3,889 research outputs found
On the K-theoretic classification of topological phases of matter
We present a rigorous and fully consistent -theoretic framework for
studying gapped topological phases of free fermions such as topological
insulators. It utilises and profits from powerful techniques in operator
-theory. From the point of view of symmetries, especially those of time
reversal, charge conjugation, and magnetic translations, operator -theory is
more general and natural than the commutative topological theory. Our approach
is model-independent, and only the symmetry data of the dynamics, which may
include information about disorder, is required. This data is completely
encoded in a suitable -superalgebra. From a representation-theoretic point
of view, symmetry-compatible gapped phases are classified by the
super-representation group of this symmetry algebra. Contrary to existing
literature, we do not use -theory to classify phases in an absolute sense,
but only relative to some arbitrary reference. -theory groups are better
thought of as groups of obstructions between homotopy classes of gapped phases.
Besides rectifying various inconsistencies in the existing literature on
-theory classification schemes, our treatment has conceptual simplicity in
its treatment of all symmetries equally. The Periodic Table of Kitaev is
exhibited as a special case within our framework, and we prove that the
phenomena of periodicity and dimension shifts are robust against disorder and
magnetic fields.Comment: 41 pages, revised version with a new abstract, introductory sections
and critique of the literatur
Bell Correlations in Quantum Field Theory
Bell correlations are the hallmark of quantum non-locality, and a rich context for analysing them is provided by the algebraic approach to quantum field theory (AQFT): the basic idea is to associate with each bounded region O of Minkowski spacetime an algebra A(O) of operators, of which a self-adjoint element P ∈ A(O) represents a physical quantity pertaining to that part of the field system lying in O, that is measurable by a procedure confined to O. The violation of Bell inequalities in AQFT is known to be "generic", as regards the choices of regions O, and of quantities P, and of states. Furthermore, they are typically "maximal" and "indestructible" in a sense that can be made mathematically precise. The prospects for “peaceful coexistence” between quantum non-locality and relativity theory’s requirement of no action-at-a-distance are also explored. The purpose of this Essay is to review the developments in these areas
Degree of separability of bipartite quantum states
We investigate the problem of finding the optimal convex decomposition of a
bipartite quantum state into a separable part and a positive remainder, in
which the weight of the separable part is maximal. This weight is naturally
identified with the degree of separability of the state. In a recent work, the
problem was solved for two-qubit states using semidefinite programming. In this
paper, we describe a procedure to obtain the optimal decomposition of a
bipartite state of any finite dimension via a sequence of semidefinite
relaxations. The sequence of decompositions thus obtained is shown to converge
to the optimal one. This provides, for the first time, a systematic method to
determine the so-called optimal Lewenstein-Sanpera decomposition of any
bipartite state. Numerical results are provided to illustrate this procedure,
and the special case of rank-2 states is also discussed.Comment: 11 pages, 7 figures, submitted to PR
T-duality simplifies bulk-boundary correspondence: some higher dimensional cases
Recently we introduced T-duality in the study of topological insulators, and
used it to show that T-duality trivialises the bulk-boundary correspondence in
2 dimensions. In this paper, we partially generalise these results to higher
dimensions and briefly discuss the 4D quantum Hall effect.Comment: 25 pages. To appear in Ann. Henri Poincar
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