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1.
We discuss three Hamiltonians, each with a central-field part H0H0 and a PT-symmetric perturbation igzigz. When H0H0 is the isotropic Harmonic oscillator the spectrum is real for all gg because HH is isospectral to H0+g2/2H0+g2/2. When H0H0 is the Hydrogen atom then infinitely many eigenvalues are complex for all gg. If the potential in H0H0 is linear in the radial variable rr then the spectrum of HH exhibits real eigenvalues for 0<g<gc0<g<gc and a PT phase transition at gcgc.  相似文献   

2.
In the present paper, patterns of diffusion-limited aggregation (DLA) grown on nonuniform substrates are investigated by means of Monte Carlo simulations. We consider a nonuniform substrate as the largest percolation cluster of dropped particles with different structures and forms that occupy more than a single site on the lattice. The aggregates are grown on such clusters, in the range the concentration, pp, from the percolation threshold, pcpc up to the jamming coverage, pjpj. At the percolation threshold, the aggregates are asymmetrical and the branches are relatively few. However, for larger values of pp, the patterns change gradually to a pure DLA. Tiny qualitative differences in this behavior are observed for different kk sizes. Correspondingly, the fractal dimension of the aggregates increases as pp raises in the same range pc≤p≤pjpcppj. This behavior is analyzed and discussed in the framework of the existing theoretical approaches.  相似文献   

3.
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 (TT), frequency (ww), Onsager coefficient (γγ) and external magnetic field (HH) near the second-order (Tc)(Tc) and first-order (Tt)(Tt) phase transition temperatures are examined for given coordination numbers qq 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)(α). Our investigations revealed that only one peak is obtained near TtTt and three peaks are found near TcTc when the Onsager coefficient is varied at a given constant frequency for q=3q=3. Fixing the Onsager coefficient and varying the frequency always leads to two peaks for q=3,4q=3,4 and 6 near TcTc. The sound attenuation peaks are observed near TtTt at lower values of external magnetic field, but as it increases the sound attenuation peaks decrease and eventually disappear.  相似文献   

4.
We discuss space-time symmetric Hamiltonian operators of the form H=H0+igHH=H0+igH, where H0H0 is Hermitian and gg real. H0H0 is invariant under the unitary operations of a point group GG while HH is invariant under transformation by elements of a subgroup GG of GG. If GG exhibits irreducible representations of dimension greater than unity, then it is possible that HH has complex eigenvalues for sufficiently small nonzero values of gg. In the particular case that HH is parity-time symmetric then it appears to exhibit real eigenvalues for all 0<g<gc0<g<gc, where gcgc is the exceptional point closest to the origin. Point-group symmetry and perturbation theory enable one to predict whether HH may exhibit real or complex eigenvalues for g>0g>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.
Generally, in literature, easy-axis single ion anisotropy and easy-axis exchange anisotropy was treated in indistinct way. In this work we propose to perform a comparative study of the effects of these two easy-axis anisotropies on the behavior of the magnetization and the critical temperature (Tc)(Tc) in the 2D classical Heisenberg antiferromagnetic model. In order to study the low-temperature thermodynamics of this model, we should consider the contribution of anisotropic spin waves, using a self-consistent harmonic approximation (SCHA) theory. We compare the predictions of SCHA with numerical simulations on L×LL×L square lattices using Monte Carlo (MC) simulations, which include effects due to all thermodynamically allowed excitations. Our SCHA results are in good agreement with our MC simulations results and have shown that the strong KK limit gives two different Ising-like behavior. In the exchange anisotropic case, the dependence of TcTc on anisotropic parameter KK becomes linear and in the single-ion anisotropic case, TcTc becomes independent of KK. Also, using MC simulations and finite size scaling, we show that the critical exponents in the two anisotropic case are compatible with the 2D Ising values α=0.125α=0.125 and γ=1.75γ=1.75.  相似文献   

