1,156 research outputs found
Finite-size left-passage probability in percolation
We obtain an exact finite-size expression for the probability that a
percolation hull will touch the boundary, on a strip of finite width. Our
calculation is based on the q-deformed Knizhnik--Zamolodchikov approach, and
the results are expressed in terms of symplectic characters. In the large size
limit, we recover the scaling behaviour predicted by Schramm's left-passage
formula. We also derive a general relation between the left-passage probability
in the Fortuin--Kasteleyn cluster model and the magnetisation profile in the
open XXZ chain with diagonal, complex boundary terms.Comment: 21 pages, 8 figure
Exactly solvable quantum spin ladders associated with the orthogonal and symplectic Lie algebras
We extend the results of spin ladder models associated with the Lie algebras
to the case of the orthogonal and symplectic algebras $o(2^n),\
sp(2^n)$ where n is the number of legs for the system. Two classes of models
are found whose symmetry, either orthogonal or symplectic, has an explicit n
dependence. Integrability of these models is shown for an arbitrary coupling of
XX type rung interactions and applied magnetic field term.Comment: 7 pages, Late
Magnetization Plateaux in Bethe Ansatz Solvable Spin-S Ladders
We examine the properties of the Bethe Ansatz solvable two- and three-leg
spin- ladders. These models include Heisenberg rung interactions of
arbitrary strength and thus capture the physics of the spin- Heisenberg
ladders for strong rung coupling. The discrete values derived for the
magnetization plateaux are seen to fit with the general prediction based on the
Lieb-Schultz- Mattis theorem. We examine the magnetic phase diagram of the
spin-1 ladder in detail and find an extended magnetization plateau at the
fractional value in agreement with the experimental observation
for the spin-1 ladder compound BIP-TENO.Comment: 11 pages, 1 figur
Magnetization Plateaus in a Solvable 3-Leg Spin Ladder
We present a solvable ladder model which displays magnetization plateaus at
fractional values of the total magnetization. Plateau signatures are also shown
to exist along special lines. The model has isotropic Heisenberg interactions
with additional many-body terms. The phase diagram can be calculated exactly
for all values of the rung coupling and the magnetic field. We also derive the
anomalous behaviour of the susceptibility near the plateau boundaries. There is
good agreement with the phase diagram obtained recently for the pure Heisenberg
ladders by numerical and perturbative techniques.Comment: 4 pages, revtex, 3 postscript figures, small changes to the text and
references update
Structure of the two-boundary XXZ model with non-diagonal boundary terms
We study the integrable XXZ model with general non-diagonal boundary terms at
both ends. The Hamiltonian is considered in terms of a two boundary extension
of the Temperley-Lieb algebra.
We use a basis that diagonalizes a conserved charge in the one-boundary case.
The action of the second boundary generator on this space is computed. For the
L-site chain and generic values of the parameters we have an irreducible space
of dimension 2^L. However at certain critical points there exists a smaller
irreducible subspace that is invariant under the action of all the bulk and
boundary generators. These are precisely the points at which Bethe Ansatz
equations have been formulated. We compute the dimension of the invariant
subspace at each critical point and show that it agrees with the splitting of
eigenvalues, found numerically, between the two Bethe Ansatz equations.Comment: 9 pages Latex. Minor correction
Non-local space-time supersymmetry on the lattice
We show that several well-known one-dimensional quantum systems possess a
hidden nonlocal supersymmetry. The simplest example is the open XXZ spin chain
with \Delta=-1/2. We use the supersymmetry to place lower bounds on the ground
state energy with various boundary conditions. For an odd number of sites in
the periodic chain, and with a particular boundary magnetic field in the open
chain, we can derive the ground state energy exactly. The supersymmetry thus
explains why it is possible to solve the Bethe equations for the ground state
in these cases. We also show that a similar space-time supersymmetry holds for
the t-J model at its integrable ferromagnetic point, where the space-time
supersymmetry and the Hamiltonian it yields coexist with a global u(1|2) graded
Lie algebra symmetry. Possible generalizations to other algebras are discussed.Comment: 12 page
Exact expressions for correlations in the ground state of the dense O(1) loop model
Conjectures for analytical expressions for correlations in the dense O
loop model on semi infinite square lattices are given. We have obtained these
results for four types of boundary conditions. Periodic and reflecting boundary
conditions have been considered before. We give many new conjectures for these
two cases and review some of the existing results. We also consider boundaries
on which loops can end. We call such boundaries ''open''. We have obtained
expressions for correlations when both boundaries are open, and one is open and
the other one is reflecting. Also, we formulate a conjecture relating the
ground state of the model with open boundaries to Fully Packed Loop models on a
finite square grid. We also review earlier obtained results about this relation
for the three other types of boundary conditions. Finally, we construct a
mapping between the ground state of the dense O loop model and the XXZ
spin chain for the different types of boundary conditions.Comment: 25 pages, version accepted by JSTA
Power spectra of TASEPs with a localized slow site
The totally asymmetric simple exclusion process (TASEP) with a localized
defect is revisited in this article with attention paid to the power spectra of
the particle occupancy N(t). Intrigued by the oscillatory behaviors in the
power spectra of an ordinary TASEP in high/low density phase(HD/LD) observed by
Adams et al. (2007 Phys. Rev. Lett. 99 020601), we introduce a single slow site
with hopping rate q<1 to the system. As the power spectrum contains
time-correlation information of the particle occupancy of the system, we are
particularly interested in how the defect affects fluctuation in particle
number of the left and right subsystems as well as that of the entire system.
Exploiting Monte Carlo simulations, we observe the disappearance of
oscillations when the defect is located at the center of the system. When the
defect is off center, oscillations are restored. To explore the origin of such
phenomenon, we use a linearized Langevin equation to calculate the power
spectrum for the sublattices and the whole lattice. We provide insights into
the interactions between the sublattices coupled through the defect site for
both simulation and analytical results.Comment: 16 pages, 6 figures; v2: Minor revision
Generalized T-Q relations and the open XXZ chain
We propose a generalization of the Baxter T-Q relation which involves more
than one independent Q(u). We argue that the eigenvalues of the transfer matrix
of the open XXZ quantum spin chain are given by such generalized T-Q relations,
for the case that at most two of the boundary parameters {\alpha_-, \alpha_+,
\beta_-, \beta_+} are nonzero, and the bulk anisotropy parameter has values
\eta = i \pi/2, i\pi/4, ...Comment: 14 pages, LaTeX; amssymb, no figure
Integrability as a consequence of discrete holomorphicity: the Z_N model
It has recently been established that imposing the condition of discrete
holomorphicity on a lattice parafermionic observable leads to the critical
Boltzmann weights in a number of lattice models. Remarkably, the solutions of
these linear equations also solve the Yang-Baxter equations. We extend this
analysis for the Z_N model by explicitly considering the condition of discrete
holomorphicity on two and three adjacent rhombi. For two rhombi this leads to a
quadratic equation in the Boltzmann weights and for three rhombi a cubic
equation. The two-rhombus equation implies the inversion relations. The
star-triangle relation follows from the three-rhombus equation. We also show
that these weights are self-dual as a consequence of discrete holomorphicity.Comment: 11 pages, 7 figures, some clarifications and a reference adde
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