727 research outputs found
A TQFT associated to the LMO invariant of three-dimensional manifolds
We construct a Topological Quantum Field Theory (in the sense of Atiyah)
associated to the universal finite-type invariant of 3-dimensional manifolds,
as a functor from the category of 3-dimensional manifolds with parametrized
boundary, satisfying some additional conditions, to an algebraic-combinatorial
category. It is built together with its truncations with respect to a natural
grading, and we prove that these TQFTs are non-degenerate and anomaly-free. The
TQFT(s) induce(s) a (series of) representation(s) of a subgroup of
the Mapping Class Group that contains the Torelli group. The N=1 truncation
produces a TQFT for the Casson-Walker-Lescop invariant.Comment: 28 pages, 13 postscript figures. Version 2 (Section 1 has been
considerably shorten, and section 3 has been slightly shorten, since they
will constitute a separate paper. Section 4, which contained only announce of
results, has been suprimated; it will appear in detail elsewhere.
Consequently some statements have been re-numbered. No mathematical changes
have been made.
Kondo effect in the isotropic Heisenberg spin chain
We investigate the boundary effects that arise when spin-
impurities interact with the edges of the antiferromagnetic spin-
Heisenberg chain through spin exchange interactions. We consider both cases
when the couplings are ferromagnetic or anti-ferromagnetic. We find that in the
case of antiferromagnetic interaction, when the impurity coupling strength is
much weaker than that in the bulk, the impurity is screened in the ground state
via the Kondo effect. The Kondo phase is characterized by the Lorentzian
density of states and dynamically generated Kondo temperature . As the
impurity coupling strength increases, increases until it reaches its
maximum value which is the maximum energy carried by a single
spinon. When the impurity coupling strength is increased further, we enter
another phase, the bound mode phase, where the impurity is screened in the
ground state by a single particle bound mode exponentially localized at the
edge to which the impurity is coupled. We find that the impurity can be
unscreened by removing the bound mode. There exists a boundary eigenstate phase
transition between the Kondo and the bound-mode phases, a transition which is
characterized by the change in the number of towers of the Hilbert space. The
transition also manifests itself in ground state quantities like local impurity
density of states and the local impurity magnetization. When the impurity
coupling is ferromagnetic, the impurity is unscreened in the ground state;
however, when the absolute value of the ratio of the impurity and bulk coupling
strengths is greater than , the impurity can be screened by adding
a bound mode that costs energy greater than . When two impurities are
considered, the phases exhibited by each impurity remain unchanged in the
thermodynamic limit, but nevertheless the system exhibits a rich phase diagram.Comment: 23 pages, 7 figures; due to the limitation "The abstract field cannot
be longer than 1,920 characters", the abstract appearing here is slightly
shorter than that in the PDF fil
Trace as an alternative decategorification functor
Categorification is a process of lifting structures to a higher categorical
level. The original structure can then be recovered by means of the so-called
"decategorification" functor. Algebras are typically categorified to additive
categories with additional structure and decategorification is usually given by
the (split) Grothendieck group. In this expository article we study an
alternative decategorification functor given by the trace or the zeroth
Hochschild--Mitchell homology. We show that this form of decategorification
endows any 2-representation of the categorified quantum sl(n) with an action of
the current algebra U(sl(n)[t]) on its center.Comment: 47 pages with tikz figures. arXiv admin note: text overlap with
arXiv:1405.5920 by other author
Theory of inelastic scattering from quantum impurities
We use the framework set up recently to compute non-perturbatively inelastic
scattering from quantum impurities [G. Zar\'and {\it et al.}, Phys. Rev. Lett.
{\bf 93}, 107204 (2004)] to study the the energy dependence of the single
particle -matrix and the inelastic scattering cross section for a number of
quantum impurity models. We study the case of the spin two-channel
Kondo model, the Anderson model, and the usual single-channel Kondo
model. We discuss the difference between non-Fermi liquid and Fermi liquid
models and study how a cross-over between the non-Fermi liquid and Fermi liquid
regimes appears in case of channel anisotropy for the two-channel Kondo
model. We show that for the most elementary non-Fermi liquid system, the
two-channel Kondo model, half of the scattering remains inelastic even at the
Fermi energy. Details of the derivation of the reduction formulas and a simple
path integral approach to connect the -matrix to local correlation functions
are also presented.Comment: published versio
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