6.
In [L. Lebtahi, Lie algebra on the transverse bundle of a decreasing family of foliations, J. Geom. Phys. 60 (2010), 122–133], we defined the transverse bundle VkVk to a decreasing family of kk foliations FiFi on a manifold MM. We have shown that there exists a (1,1)(1,1) tensor JJ of VkVk such that Jk≠0Jk0, Jk+1=0Jk+1=0 and we defined by LJ(Vk)LJ(Vk) the Lie Algebra of vector fields XX on VkVk such that, for each vector field YY on VkVk, [X,JY]=J[X,Y][X,JY]=J[X,Y].  相似文献   

7.
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9.
The magnetization reversal behavior of Permalloy nanowires has been investigated using a magneto-optic Kerr effect setup. Nanowires with various widths, w=250w=250 nm to 3 μm and a thickness of t=10t=10 nm were fabricated by electron-beam lithography and subsequent lift-off. Furthermore, similar nanowires but with a thickness gradient along the nanowire axis have been prepared to investigate the influence of the gradient on the magnetic domain wall propagation. Magnetization hysteresis loops recorded on individual nanowires without a gradient are compared to corresponding wires with a thickness gradient. The dependence of the coercive field, HcHc vs. t/wt/w shows a linear behavior for wires without a gradient. However, wires with a gradient display a more complex crossover behavior. We find a plateau in the HcHc vs. t/wt/w curve at values of ww, where a transformation from transverse to vortex domain wall type is expected.  相似文献   

10.
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 mm “ambassador” nodes and ll of each ambassador’s descendants where mm and ll are random variables selected from any choice of distributions plpl and qmqm. 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 mm and the number of selected descendants from each ambassador is the constant ll, the power-law exponent is (2l+1)/l(2l+1)/l. For this example we derive expressions for the degree distribution and clustering coefficient in terms of ll and mm. 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   

11.
12.
The Rabi model describes the simplest interaction between a cavity mode with a frequency ωcωc and a two-level system with a resonance frequency ω0ω0. 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)=0Δ=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 ΔΔ leads to non-classical discrete orthogonal polynomials ?k(?)?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 nn eigenvalues of the Rabi model arranged in increasing order can be determined as zeros of ?N(?)?N(?) of at least the degree N=n+ntN=n+nt. The value of nt>0nt>0, which is slowly increasing with nn, depends on the required precision. For instance, nt?26nt?26 for n=1000n=1000 and dimensionless interaction constant κ=0.2κ=0.2, if double precision is required. Given that the sequence of the llth zeros xnlxnl’s of ?n(?)?n(?)’s defines a monotonically decreasing discrete flow with increasing nn, 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<1κ<1, numerics and exactly solvable example suggest that the main conclusions remain to be valid also for κ≥1κ1.  相似文献   

13.
14.
In this paper, we try to propose a toy model, which follows the majority rule with the Fermi function, to uncover the role of the heterogeneous interaction between individuals in opinion formation. In order to do this, we define the impact factor IFiIFi, says individual ii, as the exponential function of its connectivity kiki with the tunable parameter ββ. ββ also shows the public information that can be collected by individuals in the system. We realize our model in scale-free networks with mean connectivity 〈k〉k. We find that much more public information (β>β2β>β2) and less public information (β<β1β<β1) cannot let either of the two opinions be the majority during the opinion formation. Furthermore, β1β1 is a constant and equal to −0.76(±0.04)0.76(±0.04), and β2β2 decreases as a power-law function of the mean connectivity 〈k〉k of the network. Our work can provide some perspectives and tools to understand the diversity of opinion in social networks.  相似文献   

15.
We study reduction of generalized complex structures. More precisely, we investigate the following question. Let JJ be a generalized complex structure on a manifold MM, which admits an action of a Lie group GG preserving JJ. Assume that M0M0 is a GG-invariant smooth submanifold and the GG-action on M0M0 is proper and free so that MG?M0/GMG?M0/G is a smooth manifold. Under what condition does JJ descend to a generalized complex structure on MGMG? 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   

