High precision determination of the low-energy constants for the two-dimensional quantum Heisenberg model on the honeycomb lattice

The low-energy constants, namely the staggered magnetization density M? _s per spin, the spin stiffness ρ _s , and the spinwave velocity c of the two-dimensional (2-d) spin-1/2 Heisenberg model on the honeycomb lattice are calculated using first principles Monte Carlo method. The spinwave velocity c is determined first through the winding numbers squared. M? _s and ρ _s are then obtained by employing the relevant volume- and temperature-dependence predictions from magnon chiral perturbation theory. The periodic boundary conditions (PBCs) implemented in our simulations lead to a honeycomb lattice covering both a rectangular and a parallelogram-shaped region. Remarkably, by appropriately utilizing the predictions of magnon chiral perturbation theory, the numerical values of M? _s , ρ _s , and c we obtain for both the considered periodic honeycomb lattice of different geometries are consistent with each other quantitatively. The numerical accuracy reached here is greatly improved. Specifically, by simulating the 2-d quantum Heisenberg model on the periodic honeycomb lattice overlaying a rectangular area, we arrive at M? _s = 0.26882(3), ρ _s  = 0.1012(2)J, and c = 1.2905(8)Ja. The results we obtain provide a useful lesson for some studies such as simulating fermion actions on hyperdiamond lattice and investigating second order phase transitions with twisted boundary conditions.     

KAM Tori for 1D Nonlinear Wave Equations¶with Periodic Boundary Conditions

In this paper, one-dimensional (1D) nonlinear wave equations

Two-dimensional Euler flows in slowly deforming domains

We consider the evolution of an incompressible two-dimensional perfect fluid as the boundary of its domain is deformed in a prescribed fashion. The flow is taken to be initially steady, and the boundary deformation is assumed to be slow compared to the fluid motion. The Eulerian flow is found to remain approximately steady throughout the evolution. At leading order, the velocity field depends instantaneously on the shape of the domain boundary, and it is determined by the steadiness and vorticity-preservation conditions. This is made explicit by reformulating the problem in terms of an area-preserving diffeomorphism g_Λ which transports the vorticity. The first-order correction to the velocity field is linear in the boundary velocity, and we show how it can be computed from the time derivative of g_Λ.The evolution of the Lagrangian position of fluid particles is also examined. Thanks to vorticity conservation, this position can be specified by an angle-like coordinate along vorticity contours. An evolution equation for this angle is derived, and the net change in angle resulting from a cyclic deformation of the domain boundary is calculated. This includes a geometric contribution which can be expressed as the integral of a certain curvature over the interior of the circuit that is traced by the parameters defining the deforming boundary.A perturbation approach using Lie series is developed for the computation of both the Eulerian flow and geometric angle for small deformations of the boundary. Explicit results are presented for the evolution of nearly axisymmetric flows in slightly deformed discs.     

Finite Size XXZ Spin Chain with Anisotropy Parameter Δ= \frac{1} {2}

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 U_q(sl(2)) symmetry of the Hamiltonian. More precisely, we find one nontrivial solution, corresponding to the ground state of the system with anisotropy parameter Δ= $\frac{1}{2}$ corresponding to q³=?1. With a view to establishing an exact representation of the ground state of the finite size XXZ spin chain in terms of elementary functions, we concentrate on the crossing parameter η dependence around η=π/3 for which there is a known solution. The approach taken involves the use of a physical solution Q of Baxter's T-Q equation, corresponding to the ground state, as well as a non-physical solution P of the same equation. The calculation of P and then of the ground state derivative is covered. Possible applications of this derivative to the theory of percolation have yet to be investigated. As far as the finite XXZ spin chain with periodic boundary conditions is concerned, we find a similar solution for an assymetric case which corresponds to the 6-vertex model with a special magnetic field. For this case we find the analytic value of the “magnetic moment” of the system in the corresponding state.     

