On finite-size Lyapunov exponents in multiscale systems

We study the effect of regime switches on finite size Lyapunov exponents (FSLEs) in determining the error growth rates and predictability of multiscale systems. We consider a dynamical system involving slow and fast regimes and switches between them. The surprising result is that due to the presence of regimes, the error growth rate can be a non-monotonic function of initial error amplitude. In particular, troughs in the large scales of FSLE spectra are shown to be a signature of slow regimes, whereas fast regimes are shown to cause large peaks in the spectra where error growth rates far exceed those estimated from the maximal Lyapunov exponent. We present analytical results explaining these signatures and corroborate them with numerical simulations. We show further that these peaks disappear in stochastic parametrizations of the fast chaotic processes, and the associated FSLE spectra reveal that large scale predictability properties of the full deterministic model are well approximated, whereas small scale features are not properly resolved.     

Field theory of finite-size effects in Ising-like systems

We propose a new perturbation approach to finitesize effects within the ⁴ field theory for a one-component order parameter with periodic boundary conditions. Our approach is applicable both above and belowT _c . Renormalization-group calculations of finite-size scaling functions in three dimensions are compared with new Monte Carlo data for theL×L×L Ising model withL=8, 16, 32. The field-theoretic predictions are in good overall agreement with the Monte Carlo data.Dedicated to Professor Herbert Wagner on the occasion of his 60th birthday     

Tunable electronic transmission gaps in a graphene superlattice

The transmission in graphene superlattices with adjustable barrier height is investigated using transfer-matrix method. It is found that one could control the angular range of transmission by changing the ratio of incidence energy and barrier height. The transmission as a function of incidence energy has more than one gaps, due to the appearance of evanescent waves in different barriers. Accordingly, more than one conductivity minimums are induced. The transmission gaps could be controlled by adjusting the incidence angle, the barrier height, and the barrier number, which gives the possibility to construct an energy-dependent wavevector filter.     

Symmetry breaking and finite-size effects in quantum many-body systems

We consider a quantum many-body system on a lattice which exhibits a spontaneous symmetry breaking in its infinite-volume ground states, but in which the corresponding order operator does not commute with the Hamiltonian. Typical examples are the Heisenberg antiferromagnet with a Néel order and the Hubbard model with a (superconducting) off-diagonal long-range order. In the corresponding finite system, the symmetry breaking is usually obscured by quantum fluctuation and one gets a symmetric ground state with a long-range order. In such a situation, Horsch and von der Linden proved that the finite system has a low-lying eigenstate whose excitation energy is not more than of orderN ^–1, whereN denotes the number of sites in the lattice. Here we study the situation where the broken symmetry is a continuous one. For a particular set of states (which are orthogonal to the ground state and with each other), we prove bounds for their energy expectation values. The bounds establish that there exist ever-increasing numbers of low-lying eigenstates whose excitation energies are bounded by a constant timesN ^–1. A crucial feature of the particular low-lying states we consider is that they can be regarded as finite-volume counterparts of the infinite-volume ground states. By forming linear combinations of these low-lying states and the (finite-volume) ground state and by taking infinite-volume limits, we construct infinite-volume ground states with explicit symmetry breaking. We conjecture that these infinite-volume ground states are ergodic, i.e., physically natural. Our general theorems not only shed light on the nature of symmetry breaking in quantum many-body systems, but also provide indispensable information for numerical approaches to these systems. We also discuss applications of our general results to a variety of interesting examples. The present paper is intended to be accessible to readers without background in mathematical approaches to quantum many-body systems.     

Scaling predictions for thermodynamical properties of finite-size liquid systems

The notion of spatial limitation of a system undergoing critical phenomena and phase transitions is presented. The scaling formulae for the singular part of the fluctuation free energy and correlation length for spatially limited liquid systems are analyzed. The conducted research supports the conclusion that reducing of the volume's size leads, after the decreasing of the correlation length, to decreasing of the susceptibility χ, which is equivalent to isothermal compressibility β_T for one-component liquid.     

Cluster analysis and finite-size scaling for Ising spin systems

Based on the connection between the Ising model and a correlated percolation model, we calculate the distribution function for the fraction (c) of lattice sites in percolating clusters in subgraphs with n percolating clusters, f(n)(c), and the distribution function for magnetization (m) in subgraphs with n percolating clusters, p(n)(m). We find that f(n)(c) and p(n)(m) have very good finite-size scaling behavior and that they have universal finite-size scaling functions for the model on square, plane triangular, and honeycomb lattices when aspect ratios of these lattices have the proportions 1:square root[3]/2:square root[3]. The complex structure of the magnetization distribution function p(m) for the system with large aspect ratio could be understood from the independent orientations of two or more percolation clusters in such a system.     

