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1.
We discuss three Hamiltonians, each with a central-field part H0 and a PT-symmetric perturbation igz. When H0 is the isotropic Harmonic oscillator the spectrum is real for all g because H is isospectral to H0+g2/2. When H0 is the Hydrogen atom then infinitely many eigenvalues are complex for all g. If the potential in H0 is linear in the radial variable r then the spectrum of H exhibits real eigenvalues for 0<g<gc and a PT phase transition at gc. 相似文献
2.
In this paper, we investigate numerically the Susceptible–Infected–Recovered–Susceptible (SIRS) epidemic model on an exponential network generated by a preferential attachment procedure. The discrete SIRS model considers two main parameters: the duration τ0 of the complete infection–recovery cycle and the duration τI of infection. A permanent source of infection I0 has also been introduced in order to avoid the vanishing of the disease in the SIRS model. The fraction of infected agents is found to oscillate with a period T≥τ0. Simulations reveal that the average fraction of infected agents depends on I0 and τI/τ0. A maximum of synchronization of infected agents, i.e. a maximum amplitude of periodic spreading oscillations, is found to occur when the ratio τI/τ0 is slightly smaller than 1/2. The model is in agreement with the general observation that an outbreak corresponds to high τI/τ0 values. 相似文献
3.
We study reduction of generalized complex structures. More precisely, we investigate the following question. Let J be a generalized complex structure on a manifold M, which admits an action of a Lie group G preserving J. Assume that M0 is a G-invariant smooth submanifold and the G-action on M0 is proper and free so that MG?M0/G is a smooth manifold. Under what condition does J descend to a generalized complex structure on MG? We describe a sufficient condition for the reduction to hold, which includes the Marsden–Weinstein reduction of symplectic manifolds and the reduction of the complex structures in Kähler manifolds as special cases. As an application, we study reduction of generalized Kähler manifolds. 相似文献
4.
We discuss space-time symmetric Hamiltonian operators of the form H=H0+igH′, where H0 is Hermitian and g real. H0 is invariant under the unitary operations of a point group G while H′ is invariant under transformation by elements of a subgroup G′ of G. If G exhibits irreducible representations of dimension greater than unity, then it is possible that H has complex eigenvalues for sufficiently small nonzero values of g. In the particular case that H is parity-time symmetric then it appears to exhibit real eigenvalues for all 0<g<gc, where gc is the exceptional point closest to the origin. Point-group symmetry and perturbation theory enable one to predict whether H may exhibit real or complex eigenvalues for g>0. We illustrate the main theoretical results and conclusions of this paper by means of two- and three-dimensional Hamiltonians exhibiting a variety of different point-group symmetries. 相似文献
5.
The aim of this paper is to develop local theory of future timelike, nonspacelike and null reachable sets from a given point q0 in the sub-Lorentzian geometry. In particular, we prove that if U is a normal neighbourhood of q0 then the three reachable sets, computed relative to U, have identical interiors and boundaries with respect to U. Further, among other things, we show that for Lorentzian metrics on contact distributions on R2n+1, n≥1, the boundary of reachable sets from q0 is, in a neighbourhood of q0, made up of null future directed curves starting from q0. Every such curve has only a finite number of non-smooth points; smooth pieces of every such curve are Hamiltonian geodesics. For general sub-Lorentzian structures, contrary to the Lorentzian case, timelike curves may appear on the boundary. It turns out that such curves are always Goh curves. We also generalize a classical result on null Lorentzian geodesics: every null future directed Hamiltonian sub-Lorentzian geodesic initiating at q0 is contained, at least to a certain moment of time, in the boundary of the reachable set from q0. 相似文献
6.
We demonstrate the emergence of non-Abelian fusion rules for excitations of a two dimensional lattice model built out of Abelian degrees of freedom. It can be considered as an extension of the usual toric code model on a two dimensional lattice augmented with matter fields. It consists of the usual C(Zp) gauge degrees of freedom living on the links together with matter degrees of freedom living on the vertices. The matter part is described by a n dimensional vector space which we call Hn. The Zp gauge particles act on the vertex particles and thus Hn can be thought of as a C(Zp) module. An exactly solvable model is built with operators acting in this Hilbert space. The vertex excitations for this model are studied and shown to obey non-Abelian fusion rules. We will show this for specific values of n and p, though we believe this feature holds for all n>p. We will see that non-Abelian anyons of the quantum double of C(S3) are obtained as part of the vertex excitations of the model with n=6 and p=3. Ising anyons are obtained in the model with n=4 and p=2. The n=3 and p=2 case is also worked out as this is the simplest model exhibiting non-Abelian fusion rules. Another common feature shared by these models is that the ground states have a higher symmetry than Zp. This makes them possible candidates for realizing quantum computation. 相似文献
7.
