1,650 research outputs found
Quantum Tetrahedra
We discuss in details the role of Wigner 6j symbol as the basic building
block unifying such different fields as state sum models for quantum geometry,
topological quantum field theory, statistical lattice models and quantum
computing. The apparent twofold nature of the 6j symbol displayed in quantum
field theory and quantum computing -a quantum tetrahedron and a computational
gate- is shown to merge together in a unified quantum-computational SU(2)-state
sum framework
Combinatorial problems of (quasi-)crystallography
Several combinatorial problems of (quasi-)crystallography are reviewed with
special emphasis on a unified approach, valid for both crystals and
quasicrystals. In particular, we consider planar sublattices, similarity
sublattices, coincidence sublattices, their module counterparts, and central
and averaged shelling. The corresponding counting functions are encapsulated in
Dirichlet series generating functions, with explicit results for the triangular
lattice and the twelvefold symmetric shield tiling. Other combinatorial
properties are briefly summarised.Comment: 12 pages, 2 PostScript figures, LaTeX using vch-book.cl
The General O(n) Quartic Matrix Model and its application to Counting Tangles and Links
The counting of alternating tangles in terms of their crossing number, number
of external legs and connected components is presented here in a unified
framework using quantum field-theoretic methods applied to a matrix model of
colored links. The overcounting related to topological equivalence of diagrams
is removed by means of a renormalization scheme of the matrix model; the
corresponding ``renormalization equations'' are derived. Some particular cases
are studied in detail and solved exactly.Comment: 21 page
Topological Resonating-Valence-Bond Spin Liquid on the Square Lattice
A one-parameter family of long-range resonating valence bond (RVB) state on
the square lattice was previously proposed to describe a critical spin liquid
(SL) phase of the spin- frustrated Heisenberg model. We provide evidence
that this RVB state in fact also realises a topological (long-range entangled)
SL, limited by two transitions to critical SL phases. The
topological phase is naturally connected to the gauge symmetry
of the local tensor. This work shows that, on one hand, spin- topological
SL with point group symmetry and spin rotation symmetry exists
on the square lattice and, on the other hand, criticality and nonbipartiteness
are compatible. We also point out that, strong similarities between our phase
diagram and the ones of classical interacting dimer models suggest both can be
described by similar Kosterlitz-Thouless transitions. This scenario is further
supported by the analysis of the one-dimensional boundary state.Comment: v2: improve presentation, present new evidence and add reference
Quantum Simulations of Lattice Gauge Theories using Ultracold Atoms in Optical Lattices
Can high energy physics be simulated by low-energy, non-relativistic,
many-body systems, such as ultracold atoms? Such ultracold atomic systems lack
the type of symmetries and dynamical properties of high energy physics models:
in particular, they manifest neither local gauge invariance nor Lorentz
invariance, which are crucial properties of the quantum field theories which
are the building blocks of the standard model of elementary particles.
However, it turns out, surprisingly, that there are ways to configure atomic
system to manifest both local gauge invariance and Lorentz invariance. In
particular, local gauge invariance can arise either as an effective, low
energy, symmetry, or as an "exact" symmetry, following from the conservation
laws in atomic interactions. Hence, one could hope that such quantum simulators
may lead to new type of (table-top) experiments, that shall be used to study
various QCD phenomena, as the confinement of dynamical quarks, phase
transitions, and other effects, which are inaccessible using the currently
known computational methods.
In this report, we review the Hamiltonian formulation of lattice gauge
theories, and then describe our recent progress in constructing quantum
simulation of Abelian and non-Abelian lattice gauge theories in 1+1 and 2+1
dimensions using ultracold atoms in optical lattices.Comment: A review; 55 pages, 14 figure
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