Spectral self-similarity in fractal one-dimensional photonic structures

Transmission spectra of one-dimensional fractal multilayer structures are found to exhibit self-similar properties. Self-similarity manifests itself in the shape of a transmission envelope (map of transmission dips) rather than in the map of resonance transmission peaks, as is commonly the case with spectra of quasiperiodic systems. To observe the self-similarity, one needs to apply a power transformation to the transmittance in addition to the usual frequency scaling. The values of this power as well as the scaling factor have been calculated analytically and found to depend on the geometrical parameters of the structure.     

Low-temperature behavior of a one-dimensional random Ising model

The random Ising chain is a very simple model with a large number of metastable states. Simple analytical calculation of the relaxation of energy and magnetization is presented. The effect of a nonzero magnetic field is discussed qualitatively. The slow relaxation in this simple model resembles that observed in spin glasses. A weak magnetic field can produce rather strong effects. The magnetization is shown to be a nonanalytic function of the field. The field also greatly alters the metastability characteristics.     

Frequency dependent conductivity of random one-dimensional chains

We examine the frequency dependent conductivity of an exactly soluble model of a random system, the one-dimensional quantum percolation model. Since this system does not fall into the category of weak scattering by the disorder, we do not find the asymptoticω ²ln²(ω/ω ₀) behavior suggested by Mott. For systems which are most disordered, we find that the conductivity spectrum consists of discrete levels, while for systems in which the disorder is much less dense, we may use a continuum approximation. The conductivity is then given by anω ^?3 cosech (2π/ωτ) law. We speculate that such behavior may be found in mixed valent planar compounds such as K₂Pt(CN₄)3(H₂O) with small amounts of Br or Cl added.     

Dynamics of entanglement in one-dimensional Ising chains with two- and three-body interactions

The evolution of entanglement in a one-dimensional Ising chain with both two-body and three-body interactions, under two types of initial states, is numerically simulated. We analyse three problems concerning the dynamics of pairwise entanglement: (i) the possibility of generating large entanglement from an initial separable state by the use of a selective irradiation scheme; (ii) the effect of three-body interaction on the generation of entanglement from an initial separable state; (iii) the effect of three-body interaction on the decay of the entanglement from a state with only (m,n)-pair maximal entangled, and the rest in product form. It is shown that a large pairwise concurrence C_mn can be obtained when the resonant, transverse radio-frequency fields are selectively switched on from the mth to nth spins. Three-body interaction will decrease the oscillation amplitude of the nearest neighbour concurrence, while the oscillation amplitude of remote pairwise concurrence will be greatly increased with the consideration of three-body interactions. For an initial (m,n)-pair maximal entangled state, a slow decay of the pairwise concurrence C_mn is found with the introduction of three-body interactions.     

Dynamics of remote entanglement in one-dimensional Ising chains with Dzyaloshinskii-Moriya interactions

With the introduction of Dzyaloshinskii-Moriya (DM) interaction, dynamics of the remote entanglement in one-dimensional Ising chains is investigated. It is found that the DM interaction can excite the remote entanglement from an initial Néel state. For a given strength of DM interaction, the concurrence between the end spins oscillates and decreases simultaneously with the increase of the chain’s length, and drops to zero at a critical length. For the chains with two and three spins, it is very interesting that the dynamics of the staggered magnetization (or the chiral parameter) can be used to qualitatively estimate the evolution of the remote concurrence between the end spins. At last, we discuss the generation of W state from the Ising chain with DM interactions, and it is obtained that W state can only be prepared in the three-qubit and four-qubit chains with a specific strength of DM interaction.     

Ground states versus low-temperature equilibria in random field Ising chains

Random walk arguments and exact numerical computations are used to study one-dimensional random field chains. The ground state structure is described with absorbing and non-absorbing random walk excursions. At low temperatures, the local magnetization follows the ground state except at regions where a local random field fluctuation makes thermal excitations easier. This is explained by the random walk picture, implying that the magnetization lengthscale is a product of the domain size and the thermal excitation scale. Received 16 October 2000 and Received in final form 7 June 2001     

Dynamics of the one-dimensional random transverse Ising model with next-nearest-neighbor interactions

The dynamics of the one-dimensional random transverse Ising model with both nearest-neighbor (NN) and next-nearest-neighbor (NNN) interactions is studied in the high-temperature limit by the method of recurrence relations. Both the time-dependent transverse correlation function and the corresponding spectral density are calculated for two typical disordered states. We find that for the case of bimodal disorder the dynamics of the system undergoes a crossover from a collective-mode behavior to a central-peak one and for the case of Gaussian disorder the dynamics is complex. For both cases, it is found that the central-peak behavior becomes more obvious and the collective-mode behavior becomes weaker as K_i increase, especially when K_i>J_i/2 (J_i and K_i are the exchange couplings of the NN and NNN interactions, respectively). However, the effects are small when the NNN interactions are weak (K_i<J_i/2).     