We have studied the anisotropic two-dimensional nearest-neighbor Ising model with competitive interactions in both uniform longitudinal field H and transverse magnetic field Ω. Using the effective-field theory (EFT) with correlation in cluster with N=1 spin we calculate the thermodynamic properties as a function of temperature with values H and Ω fixed. The model consists of ferromagnetic interaction Jx in the x direction and antiferromagnetic interaction Jy in the y direction, and it is found that for H/Jy∈[0,2] the system exhibits a second-order phase transition. The thermodynamic properties are obtained for the particular case of λ=Jx/Jy=1 (isotropic square lattice). 相似文献
8.
We analyse the phase diagram of a quantum mean spherical model in terms of the temperature T, a quantum parameter g, and the ratio p=−J2/J1, where J1>0 refers to ferromagnetic interactions between first-neighbour sites along the d directions of a hypercubic lattice, and J2<0 is associated with competing antiferromagnetic interactions between second neighbours along m≤d directions. We regain a number of known results for the classical version of this model, including the topology of the critical line in the g=0 space, with a Lifshitz point at p=1/4, for d>2, and closed-form expressions for the decay of the pair correlations in one dimension. In the T=0 phase diagram, there is a critical border, gc=gc(p) for d≥2, with a singularity at the Lifshitz point if d<(m+4)/2. We also establish upper and lower critical dimensions, and analyse the quantum critical behavior in the neighborhood of p=1/4. 相似文献
9.
We have studied the barrier crossing dynamics in presence of non-Gaussian noises. It has been observed that multiplicative colored non-Gaussian noise can induce resonant activation (RA). The conspicuous dependence of mean first passage time (MFPT) on correlation time (τ2) of additive colored noise having fixed variance have been analyzed. Beyond a critical value of τ2 the MFPT increases for a given rate of increase of noise strength with τ2 if the additive colored noise is non-Gaussian. The MFPT first decreases with increase of the non-Gaussian parameter (measures deviation from Gaussian character) of multiplicative colored noise followed by an increase exhibiting a minimum. The appearance of the minimum critically depends on the additive noise. 相似文献
10.
Let M be a connected compact quantizable Kähler manifold equipped with a Hamiltonian action of a connected compact Lie group G. Let M//G=?−1(0)/G=M0 be the symplectic quotient at value 0 of the moment map ?. The space M0 may in general not be smooth. It is known that, as vector spaces, there is a natural isomorphism between the quantum Hilbert space over M0 and the G-invariant subspace of the quantum Hilbert space over M. In this paper, without any regularity assumption on the quotient M0, we discuss the relation between the inner products of these two quantum Hilbert spaces under the above natural isomorphism; we establish asymptotic unitarity to leading order in Planck’s constant of a modified map of the above isomorphism under a “metaplectic correction” of the two quantum Hilbert spaces. 相似文献
11.
The sound attenuation phenomena is investigated for a spin- 3/2 Ising model on the Bethe lattice in terms of the recursion relations by using the Onsager theory of irreversible thermodynamics. The dependencies of sound attenuation on the temperature (T), frequency (w), Onsager coefficient (γ) and external magnetic field (H) near the second-order (Tc) and first-order (Tt) phase transition temperatures are examined for given coordination numbers q on the Bethe lattice. It is assumed that the sound wave couples to the order-parameter fluctuations which decay mainly via the order-parameter relaxation process, thus two relaxation times are obtained and which are used to obtain an expression for the sound attenuation coefficient (α). Our investigations revealed that only one peak is obtained near Tt and three peaks are found near Tc when the Onsager coefficient is varied at a given constant frequency for q=3. Fixing the Onsager coefficient and varying the frequency always leads to two peaks for q=3,4 and 6 near Tc. The sound attenuation peaks are observed near Tt at lower values of external magnetic field, but as it increases the sound attenuation peaks decrease and eventually disappear. 相似文献
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13.
We introduce a model for the Contact Process with relaxing immunization CPRI . In this model, local memory is introduced by a time and space dependence of the contamination probability. The model has two parameters: a typical immunization time τ and a maximum contamination probability a. The system presents an absorbing state phase transition whenever the contamination probability a is above a minimum threshold. For short immunization times, the system evolves to a statistically stationary active state. Above τc(a), immunization predominates and the system evolves to the absorbing vacuum state. We employ a finite-size scaling analysis to show that the transition belongs to the standard directed percolation universality class. The critical immunization time diverges in the limit of a→1. In this regime, the density of active sites decays exponentially as τ increases, but never reaches the vacuum state in the thermodynamic limit. 相似文献
14.
We introduce a network evolution process motivated by the network of citations in the scientific literature. In each iteration of the process a node is born and directed links are created from the new node to a set of target nodes already in the network. This set includes m “ambassador” nodes and l of each ambassador’s descendants where m and l are random variables selected from any choice of distributions pl and qm. The process mimics the tendency of authors to cite varying numbers of papers included in the bibliographies of the other papers they cite. We show that the degree distributions of the networks generated after a large number of iterations are scale-free and derive an expression for the power-law exponent. In a particular case of the model where the number of ambassadors is always the constant m and the number of selected descendants from each ambassador is the constant l, the power-law exponent is (2l+1)/l. For this example we derive expressions for the degree distribution and clustering coefficient in terms of l and m. We conclude that the proposed model can be tuned to have the same power law exponent and clustering coefficient of a broad range of the scale-free distributions that have been studied empirically. 相似文献
15.
