166 research outputs found
Spin chains and combinatorics: twisted boundary conditions
The finite XXZ Heisenberg spin chain with twisted boundary conditions was
considered. For the case of even number of sites , anisotropy parameter -1/2
and twisting angle the Hamiltonian of the system possesses an
eigenvalue . The explicit form of the corresponding eigenvector was
found for . Conjecturing that this vector is the ground state of the
system we made and verified several conjectures related to the norm of the
ground state vector, its component with maximal absolute value and some
correlation functions, which have combinatorial nature. In particular, the
squared norm of the ground state vector is probably coincides with the number
of half-turn symmetric alternating sign matrices.Comment: LaTeX file, 7 page
SM(2,4k) fermionic characters and restricted jagged partitions
A derivation of the basis of states for the superconformal minimal
models is presented. It relies on a general hypothesis concerning the role of
the null field of dimension . The basis is expressed solely in terms of
modes and it takes the form of simple exclusion conditions (being thus a
quasi-particle-type basis). Its elements are in correspondence with
-restricted jagged partitions. The generating functions of the latter
provide novel fermionic forms for the characters of the irreducible
representations in both Ramond and Neveu-Schwarz sectors.Comment: 12 page
On the domain wall partition functions of level-1 affine so(n) vertex models
We derive determinant expressions for domain wall partition functions of
level-1 affine so(n) vertex models, n >= 4, at discrete values of the crossing
parameter lambda = m pi / 2(n-3), m in Z, in the critical regime.Comment: 14 pages, 13 figures included in latex fil
The Razumov-Stroganov conjecture: Stochastic processes, loops and combinatorics
A fascinating conjectural connection between statistical mechanics and
combinatorics has in the past five years led to the publication of a number of
papers in various areas, including stochastic processes, solvable lattice
models and supersymmetry. This connection, known as the Razumov-Stroganov
conjecture, expresses eigenstates of physical systems in terms of objects known
from combinatorics, which is the mathematical theory of counting. This note
intends to explain this connection in light of the recent papers by Zinn-Justin
and Di Francesco.Comment: 6 pages, 4 figures, JSTAT News & Perspective
A refined Razumov-Stroganov conjecture
We extend the Razumov-Stroganov conjecture relating the groundstate of the
O(1) spin chain to alternating sign matrices, by relating the groundstate of
the monodromy matrix of the O(1) model to the so-called refined alternating
sign matrices, i.e. with prescribed configuration of their first row, as well
as to refined fully-packed loop configurations on a square grid, keeping track
both of the loop connectivity and of the configuration of their top row. We
also conjecture a direct relation between this groundstate and refined totally
symmetric self-complementary plane partitions, namely, in their formulation as
sets of non-intersecting lattice paths, with prescribed last steps of all
paths.Comment: 20 pages, 4 figures, uses epsf and harvmac macros a few typos
correcte
Bethe roots and refined enumeration of alternating-sign matrices
The properties of the most probable ground state candidate for the XXZ spin
chain with the anisotropy parameter equal to -1/2 and an odd number of sites is
considered. Some linear combinations of the components of the considered state,
divided by the maximal component, coincide with the elementary symmetric
polynomials in the corresponding Bethe roots. It is proved that those
polynomials are equal to the numbers providing the refined enumeration of the
alternating-sign matrices of order M+1 divided by the total number of the
alternating-sign matrices of order M, for the chain of length 2M+1.Comment: LaTeX 2e, 12 pages, minor corrections, references adde
Conformal invariance and its breaking in a stochastic model of a fluctuating interface
Using Monte-Carlo simulations on large lattices, we study the effects of
changing the parameter (the ratio of the adsorption and desorption rates)
of the raise and peel model. This is a nonlocal stochastic model of a
fluctuating interface. We show that for the system is massive, for
it is massless and conformal invariant. For the conformal
invariance is broken. The system is in a scale invariant but not conformal
invariant phase. As far as we know it is the first example of a system which
shows such a behavior. Moreover in the broken phase, the critical exponents
vary continuously with the parameter . This stays true also for the critical
exponent which characterizes the probability distribution function of
avalanches (the critical exponent staying unchanged).Comment: 22 pages and 20 figure
Six - Vertex Model with Domain wall boundary conditions. Variable inhomogeneities
We consider the six-vertex model with domain wall boundary conditions. We
choose the inhomogeneities as solutions of the Bethe Ansatz equations. The
Bethe Ansatz equations have many solutions, so we can consider a wide variety
of inhomogeneities. For certain choices of the inhomogeneities we study arrow
correlation functions on the horizontal line going through the centre. In
particular we obtain a multiple integral representation for the emptiness
formation probability that generalizes the known formul\ae for XXZ
antiferromagnets.Comment: 12 pages, 1 figur
Higher spin vertex models with domain wall boundary conditions
We derive determinant expressions for the partition functions of spin-k/2
vertex models on a finite square lattice with domain wall boundary conditions.Comment: 14 pages, 12 figures. Minor corrections. Version to appear in JSTA
Phase-Space Networks of Geometrical Frustrated Systems
Geometric frustration leads to complex phases of matter with exotic
properties. Antiferromagnets on triangular lattices and square ice are two
simple models of geometrical frustration. We map their highly degenerated
ground-state phase spaces as discrete networks such that network analysis tools
can be introduced to phase-space studies. The resulting phase spaces establish
a novel class of complex networks with Gaussian spectral densities. Although
phase-space networks are heterogeneously connected, the systems are still
ergodic except under periodic boundary conditions. We elucidate the boundary
effects by mapping the two models as stacks of cubes and spheres in higher
dimensions. Sphere stacking in various containers, i.e. square ice under
various boundary conditions, reveals challenging combinatorial questions. This
network approach can be generalized to phase spaces of some other complex
systems
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