Rigorous bounds of the Lyapunov exponents of the one-dimensional random Ising model

We find analytic upper and lower bounds of the Lyapunov exponents of the product of random matrices related to the one-dimensional disordered Ising model, using a deterministic map which transforms the original system into a new one with smaller average couplings and magnetic fields. The iteration of the map gives bounds which estimate the Lyapunov exponents with increasing accuracy. We prove, in fact, that both the upper and the lower bounds converge to the Lyapunov exponents in the limit of infinite iterations of the map. A formal expression of the Lyapunov exponents is thus obtained in terms of the limit of a sequence. Our results allow us to introduce a new numerical procedure for the computation of the Lyapunov exponents which has a precision higher than Monte Carlo simulations.     

Effect of the Dzyaloshinskii-Moriya interaction on heat conductivity in one-dimensional quantum Ising chains

The effect of the Dzyaloshinskii-Moriya (DM) interaction on the heat conduction in the quantum Ising chain has been studied by solving the Lindblad master equation. The chain is subject to a uniform transverse field h, while the exchange couplings {J _m } between the nearest-neighbor spins are either uniform, random or quasi-periodic. The average energy-density profile and the average energy current in the non-equilibrium steady state have been numerically calculated. The ballistic transport is observed in the uniform Ising chain with DM interaction. For the random Ising chain with DM interaction, the energy gradient is observed in the bulk of the spin chain whose energy current appears to scale as the system size ⟨Q⟩ ∼ exp(βN) with β < 0. For the quasi-periodic Ising chain with DM interaction, the J _m takes the two values J _A and J _B arranged in the Fibonacci sequence. The energy gradient also exists in the spin chain and the energy current behaves as ⟨Q⟩ ∼ N ^α with α < 0. By increasing the strength of the DM interaction D, a non-trivial transition from the thermal insulator heat transport to anomalous heat conduction is found in the Fibonacci Ising chain with large ratio of couplings λ = J _A /J _B . A rough phase diagram of λ vs. D is given in this paper as well.     

Metastability of Ising spin chains with nearest—neighbour and next—nearest—neighbour interactions in random fields

One-dimensional Ising systems in random fields (RFs) are studied taking into account the nearest-neighbour and next-nearest-neighbour interactions. We investigate two distributions of RFs: binary and Gaussian distributions. We consider four cases of the exchange couplings: ferro－ferromagnetic (F－F), ferro－antiferromagnetic (F－AF), antiferro－ferromagnetic (AF－F) and antiferro－antiferromagnetic (AF－AF). The energy minima of chains of no more than 30 spins with periodic boundary conditions are analysed exactly. We found that the average number of energy minima grows exponentially with the number of spins in both cases of RFs. The energy distributions across the corresponding energy minima are shown. The effects of RFs on both the average and density of metastable states are explained. For a weak RF, the energy distributions display a multipartitioned structure. We also discuss the frustration effect due to RFs and exchange fields. Finally, the distributions of magnetization are calculated. The absolute value of magnetization averaged over all metastable states decreases logarithmically with the number of spins.     

Thermodynamic limit for the one-dimensional Ising systems with random interaction of finite range

The existence of the thermodynamic limit is proved for the random one-dimensional Ising systems under the assumption that the interaction energies are random variables taking on continuous values and the distribution of the random variables is given by a continuous function. It is assumed that the total number of possible configurations for each lattice site is finite and the range of interaction is finite.     

Entanglement in quantum spin chains, symmetry classes of random matrices, and conformal field theory

We compute the entropy of entanglement between the first N spins and the rest of the system in the ground states of a general class of quantum spin chains. We show that under certain conditions the entropy can be expressed in terms of averages over ensembles of random matrices. These averages can be evaluated, allowing us to prove that at critical points the entropy grows like kappalog(2N+kappa as N-->infinity, where kappa and kappa are determined explicitly. In an important class of systems, kappa is equal to one-third of the central charge of an associated Virasoro algebra. Our expression for kappa therefore provides an explicit formula for the central charge.     

