Gauge invariance and current algebra in nonrelativistic many-body theory

Abstract

The main purpose of this paper is to further our theoretical understanding of the fractional quantum Hall effect, in particular of spin effects, in two-dimensional incompressible electron fluids subject to a strong, transverse magnetic field. As a prerequisite for an analysis of the quantum Hall effect, the authors develop a general formulation of the many-body theory of spinning particles coupled to external electromagnetic fields and moving through a general, geometrically nontrivial background. Their formulation is based on a Lagrangian path-integral quantization and is valid in arbitrary coordinates, including coordinates moving according to a volume-preserving flow. It is found that nonrelativistic quantum theory exhibits a fundamental, local U(1)\ifmmode\times\else\texttimes\fi{}SU(2) gauge invariance, and the corresponding gauge fields are identified with physical, external fields. To illustrate the usefulness of their formalism, the authors prove a general form of the quantum-mechanical Larmor theorem and discuss some well-known effects, including the Barnett-Einstein-de Haas effect and superconductivity, emphasizing the implications of U(1)\ifmmode\times\else\texttimes\fi{}SU(2) gauge invariance. They then consider two-dimensional, incompressible quantum fluids in more detail. Exploiting U(1)\ifmmode\times\else\texttimes\fi{}SU(2) gauge invariance, they calculate the leading terms in the effective actions of such systems as functionals of the U(1) and SU(2) gauge fields, on large-distance and low-frequency scales. Among the applications of these results are a simple proof of the Goldstone theorem for spin waves and the linearresponse theory of two-dimensional, incompressible Hall fluids, including a Hall effect for spin currents and sum rules for the response coefficients. For two-dimensional, incompressible systems with broken parity and time-reversal symmetry, a particularly significant implication of U(1)\ifmmode\times\else\texttimes\fi{}SU(2) gauge invariance is a duality between the physics inside the bulk of the system and the physics of gapless, chiral modes propagating along the boundary of the system. These modes form chiral $\stackrel{^}{\mathrm{u}}(1)$ and $\mathrm{s}\stackrel{^}{\mathrm{u}}(2)$ current algebras. The representation theory of these current algebras, combined with natural physical constraints, permits one to derive the quantization of the response coefficients, such as the Hall conductivity. A classification of incompressible Hall fluids is outlined, and many examples, including one concerning a superfluid $^{3}\mathrm{He}$-$\frac{A}{B}$ interface, are discussed.