Excess entropy for a vacancy in a crystal

The following topics are discussed: (1) Within the harmonic theory a general expression is given for excess entropy of a localized detect in any crystal with cyclic boundary conditions and in the small perturbation limit. For a single vacancy defect Stripp and Kirkwood's formula is obtained, ΔS/k_B = d/2,d being the dimension of the space. For other localized defects the perturbation formula requires calculation of the unperturbed Green's matrix. (2) It is argued that the effect of free surface boundary conditions, as well as other surface changes, leaves ΔS effectively invariant. (3) Using the vacancy perturbation result for ΔS and Lawson's formula, a value for the vacancy formation volume is obtained only in terms of known measured parameters; this result is compared to experimental and other calculated values of ΔV_f. (4) If localized vibrational modes are present due to the vacancy, what effect these might have on ΔS, and, from this, what materials might be candidates for observing localized modes.     

A Novel Interpretation of the Klein-Gordon Equation

The covariant Klein-Gordon equation requires twice the boundary conditions of the Schrödinger equation and does not have an accepted single-particle interpretation. Instead of interpreting its solution as a probability wave determined by an initial boundary condition, this paper considers the possibility that the solutions are determined by both an initial and a final boundary condition. By constructing an invariant joint probability distribution from the size of the solution space, it is shown that the usual measurement probabilities can nearly be recovered in the non-relativistic limit, provided that neither boundary constrains the energy to a precision near ?/t ₀ (where t ₀ is the time duration between the boundary conditions). Otherwise, deviations from standard quantum mechanics are predicted.     

Critical Behavior in a Quasi D Dimensional Spin Model

We study a classical spin model (more precisely a class of models) with O(N) symmetry that can be viewed as a simplified D dimensional lattice model. It is equivalent to a non-translationinvariant one dimensional model and contains the dimensionality D as a parameter that need not be an integer. The critical dimension turns out to be 2, just as in the usual translation invariant models. We study the phase structure, critical phenomena and spontaneous symmetry breaking. Furthermore we compute the perturbation expansion to low order with various boundary conditions. In our simplified models a number of questions can be answered that remain controversial in the translation invariant models, such as the asymptoticity of the perturbation expansion and the role of super-instantons. We find that perturbation theory produces the right asymptotic expansion in dimension D2 only with special boundary conditions. Finally the model allows a test of the percolation ideas of Patrascioiu and Seiler.     

A KAM Theorem for Hamiltonian Partial Differential Equations in Higher Dimensional Spaces

In this paper, we give a KAM theorem for a class of infinite dimensional nearly integrable Hamiltonian systems. The theorem can be applied to some Hamiltonian partial differential equations in higher dimensional spaces with periodic boundary conditions to construct linearly stable quasi–periodic solutions and its local Birkhoff normal form. The applications to the higher dimensional beam equations and the higher dimensional Schrödinger equations with nonlocal smooth nonlinearity are also given in this paper.     

Perturbation theory of scattering from irregular bodies

A perturbation solution is derived for the following problem: A time harmonic wave of amplitude ψ, propagating in a medium with wave number k, is incident on an irregular volume V, inside of which the propagation constant k′(r) can be an arbitrary function of | r |, where r is a position vector with origin inside V. The boundary conditions are that both ψ and its normal derivative ∂ψ/∂n may be discontinuous across the surface of V. Special cases of these conditions correspond to acoustic scattering, to B-wave scattering from a dielectric cylinder, or to the classical Dirichlet (ψ = 0) or Neumann (∂ψ/∂n = 0) surface conditions. An integral equation is derived that satisfies the appropriate differential equations both outside and inside the body, and satisfies the boundary conditions as well. This equation is reduced to a set of linear algebraic equations by expansion in a certain basis set and these linear equations are then solved in a perturbation approximation for the case that the surface of the body differs from a sphere or cylinder by a small parameter λ. Comparison is made with formulae in the literature, and except for some minor discrepancies, which are here corrected, there is general agreement.     

Cosmic Acceleration and f (R) Theory: Perturbed Solution in a Matter FLRW Model

In the present paper we consider f (R) gravity theories in the metric approach and we derive the equations of motion, focusing also on the boundary conditions. In such a way we apply the general equations to a first order perturbation expansion of the Lagrangian. We present a model able to fit supernovae data without introducing dark energy.     

