26,679 research outputs found
Non-Abelian Holonomy of BCS and SDW Quasi-particles
In this work we investigate properties of fermions in the SO(5) theory of
high Tc superconductivity. We show that the adiabatic time evolution of a SO(5)
superspin vector leads to a non-Abelian SU(2) holonomy of the SO(5) spinor
states. Physically, this non-trivial holonomy arises from the non-zero overlap
between the SDW and BCS quasi-particle states. While the usual Berry's phase of
a SO(3) spinor is described by a Dirac magnetic monopole at the degeneracy
point, the non-Abelian holonomy of a SO(5) spinor is described by a Yang
monopole at the degeneracy point, and is deeply related to the existence of the
second Hopf map from to . We conclude this work by extending the
bosonic SO(5) nonlinear sigma model to include the fermionic states around the
gap nodes as 4 component Dirac fermions coupled to SU(2) gauge fields in 2+1
dimensions.Comment: 45 pages, 5 figures, typos corrected, references adde
Hopf algebras and finite tensor categories in conformal field theory
In conformal field theory the understanding of correlation functions can be
divided into two distinct conceptual levels: The analytic properties of the
correlators endow the representation categories of the underlying chiral
symmetry algebras with additional structure, which in suitable cases is the one
of a finite tensor category. The problem of specifying the correlators can then
be encoded in algebraic structure internal to those categories. After reviewing
results for conformal field theories for which these representation categories
are semisimple, we explain what is known about representation categories of
chiral symmetry algebras that are not semisimple. We focus on generalizations
of the Verlinde formula, for which certain finite-dimensional complex Hopf
algebras are used as a tool, and on the structural importance of the presence
of a Hopf algebra internal to finite tensor categories.Comment: 46 pages, several figures. v2: missing text added after (4.5),
references added, and a few minor changes. v3: typos corrected, bibliography
update
Classical mappings of the symplectic model and their application to the theory of large-amplitude collective motion
We study the algebra Sp(n,R) of the symplectic model, in particular for the
cases n=1,2,3, in a new way. Starting from the Poisson-bracket realization we
derive a set of partial differential equations for the generators as functions
of classical canonical variables. We obtain a solution to these equations that
represents the classical limit of a boson mapping of the algebra. The
relationship to the collective dynamics is formulated as a theorem that
associates the mapping with an exact solution of the time-dependent Hartree
approximation. This solution determines a decoupled classical symplectic
manifold, thus satisfying the criteria that define an exactly solvable model in
the theory of large amplitude collective motion. The models thus obtained also
provide a test of methods for constructing an approximately decoupled manifold
in fully realistic cases. We show that an algorithm developed in one of our
earlier works reproduces the main results of the theorem.Comment: 23 pages, LaTeX using REVTeX 3.
Dynamical Properties of the Mukhanov-Sasaki Hamiltonian in the context of adiabatic vacua and the Lewis-Riesenfeld invariant
We use the method of the Lewis-Riesenfeld invariant to analyze the dynamical
properties of the Mukhanov-Sasaki Hamiltonian and, following this approach,
investigate whether we can obtain possible candidates for initial states in the
context of inflation considering a quasi-de Sitter spacetime. Our main interest
lies in the question to which extent these already well-established methods at
the classical and quantum level for finitely many degrees of freedom can be
generalized to field theory. As our results show, a straightforward
generalization does in general not lead to a unitary operator on Fock space
that implements the corresponding time-dependent canonical transformation
associated with the Lewis-Riesenfeld invariant. The action of this operator can
be rewritten as a time-dependent Bogoliubov transformation and we show that its
generalization to Fock space has to be chosen appropriately in order that the
Shale-Stinespring condition is not violated, where we also compare our results
to already existing ones in the literature. Furthermore, our analysis relates
the Ermakov differential equation that plays the role of an auxiliary equation,
whose solution is necessary to construct the Lewis-Riesenfeld invariant, as
well as the corresponding time-dependent canonical transformation to the
defining differential equation for adiabatic vacua. Therefore, a given solution
of the Ermakov equation directly yields a full solution to the differential
equation for adiabatic vacua involving no truncation at some adiabatic order.
As a consequence, we can interpret our result obtained here as a kind of
non-squeezed Bunch-Davies mode, where the term non-squeezed refers to a
possible residual squeezing that can be involved in the unitary operator for
certain choices of the Bogoliubov map.Comment: 40 pages, 5 figures, minor changes: slightly rewrote the
introduction, extended the discussion on the infrared modes, corrected typos
and added reference
Function spaces and classifying spaces of algebras over a prop
The goal of this paper is to prove that the classifying spaces of categories
of algebras governed by a prop can be determined by using function spaces on
the category of props. We first consider a function space of props to define
the moduli space of algebra structures over this prop on an object of the base
category. Then we mainly prove that this moduli space is the homotopy fiber of
a forgetful map of classifying spaces, generalizing to the prop setting a
theorem of Rezk. The crux of our proof lies in the construction of certain
universal diagrams in categories of algebras over a prop. We introduce a
general method to carry out such constructions in a functorial way.Comment: 28 pages, modifications mainly in section 2 (more details in some
proofs and additional explanations), typo corrections. Final version, to
appear in Algebr. Geom. Topo
Algebraic Approach to Interacting Quantum Systems
We present an algebraic framework for interacting extended quantum systems to
study complex phenomena characterized by the coexistence and competition of
different states of matter. We start by showing how to connect different
(spin-particle-gauge) {\it languages} by means of exact mappings (isomorphisms)
that we name {\it dictionaries} and prove a fundamental theorem establishing
when two arbitrary languages can be connected. These mappings serve to unravel
symmetries which are hidden in one representation but become manifest in
another. In addition, we establish a formal link between seemingly unrelated
physical phenomena by changing the language of our model description. This link
leads to the idea of {\it universality} or equivalence. Moreover, we introduce
the novel concept of {\it emergent symmetry} as another symmetry guiding
principle. By introducing the notion of {\it hierarchical languages}, we
determine the quantum phase diagram of lattice models (previously unsolved) and
unveil hidden order parameters to explore new states of matter. Hierarchical
languages also constitute an essential tool to provide a unified description of
phases which compete and coexist. Overall, our framework provides a simple and
systematic methodology to predict and discover new kinds of orders. Another
aspect exploited by the present formalism is the relation between condensed
matter and lattice gauge theories through quantum link models. We conclude
discussing applications of these dictionaries to the area of quantum
information and computation with emphasis in building new models of computation
and quantum programming languages.Comment: 44 pages, 14 psfigures. Advances in Physics 53, 1 (2004
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