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One-dimensional fermions as two-dimensional droplets via Chern-Simons theory

Iso, S, Dimitra Karabali A, BUNJI SAKITA

Published Jan 1, 1992• 47 citations
OpenAlex

Abstract

Based on the observation that a particle motion in one dimension maps to a two-dimensional motion of a charged particle in a uniform magnetic field, constrained in the lowest Landau level, we formulate a system of one-dimen- sional nonrelativistic fermions by using a Chern-Simons field theory in 2+1 dimensions. Using a hydrodynamical formulation we obtain a two-dimensional droplet picture of one-dimensional fermions. The dynamics involved is that of the boundary between a uniform density of particles and vortices. We use the sharp boundary approximation. In the case of well separated boundaries we derive the one-dimensional collective field Hamiltonian. Symmetries of the theory are also discussed as properties of curves in two dimensions.

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A novel representation is derived for the microscopic partition function of two-dimensional electrons confined to the lowest Landau level. The dynamics is described by the two-component quantum plasma of particles and holes. We find a topological term in the Euclidean microscopic action to be dynamically generated in this representation. Starting from this partition function we find uniform solutions of the equations of motion only at \ensuremath{\nu}=(2k+1${)}^{\mathrm{\ensuremath{-}}1}$. At these special filling fractions the Chern-Simons theory arises as a saddle-point approximation to the microscopic action.
The discovery of many strongly correlated metallic phases has inspired different routes to generalize or go beyond the celebrated Landau Fermi liquid theory. To this end, from universal consideration of symmetries and anomalies, Else, Thorngren, and Senthil (ETS) have introduced a class of theories called ersatz Fermi liquids which possess a Fermi surface and satisfy a generalized Luttinger's theorem. In this work, we view all such fermion liquids obeying the Luttinger theorem as incompressible quantum Hall liquids in higher-dimensional phase space and use it as the starting point to derive their effective low-energy field theory. The noncommutativity of phase space motivates us to use the Seiberg-Witten map to derive the field theory in an ordinary (commutative) space and naturally leads to terms that correspond to the correct topological Chern-Simons action postulated by ETS in one, two, and three dimensions. Additionally, our approach also reproduces all the nontopological terms that characterize important contributions to the response, including the semiclassical equations of motion. Finally, our derivations of Chern-Simons terms from the Seiberg-Witten map also verify a longstanding conjecture in noncommutative field theory.
We derive the exact form of the bosonized Hamiltonian for a many-body fermion system in one spatial dimension with arbitrary dispersion relations, using the droplet bosonization method. For a single-particle Hamiltonian polynomial in the momentum, the bosonized Hamiltonian is a polynomial of one degree higher in the bosonic ``boundary'' field and includes subleading lower-order and derivative terms. This generalizes the known results for massless relativistic and nonrelativistic fermions (quadratic and cubic bosonic Hamiltonians, respectively). We also consider two-body interactions and demonstrate that they lead to interesting collective behavior and phase transitions in the Fermi sea.