31,969 research outputs found
Anomalies in the Entanglement Properties of the Square Lattice Heisenberg Model
We compute the bipartite entanglement properties of the spin-half
square-lattice Heisenberg model by a variety of numerical techniques that
include valence bond quantum Monte Carlo (QMC), stochastic series expansion
QMC, high temperature series expansions and zero temperature coupling constant
expansions around the Ising limit. We find that the area law is always
satisfied, but in addition to the entanglement entropy per unit boundary
length, there are other terms that depend logarithmically on the subregion
size, arising from broken symmetry in the bulk and from the existence of
corners at the boundary. We find that the numerical results are anomalous in
several ways. First, the bulk term arising from broken symmetry deviates from
an exact calculation that can be done for a mean-field Neel state. Second, the
corner logs do not agree with the known results for non-interacting Boson
modes. And, third, even the finite temperature mutual information shows an
anomalous behavior as T goes to zero, suggesting that T->0 and L->infinity
limits do not commute. These calculations show that entanglement entropy
demonstrates a very rich behavior in d>1, which deserves further attention.Comment: 12 pages, 7 figures, 2 tables. Numerical values in Table I correcte
MUBs inequivalence and affine planes
There are fairly large families of unitarily inequivalent complete sets of
N+1 mutually unbiased bases (MUBs) in C^N for various prime powers N. The
number of such sets is not bounded above by any polynomial as a function of N.
While it is standard that there is a superficial similarity between complete
sets of MUBs and finite affine planes, there is an intimate relationship
between these large families and affine planes. This note briefly summarizes
"old" results that do not appear to be well-known concerning known families of
complete sets of MUBs and their associated planes.Comment: This is the version of this paper appearing in J. Mathematical
Physics 53, 032204 (2012) except for format changes due to the journal's
style policie
Matter in Toy Dynamical Geometries
One of the objectives of theories describing quantum dynamical geometry is to
compute expectation values of geometrical observables. The results of such
computations can be affected by whether or not matter is taken into account. It
is thus important to understand to what extent and to what effect matter can
affect dynamical geometries. Using a simple model, it is shown that matter can
effectively mold a geometry into an isotropic configuration. Implications for
"atomistic" models of quantum geometry are briefly discussed.Comment: 8 pages, 1 figure, paper presented at DICE 200
Projective geometries arising from Elekes-Szab\'o problems
We generalise the Elekes-Szab\'o theorem to arbitrary arity and dimension and
characterise the complex algebraic varieties without power saving. The
characterisation involves certain algebraic subgroups of commutative algebraic
groups endowed with an extra structure arising from a skew field of
endomorphisms. We also extend the Erd\H{o}s-Szemer\'edi sum-product phenomenon
to elliptic curves. Our approach is based on Hrushovski's framework of
pseudo-finite dimensions and the abelian group configuration theorem.Comment: 48 pages. Minor improvements in presentation. To appear in ASEN
Spatial Hypersurfaces in Causal Set Cosmology
Within the causal set approach to quantum gravity, a discrete analog of a
spacelike region is a set of unrelated elements, or an antichain. In the
continuum approximation of the theory, a moment-of-time hypersurface is well
represented by an inextendible antichain. We construct a richer structure
corresponding to a thickening of this antichain containing non-trivial
geometric and topological information. We find that covariant observables can
be associated with such thickened antichains and transitions between them, in
classical stochastic growth models of causal sets. This construction highlights
the difference between the covariant measure on causal set cosmology and the
standard sum-over-histories approach: the measure is assigned to completed
histories rather than to histories on a restricted spacetime region. The
resulting re-phrasing of the sum-over-histories may be fruitful in other
approaches to quantum gravity.Comment: Revtex, 12 pages, 2 figure
Deconfinement and Thermodynamics in 5D Holographic Models of QCD
We review 5D holographic approaches to finite temperature QCD. Thermodynamic
properties of the "hard-wall" and the "soft-wall" models are derived. Various
non-realistic features in these models are cured by the set-up of improved
holographic QCD, that we review here.Comment: Invited review paper for Mod. Phys. Let
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