90 research outputs found
Solutions to generalized Yang-Baxter equations via ribbon fusion categories
Inspired by quantum information theory, we look for representations of the
braid groups on for some fixed vector space
such that each braid generator acts on consecutive
tensor factors from through . The braid relation for is
essentially the Yang-Baxter equation, and the cases for are called
generalized Yang-Baxter equations. We observe that certain objects in ribbon
fusion categories naturally give rise to such representations for the case
. Examples are given from the Ising theory (or the closely related
), for odd, and . The solution from the
Jones-Kauffman theory at a root of unity, which is closely related to
or , is explicitly described in the end.Comment: Some minor change
The soft mode in the Sachdev-Ye-Kitaev model and its gravity dual
We give an exposition of the SYK model with several new results. A non-local
correction to the Schwarzian effective action is found. The same action is
obtained by integrating out the bulk degrees of freedom in a certain variant of
dilaton gravity. We also discuss general properties of out-of-time-order
correlators.Comment: 57 pages plus appendices, 6 figures. v.2: minor correction,
additional references. v.3: minor addition, additional references; v.4: minor
corrections; v.5: JHEP versio
Statistical mechanics of a two-dimensional black hole
The dynamics of a nearly-AdS2 spacetime with boundaries is reduced to two
particles in the anti-de Sitter space. We determine the class of physically
meaningful wavefunctions, and prescribe the statistical mechanics of a black
hole. We demonstrate how wavefunctions for a two-sided black hole and a
regularized notion of trace can be used to construct thermal partition
functions, and more generally, arbitrary density matrices. We also obtain
correlation functions of external operators.Comment: 33 pages + appendices; v.2: additional reference, explicit formula
for two-point correlators, and other minor improvement
Periodic table for topological insulators and superconductors
Gapped phases of noninteracting fermions, with and without charge
conservation and time-reversal symmetry, are classified using Bott periodicity.
The symmetry and spatial dimension determines a general universality class,
which corresponds to one of the 2 types of complex and 8 types of real Clifford
algebras. The phases within a given class are further characterized by a
topological invariant, an element of some Abelian group that can be 0, Z, or
Z_2. The interface between two infinite phases with different topological
numbers must carry some gapless mode. Topological properties of finite systems
are described in terms of K-homology. This classification is robust with
respect to disorder, provided electron states near the Fermi energy are absent
or localized. In some cases (e.g., integer quantum Hall systems) the
K-theoretic classification is stable to interactions, but a counterexample is
also given.Comment: To appear in the Proceedings of the L.D.Landau Memorial Conference
"Advances in Theoretical Physics", June 22-26, 2008, Chernogolovka, Moscow
region, Russia (v2: arXiv hyperlinks fixed
Anyons in an exactly solved model and beyond
A spin 1/2 system on a honeycomb lattice is studied. The interactions between
nearest neighbors are of XX, YY or ZZ type, depending on the direction of the
link; different types of interactions may differ in strength. The model is
solved exactly by a reduction to free fermions in a static
gauge field. A phase diagram in the parameter space is obtained. One of the
phases has an energy gap and carries excitations that are Abelian anyons. The
other phase is gapless, but acquires a gap in the presence of magnetic field.
In the latter case excitations are non-Abelian anyons whose braiding rules
coincide with those of conformal blocks for the Ising model. We also consider a
general theory of free fermions with a gapped spectrum, which is characterized
by a spectral Chern number . The Abelian and non-Abelian phases of the
original model correspond to and , respectively. The anyonic
properties of excitation depend on , whereas itself governs
edge thermal transport. The paper also provides mathematical background on
anyons as well as an elementary theory of Chern number for quasidiagonal
matrices.Comment: 113 pages. LaTeX + 299 .eps files (see comments in hexagon.tex for
known-good compilation environment). VERSION 3: some typos fixed, one
reference adde
Topological entanglement entropy
We formulate a universal characterization of the many-particle quantum
entanglement in the ground state of a topologically ordered two-dimensional
medium with a mass gap. We consider a disk in the plane, with a smooth boundary
of length L, large compared to the correlation length. In the ground state, by
tracing out all degrees of freedom in the exterior of the disk, we obtain a
marginal density operator \rho for the degrees of freedom in the interior. The
von Neumann entropy S(\rho) of this density operator, a measure of the
entanglement of the interior and exterior variables, has the form S(\rho)=
\alpha L -\gamma + ..., where the ellipsis represents terms that vanish in the
limit L\to\infty. The coefficient \alpha, arising from short wavelength modes
localized near the boundary, is nonuniversal and ultraviolet divergent, but
-\gamma is a universal additive constant characterizing a global feature of the
entanglement in the ground state. Using topological quantum field theory
methods, we derive a formula for \gamma in terms of properties of the
superselection sectors of the medium.Comment: 4 pages, 3 eps figures. v2: reference adde
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