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arXiv:1412.8716 · Physical Review B

Collective field theory for quantum Hall states

Michael Laskin, Tankut Can, Paul B. Wiegmann

Published Dec 23, 2015• 35 citations
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Abstract

We develop a collective field theory for fractional quantum Hall (FQH) states. We show that in the leading approximation for a large number of particles, the properties of Laughlin states are captured by a Gaussian free field theory with a background charge. Gradient corrections to the Gaussian field theory arise from the covariant ultraviolet regularization of the theory, which produces the gravitational anomaly. These corrections are described by a theory closely related to the Liouville theory of quantum gravity. The field theory simplifies the computation of correlation functions in FQH states and makes manifest the effect of quantum anomalies.

DOI
10.1103/physrevb.92.235141

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We develop a collective field theory for fractional quantum Hall (FQH) states. We show that in the leading approximation for a large number of particles, the properties of Laughlin states are captured by a Gaussian free field theory with a background charge. Gradient corrections to the Gaussian field theory arise from the covariant ultraviolet regularization of the theory, which produces the gravitational anomaly. These corrections are described by a theory closely related to the Liouville theory of quantum gravity. The field theory simplifies the computation of correlation functions in FQH states and makes manifest the effect of quantum anomalies.
We develop a field theory description of fractional quantum Hall (FQH) states. We show that in the leading approximation in a gradient expansion, Laughlin states are described by a Gaussian free field theory with a background charge which is identified with the anomalous viscosity of the states. The background charge increases the central charge of the corresponding conformal field theory above 1, similar to that of the theory of 2D quantum gravity. Gradient corrections to the Gaussian field theory arising from ultraviolet regularization reflect the gravitational anomaly. They are also related to the Liouville theory of quantum gravity. We show how the gradient expansion of the field action encodes the universal features of the FQH effect beyond that of Hall conductance. This method provides a more transparent and useful alternative for computing the gravitational anomaly and correlation functions of the FQHE in a curved space than the method of iterating the Ward identity employed by the authors in previous papers.
We derive an effective field theory for the isotropic-nematic quantum phase transition of fractional quantum Hall (FQH) states. We demonstrate that for a system with an isotropic background the low-energy effective theory of the nematic order parameter has z=2z=2z=2 dynamical scaling exponent, due to a Berry phase term of the order parameter, which is related to the non-dissipative Hall viscosity. Employing the composite fermion theory with a quadrupolar interaction between electrons, we show that a sufficiently attractive quadrupolar interaction triggers a phase transition from the isotropic FQH fluid into a nematic fractional quantum Hall phase. By investigating the spectrum of collective excitations, we demonstrate that the mass gap of Girvin-MacDonald-Platzman (GMP) mode collapses at the isotropic-nematic quantum phase transition. On the other hand, Laughlin quasiparticles and the Kohn collective mode remain gapped at this quantum phase transition, and Kohn's theorem is satisfied. The leading couplings between the nematic order parameter and the gauge fields include a term of the same form as Wen-Zee term. A disclination of the nematic order parameter carries an unquantized electric charge. We also discuss the relation between nematic degrees of freedom and the geometrical response of the fractional quantum Hall fluid.
We present an application of collective field theory to the fractional quantum Hall effect (FQHE). We first express the condition, that the electrons are all in the lowest Landau level, as a constraint equation for the state functional. We then derive the fractional filling factor from this equation together with the no-free-vortex assumption. A hierarchy of filling factors is derived by using the particle-vortex dual transformations. In the final section we discuss an attempt at a dynamical theory of FQHE, which would justify the no-free-vortex assumption. A derivation of Laughlin’s wave function with and without quasi-hole excitations is also given.
It has long been known that Laughlin's wave functions, describing the fractional quantum Hall effect at filling fractions ν=1/(2k+1), can be obtained as correlation functions in conformal field theory. We show how to generalize this approach to construct explicit trial wave functions for all states in the quantum Hall hierarchy corresponding to quasiparticle (as opposed to quasihole) condensates, including the recently observed state at ν= 4/11. At the filling fractions ν= n/(2np+1) this construction exactly reproduces Jain's composite fermion wave functions. An explicit connection is made to Wen's topological classification of FQH states.
The low-energy physics of fractional quantum Hall (FQH) states-a paradigm of strongly correlated topological phases of matter-to a large extent is captured by weakly interacting quasiparticles known as composite fermions.In this paper, based on numerical simulations and effective field theory, we argue that some high-energy states in the FQH spectra necessitate a different description based on parton quasiparticles.We show that Jain states at filling factor ν ¼ n=ð2pn Æ 1Þ with integers n, p ≥ 2 support two kinds of collective modes: In addition to the well-known Girvin-MacDonald-Platzman (GMP) mode, they host a high-energy collective mode, which we interpret as the GMP mode of partons.We elucidate observable signatures of the parton mode in the dynamics following a geometric quench.We construct a microscopic wave function for the parton mode and demonstrate agreement between its variational energy and exact diagonalization.Using the parton construction, we derive a field theory of the Jain states and show that the previously proposed effective theories follow from our approach.Our results point to partons being "real" quasiparticles which, in a way reminiscent of quarks, become observable only at sufficiently high energies.
We show that the wave functions of the fluid ground states of fractional quantum Hall systems, in the thermodynamic limit, are universal at long distances and that they have a generalized Laughlin form. This universality is a consequence of the analytic properties of the equal-time density correlation functions at long distances. The correlation functions calculated from the field theoretic approach to the fractional quantum Hall effect have the correct analytic properties and the wave function calculated in the Gaussian approximation becomes exact in the asymptotic limit.