On the thermodynamics of the random one-dimensional Ising chain in a transverse field

For three simple one-dimensional disordered models: (a) the Ising chain with random magnetic moments in a transverse field, (b) the Ising chain with random coupling constants in a transverse field, and (c) the X-Y model with a special type of disorder, the asymptotic equivalence in the thermodynamic limit is proved and some of its consequences are discussed. The spectral density of the finite chain for the model (a) is calculated by Dean's method for several representative cases and the presence of the local modes is indicated. The expressions for the initial susceptibilities for the models (a) and (b) are reviewed and (in two cases) the derivations are simplified.     

Spectral density of mixtures of random density matrices for qubits

We derive the spectral density of the equiprobable mixture of two random density matrices of a two-level quantum system. We also work out the spectral density of mixture under the so-called quantum addition rule. We use the spectral densities to calculate the average entropy of mixtures of random density matrices, and show that the average entropy of the arithmetic-mean-state of n qubit density matrices randomly chosen from the Hilbert–Schmidt ensemble is never decreasing with the number n. We also get the exact value of the average squared fidelity. Some conjectures and open problems related to von Neumann entropy are also proposed.     

Compressible Ising double chains

We obtain exact expressions for the free energy and the magnetic susceptibility in zero field of a compressible double Ising chain with first and second neighbour interactions. The chain is supposed to be made of rigid rods which move like dumbbells in an elastic harmonic potential. The exchange interactions along the direction of the chain are linear functions of the spacing between rods. The effective spin hamiltonian of the double chain involves short-range two and four-spin interactions. Due to the existence of compensation points, we obtain regions of peculiar thermodynamic properties in the pressure-temperature phase diagram.     

Singularity of the density of states for one-dimensional chains with random couplings

We prove that the density of states for the tight-binding model with off-diagonal disorder under general conditions diverges forR0 at least as . This result is established through the study of the recurrence properties of an associated Markov chain.Partial financial support by GNAFA (CNR)Partial financial support by CNPq, grant n.303795-77FA     

Multiscaling and multifractality in an one-dimensional Ising model

Scaling properties of the Gibbs distribution of a finite-size one-dimensional Ising model are investigated as the thermodynamic limit is approached. It is shown that, for each nonzero temperature, coarse-grained probabilities of the appearance of particular energy levels display multiscaling with the scaling length ℓ = 1/M _n, where n denotes the number of spins and M_n is the total number of energy levels. Using the multifractal formalism, the probabilities are argued to reveal also multifractal properties. Received 10 July 2000 and Received in final form 6 November 2000     

The random field Ising model

We discuss the current status of random field systems, particularly those with Ising symmetry. Both theory and experiment agree that, in the equilibrium state, there is a transition to an ordered state in three dimensions and no such transition in two dimensions. The critical behavior in three dimensions is, however, not very well understood. More work remains to be done to understand the dynamics, both in the critical region and the low temperature phase.     

Ising model on self-avoiding walk chains

The phase transitions of nearest-neighbour interacting Ising models on self-avoiding walk (SAW) chains on square and triangular lattices have been studied using Monte Carlo technique. To estimate the transition temperature (T _c) bounds, the average number of nearest-neighbours (Z _eff) of spins on SAWs have been determined using the computer simulation results, and the percolation thresholds (p _c) for site dilution on SAWs have been determined using Monte Carlo methods. We find, for SAWs on square and triangular lattices respectively,Z _eff=2.330 and 3.005 (which compare very well with our previous theoretically estimated values) andp _c=0.022±0.003 and 0.045±0.005. When put in Bethe-Peierls approximations, the above values ofZ _eff givekT _c/J<1.06 and 1.65 for Ising models on SAWs on square and triangular lattices respectively, while, using the semi-empirical relation connecting the Ising modelT _c's andp _c's for the same lattices, we findkT _c/J0.57 and 0.78 for the respective models. Using the computer simulation results for the shortest connecting path lengths in SAWs on both kinds of lattices, and integrating the spin correlations on them, we find the susceptibility exponent =1.024±0.007, for the model on SAWs on two dimensional lattices.