Peierls Argument and Long-Range Order Behavior of Quantum Lattice Systems with Unbounded Spins

Results on long-range order behavior are obtained for systems in arbitrary dimension (v2) with a wide class of spin–spin long-range interactions, without assuming the reflection positivity property.     

Staggered Phases in Diluted Systems with Continuous Spins

We consider systems with continuous spins and annealed dilution. We show that, as in the discrete case, such systems often undergo a phase transition, which is manifested in the appearance of a staggered intermediate phase. In particular, these phases appear in systems such as the massive Gaussian model where there is no phase transition in the undiluted system. Received: 2 January 1997 / Accepted: 1 February 1997     

Propagation of Correlations in Quantum Lattice Systems

We provide a simple proof of the Lieb-Robinson bound and use it to prove the existence of the dynamics for interactions with polynomial decay. We then use our results to demonstrate that there is an upper bound on the rate at which correlations between observables with separated support can accumulate as a consequence of the dynamics.     

Uniqueness Problem for Quantum Lattice Systems with Compact Spins

Quantum lattice systems with compact spins and nearest-neighbour interactions are considered. Uniqueness of the corresponding Euclidean Gibbs states is proved uniformly with respect to the temperature, in the case where the particles have a sufficiently small mass.     

Liapunov Multipliers and Decay of Correlations in Dynamical Systems

The essential decorrelation rate of a hyperbolic dynamical system is the decay rate of time-correlations one expects to see stably for typical observables once resonances are projected out. We define and illustrate these notions and study the conjecture that for observables in $\mathcal{C}^1$ , the essential decorrelation rate is never faster than what is dictated by the smallest unstable Liapunov multiplier.     

Long-Range Temporal Correlations in Kinetic Roughening

Journal of Statistical Physics - We introduce a class of generalized Langevin-type growth equations exhibiting long-ranged temporal correlations in kinetic roughening, and investigate memory...     

Efficient Localization Bounds in a Continuous N-Particle Anderson Model with Long-Range Interaction

We establish strong dynamical and exponential spectral localization for a class of multi-particle Anderson models in a Euclidean space with an alloy-type random potential and a sub-exponentially decaying interaction of infinite range. For the first time in the mathematical literature, the uniform decay bounds on the eigenfunction correlators (EFCs) at low energies are proved, in the multi-particle continuous configuration space, in the (symmetrized) norm-distance, which is a natural distance in the multi-particle configuration space, and not in the Hausdorff distance. This results in uniform bounds on the EFCs in arbitrarily large but bounded domains in the physical configuration space, and not only in the actually infinite space, as in prior works on multi-particle localization in Euclidean spaces.     

Asymptotic Correlations in Gapped and Critical Topological Phases of 1D Quantum Systems

Journal of Statistical Physics - Topological phases protected by symmetry can occur in gapped and—surprisingly—in critical systems. We consider non-interacting fermions in one dimension...     

On Decay of Correlations for Unbounded Spin Systems with Arbitrary Boundary Conditions

We propose a method based on cluster expansion to study the truncated correlations of unbounded spin systems uniformly in the boundary condition and in a possible external field. By this method we study the spin–spin truncated correlations of various systems, including the case of infinite range simply integrable interactions, and we show how suitable boundary conditions and/or external fields may improve the decay of the correlations.     

Decay of Correlations in Random-Cluster Models

We prove exponential decay for the tail of the radius R of the cluster at the origin, for subcritical random-cluster models, under an assumption slightly weaker than that (here, d is the number of dimensions). Specifically, if throughout the subcritical phase, then for some α > 0. This implies the exponential decay of the two-point correlation function of subcritical Potts models, subject to a hypothesis of (at least) polynomial decay of this function. Similar results are known already for percolation and Ising models, and for Potts models when the number q of available states is sufficiently large; indeed the hypothesis of polynomial decay has been proved rigorously for these cases. In two dimensions, the hypothesis that is weaker than requiring that the susceptibility be finite, i.e., that the two-point function be summable. The principal new technique is a form of Russo's formula for random-cluster models reported by Bezuidenhout, Grimmett, and Kesten. For the current application, this leads to an analysis of a first-passage problem for random-cluster models, and a proof that the associated time constant is strictly positive if and only if the tail of R decays exponentially. Received: 25 September 1996 / Accepted: 21 February 1997     

Decay of Correlations and Dispersing Billiards

We give a rigorous proof of exponential decay of correlations for all major classes of planar dispersing billiards: periodic Lorentz gases with and without horizon and dispersing billiard tables with corner points     

Dobrushin's Uniqueness for Quantum Lattice Systems with Nonlocal Interaction

Based on Dobrushin's fundamental criterion, we prove uniqueness of Euclidean Gibbs states for a certain class of quantum lattice systems with unbounded spins, nonharmonic pair potentials and infinite radius of interaction. The necessary estimates on Dobrushin's coefficients are obtained from the Log-Sobolev inequality which holds for the one-point conditional distributions on the infinite dimensional single spin (= loop) spaces. Received: 25 October 1996 / Accepted: 3 March 1997     

Continuous Versus Discrete Spins in the Hyperbolic Plane

We study the O(n) model on graphs quasi-isometric to the hyperbolic plane, with free boundary conditions. We observe that the pair correlation decays exponentially with distance, for all temperatures, if and only if \(n>1.\)     

Spectral Gap and Exponential Decay of Correlations

We study the relation between the spectral gap above the ground state and the decay of the correlations in the ground state in quantum spin and fermion systems with short-range interactions on a wide class of lattices. We prove that, if two observables anticommute with each other at large distance, then the nonvanishing spectral gap implies exponential decay of the corresponding correlation. When two observables commute with each other at large distance, the connected correlation function decays exponentially under the gap assumption. If the observables behave as a vector under the U(1) rotation of a global symmetry of the system, we use previous results on the large distance decay of the correlation function to show the stronger statement that the correlation function itself, rather than just the connected correlation function, decays exponentially under the gap assumption on a lattice with a certain self-similarity in (fractal) dimensions D < 2. In particular, if the system is translationally invariant in one of the spatial directions, then this self-similarity condition is automatically satisfied. We also treat systems with long-range, power-law decaying interactions.     

周期轨道的量子化与量子能谱中的长程关联

用二维可积系统的半经典量子化方案和二维无关联振子系统的量子能级与周期轨道之间的对应关系,讨论了一组量子能级之间具有长程关联的内在机制,在二维无关联振子系统中,发现了具有相同拓扑M(M1,M2)的周期轨道相对应的量子能级之间存在着长程关联,并以二维4次无关联振子系统为例做了具体说明.     

Quantum Stabilization and Decay of Correlations in Anharmonic Crystals

Letters in Mathematical Physics -     

Robust Exponential Decay of Correlations for Singular-Flows

We construct open sets of C ^k (k ≥ 2) vector fields with singularities that have robust exponential decay of correlations with respect to the unique physical measure. In particular we prove that the geometric Lorenz attractor has exponential decay of correlations with respect to the unique physical measure.     

Mesoscopic Modeling for Continuous Spin Lattice Systems: Model Problems and Micromagnetics Applications

In this paper we derive deterministic mesoscopic theories for model continuous spin lattice systems both at equilibrium and non-equilibrium in the presence of thermal fluctuations. The full magnetic Hamiltonian that includes singular integral (dipolar) interactions is also considered at equilibrium. The non-equilibrium microscopic models we consider are relaxation-type dynamics arising in kinetic Monte Carlo or Langevin-type simulations of lattice systems. In this context we also employ the derived mesoscopic models to study the relaxation of such algorithms to equilibrium