17,527 research outputs found
Fractional Laplacian matrix on the finite periodic linear chain and its periodic Riesz fractional derivative continuum limit
The 1D discrete fractional Laplacian operator on a cyclically closed
(periodic) linear chain with finitenumber of identical particles is
introduced. We suggest a "fractional elastic harmonic potential", and obtain
the -periodic fractionalLaplacian operator in the form of a power law matrix
function for the finite chain ( arbitrary not necessarily large) in explicit
form.In the limiting case this fractional Laplacian
matrix recovers the fractional Laplacian matrix ofthe infinite chain.The
lattice model contains two free material constants, the particle mass and
a frequency.The "periodic string continuum limit" of the
fractional lattice model is analyzed where lattice constant and
length of the chain ("string") is kept finite: Assuming finiteness of
the total mass and totalelastic energy of the chain in the continuum limit
leads to asymptotic scaling behavior for of thetwo material
constants,namely and . In
this way we obtain the -periodic fractional Laplacian (Riesz fractional
derivative) kernel in explicit form.This -periodic fractional Laplacian
kernel recovers for the well known 1D infinite space
fractional Laplacian (Riesz fractional derivative) kernel. When the scaling
exponentof the Laplacian takesintegers, the fractional Laplacian kernel
recovers, respectively, -periodic and infinite space (localized)
distributionalrepresentations of integer-order Laplacians.The results of this
paper appear to beuseful for the analysis of fractional finite domain problems
for instance in anomalous diffusion (L\'evy flights), fractional Quantum
Mechanics,and the development of fractional discrete calculus on finite
lattices
Extending Luttinger's theorem to Z(2) fractionalized phases of matter
Luttinger's theorem for Fermi liquids equates the volume enclosed by the
Fermi surface in momentum space to the electron filling, independent of the
strength and nature of interactions. Motivated by recent momentum balance
arguments that establish this result in a non-perturbative fashion [M.
Oshikawa, Phys. Rev. Lett. {\bf 84}, 3370 (2000)], we present extensions of
this momentum balance argument to exotic systems which exhibit quantum number
fractionalization focussing on fractionalized insulators, superfluids and
Fermi liquids. These lead to nontrivial relations between the particle filling
and some intrinsic property of these quantum phases, and hence may be regarded
as natural extensions of Luttinger's theorem. We find that there is an
important distinction between fractionalized states arising naturally from half
filling versus those arising from integer filling. We also note how these
results can be useful for identifying fractionalized states in numerical
experiments.Comment: 24 pages, 5 eps figure
Confinement- Deconfinement Phase Transition and Fractional Instanton Quarks in Dense Matter
We present arguments suggesting that large size overlapping instantons are
the driving mechanism of the confinement-deconfinement phase transition at
nonzero chemical potential mu. The arguments are based on the picture that
instantons at very large chemical potential in the weak coupling regime are
localized configurations with finite size \rho\sim\mu^{-1}. At the same time,
the same instantons at smaller chemical potential in the strong coupling regime
are well represented by the so-called instanton-quarks with fractional
topological charge 1/N_c. We estimate the critical chemical potential mu_c(T)
where transition between these two regimes takes place. We identify this
transition with confinement- deconfinement phase transition.
We also argue that the instanton quarks carry magnetic charges. As a
consequence of it, there is a relation between our picture and the standard
t'Hooft and Mandelstam picture of the confinement. We also comment on possible
relations of instanton-quarks with "periodic instantons", " center vortices",
and "fractional instantons" in the brane construction. We also argue that the
variation of the external parameter mu, which plays the role of the vacuum
expectation value of a "Higgs" field at mu >> \Lambda_{QCD}, allows to study
the transition from a "Higgs -like" gauge theory (weak coupling regime, mu>>
\Lambda_{QCD}) to ordinary QCD (strong coupling regime, mu<< \Lambda_{QCD}). We
also comment on some recent lattice results on topological charge density
distribution which support our picture.Comment: Invited talk delivered at the Light Cone Workshop, July 7-15, 2005,
Cairns, Australi
Lattice Chern-Simons Gravity via Ponzano-Regge Model
We propose a lattice version of Chern-Simons gravity and show that the
partition function coincides with Ponzano-Regge model and the action leads to
the Chern-Simons gravity in the continuum limit. The action is explicitly
constructed by lattice dreibein and spin connection and is shown to be
invariant under lattice local Lorentz transformation and gauge diffeomorphism.
The action includes the constraint which can be interpreted as a gauge fixing
condition of the lattice gauge diffeomorphism.Comment: LaTeX, 26 pages, 6 eps figure
Galois Modular Invariants of WZW Models
The set of modular invariants that can be obtained from Galois
transformations is investigated systematically for WZW models. It is shown that
a large subset of Galois modular invariants coincides with simple current
invariants. For algebras of type B and D infinite series of previously unknown
exceptional automorphism invariants are found.Comment: phyzzx macros, 38 pages. NIKHEF-H/94-3
Parafermionic theory with the symmetry Z_N, for N odd
We construct a parafermionic conformal theory with the symmetry Z_N, for N
odd, based on the second solution of Fateev-Zamolodchikov for the corresponding
parafermionic chiral algebra. Primary operators are classified according to
their transformation properties under the dihedral group D_N, as singlet,
doublet 1,2,...,(N-1)/2, and disorder operators. In an assumed Coulomb gas
scenario, the corresponding vertex operators are accommodated by the weight
lattice of the Lie algebra B_(N-1)/2. The unitary theories are representations
of the coset SO_n(N) x SO_2(N) / SO_{n+2}(N), with n=1,2,... . Physically, they
realise the series of multicritical points in statistical theories having a D_N
symmetry.Comment: 34 pages, 1 figure. v2: note added in proo
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