76 research outputs found
A generalisation of Schramm's formula for SLE(2)
The scaling limit of planar loop-erased random walks is described by a
stochastic Loewner evolution with parameter kappa=2. In this note SLE(2) in the
upper half-plane H minus a simply-connected compact subset K of H is studied.
As a main result, the left-passage probability with respect to K is explicitly
determined.Comment: 16 pages, 3 figures, Tik
Open spin chains with dynamic lattice supersymmetry
The quantum spin XXZ chain with anisotropy parameter
possesses a dynamic supersymmetry on the lattice. This supersymmetry and a
generalisation to higher spin are investigated in the case of open spin chains.
A family of non-diagonal boundary interactions that are compatible with the
lattice supersymmetry and depend on several parameters is constructed. The
cohomology of the corresponding supercharges is explicitly computed as a
function of the parameters and the length of the chain. For certain specific
values of the parameters, this cohomology is shown to be non-trivial. This
implies that the spin-chain ground states are supersymmetry singlets. Special
scalar products involving an arbitrary number of these supersymmetry singlets
for chains of different lengths are exactly computed. As a physical
application, the logarithmic bipartite fidelity of the open quantum spin
XXZ chain with and special diagonal boundary interactions is
determined.Comment: 33 pages, 2 figure
Symmetry classes of alternating sign matrices in the nineteen-vertex model
The nineteen-vertex model on a periodic lattice with an anti-diagonal twist
is investigated. Its inhomogeneous transfer matrix is shown to have a simple
eigenvalue, with the corresponding eigenstate displaying intriguing
combinatorial features. Similar results were previously found for the same
model with a diagonal twist. The eigenstate for the anti-diagonal twist is
explicitly constructed using the quantum separation of variables technique. A
number of sum rules and special components are computed and expressed in terms
of Kuperberg's determinants for partition functions of the inhomogeneous
six-vertex model. The computations of some components of the special eigenstate
for the diagonal twist are also presented. In the homogeneous limit, the
special eigenstates become eigenvectors of the Hamiltonians of the integrable
spin-one XXZ chain with twisted boundary conditions. Their sum rules and
special components for both twists are expressed in terms of generating
functions arising in the weighted enumeration of various symmetry classes of
alternating sign matrices (ASMs). These include half-turn symmetric ASMs,
quarter-turn symmetric ASMs, vertically symmetric ASMs, vertically and
horizontally perverse ASMs and double U-turn ASMs. As side results, new
determinant and pfaffian formulas for the weighted enumeration of various
symmetry classes of alternating sign matrices are obtained.Comment: 61 pages, 13 figure
SLE on doubly-connected domains and the winding of loop-erased random walks
Two-dimensional loop-erased random walks (LERWs) are random planar curves
whose scaling limit is known to be a Schramm-Loewner evolution SLE_k with
parameter k = 2. In this note, some properties of an SLE_k trace on
doubly-connected domains are studied and a connection to passive scalar
diffusion in a Burgers flow is emphasised. In particular, the endpoint
probability distribution and winding probabilities for SLE_2 on a cylinder,
starting from one boundary component and stopped when hitting the other, are
found. A relation of the result to conditioned one-dimensional Brownian motion
is pointed out. Moreover, this result permits to study the statistics of the
winding number for SLE_2 with fixed endpoints. A solution for the endpoint
distribution of SLE_4 on the cylinder is obtained and a relation to reflected
Brownian motion pointed out.Comment: 22 pages, 4 figure
Spin Chains with Dynamical Lattice Supersymmetry
Spin chains with exact supersymmetry on finite one-dimensional lattices are considered. The supercharges are nilpotent operators on the lattice of dynamical nature: they change the number of sites. A local criterion for the nilpotency on periodic lattices is formulated. Any of its solutions leads to a supersymmetric spin chain. It is shown that a class of special solutions at arbitrary spin gives the lattice equivalents of the superconformal minimal models. The case of spin one is investigated in detail: in particular, it is shown that the Fateev-Zamolodchikov chain and its off-critical extension possess a lattice supersymmetry for all its coupling constants. Its supersymmetry singlets are thoroughly analysed, and a relation between their components and the weighted enumeration of alternating sign matrices is conjecture
Bethe ansatz solvability and supersymmetry of the model of single fermions and pairs
A detailed study of a model for strongly-interacting fermions with exclusion
rules and lattice supersymmetry is presented. A submanifold in
the space of parameters of the model where it is Bethe-ansatz solvable is
identified. The relation between this manifold and the existence of additional,
so-called dynamic, supersymmetries is discussed. The ground states are analysed
with the help of cohomology techniques, and their exact finite-size Bethe roots
are found. Moreover, through analytical and numerical studies it is argued that
the model provides a lattice version of the super-sine-Gordon
model at a particular coupling where an additional
supersymmetry is present. The dynamic supersymmetry is shown to allow an exact
determination of the gap scaling function of the model.Comment: 46 pages, 10 figure
The Eight-Vertex Model and Lattice Supersymmetry
We show that the XYZ spin chain along the special line of couplings J x J y +J x J z +J y J z =0 possesses a hidden supersymmetry. This lattice supersymmetry is non-local and changes the number of sites. It extends to the full transfer matrix of the corresponding eight-vertex model. In particular, it is shown how to derive the supercharges from Baxter's Bethe ansatz. This analysis leads to new conjectures concerning the ground state for chains of odd length. We also discuss a correspondence between the spectrum of this XYZ chain and that of a manifestly supersymmetric staggered fermion chai
The open XXZ chain at and the boundary quantum Knizhnik-Zamolodchikov equations
The open XXZ spin chain with the anisotropy parameter and
diagonal boundary magnetic fields that depend on a parameter is studied.
For real , the exact finite-size ground-state eigenvalue of the spin-chain
Hamiltonian is explicitly computed. In a suitable normalisation, the
ground-state components are characterised as polynomials in with integer
coefficients. Linear sum rules and special components of this eigenvector are
explicitly computed in terms of determinant formulas. These results follow from
the construction of a contour-integral solution to the boundary quantum
Knizhnik-Zamolodchikov equations associated with the -matrix and diagonal
-matrices of the six-vertex model. A relation between this solution and a
weighted enumeration of totally-symmetric alternating sign matrices is
conjectured.Comment: 36 pages, no figure
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