6,024 research outputs found
On the Lagrangian structure of integrable hierarchies
We develop the concept of pluri-Lagrangian structures for integrable
hierarchies. This is a continuous counterpart of the pluri-Lagrangian (or
Lagrangian multiform) theory of integrable lattice systems. We derive the
multi-time Euler Lagrange equations in their full generality for hierarchies of
two-dimensional systems, and construct a pluri-Lagrangian formulation of the
potential Korteweg-de Vries hierarchy.Comment: 29 page
Bethe Ansatz Equations for the Broken -Symmetric Model
We obtain the Bethe Ansatz equations for the broken -symmetric
model by constructing a functional relation of the transfer matrix of
-operators. This model is an elliptic off-critical extension of the
Fateev-Zamolodchikov model. We calculate the free energy of this model on the
basis of the string hypothesis.Comment: 43 pages, latex, 11 figure
Gaudin Hypothesis for the XYZ Spin Chain
The XYZ spin chain is considered in the framework of the generalized
algebraic Bethe ansatz developed by Takhtajan and Faddeev. The sum of norms of
the Bethe vectors is computed and expressed in the form of a Jacobian. This
result corresponds to the Gaudin hypothesis for the XYZ spin chain.Comment: 12 pages, LaTeX2e (+ amssymb, amsthm); to appear in J. Phys.
Avalanche Collapse of Interdependent Network
We reveal the nature of the avalanche collapse of the giant viable component
in multiplex networks under perturbations such as random damage. Specifically,
we identify latent critical clusters associated with the avalanches of random
damage. Divergence of their mean size signals the approach to the hybrid phase
transition from one side, while there are no critical precursors on the other
side. We find that this discontinuous transition occurs in scale-free multiplex
networks whenever the mean degree of at least one of the interdependent
networks does not diverge.Comment: 4 pages, 5 figure
Critical dynamics of the k-core pruning process
We present the theory of the k-core pruning process (progressive removal of
nodes with degree less than k) in uncorrelated random networks. We derive exact
equations describing this process and the evolution of the network structure,
and solve them numerically and, in the critical regime of the process,
analytically. We show that the pruning process exhibits three different
behaviors depending on whether the mean degree of the initial network is
above, equal to, or below the threshold _c corresponding to the emergence of
the giant k-core. We find that above the threshold the network relaxes
exponentially to the k-core. The system manifests the phenomenon known as
"critical slowing down", as the relaxation time diverges when tends to
_c. At the threshold, the dynamics become critical characterized by a
power-law relaxation (1/t^2). Below the threshold, a long-lasting transient
process (a "plateau" stage) occurs. This transient process ends with a collapse
in which the entire network disappears completely. The duration of the process
diverges when tends to _c. We show that the critical dynamics of the
pruning are determined by branching processes of spreading damage. Clusters of
nodes of degree exactly k are the evolving substrate for these branching
processes. Our theory completely describes this branching cascade of damage in
uncorrelated networks by providing the time dependent distribution function of
branching. These theoretical results are supported by our simulations of the
-core pruning in Erdos-Renyi graphs.Comment: 12 pages, 10 figure
Directed-loop Monte Carlo simulations of vertex models
We show how the directed-loop Monte Carlo algorithm can be applied to study
vertex models. The algorithm is employed to calculate the arrow polarization in
the six-vertex model with the domain wall boundary conditions (DWBC). The model
exhibits spatially separated ordered and ``disordered'' regions. We show how
the boundary between these regions depends on parameters of the model. We give
some predictions on the behavior of the polarization in the thermodynamic limit
and discuss the relation to the Arctic Circle theorem.Comment: Extended version with autocorrelations and more figures. Added 2
reference
Bootstrap Percolation on Complex Networks
We consider bootstrap percolation on uncorrelated complex networks. We obtain
the phase diagram for this process with respect to two parameters: , the
fraction of vertices initially activated, and , the fraction of undamaged
vertices in the graph. We observe two transitions: the giant active component
appears continuously at a first threshold. There may also be a second,
discontinuous, hybrid transition at a higher threshold. Avalanches of
activations increase in size as this second critical point is approached,
finally diverging at this threshold. We describe the existence of a special
critical point at which this second transition first appears. In networks with
degree distributions whose second moment diverges (but whose first moment does
not), we find a qualitatively different behavior. In this case the giant active
component appears for any and , and the discontinuous transition is
absent. This means that the giant active component is robust to damage, and
also is very easily activated. We also formulate a generalized bootstrap
process in which each vertex can have an arbitrary threshold.Comment: 9 pages, 3 figure
Exact and simple results for the XYZ and strongly interacting fermion chains
We conjecture exact and simple formulas for physical quantities in two
quantum chains. A classic result of this type is Onsager, Kaufman and Yang's
formula for the spontaneous magnetization in the Ising model, subsequently
generalized to the chiral Potts models. We conjecture that analogous results
occur in the XYZ chain when the couplings obey J_xJ_y + J_yJ_z + J_x J_z=0, and
in a related fermion chain with strong interactions and supersymmetry. We find
exact formulas for the magnetization and gap in the former, and the staggered
density in the latter, by exploiting the fact that certain quantities are
independent of finite-size effects
Exact Solution of a Three-Dimensional Dimer System
We consider a three-dimensional lattice model consisting of layers of vertex
models coupled with interlayer interactions. For a particular non-trivial
interlayer interaction between charge-conserving vertex models and using a
transfer matrix approach, we show that the eigenvalues and eigenvectors of the
transfer matrix are related to those of the two-dimensional vertex model. The
result is applied to analyze the phase transitions in a realistic
three-dimensional dimer system.Comment: 11 pages in REVTex with 2 PS figure
Ground State of the Quantum Symmetric Finite Size XXZ Spin Chain with Anisotropy Parameter
We find an analytic solution of the Bethe Ansatz equations (BAE) for the
special case of a finite XXZ spin chain with free boundary conditions and with
a complex surface field which provides for symmetry of the
Hamiltonian. More precisely, we find one nontrivial solution, corresponding to
the ground state of the system with anisotropy parameter
corresponding to .Comment: 6 page
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