61 research outputs found
Correlations in the impenetrable electron gas
We consider non-relativistic electrons in one dimension with infinitely
strong repulsive delta function interaction. We calculate the long-time,
large-distance asymptotics of field-field correlators in the gas phase. The gas
phase at low temperatures is characterized by the ideal gas law. We calculate
the exponential decay, the power law corrections and the constant factor of the
asymptotics. Our results are valid at any temperature. They simplify at low
temperatures, where they are easily recognized as products of free fermionic
correlation functions with corrections arising due to the interaction.Comment: 17 pages, Late
Determinant Representations for Correlation Functions of Spin-1/2 XXX and XXZ Heisenberg Magnets
We consider correlation functions of the spin-\half XXX and XXZ Heisenberg
chains in a magnetic field. Starting from the algebraic Bethe Ansatz we derive
representations for various correlation functions in terms of determinants of
Fredholm integral operators.Comment: 23 pages, TeX, BONN-TH-94-14, revised version: typos correcte
Entanglement entropy in quantum spin chains with finite range interaction
We study the entropy of entanglement of the ground state in a wide family of
one-dimensional quantum spin chains whose interaction is of finite range and
translation invariant. Such systems can be thought of as generalizations of the
XY model. The chain is divided in two parts: one containing the first
consecutive L spins; the second the remaining ones. In this setting the entropy
of entanglement is the von Neumann entropy of either part. At the core of our
computation is the explicit evaluation of the leading order term as L tends to
infinity of the determinant of a block-Toeplitz matrix whose symbol belongs to
a general class of 2 x 2 matrix functions. The asymptotics of such determinant
is computed in terms of multi-dimensional theta-functions associated to a
hyperelliptic curve of genus g >= 1, which enter into the solution of a
Riemann-Hilbert problem. Phase transitions for thes systems are characterized
by the branch points of the hyperelliptic curve approaching the unit circle. In
these circumstances the entropy diverges logarithmically. We also recover, as
particular cases, the formulae for the entropy discovered by Jin and Korepin
(2004) for the XX model and Its, Jin and Korepin (2005,2006) for the XY model.Comment: 75 pages, 10 figures. Revised version with minor correction
Riemann-Hilbert problem for Hurwitz Frobenius manifolds: regular singularities
In this paper we study the Fuchsian Riemann-Hilbert (inverse monodromy)
problem corresponding to Frobenius structures on Hurwitz spaces. We find a
solution to this Riemann-Hilbert problem in terms of integrals of certain
meromorphic differentials over a basis of an appropriate relative homology
space, study the corresponding monodromy group and compute the monodromy
matrices explicitly for various special cases.Comment: final versio
On the calculation of finite-gap solutions of the KdV equation
A simple and general approach for calculating the elliptic finite-gap
solutions of the Korteweg-de Vries (KdV) equation is proposed. Our approach is
based on the use of the finite-gap equations and the general representation of
these solutions in the form of rational functions of the elliptic Weierstrass
function. The calculation of initial elliptic finite-gap solutions is reduced
to the solution of the finite-band equations with respect to the parameters of
the representation. The time evolution of these solutions is described via the
dynamic equations of their poles, integrated with the help of the finite-gap
equations. The proposed approach is applied by calculating the elliptic 1-, 2-
and 3-gap solutions of the KdV equations
Lam\'e polynomials, hyperelliptic reductions and Lam\'e band structure
The band structure of the Lam\'e equation, viewed as a one-dimensional
Schr\"odinger equation with a periodic potential, is studied. At integer values
of the degree parameter l, the dispersion relation is reduced to the l=1
dispersion relation, and a previously published l=2 dispersion relation is
shown to be partially incorrect. The Hermite-Krichever Ansatz, which expresses
Lam\'e equation solutions in terms of l=1 solutions, is the chief tool. It is
based on a projection from a genus-l hyperelliptic curve, which parametrizes
solutions, to an elliptic curve. A general formula for this covering is
derived, and is used to reduce certain hyperelliptic integrals to elliptic
ones. Degeneracies between band edges, which can occur if the Lam\'e equation
parameters take complex values, are investigated. If the Lam\'e equation is
viewed as a differential equation on an elliptic curve, a formula is
conjectured for the number of points in elliptic moduli space (elliptic curve
parameter space) at which degeneracies occur. Tables of spectral polynomials
and Lam\'e polynomials, i.e., band edge solutions, are given. A table in the
older literature is corrected.Comment: 38 pages, 1 figure; final revision
Dual Isomonodromic Deformations and Moment Maps to Loop Algebras
The Hamiltonian structure of the monodromy preserving deformation equations
of Jimbo {\it et al } is explained in terms of parameter dependent pairs of
moment maps from a symplectic vector space to the dual spaces of two different
loop algebras. The nonautonomous Hamiltonian systems generating the
deformations are obtained by pulling back spectral invariants on Poisson
subspaces consisting of elements that are rational in the loop parameter and
identifying the deformation parameters with those determining the moment maps.
This construction is shown to lead to ``dual'' pairs of matrix differential
operators whose monodromy is preserved under the same family of deformations.
As illustrative examples, involving discrete and continuous reductions, a
higher rank generalization of the Hamiltonian equations governing the
correlation functions for an impenetrable Bose gas is obtained, as well as dual
pairs of isomonodromy representations for the equations of the Painleve
transcendents and .Comment: preprint CRM-1844 (1993), 28 pgs. (Corrected date and abstract.
Proofs of Two Conjectures Related to the Thermodynamic Bethe Ansatz
We prove that the solution to a pair of nonlinear integral equations arising
in the thermodynamic Bethe Ansatz can be expressed in terms of the resolvent
kernel of the linear integral operator with kernel
exp(-u(theta)-u(theta'))/cosh[(1/2)(theta-theta')]Comment: 16 pages, LaTeX file, no figures. Revision has minor change
Strong resonant tunneling, level repulsion and spectral type for one-dimensional adiabatic quasi-periodic Schr\"{o}dinger operators
In this paper, we consider one dimensional adiabatic quasi-periodic
Schr\"{o}dinger operators in the regime of strong resonant tunneling. We show
the emergence of a level repulsion phenomenon which is seen to be very
naturally related to the local spectral type of the operator: the more singular
the spectrum, the weaker the repulsion
Fredholm determinants and pole-free solutions to the noncommutative Painleve' II equation
We extend the formalism of integrable operators a' la
Its-Izergin-Korepin-Slavnov to matrix-valued convolution operators on a
semi-infinite interval and to matrix integral operators with a kernel of the
form E_1^T(x) E_2(y)/(x+y) thus proving that their resolvent operators can be
expressed in terms of solutions of some specific Riemann-Hilbert problems. We
also describe some applications, mainly to a noncommutative version of
Painleve' II (recently introduced by Retakh and Rubtsov), a related
noncommutative equation of Painleve' type. We construct a particular family of
solutions of the noncommutative Painleve' II that are pole-free (for real
values of the variables) and hence analogous to the Hastings-McLeod solution of
(commutative) Painleve' II. Such a solution plays the same role as its
commutative counterpart relative to the Tracy-Widom theorem, but for the
computation of the Fredholm determinant of a matrix version of the Airy kernel.Comment: 46 pages, no figures (oddly
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