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arXiv:cond-mat/9604040

Chiral sound waves from a gauge theory of 1D generalized statistics

Silvio Rabello

Published Apr 6, 1996• 0 citations
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Abstract

A topological gauge field theory in one spatial dimension is studied, with the gauge fields as generators of two commuting U(1) Kauc-Moody algebras. Coupling of these gauge fields to nonrelativistic bosonic matter fields, produces a statistical transmutation of the later, as in the Chern-Simons theory in two dimensions. The sound waves of the model are investigated and proven to be chiral bosonic excitations, with the same spectrum as the density fluctuations of the Luttinger model.

DOI
10.48550/arxiv.cond-mat/9604040

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A topological gauge field theory in one spatial dimension is studied, with the gauge fields as generators of two commuting U(1) Kauc-Moody algebras. Coupling of these gauge fields to nonrelativistic bosonic matter fields, produces a statistical transmutation of the later, as in the Chern-Simons theory in two dimensions. The sound waves of the model are investigated and proven to be chiral bosonic excitations, with the same spectrum as the density fluctuations of the Luttinger model.
A field theory with generalized statistics in one space dimension is introduced. The statistics enters the scene through the coupling of the matter fields to a statistical gauge field, as it happens in the Chern-Simons theory in two dimensions. We study the particle-hole excitations and show that the long wavelength physics of this model describes a gas obeying the Haldane generalized exclusion statistics. The statistical interaction is found to provide a way to describe the low-T critical properties of one-dimensional non-Fermi-liquids.
The identity of quantum matter can be effectively altered by means of gauge fields. In two spatial dimensions this is illustrated by the Chern-Simons flux-attachment mechanism, but such a mechanism is not possible in lower dimensions. Here, we study a one-dimensional interacting Bose gas in the presence of a gauge field. This model can be explicitly mapped into an interacting anyonic system by a large gauge transformation, indicating a statistical transmutation analogous to that of Chern-Simons. The Bogoliubov spectrum in the weakly-interacting limit reveals the presence of a roton minimum arising from the statistical interaction. At a mean-field level chiral solitons are recovered. Should these be understood as quantum bound states, it is natural to interpret them as corresponding to localised anyonic quasiparticles. Hydrodynamic arguments highlight the presence of dispersive chiral shock waves in the propagation of a wavepacket due to a Riemann-Hopf nonlinearity. Numerical calculations show the presence of both chiral soliton trains and shock waves.
Topological gauge theories provide powerful effective descriptions of certain strongly correlated systems, a prime example being the Chern-Simons gauge theory of fractional quantum Hall states.Engineering topological gauge theories in controlled quantum systems is of both conceptual and practical importance, as it would provide access to systems with exotic excitations such as anyons without the need for strong correlations.Here, we discuss a scheme to engineer the chiral BF theory, a minimal model of a topological gauge theory corresponding to a one-dimensional reduction of the Chern-Simons theory, with ultracold atoms.Using the local conservation laws of the theory, we encode its quantum Hamiltonian into an ultracold quantum gas with chiral interactions.Building on a seminal proposal by Edmonds et al. [Phys. Rev. Lett. 110, 085301 (2013)], we show how to implement it in a Raman-coupled Bose-Einstein condensate with imbalanced scattering lengths, as we have recently realized experimentally [Frölian et al., Nature (London) 608, 293 (2022)]. We discuss the properties of the chiral condensate from a gauge theory perspective and assess the validity of the effective quantum description for accessible experimental parameters via numerical simulations.Our approach lays the foundation for realizing topological gauge theories in higher dimensions with Bose-Einstein condensates.
We consider the large-NNN Chern-Simons theory for attractive bosonic matter (Jackiw-Pi model) in the Hamiltonian collective-field approach based on the 1/N1/N1/N expansion. We show that the dynamics of low-lying density excitations around the ground-state vortex configuration is equivalent to that of the Sutherland model. The correspondence between the Chern-Simons coupling constant $\ensuremath{\lambda}$ and the Calogero-Sutherland statistical parameter ${\ensuremath{\lambda}}_{s}$ signals some sort of statistical transmutation accompanying the dimensional reduction of the initial problem.
We propose a generic construction of exactly soluble local bosonic models that realize various topological orders with gappable boundaries. In particular, we construct an exactly soluble bosonic model that realizes a (3+1)-dimensional [(3+1)D] Z2{Z}_{2}Z2​-gauge theory with emergent fermionic Kramers doublet. We show that the emergence of such a fermion will cause the nucleation of certain topological excitations in space-time without pin+{\mathrm{pin}}^{+}pin+ structure. The exactly soluble model also leads to a statistical transmutation in (3+1)D. In addition, we construct exactly soluble bosonic models that realize 2 types of time-reversal symmetry-enriched Z2{Z}_{2}Z2​ topological orders in 2+1 dimensions, and 20 types of simplest time-reversal symmetry-enriched topological (SET) orders which have only one nontrivial pointlike and stringlike topological excitation. Many physical properties of those topological states are calculated using the exactly soluble models. We find that some time-reversal SET orders have pointlike excitations that carry Kramers doublet, a fractionalized time-reversal symmetry. We also find that some Z2{Z}_{2}Z2​ SET orders have stringlike excitations that carry anomalous (nononsite) Z2{Z}_{2}Z2​ symmetry, which can be viewed as a fractionalization of Z2{Z}_{2}Z2​ symmetry on strings. Our construction is based on cochains and cocycles in algebraic topology, which is very versatile. In principle, it can also realize emergent topological field theory beyond the twisted gauge theory.