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arXiv:cond-mat/9812254 · Physical review. B, Condensed matter

Bulk and edge correlations in the compressible half-filled quantum Hall state

Milica V. Milovanović, Efrat Shimshoni

Published Apr 15, 1999• 7 citations
arXivHTMLPDF

Abstract

We study bulk and edge correlations in the compressible half-filled state [B. I. Halperin et al., Phys. Rev. B 47, 7312 (1993) and, E. Rezayi and N. Read, Phys. Rev. Lett. 72, 900 (1994); 73, 1052 (1994)] using a modified version of the plasma analogy. The corresponding plasma has anomalously weak screening properties, and as a consequence, we find that the correlations along the edge do not decay algebraically as in the Laughlin (incompressible) case, while the bulk correlations decay in the same way. The results suggest that due to the strong coupling between charged modes on the edge and the neutral fermions in the bulk, reflected by the weak screening in the plasma analog, the (attractive) correlation hole is not well defined on the edge. Hence, the system there can be modeled as a free Fermi gas of electrons (with an appropriate boundary condition). We finally comment on a possible scenario in which the Laughlin-like dynamical edge correlations may nevertheless be realized.

DOI
10.1103/physrevb.59.10757

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We study bulk and edge correlations in the compressible half-filled state [B. I. Halperin et al., Phys. Rev. B 47, 7312 (1993) and, E. Rezayi and N. Read, Phys. Rev. Lett. 72, 900 (1994); 73, 1052 (1994)] using a modified version of the plasma analogy. The corresponding plasma has anomalously weak screening properties, and as a consequence, we find that the correlations along the edge do not decay algebraically as in the Laughlin (incompressible) case, while the bulk correlations decay in the same way. The results suggest that due to the strong coupling between charged modes on the edge and the neutral fermions in the bulk, reflected by the weak screening in the plasma analog, the (attractive) correlation hole is not well defined on the edge. Hence, the system there can be modeled as a free Fermi gas of electrons (with an appropriate boundary condition). We finally comment on a possible scenario in which the Laughlin-like dynamical edge correlations may nevertheless be realized.
For a meaningful numerical investigation of thermodynamic properties, systems bigger than the correlation length need to be studied. Fortunately, due to the existence of a gap, the incompressible fractional quantum Hall states have a finite correlation length, which, for some states (e.g., 1/3, 2/5) happens to be smaller than the system sizes that can be studied. On the other hand, since the Fermi liquid is a critical state with power law correlations, one might expect that finite system studies are less likely to provide any conclusive results. Rezayi and Read (RR) [1] have recently claimed that their numerical study provides “convincing evidence for the correctness of the Halperin-Lee-Read [2] theory of a compressible fermi-liquid-like state ” at ν = 1/2. We argue below that their study sheds little light on the on the thermodynamic nature of the compressible state at ν = 1/2. RR have studied N-electron systems at flux Nφ = 2(N − 1) in the spherical geometry. Their numerical calculations test a special case of the composite fermion theory [3], which provides a general relationship between the low-energy spectra and eigenstates of the systems at Nφ and N ∗ φ = Nφ − 2(N − 1). Extensive calculations of this type have been performed in the past, including for Nφ = 2(N −1) [4–6]. RR merely interpret the results differently. For
The nature of the fractional quantum Hall state at Landau level filling factor 5/2 remains elusive despite intensive experimental and theoretical work. While the leading theoretical candidates are Moore-Read Pfaffian (Pf) and its particle-hole conjugate anti-Pfaffian (APf), neither received unambiguous experimental support. We show that a state that is intermediate between them, made of alternating stripes of Pf and APf in the bulk, is a viable candidate. Such a state is shown to be incompressible and thus a charge insulator in the bulk, but a heat conductor due to the presence of gapless neutral bulk modes. We argue that the properties of such a state are consistent with existing numerical and experimental work, and discuss possible experimental probes of its presence.
