39 research outputs found
Norms of Bethe Wave Functions
Bethe Ansatz solvable models are considered, like XXZ Heisenberg
anti-ferromagnet and Bose gas with delta interaction. Periodic boundary conditions lead to Bethe
equation. The square of the norm of Bethe wave function is equal to a determinant of linearized
system of Bethe equations (determinant of matrix of second derivatives of Yang action). The proof
was first published in Communications in Mathematical Physics, vol 86, page 391 in l982. Also
domain wall boundary conditions for 6 vertex model were discovered in the same paper [see
Appendix D]. These play an important role for algebraic combinatorics: alternating sign matrices,
domino tiling and plane partition
On the correlation functions of the domain wall six vertex model
We propose an (essentially combinatorial) approach to the correlation
functions of the domain wall six vertex model. We reproduce the boundary
1-point function determinant expression of Bogoliubov, Pronko and Zvonarev,
then use that as a building block to obtain analogous expressions for boundary
2-point functions. The latter can be used, at least in principle, to express
more general boundary (and bulk) correlation functions as sums over (products
of) determinants.Comment: LaTeX2e, requires eepic, 25 pages, including 29 figure
Boundary correlation functions of the six-vertex model
We consider the six-vertex model on an square lattice with the
domain wall boundary conditions. Boundary one-point correlation functions of
the model are expressed as determinants of matrices, generalizing
the known result for the partition function. In the free fermion case the
explicit answers are obtained. The introduced correlation functions are closely
related to the problem of enumeration of alternating sign matrices and domino
tilings.Comment: 20 pages, 2 figures, typos correcte
On the problem of calculation of correlation functions in the six-vertex model with domain wall boundary conditions
The problem of calculation of correlation functions in the six-vertex model
with domain wall boundary conditions is addressed by considering a particular
nonlocal correlation function, called row configuration probability. This
correlation function can be used as building block for computing various (both
local and nonlocal) correlation functions in the model. The row configuration
probability is calculated using the quantum inverse scattering method; the
final result is given in terms of a multiple integral. The connection with the
emptiness formation probability, another nonlocal correlation function which
was computed elsewhere using similar methods, is also discussed.Comment: 15 pages, 2 figure
On the partition function of the six-vertex model with domain wall boundary conditions
The six-vertex model on an square lattice with domain wall
boundary conditions is considered. A Fredholm determinant representation for
the partition function of the model is given. The kernel of the corresponding
integral operator is of the so-called integrable type, and involves classical
orthogonal polynomials. From this representation, a ``reconstruction'' formula
is proposed, which expresses the partition function as the trace of a suitably
chosen quantum operator, in the spirit of corner transfer matrix and vertex
operator approaches to integrable spin models.Comment: typos correcte
On two-point boundary correlations in the six-vertex model with DWBC
The six-vertex model with domain wall boundary conditions (DWBC) on an N x N
square lattice is considered. The two-point correlation function describing the
probability of having two vertices in a given state at opposite (top and
bottom) boundaries of the lattice is calculated. It is shown that this
two-point boundary correlator is expressible in a very simple way in terms of
the one-point boundary correlators of the model on N x N and (N-1) x (N-1)
lattices. In alternating sign matrix (ASM) language this result implies that
the doubly refined x-enumerations of ASMs are just appropriate combinations of
the singly refined ones.Comment: v2: a reference added, typos correcte
The arctic curve of the domain-wall six-vertex model in its anti-ferroelectric regime
An explicit expression for the spatial curve separating the region of
ferroelectric order (`frozen' zone) from the disordered one (`temperate' zone)
in the six-vertex model with domain wall boundary conditions in its
anti-ferroelectric regime is obtained.Comment: 12 pages, 1 figur
Excited states in the twisted XXZ spin chain
We compute the finite size spectrum for the spin 1/2 XXZ chain with twisted
boundary conditions, for anisotropy in the regime , and
arbitrary twist . The string hypothesis is employed for treating
complex excitations. The Bethe Ansatz equtions are solved within a coupled
non-linear integral equation approach, with one equation for each type of
string. The root-of-unity quantum group invariant periodic chain reduces to the
XXZ_1/2 chain with a set of twist boundary conditions (,
an integer multiple of ). For this model, the restricted
Hilbert space corresponds to an unitary conformal field theory, and we recover
all primary states in the Kac table in terms of states with specific twist and
strings.Comment: 16 pages, Latex; added discussion on quantum group invariance and
arbitrary magnon numbe
The role of orthogonal polynomials in the six-vertex model and its combinatorial applications
The Hankel determinant representations for the partition function and
boundary correlation functions of the six-vertex model with domain wall
boundary conditions are investigated by the methods of orthogonal polynomial
theory. For specific values of the parameters of the model, corresponding to
1-, 2- and 3-enumerations of Alternating Sign Matrices (ASMs), these
polynomials specialize to classical ones (Continuous Hahn, Meixner-Pollaczek,
and Continuous Dual Hahn, respectively). As a consequence, a unified and
simplified treatment of ASMs enumerations turns out to be possible, leading
also to some new results such as the refined 3-enumerations of ASMs.
Furthermore, the use of orthogonal polynomials allows us to express, for
generic values of the parameters of the model, the partition function of the
(partially) inhomogeneous model in terms of the one-point boundary correlation
functions of the homogeneous one.Comment: Talk presented by F.C. at the Short Program of the Centre de
Recherches Mathematiques: Random Matrices, Random Processes and Integrable
Systems, Montreal, June 20 - July 8, 200
The correlation functions of the XX Heisenberg magnet and random walks of vicious walkers
A relationship of the random walks on one-dimensional periodic lattice and
the correlation functions of the XX Heisenberg spin chain is investigated. The
operator averages taken over the ferromagnetic state play a role of generating
functions of the number of paths made by the so-called "vicious" random walkers
(the vicious walkers annihilate each other provided they arrive at the same
lattice site). It is shown that the two-point correlation function of spins,
calculated over eigen-states of the XX magnet, can be interpreted as the
generating function of paths made by a single walker in a medium characterized
by a non-constant number of vicious neighbors. The answers are obtained for a
number of paths made by the described walker from some fixed lattice site to
another sufficiently remote one. Asymptotical estimates for the number of paths
are provided in the limit, when the number of steps is increased.Comment: 16 pages, 1 figure, LaTeX. Extended talk at the Conference "Classical
and Quantum Integrable Systems CQIS-08" (Protvino, Russia, January 21-24,
2008). To appear in Theoretical and Mathematical Physics in 2009 (In Russian