16.
We studied damage spreading in a Driven Lattice Gas (DLG) model as a function of the temperature TT, the magnitude of the external driving field EE, and the lattice size. The DLG model undergoes an order–disorder second-order phase transition at the critical temperature Tc(E)Tc(E), such that the ordered phase is characterized by high-density strips running along the direction of the applied field; while in the disordered phase one has a lattice-gas-like behavior. It is found that the damage always spreads for all the investigated temperatures and reaches a saturation value DsatDsat that depends only on TT. DsatDsat increases for T<Tc(E=∞)T<Tc(E=), decreases for T>Tc(E=∞)T>Tc(E=) and is free of finite-size effects. This behavior can be explained as due to the existence of interfaces between the high-density strips and the lattice-gas-like phase whose roughness depends on TT. Also, we investigated damage spreading for a range of finite fields as a function of TT, finding a behavior similar to that of the case with E=∞E=.  相似文献   

17.
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)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 nn dimensional vector space which we call HnHn. The ZpZp gauge particles act on the vertex particles and thus HnHn can be thought of as a C(Zp)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 nn and pp, though we believe this feature holds for all n>pn>p. We will see that non-Abelian anyons of the quantum double of C(S3)C(S3) are obtained as part of the vertex excitations of the model with n=6n=6 and p=3p=3. Ising anyons are obtained in the model with n=4n=4 and p=2p=2. The n=3n=3 and p=2p=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 ZpZp. This makes them possible candidates for realizing quantum computation.  相似文献   

18.
In Single Gate HEMT (SGHEMT) shortening of gate length (Lg)(Lg) below 100 nm leads to reduction in Transconductance (gm)(gm), which reduces the unloaded voltage gain (gm/gd)(gm/gd) of the device, thereby reducing the maximum frequency of oscillation (fmax)(fmax). The main reason for this reduction in gmgm with LgLg in the Single Gate HEMT (SGHEMT) is its inability to maintain the desired channel aspect ratio (αα). At such a miniaturization level, αα not only depends on the channel depth (d)(d) but also on the channel thickness (dc)(dc) of the device [5]. Moreover, the variation of dcdc may switch the device characteristics from quantum regime to classical regime  and . The Double Gate HEMT (DGHEMT)  and  has emerged as a solution for further reduction in LgLg and provides enhancements over SGHEMT by virtue of its double gate and also for same dcdc due to double heterojunctions, which virtually increases the value of αα. In the present work, extensive simulation work has been carried out using ATLAS device simulator [35] in order to study the effect of dcdc and LgLg on DGHEMT and SGHEMT. An analytical model has also been proposed for SGHEMT and DGHEMT to incorporate the effect of variation of dcdc and LgLg.  相似文献   

19.
We have studied the anisotropic two-dimensional nearest-neighbor Ising model with competitive interactions in both uniform longitudinal field HH and transverse magnetic field ΩΩ. Using the effective-field theory (EFT) with correlation in cluster with N=1N=1 spin we calculate the thermodynamic properties as a function of temperature with values HH and ΩΩ fixed. The model consists of ferromagnetic interaction JxJx in the xx direction and antiferromagnetic interaction JyJy in the yy direction, and it is found that for H/Jy∈[0,2]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=1λ=Jx/Jy=1 (isotropic square lattice).  相似文献   

20.
We analyse the phase diagram of a quantum mean spherical model in terms of the temperature TT, a quantum parameter gg, and the ratio p=−J2/J1p=J2/J1, where J1>0J1>0 refers to ferromagnetic interactions between first-neighbour sites along the dd directions of a hypercubic lattice, and J2<0J2<0 is associated with competing antiferromagnetic interactions between second neighbours along m≤dmd 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=0g=0 space, with a Lifshitz point at p=1/4p=1/4, for d>2d>2, and closed-form expressions for the decay of the pair correlations in one dimension. In the T=0T=0 phase diagram, there is a critical border, gc=gc(p)gc=gc(p) for d≥2d2, with a singularity at the Lifshitz point if d<(m+4)/2d<(m+4)/2. We also establish upper and lower critical dimensions, and analyse the quantum critical behavior in the neighborhood of p=1/4p=1/4.  相似文献   

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