Abstract The relation between braid and exclusion statistics is examined in one-dimensional systems, within the framework of Chern–Simons statistical transmutation in gauge invariant form with an appropriate dimensional reduction. If the matter action is anomalous, as for chiral fermions, a relation between braid and exclusion statistics can be established explicitly for both mutual and nonmutual cases. However, if it is not anomalous, the exclusion statistics of emergent low energy excitations is not necessarily connected to the braid statistics of the physical charged fields of the system. Finally, we also discuss the bosonization of one-dimensional anyonic systems through T-duality.
Braiding phenomena, from the charge-flux Aharonov-Bohm effect to anyonic statistics in fractional quantum Hall systems, are paradigmatic manifestations of topology in quantum physics. Ordinary mutual braiding between
p- and
q-dimensional excitations occurs in
d=p+q+2 spatial dimensions. In this work, we introduce a universal construction of mutual statistics in the adjacent dimension
d=p+q+1, applicable to excitations obeying
ZN fusion for arbitrary
N and all excitation dimensions
p and
q. The corresponding invariant is the Berry phase accumulated in a simple
4N-step microscopic unitary process built from local excitation operators on lattices. This process measures the linking of one excitation with the
N-fold fusion junction of the other, encompassing particle-particle statistics in one dimension, particle-loop statistics in two dimensions, and loop-loop or particle-membrane statistics in three dimensions. We establish the quantization and bilinearity of the invariant and show that its field-theory response is governed by the Bockstein homomorphism, motivating the name Bockstein braiding statistics. Interpreting the excitation operators as open symmetry operators turns the same invariant into a direct microscopic diagnostic of mixed anomalies between symmetries. We demonstrate this diagnostic in a (1+1)D spin chain, where the nontrivial Bockstein braiding phase proves the mixed anomaly between the spin-flip symmetry
∏X and the nearest-neighbor controlled-
Z symmetry
∏CZ. We construct explicit (2+1)D and (3+1)D lattice analogs, yielding new anomalous symmetry pairs, and apply the framework to strongly coupled (3+1)D continuum gauge theories. Nontrivial Bockstein braiding rules out a fully symmetric gapped phase, obstructs simultaneous condensation of the two excitations, and implies fractionalization of higher-form symmetries.
The book presents the wide range of topics in two-dimensional physics of quantum Hall systems, especially fractional quantum Hall states. It starts with the fundamental problems of quantum statistics in two dimensions and the corresponding braid group formalism. The braid group formalism of anyons (previously known) is developed for composite fermions. The main formalism used in many-body quantum Hall theories -- the Chern-Simons theory is also presented. The Chern-Simons theory of anyons (particles obeying fractional statistics) and composite fermions (related to Hall systems) is given, in detail. Numerical studies, which play the important role in quantum Hall theories, are presented for spherical systems (Haldane sphere). The composite fermion theory is tested in numerical studies. The concept of the hierarchy of condensed states of composite fermion excitations is introduced (in analogy to the Haldane hierarchy)1). The hierarchies of odd-denominator states and even-denominator states are presented. The BCS paired Hall state is also discussed. The introduction into multi-component quantum Hall systems and spin quantum Hall systems is sketched. 1)First condensed states of composite fermion excitations have been very recently confirmed in the experiment (Pan et al. Phys. Rev. Lett. 90 (2003) 016801). a sample of this book is available at http://www.oup.co.uk/isbn/0-19-852870-1
Abstract The book presents the wide range of topics in two-dimensional physics of quantum Hall systems, especially fractional quantum Hall states. It starts with the fundamental problems of quantum statistics in two dimensions and the corresponding braid group formalism. The braid group formalism of anyons (previously known) is developed for composite fermions. The effective formalism used in many-body quantum Hall theories — the Chern–Simons theory is also presented. The Chern–Simons theory of anyons (particles obeying fractional statistics) and composite fermions (related to Hall systems) is given, in detail. Numerical studies, which play the important role in quantum Hall analyses, are presented for spherical systems (Haldane sphere). The composite fermion theory is tested in numerical studies. The concept of the hierarchy of condensed states of composite fermion excitations is introduced (in analogy to the Haldane hierarchy). The hierarchies of odd-denominator states and even-denominator states are presented. The BCS paired Hall state is also discussed. An introduction to multi-component quantum Hall systems and spin quantum Hall systems is sketched.
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.
We clarify the relation between two kinds of statistics for particle excitations in planar systems: the braid statistics of anyons and the Haldane exclusion statistics (HES). It is shown nonperturbatively that the HES exists for incompressible anyon liquid in the presence of a Hall response. We also study the statistical properties of a specific quantum anomalous Hall model with Chern-Simons term by perturbation in both compressible and incompressible regimes, where the crucial role of edge states to the HES is shown.
The idea of fractional exclusion statistics proposed by Haldane is applied to systems with internal degrees of freedom, and its thermodynamics is examined. In the case of one dimension, various calculated bulk quantities show that the critical behavior of such systems can be described by c=1 conformal field theories and conformal dimensions are completely characterized by statistical interactions. It is also found that statistical interactions have an intimate relationship with a topological order matrix in Chern-Simons theory for the fractional quantum Hall effect.