Anomalous hydrodynamics of fractional quantum Hall states
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nomalous Hydrodynamics of Fractional Quantum Hall States1 P. Wiegmann Department of Physics, University of Chicago Chicago, IL 60637, USA email: [email protected] Received May 16, 2013
Dedicated to the memory of Professor Anatoly Larkin Abstract—We propose a comprehensive framework for quantum hydrodynamics of the fractional quantum Hall (FQH) states. We suggest that the electronic fluid in the FQH regime can be phenomenologically described by the quantized hydrodynamics of vortices in an incompressible rotating liquid. We demonstrate that such hydrodynamics captures all major features of FQH states, including the subtle effect of the Lorentz shear stress. We present a consistent quantization of the hydrodynamics of an incompressible fluid, providing a powerful framework to study the FQH effect and superfluids. We obtain the quantum hydrodynamics of the vortex flow by quantizing the Kirchhoff equations for vortex dynamics. DOI: 10.1134/S1063776113110162 1
1. INTRODUCTION
Quantum systems with the effectively strong inter action form liquids whose flows are coherent quantum collective motions. Among them, there are interesting notable cases where such liquids allow a hydrodynam ics description. That is when the longwave, slow flows can be effectively described solely in terms of a macro scopic, but quantum, pair of canonical fields of den sity ρ(r, t) and velocity v(r, t). Such quantum flows are the subject of quantum hydrodynamics. In the classi cal case, the principle of local equilibrium reduces the Boltzmann kinetic equation for the distribution func tion to the hydrodynamics equations for the density and the velocity (see, e.g., [1]). Local equilibrium occurs when the characteristic time of the flow exceeds the characteristic time of collisions, and the characteristic scale of the flow exceeds the mean free path of particles. A quantum analog of the principle of local equilibrium is yet to be understood, but when it comes to effect, it involves longrange coherent effects. Strong coherence emerges as a result of inter actions. Notable examples of quantum hydrodynam ics are superfluid helium, superconductors, trapped cooled atomic gases, and Luttinger liquids. A frac tional quantum Hall (FQH) liquid is yet another case. Electronic states confined within the lowest Lan dau level by the quantizing magnetic field are holo morphic. The holomorphic nature of states makes the hydrodynamics description possible. A quest for the hydrodynamics of a FQH liquid originated in a seminal paper [2]. Earlier approaches to FQH states in Refs. [3–5] are somewhat related to 1 The article is published in the original.
the hydrodynamics, as noted in Ref. [6]. Hydrody namics of FQH states is in the focus of a renewed interest. In hydrodynamics, a few basic principles, symme tries, and a few phenomenological parameters are suf ficient to formulate the fundamental equations. In the case of the FQH effect (FQHE), we already possess sufficient characterizations of states. They can be used as a basis of the hydrodynam
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