1,253 research outputs found
Elliptic solutions to difference non-linear equations and related many-body problems
We study algebro-geometric (finite-gap) and elliptic solutions of fully
discretized KP or 2D Toda equations. In bilinear form they are Hirota's
difference equation for -functions. Starting from a given algebraic
curve, we express the -function and the Baker-Akhiezer function in terms
of the Riemann theta function. We show that the elliptic solutions, when the
-function is an elliptic polynomial, form a subclass of the general
algebro-geometric solutions. We construct the algebraic curves of the elliptic
solutions. The evolution of zeros of the elliptic solutions is governed by the
discrete time generalization of the Ruijsenaars-Schneider many body system. The
zeros obey equations which have the form of nested Bethe-Ansatz equations,
known from integrable quantum field theories. We discuss the Lax representation
and the action-angle-type variables for the many body system. We also discuss
elliptic solutions to discrete analogues of KdV, sine-Gordon and 1D Toda
equations and describe the loci of the zeros.Comment: 22 pages, Latex with emlines2.st
Geometrical phases and quantum numbers of solitons in nonlinear sigma-models
Solitons of a nonlinear field interacting with fermions often acquire a
fermionic number or an electric charge if fermions carry a charge. We show how
the same mechanism (chiral anomaly) gives solitons statistical and rotational
properties of fermions. These properties are encoded in a geometrical phase,
i.e., an imaginary part of a Euclidian action for a nonlinear sigma-model. In
the most interesting cases the geometrical phase is non-perturbative and has a
form of an integer-valued theta-term.Comment: 5 pages, no figure
Fusion rules for Quantum Transfer Matrices as a Dynamical System on Grassmann Manifolds
We show that the set of transfer matrices of an arbitrary fusion type for an
integrable quantum model obey these bilinear functional relations, which are
identified with an integrable dynamical system on a Grassmann manifold (higher
Hirota equation). The bilinear relations were previously known for a particular
class of transfer matrices corresponding to rectangular Young diagrams. We
extend this result for general Young diagrams. A general solution of the
bilinear equations is presented.Comment: LaTex (MPLA macros included) 10 pages, 1 figure, included in the tex
Quantum Integrable Systems and Elliptic Solutions of Classical Discrete Nonlinear Equations
Functional relation for commuting quantum transfer matrices of quantum
integrable models is identified with classical Hirota's bilinear difference
equation. This equation is equivalent to the completely discretized classical
2D Toda lattice with open boundaries. The standard objects of quantum
integrable models are identified with elements of classical nonlinear
integrable difference equation. In particular, elliptic solutions of Hirota's
equation give complete set of eigenvalues of the quantum transfer matrices.
Eigenvalues of Baxter's -operator are solutions to the auxiliary linear
problems for classical Hirota's equation. The elliptic solutions relevant to
Bethe ansatz are studied. The nested Bethe ansatz equations for -type
models appear as discrete time equations of motions for zeros of classical
-functions and Baker-Akhiezer functions. Determinant representations of
the general solution to bilinear discrete Hirota's equation and a new
determinant formula for eigenvalues of the quantum transfer matrices are
obtained.Comment: 32 pages, LaTeX file, no figure
Gauging of Geometric Actions and Integrable Hierarchies of KP Type
This work consist of two interrelated parts. First, we derive massive
gauge-invariant generalizations of geometric actions on coadjoint orbits of
arbitrary (infinite-dimensional) groups with central extensions, with gauge
group being certain (infinite-dimensional) subgroup of . We show that
there exist generalized ``zero-curvature'' representation of the pertinent
equations of motion on the coadjoint orbit. Second, in the special case of
being Kac-Moody group the equations of motion of the underlying gauged WZNW
geometric action are identified as additional-symmetry flows of generalized
Drinfeld-Sokolov integrable hierarchies based on the loop algebra {\hat \cG}.
For {\hat \cG} = {\hat {SL}}(M+R) the latter hiearchies are equivalent to a
class of constrained (reduced) KP hierarchies called {\sl cKP}_{R,M}, which
contain as special cases a series of well-known integrable systems (mKdV, AKNS,
Fordy-Kulish, Yajima-Oikawa etc.). We describe in some detail the loop algebras
of additional (non-isospectral) symmetries of {\sl cKP}_{R,M} hierarchies.
Apart from gauged WZNW models, certain higher-dimensional nonlinear systems
such as Davey-Stewartson and -wave resonant systems are also identified as
additional symmetry flows of {\sl cKP}_{R,M} hierarchies. Along the way we
exhibit explicitly the interrelation between the Sato pseudo-differential
operator formulation and the algebraic (generalized) Drinfeld-Sokolov
formulation of {\sl cKP}_{R,M} hierarchies. Also we present the explicit
derivation of the general Darboux-B\"acklund solutions of {\sl cKP}_{R,M}
preserving their additional (non-isospectral) symmetries, which for R=1 contain
among themselves solutions to the gauged WZNW field
equations.Comment: LaTeX209, 47 page
Chiral non-linear sigma-models as models for topological superconductivity
We study the mechanism of topological superconductivity in a hierarchical
chain of chiral non-linear sigma-models (models of current algebra) in one,
two, and three spatial dimensions. The models have roots in the 1D
Peierls-Frohlich model and illustrate how the 1D Frohlich's ideal conductivity
extends to a genuine superconductivity in dimensions higher than one. The
mechanism is based on the fact that a point-like topological soliton carries an
electric charge. We discuss a flux quantization mechanism and show that it is
essentially a generalization of the persistent current phenomenon, known in
quantum wires. We also discuss why the superconducting state is stable in the
presence of a weak disorder.Comment: 5 pages, revtex, no figure
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