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arXiv:cond-mat/0612234 · Journal of Experimental and Theoretical Physics Letters

Modulation theory of quantum tunneling into a Calogero-Sutherland fluid

D. B. Gutman

Published Sep 1, 2007• 4 citations
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Abstract

Quantum hydrodynamics of interacting electrons with a parabolic single-particle spectrum is studied using the Calogero-Sutherland model. The effective action and modulation equations, describing the evolution of periodic excitations in the fluid, are derived. Applications to the problem of a single-electron tunneling into the FQHE edge state are discussed.

DOI
10.1134/s0021364007130140

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Quantum hydrodynamics of interacting electrons with a parabolic single-particle spectrum is studied using the Calogero-Sutherland model. The effective action and modulation equations, describing the evolution of periodic excitations in the fluid, are derived. Applications to the problem of a single-electron tunneling into the FQHE edge state are discussed.
We consider the known effective field theory of the Calogero-Sutherland model in the thermodynamic limit of large number of particles, obtained from the standard procedure in conformal field theory: the Hilbert space is constructed a priori in terms of irreducible representations of the symmetry algebra, and not by diagonalization of the hamiltonian, which is given in terms of fields that carry representations of the W 1 + ∞ algebra (representing the incompressibility of the Fermi sea). Nevertheless, the role of the effective hamiltonian of the theory is to establish a specific dynamics, which deserves further consideration. We show that the time evolution of the (chiral or antichiral) density field is given by the quantum Benjamin-Ono equation, in agreement with previous results obtained from the alternative description of the continuous limit of the model, based on quantum hydrodynamics. In this study, all calculations are performed at the quantum operator level, without making any assumption on the semiclassical limit of the fields and their equations of motion. This result may be considered as a reliable indication of the equivalence between the quantum field theoretic and quantum hydrodynamical formulations of the effective theories of the model. A one-dimensional quantum compressible fluid that includes both chiralities is the physical picture that emerges for the continuous limit of the Calogero-Sutherland model. • The chiral density field of the Calogero-Sutherland EFT satisfies a quantum Benjamin-Ono equation. • Agrees with quantum hydrodynamics results by Abanov and Wiegmann, extending their validity to quantum fields themselves. • Findings could be of interest to the QFT community, with new tools to deal with time non-locality within the realm of CFT. • Equivalence between the EFT and the hydrodynamic formulations of the thermodynamic limit of the Calogero-Sutherland model is supported.
The effective field theory of the Calogero–Sutherland model represents a universality class of quantum hydrodynamic fluids in one spatial dimension. It describes quantum compressible fluids involving both chiralities in which the chiral density field obeys the quantum Benjamin–Ono equation. An extension of this theory to describe a laminar flow of the Calogero–Sutherland fluids in a rectangular geometry with small transverse width and the topology of a ribbon, is considered here. The physical picture is based on the edge states in the hierarchical quantum Hall effect, which may be seen as a collection of parallel one-dimensional quantum incompressible fluids moving along but confined within the transverse microscopic width of the edge of the sample. The effective theory is thus defined as the direct product of two one-dimensional theories of the Calogero–Sutherland class so that one involves motion while the other is confining. Charge transport may be induced by coupling the system to an external electromagnetic field that yields a global translation of the ground state. The effective theory describes quantum solitonic excitations along the direction of the flow and possesses a two-dimensional electric current density which shows a Wigner semicircle law profile in the transverse direction, suggesting a Poiseuille-like behavior but without dissipative viscous effects since the velocity of the fluid is not a well-defined quantum field. This simple physical picture predicts interesting phenomena with distinctive signatures that may be tested in real samples.
The Calogero-Sutherland model represents a paradigmatic example of an integrable quantum system with applications ranging from cold atoms to random matrix theory. Combining sum rules with the Monte Carlo technique, we introduce a stochastic method that allows one to compute the dynamic structure factor and obtain an exact description of excitations beyond the conventional Luttinger liquid regime. We explore a broad range of interaction regimes, including weak interactions, where a Bogoliubov-type spectrum emerges, the Tonks-Girardeau regime, where excitations resemble those of an ideal Fermi gas, and strong interactions, where umklapp scattering leads to a Brillouin zone structure, typical of a crystal. Additionally, we discuss the connection between the hydrodynamic description of one-dimensional quantum gases, liquids, and solids with the Calogero-Sutherland wave function. The model's universality extends beyond atoms in waveguides, with implications for disordered systems and random matrix theory.
We show that the single quasi-particle Schrödinger equation for a certain form of one-body potential yields a stationary one soliton solution. The one-body potential is assumed to arise from the self- interacting charge distribution with the singular kernel of the Calogero-Sutherland model. The quasi-particle has negative or positive charge for negative or positive coupling constant of the interaction. The magnitude of the charge is unity only for the semion. It is also pointed out that for repulsive coupling, our equation is mathematically the same as the steady-state Smoluchowski equation of Dyson's Coulomb gas model.