Invariants of the length spectrum and spectral invariants of planar convex domains

This paper is concerned with a conjecture of Guillemin and Melrose that the length spectrum of a strictly convex bounded domain together with the spectra of the linear Poincaré maps corresponding to the periodic broken geodesics in determine uniquely the billiard ball map up to a symplectic conjugation. We consider continuous deformations of bounded domains _s ,s[0, 1], with smooth boundaries and suppose that ₀ is strictly convex and that the length spectrum does not change along the deformation. We prove that ₀ is strictly convex for anys along the deformation and that for different values of the parameters the corresponding billiard ball maps are symplectically equivalent to each other on the union of the invariant KAM circles. We prove as well that the KAM circles and the restriction of the billiard ball map on them are spectral invariants of the Laplacian with Dirichlet (Neumann) boundary conditions for suitable deformations of strictly convex domains.Supported by Alexander von Humboldt foundation     

A Quantum Correction to the Momentum Distribution Function of Particles

On the basis of the solution to the equation for a single-particle density matrix, the momentum distribution is obtained for a light-particle impurity placed in an ordered system of heavy particles and interacting with them with the amplitude U₀. The effect of the value of U₀/T₀ on the functional form of the momentum distribution is investigated. It is shown that the momentum-distribution function obtained within the perturbation theory up to the terms ~(U₀/T₀)² retains its form also outside the region of applicability of the perturbation theory; however, the relative magnitude of the correction is much smaller than that given by the perturbation theory.     

An exponentially local spectral flow for possibly non-self-adjoint perturbations of non-interacting quantum spins,inspired by KAM theory

Since its introduction by Hastings (Phys Rev B 69:104431, 2004), the technique of quasi-adiabatic continuation has become a central tool in the discussion and classification of ground-state phases. It connects the ground states of self-adjoint Hamiltonians in the same phase by a unitary quasi-local transformation. This paper takes a step towards extending this result to non-self-adjoint perturbations, though, for technical reason, we restrict ourselves here to weak perturbations of non-interacting spins. The extension to non-self-adjoint perturbation is important for potential applications to Glauber dynamics (and its quantum analogues). In contrast to the standard quasi-adiabatic transformation, the transformation constructed here is exponentially local. Our scheme is inspired by KAM theory, with frustration-free operators playing the role of integrable Hamiltonians.     

A mixed basis approach in the SGP-limit

A perturbation method for computing quick estimates of the echo decay in pulsed spin echo gradient NMR diffusion experiments in the short gradient pulse limit is presented. The perturbation basis involves (relatively few) dipole distributions on the boundaries generating a small perturbation matrix in O(s²) time, where s denotes the number of boundary elements. Several approximate eigenvalues and eigenfunctions to the diffusion operator are retrieved. The method is applied to 1D and 2D systems with Neumann boundary conditions.     

Perturbation determinants for singular perturbations

Let A be a densely defined symmetric operator and let {Ã′, Ã} be an ordered pair of proper extensions of A such that their resolvent difference is of trace class. We study the perturbation determinant Δ_Ã′/Ã(·) of the singular pair {Ã′, Ã} by using the boundary triplet approach. We show that, under additional mild assumptions on {Ã′, Ã, the perturbation determinant Δ_Ã′/Ã(·) is the ratio of two ordinary determinants involving the Weyl function and boundary operators. In particular, if the deficiency indices of A are finite, then we obtain Δ_Ã′/Ã(z) = det (B′ - M(z))/det (B - M (z)), z ∈ ρ(Ã), where M(·) stands for the Weyl function and B′ and B for the boundary operators corresponding to Ã′ and à with respect to a chosen boundary triplet Π. The results are applied to ordinary differential operators and to second-order elliptic operators.     

Absence of diffusion in the Anderson tight binding model for large disorder or low energy

We prove that the Green's function of the Anderson tight binding Hamiltonian decays exponentially fast at long distances on ? ^v , with probability 1. We must assume that either the disorder is large or the energy is sufficiently low. Our proof is based on perturbation theory about an infinite sequence of block Hamiltonians and is related to KAM methods.