31,298 research outputs found
Yangian Symmetry and Quantum Inverse Scattering Method for the One-Dimensional Hubbard Model
We develop the quantum inverse scattering method for the one-dimensional
Hubbard model on the infinite interval at zero density. -matrix and
monodromy matrix are obtained as limits from their known counterparts on the
finite interval. The -matrix greatly simplifies in the considered limit. The
new -matrix contains a submatrix which turns into the rational -matrix of
the XXX-chain by an appropriate reparametrization. The corresponding submatrix
of the monodromy matrix thus provides a representation of the Y(su(2)) Yangian.
From its quantum determinant we obtain an infinite series of mutually commuting
Yangian invariant operators which includes the Hamiltonian.Comment: 14 pages, LaTeX, no figures, minor change
Duality and free energy analyticity bounds for few-body Ising models with extensive homology rank
We consider pairs of few-body Ising models where each spin enters a bounded number of interaction terms (bonds) such that each model can be obtained from the dual of the other after freezing k spins on large-degree sites. Such a pair of Ising models can be interpreted as a two-chain complex with k being the rank of the first homology group. Our focus is on the case where k is extensive, that is, scales linearly with the number of bonds n. Flipping any of these additional spins introduces a homologically nontrivial defect (generalized domain wall). In the presence of bond disorder, we prove the existence of a low-temperature weak-disorder region where additional summation over the defects has no effect on the free energy density f(T) in the thermodynamical limit and of a high-temperature region where an extensive homological defect does not affect f(T). We also discuss the convergence of the high- and low-temperature series for the free energy density, prove the analyticity of limiting f(T) at high and low temperatures, and construct inequalities for the critical point(s) where analyticity is lost. As an application, we prove multiplicity of the conventionally defined critical points for Ising models on all { f, d} tilings of the infinite hyperbolic plane, where df/(d + f) \u3e 2. Namely, for these infinite graphs, we show that critical temperatures with free and wired boundary conditions differ, Tc(f)T(f)
Opinion Dynamics in Heterogeneous Networks: Convergence Conjectures and Theorems
Recently, significant attention has been dedicated to the models of opinion
dynamics in which opinions are described by real numbers, and agents update
their opinions synchronously by averaging their neighbors' opinions. The
neighbors of each agent can be defined as either (1) those agents whose
opinions are in its "confidence range," or (2) those agents whose "influence
range" contain the agent's opinion. The former definition is employed in
Hegselmann and Krause's bounded confidence model, and the latter is novel here.
As the confidence and influence ranges are distinct for each agent, the
heterogeneous state-dependent interconnection topology leads to a
poorly-understood complex dynamic behavior. In both models, we classify the
agents via their interconnection topology and, accordingly, compute the
equilibria of the system. Then, we define a positive invariant set centered at
each equilibrium opinion vector. We show that if a trajectory enters one such
set, then it converges to a steady state with constant interconnection
topology. This result gives us a novel sufficient condition for both models to
establish convergence, and is consistent with our conjecture that all
trajectories of the bounded confidence and influence models eventually converge
to a steady state under fixed topology.Comment: 22 pages, Submitted to SIAM Journal on Control and Optimization
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