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arXiv:cond-mat/0402562 · Physical Review B

Microscopic construction of the chiral Luttinger liquid theory of the quantum Hall edge

Alexey Mikhajlovich Boyarsky, Vadim Cheianov, Oleg Ruchayskiy C

Published Dec 8, 2004• 20 citations
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Abstract

We give a microscopic derivation of the chiral Luttinger liquid theory $(\ensuremath{\chi}LL)$ for the Laughlin states. Starting from the wave function describing an arbitrary incompressibly deformed Laughlin state (IDLS) we quantize these deformations. In this way we obtain the low-energy projections of local microscopic operators and derive the quantum field theory of edge excitations directly from quantum mechanics of electrons. This shows that to describe experimental and numeric deviations from $\ensuremath{\chi}LL$ one needs to go beyond Laughlin's approximation. We show that in the large $N$ limit the IDLS is described by the dispersionless Toda hierarchy.

DOI
10.1103/physrevb.70.235309

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We give a microscopic derivation of the chiral Luttinger liquid theory (\ensuremathχLL)(\ensuremath{\chi}LL)(\ensuremathχLL) for the Laughlin states. Starting from the wave function describing an arbitrary incompressibly deformed Laughlin state (IDLS) we quantize these deformations. In this way we obtain the low-energy projections of local microscopic operators and derive the quantum field theory of edge excitations directly from quantum mechanics of electrons. This shows that to describe experimental and numeric deviations from \ensuremathχLL\ensuremath{\chi}LL\ensuremathχLL one needs to go beyond Laughlin's approximation. We show that in the large NNN limit the IDLS is described by the dispersionless Toda hierarchy.
Based on the composite fermion approach, we derive a microscopic theory describing the low-lying edge excitations in the fractional quantum Hall liquid with ν=ν∗ϕ~ν∗+1ν=\frac{ν^*}{\tildeϕν^*+1}ν=ϕ~​ν∗+1ν∗​. For ν∗>0ν^*>0ν∗>0, it is found that the composite fermion model reduces to an SU(ν∗)(ν^*)(ν∗) Calogero-Sutherland model in the one-dimensional limit, whereas it is not exact soluble for ν∗<0ν^*<0ν∗<0. However, the ground states in both cases can be found and the low-lying excitations can be shown the chiral Luttinger liquid behaviors since a gap exists between the right- and left-moving sectors in each branch of the azimuthal excitations.
The effective field theory of the fractional quantum Hall edge is reformulated from microscopic dynamics. Noncommutative Chern-Simons theory is a microscopic description for the quantum Hall fluid. We use it for reference. Considering relabeling symmetry of the electrons and incompressibility of the fluid, we obtain a constraint and derive a chiral Tomonaga-Luttinger liquid theory containing interaction terms. We calculate one-loop corrections to the phonon and electron propagators and get a new tunneling exponent. It agrees with experiments.
We consider a series of problems regarding quantum Hall edges, focusing on both dynamics and the mathematical structure of edge states. We begin in Chapter 3 with a limiting case of the Laughlin state placed in a very steep confining potential, but which is weak compared to the interactions. We find that the eigenstates have a Jack polynomial structure and an energy spectrum which is extremely different from the well-known Luttinger liquid edge. In Chapter 5 we analyse the inner products of edge state wavefunctions, using an effective description given by a large-N expansion ansatz proposed by J. Dubail, N. Read and E. Rezayi, PRB 86, 245310 (2012). As noted by these authors, the terms in this ansatz can be constrained using symmetry, a procedure we perform to high orders. We then check the conjecture by calculating overlaps exactly for small system sizes and comparing the numerics with our high-order expansion to find excellent agreement. Finally, Chapter 6 considers the behaviour of quantum Hall edges close to the Luttinger liquid fixed point that occurs in the low energy, large system limit. We construct effective Hamiltonians using a local field theory description and then consider the effect of bulk symmetries on this edge. The symmetry analysis produces remarkable simplifications which allow for very accurate descriptions of the low-energy edge physics even relatively far away from the Luttinger liquid fixed point.
We derive a microscopic model describing low-lying edge excitations in a fractional quantum Hall liquid with $\ensuremath{\nu}={\ensuremath{\nu}}^{*}/(\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\varphi}}{\ensuremath{\nu}}^{*}+1).$ For ${\ensuremath{\nu}}^{*}>0,$ it is found that the composite-fermion model reduces to an $\mathrm{SU}({\ensuremath{\nu}}^{*})$ Calogero-Sutherland model in a dimension reduction, whereas it is not exactly soluble for ${\ensuremath{\nu}}^{*}<0.$ However, the ground states in both cases can be found, and the low-lying excitations can show chiral Luttinger-liquid behaviors. On the other hand, we show that the finite-temperature behavior of the $G\ensuremath{-}T$ curve will deviate from the prediction of the chiral Luttinger liquid. We also point out that the suppression of the ``spin'' degrees of freedom agrees with very recent experiments by Chang et al. [Phys. Rev. Lett. 77, 2538 (1996)]. The two-boson model of Lee and Wen (preprint, cond-mat/9801318) is described microscopically.
We derive a microscopic theory of the composite particles describing the low-lying edge excitations in the fractional quantum Hall liquid with ν=1/m. Using the composite-particle transformation, one finds that the edge states of the system in a disc sample are described by the Calogero–Sutherland model with other interactions between the composite particles as perturbations. The effects of both the interactions and the confinement potential are considered, and proved unable to change the exponent g=ν=1/m. We also obtain the important chiral condition at the mean field level, and as a result justify microscopically the Chiral Luttinger Liquid model of the FQHE edge states.