The energy spectra and wave functions of up to 14 interacting quasielectrons (QE's) in the Laughlin $\ensuremath{\nu}=\frac{1}{3}$ fractional quantum Hall (FQH) state are investigated using exact numerical diagonalization. It is shown that at sufficiently high density the QE's form pairs or larger clusters. This behavior, opposite to Laughlin correlations, invalidates the (sometimes invoked) reapplication of the composite fermion picture to the individual QE's. The series of finite-size incompressible ground states are identified at the QE filling factors ${\ensuremath{\nu}}_{\mathrm{QE}}=\frac{1}{2},$ 13,\frac{1}{3},31​, 23,\frac{2}{3},32​, corresponding to the electron fillings $\ensuremath{\nu}=\frac{3}{8},$ 411,\frac{4}{11},114​, 513.\frac{5}{13}.135​. The equivalent quasihole (QH) states occur at ${\ensuremath{\nu}}_{\mathrm{QH}}=\frac{1}{4},$ 15,\frac{1}{5},51​, 27,\frac{2}{7},72​, corresponding to $\ensuremath{\nu}=\frac{3}{10},$ 413,\frac{4}{13},134​, 517.\frac{5}{17}.175​. All these six novel FQH states were recently discovered experimentally. Detailed analysis indicates that QE or QH correlations in these states are different from those of well-known FQH electron states (e.g., Laughlin or Moore--Read states), leaving the origin of their incompressibility uncertain. Halperin's idea of Laughlin states of QP pairs is also explored, but is does not seem adequate.
We study the Laughlin wave function on the cylinder. We find it only describes an incompressible fluid when the two lengths of the cylinder are comparable. As the radius is made smaller at fixed area, we observe a continuous transition to the charge-density-wave Tao-Thouless state. We also present some exact properties of the wave function in its polynomial form. We then study the edge excitations of the quantum Hall incompressible fluid modeled by the Laughlin wave function. The exponent describing the fluctuation of the edge predicted by recent theories is shown to be identical with numerical calculations. In particular, we obtain the occupation amplitudes of edge state n(k) for four to ten electron-size systems. When plotted as a function of the scaled wave vector they become essentially free of finite-size effects. The resulting curve obtains a very good agreement with the appropriate infinite-size Calogero-Sutherland model occupation numbers. Finally, we numerically obtain n(k) of the edge excitations for some pairing states which may be relevant to the \ensuremath{\nu}=5/2 incompressible Hall state.
This paper reviews the authors' theoretical understanding for the quantum Hall anomalies recently observed at fractional fillings with even denominators, more particularly at filling 1/2, where the electron spins are still fully polarized. After a brief reminder of the basic experimental evidence dip of {rho}{sub xx}, lack of plateau in {rho}{sub xx}, temperature behavior, the author discusses the important role of electron-hole (e-h) symmetry at half-filling. In particular the two alternative possibilities of spontaneous e-h symmetry breaking or non-breaking are discussed along with their physical implications. Hartree-Fock charge density waves are presented as an explicit example of spontaneous e-h symmetry breaking. Next, several different scenarios are discussed for the ground state at half-filling. One, largely discussed in the experimental papers is that of Halperin, involving phase separation. Another is that of Fano et al. where the half-filling ground state is supposed to be incompressible and homogeneous, but deformable. Pursuing further the second scenario, numerical evidence from the recent work of Chen is presented, indicating incompressibility and deformability. This work also demonstrates a rather abrupt switching of short-range electron correlations. Just below half-filling, an electron is typically surrounded by six close-by electrons (like a triangular lattice).
We consider the Hall conductivity of composite fermions in the theory of Halperin, Lee, and Read (HLR) [B. I. Halperin, P. A. Lee, and N. Read, Phys. Rev. B 47, 7312 (1993)]. We present a fully quantum-mechanical numerical calculation that shows, under suitable conditions, the HLR theory exhibits a particle-hole symmetric dc electrical Hall response in the presence of quenched disorder. Remarkably, this response of the HLR theory remains robust even when the disorder range is of the order of the Fermi wavelength. We find that deviations from the particle-hole symmetric response can appear in the ac Hall conductivity at frequencies sufficiently large compared to the inverse system size. Our results agree with a recent semiclassical analysis by Wang et al. [C. Wang, N. R. Cooper, B. I. Halperin, and A. Stern, Phys. Rev. X 7, 031029 (2017)] and complement the arguments based on the fully quantum-mechanical model by Kumar et al. [P. Kumar, M. Mulligan, and S. Raghu, Phys. Rev. B 98, 115105 (2018)]. These results provide further evidence that the HLR theory is compatible with an emergent particle-hole symmetry.