Alexander Moroz 《Annals of Physics》2014,340(1):252-266
The Rabi model describes the simplest interaction between a cavity mode with a frequency ωc and a two-level system with a resonance frequency ω0. It is shown here that the spectrum of the Rabi model coincides with the support of the discrete Stieltjes integral measure in the orthogonality relations of recently introduced orthogonal polynomials. The exactly solvable limit of the Rabi model corresponding to Δ=ω0/(2ωc)=0, which describes a displaced harmonic oscillator, is characterized by the discrete Charlier polynomials in normalized energy ?, which are orthogonal on an equidistant lattice. A non-zero value of Δ leads to non-classical discrete orthogonal polynomials ?k(?) and induces a deformation of the underlying equidistant lattice. The results provide a basis for a novel analytic method of solving the Rabi model. The number of ca. 1350 calculable energy levels per parity subspace obtained in double precision (cca 16 digits) by an elementary stepping algorithm is up to two orders of magnitude higher than is possible to obtain by Braak’s solution. Any first n eigenvalues of the Rabi model arranged in increasing order can be determined as zeros of ?N(?) of at least the degree N=n+nt. The value of nt>0, which is slowly increasing with n, depends on the required precision. For instance, nt?26 for n=1000 and dimensionless interaction constant κ=0.2, if double precision is required. Given that the sequence of the lth zeros xnl’s of ?n(?)’s defines a monotonically decreasing discrete flow with increasing n, the Rabi model is indistinguishable from an algebraically solvable model in any finite precision. Although we can rigorously prove our results only for dimensionless interaction constant κ<1, numerics and exactly solvable example suggest that the main conclusions remain to be valid also for κ≥1. 相似文献
16.
A multi-parametric version of the nonadditive entropy Sq is introduced. This new entropic form, denoted by Sa,b,r, possesses many interesting statistical properties, and it reduces to the entropy Sq for b=0, a=r:=1−q (hence Boltzmann–Gibbs entropy SBG for b=0, a=r→0). The construction of the entropy Sa,b,r is based on a general group-theoretical approach recently proposed by one of us, Tempesta (2016). Indeed, essentially all the properties of this new entropy are obtained as a consequence of the existence of a rational group law, which expresses the structure of Sa,b,r with respect to the composition of statistically independent subsystems. Depending on the choice of the parameters, the entropy Sa,b,r can be used to cover a wide range of physical situations, in which the measure of the accessible phase space increases say exponentially with the number of particles N of the system, or even stabilizes, by increasing N, to a limiting value. 相似文献
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18.
A protocol for transferring an unknown single qubit state evidences quantum features when the average fidelity of the outcomes is, in principle, greater than 2/3. We propose to use the probabilistic and unambiguous state extraction scheme as a mechanism to redistribute the fidelity in the outcome of the standard teleportation when the process is performed with an X-state as a noisy quantum channel. We show that the entanglement of the channel is necessary but not sufficient in order for the average fidelity fX to display quantum features, i.e., we find a threshold CX for the concurrence of the channel. On the other hand, if the mechanism for redistributing fidelity is successful then we find a filterable outcome with average fidelity fX,0 that can be greater than fX. In addition, we find the threshold concurrence of the channel CX,0 in order for the average fidelity fX,0 to display quantum features and surprisingly, the threshold concurrence CX,0 can be less than CX. Even more, we find some special cases for which the threshold values become zero. 相似文献
19.
In order to explain the occurrence of a minimum in firing rate which occurs for certain mean input levels μ as noise level σ increases (inverse stochastic resonance, ISR) in Hodgkin–Huxley (HH) systems, we analyze the underlying transitions from a stable equilibrium point to limit cycle and vice-versa. For a value of μ at which ISR is pronounced, properties of the corresponding stable equilibrium point are found. A linearized approximation around this point has oscillatory solutions from whose maxima spikes tend to occur. A one dimensional diffusion is also constructed for small noise. Properties of the basin of attraction of the limit cycle (spike) are investigated heuristically. Long term trials of duration 500000 ms are carried out for values of σ from 0 to 2.0. The graph of mean spike count versus σ is divided into 4 regions R1,…,R4, where R3 contains the minimum associated with ISR. In R1 transitions to the basin of attraction of the rest point are not observed until a small critical value of σ=σc1 is reached, at the beginning of R2. The sudden decline in firing rate when σ is just greater than σc1 implies that there is only a small range of noise levels 0<σ<σc1 where repetitive spiking is safe from annihilation by noise. The firing rate remains small throughout R3. At a larger critical value σ=σc2 which signals the beginning of R4, the probability of transitions from the basin of attraction of the equilibrium point to that of the limit cycle apparently becomes greater than zero and the spike rate thereafter increases with increasing σ. The quantitative scheme underlying the ISR curve is outlined in terms of the properties of exit time random variables. In the final subsection, several statistical properties of the main random variables associated with long term spiking activity are given, including distributions of exit times from the two relevant basins of attraction and the interspike interval. 相似文献