lowest Landau level. 73.50.Jt, 05.30.-d, 74.20.-z Typeset using REVTEX 1 The discovery of an anomaly in acoustic wave propagation attracted much interest in the physics of a two-dimensional electron gas (2DEG) near filling factor ν = 1/2 [1]. Shortly after the experimental discovery an intriguing idea, composite fermion theory, was put forward in an extensive paper by Halperin, Lee and Read (HLR) [2]. In the HLR theory, an electron is represented as a Chern-Simons (CS) fermion carrying two quanta of fictitious magnetic flux pointing in the direction opposite to the external magnetic field [3]. The advantage of such representation is that on average the fictitious magnetic field cancels the real one so that CS fermions see no net field. HLR theory asserts that under such a condition CS fermions form a Fermi liquid, and the observed anomaly is the manifestation of that fact. This idea obtained wide acceptance when experimental evidence indicating the existence of a Fermi surface at ν = 1/2 were obtained [4]. Since in the CS approach the fictitious flux is dynamic, it is important to estimate the corrections caused by its fluctuations. The hope is that the latter will just renormalize the parameters of the Fermi liquid. When calculation of such correction was done, it is found that the correction diverges [2,5]. Attempts to resum the divergence have not led to a conclusive result. Before going any further, let us first define what we mean by the CS Fermi liquid. It is possible to prove that the electron resistivity tensor ραβ can be written as the sum of two terms: ραβ = ρ CS αβ + ǫαβ 2h e2 . (1) In the above ρ αβ = (σ CS) αβ is the CS fermion resistivity tensor. To calculate σ CS αβ , one needs to sum all CS fermion particle-hole diagrams that are irreducible with respect to cutting a CS gauge line [2,6]. CS Fermi liquid theory asserts that the correct ρ αβ in Eq.(1) should have the form characteristic of a zero field Fermi liquid. In a recent paper Lee, Krotov, Gan and Kivelson (LKGK) raised a question about the compatibility between the CS Fermi liquid theory and the particle-hole symmetry in the lowest Landau level [7]. In particular, they showed that particle-hole symmetry implies the 2 electron Hall conductivity must be e/2h. (The latter has been experimentally confirmed in Ref. [8].) When combined with Eq.(1), they showed that this implies that σ xy = −e2/2h as long as σ xx 6= 0. LKGK then used the fact that σ xy 6= 0 to question the assertion that the CS fermions form a zero-field Fermi liquid. Disorder poses another challenge on the CS Fermi liquid description. For example, let us consider a particle-hole symmetric disorder potential in the lowest Landau level. When this potential overwhelms the interaction energy, the system breaks up into puddles of the ν = 1 quantum Hall liquid and vacuum. Due to the particle-hole symmetry, both types of puddles percolate. This percolating state is the critical state of the ν = 0→ν = 1 plateau transition. So far the CS Fermi liquid picture has been unable to account for it. The following is a qualitative summary of our results. 1) σxy = e /2h suggests that a CS fermion must be dressed by a void of unit charge. [9] We call the resulting dipole-like object a neutral fermion. 2) The neutral fermions obey a constrained dynamics [10,11], and violate the time reversal symmetry. 3) The neutral fermion contribute to electrical conduction through its polarization current. The conductivity tensor σ αβ obtained from the correlation function of such polarization current is related to the electron σαβ via σαβ = e 2h ǫαβ + σ N αβ . 4) In the limit of very strong disorder, the neutral fermions are broken apart. When that happens, the system realizes the critical state of the ν = 0→ν = 1 plateau transition. To begin, let us first discuss the implications of σxy = e /2h [7]. Imagine inserting a CS fermion at a fixed point in space. Since the CS fermion can be viewed as an electron carrying two quanta of fictitious flux, an excess of two magnetic flux quanta develops at that point. The EMF produced by this time-dependent flux induces an outflow of electrons because of the Hall effect. Since σxy = e /2h, the total charge that flows out is 2 × 1 2 e = e. As the result the CS fermion is dressed by a void of unit charge. The resulting object is neutral relative to an uniformly charged background. It was called a “composite fermion” in an insightful paper by Read [9]. Since in the literature the term “composite fermion” is often used interchangeably with “CS fermion”, we decide to call Read’s neutral particle “neutral fermion” to avoid confusion. At present there are several ongoing attempts at describing 3 the neutral fermions [10–12]. Our main objective are two folds: 1) describe neutral fermions in ideal system, and 2) understand how does the critical state of plateau transition appears in the limit of strong disorder. Our physical picture is as follow. We view each electron as a CS boson carrying a quantum of fictitious flux. Were it not for the fictitious flux, the bosons would condense into the ideal ν = 1/2 Laughlin state Ψ1/2 = ∏ (ij)(zi − zj) exp{− ∑ k |zk|/4l 0}. (Here l0 = √ h̄c/Be is the magnetic length.) Each fictitious flux quantum creates and binds to a charge −1/2 quasiparticle of the 1/2 liquid. (Because of the fictitious magnetic flux the statistics angle of this quasiparticle is −π/2 instead of π/2.) To maintain charge neutrality, an equal number of charge 1/2 (statistics −π/2) quasiholes are nucleated. The fermionic bound state of two such opposite charged quasiparticles is the neutral fermion. The above picture is consistent with the following wavefunction consideration. It is well known that the wavefunction Ψ = PLdet(zj , z̄j)Ψ1/2 has an almost perfect overlap with the exact numerical ground state of systems with a small number of particles. [10,13] In the above PL is the lowest Landau level projection operator, and det is the Slater determinant of a filled Fermi sea. The projection operator causes zi→2 ∂ ∂zi in the Slater determinant. As the result, PL[det] displaces the zeroes of Ψ1/2. [9] However, since the resulting Ψ is a fermion wavefunction in the lowest Landau level, it can always be written as Ψ = S({zk}) ∏ (ij)(zi − zj) exp{− ∑ k |zk|/4l 0}, where S is a homogeneous symmetric polynomial of degree N − 1 (N= the total electron number). Thus the net effect of PL[det] on Ψ1/2 is to displace only one out of the two vortices associated with every particle. The consequence of that is to create charge ±1/2 quasiparticles and quasiholes. The CS boson action is given by