Many-body localization in disorder-free systems: The importance of finite-size constraints

Recently it has been suggested that many-body localization (MBL) can occur in translation-invariant systems, and candidate 1D models have been proposed. We find that such models, in contrast to MBL systems with quenched disorder, typically exhibit much more severe finite-size effects due to the presence of two or more vastly different energy scales. In a finite system, this can artificially split the density of states (DOS) into bands separated by large gaps. We argue for such models to faithfully represent the thermodynamic limit behavior, the ratio of relevant coupling must exceed a certain system-size depedent cutoff, chosen such that various bands in the DOS overlap one another. Setting the parameters this way to minimize finite-size effects, we study several translation-invariant MBL candidate models using exact diagonalization. Based on diagnostics including entanglement and local observables, we observe thermal (ergodic), rather than MBL-like behavior. Our results suggest that MBL in translation-invariant systems with two or more very different energy scales is less robust than perturbative arguments suggest, possibly pointing to the importance of non-perturbative effects which induce delocalization in the thermodynamic limit.     

Suppression of ground-state magnetization in finite-size systems due to off-diagonal interaction fluctuations

We study a generic model of interacting fermions in a finite-size disordered system. We show that the off-diagonal interaction matrix elements induce density of state fluctuations which generically favor a minimum spin ground state at large interaction amplitude, U. This effect competes with the exchange effect which favors large magnetization at large U, and it suppresses this exchange magnetization in a large parameter range. When off-diagonal fluctuations dominate, the model predicts a spin gap which is larger for odd-spin ground states as for even spin, suggesting a simple experimental signature of this off-diagonal effect in Coulomb blockade transport measurements.     

Omnidirectional transmission and reflection of pseudospin-1 Dirac fermions in a Lieb superlattice

We study the behavior of pseudospin-1 Dirac fermions in a Lieb lattice subjected to an external periodic potential. It is found that there exists a zero-averaged wave-number passband at the incident energy corresponding to half of the potential step, in contrast to zero-averaged wave-number gap in graphene superlattices. By tuning the sublattice site-energy, the passband can be turned into an omnidirectional gap. Consequently, a transformation from omnidirectional transmission to reflection, accompanied with a switch of conductance from maximum to zero can be realized easily. It is expected that the controllable properties are useful for some applications in optical or electronic devices.     

Theory of average pulse propagation in high-bit-rate optical transmission systems with strong dispersion management

A theory of signal transmission in high-bit-rate optical communication systems with large variations of dispersion (strong dispersion management) is presented. It is found that the averaged propagation of a chirped breathing optical pulse along the line is described by the nonlinear Schrödinger equation with an additional parabolic potential. The shape of the averaged pulse is intermediate between the sech-type soliton and a Gaussian pulse. The rapidly decaying Gaussian wings of such pulses allow denser information packing in comparison with the use of sech-type fundamental solitons.     

Compression of terahertz radiation in resonant systems with a quantum superlattice

It is shown theoretically that ultrashort terahertz electromagnetic pulses can be generated in resonant systems containing a quantum semiconductor superlattice. This effect is due to the compression of radiation incident on a cavity as a result of ultrafast self-modulation of the cavity Q factor upon cyclic switching of the superlattice from the screening to the transparent state in a pump field.     

Nonlinear operation of a finite-size tunnel junction between 2D electron systems

A nonlinear regime of electron tunneling into double quantum wells with independent contacts to each well is investigated theoretically. The nonlinear current–voltage response in these systems takes place when the voltageV applied to the contacts exceeds the tunneling resonance width. It is found that such a nonlinearity acquires essentially new features when the length of the tunneling area is comparable with the square root of the ratio of the averaged in-plane conductivity of the electrons to the resonance tunneling conductance. Under these conditions, the local interlayer voltagev does not coincide with V and depends on the in-plane coordinate x. The local tunneling conductance also depends on x in the nonlinear regime. This dependence creates a positive feedback, which leads to a bistable behavior of the system (two stable patterns of the local voltage at a fixed V). The bistability is reflected by Z -like regions of the current–voltage characteristics.     

An investigation of finite-size scaling for systems with long-range interaction: The spherical model

A method is suggested for the derivation of finite-size corrections in the thermodynamic functions of systems with pair interaction potential decaying at large distancesr asr ^{–d –} , whered is the space dimensionality and>0. It allows for a unified treatment of short-range (=2) and long-range (<2) interaction. The asymptotic analysis is illustrated by the mean spherical model of general geometryL ^d–d× ^d subject to periodic boundary conditions. The Fisher-Privman equation of state is generalized to arbitrary real values ofd, 0d. It is shown that the-expansion may be used to study the breakdown of standard finite-size scaling at the borderline dimensionalities.