| Ref | Ballistic versus diffusive transport in graphene |
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| 摘要 | We investigate the transport of electrons in disordered and pristine graphene devices. Fano shot noise, a standard metric to assess the mechanism for electronic transport in mesoscopic devices, has been shown to produce almost the same magnitude (≈1/3) in ballistic and diffusive graphene devices and is therefore of limited applicability. We consider a two-terminal geometry where the graphene flake is contacted by narrow metallic leads. We propose that the dependence of the conductance on the position of one of the leads, a conductance profile, can give us insight into the charge flow, which can in turn be used to analyze the transport mechanism. Moreover, we simulate scanning probe microscopy (SPM) measurements for the same devices, which can visualize the flow of charge inside the device, thus complementing the transport calculations. From our simulations, we find that both the conductance profile and SPM measurements are excellent tools to assess the transport mechanism differentiating ballistic and diffusive graphene systems. |
| Ref | Spin–spin model for two-level system/bath problems: A numerical study |
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| 摘要 | We study a new model for treating quantum dissipative systems, in which the bath is modeled as a collection of spins coupled to the system of interest. We develop a quasiclassical method to study this model, approximating the quantum Heisenberg equations by the classical ones, supplemented with stochastic initial conditions carefully chosen so that the results obtained from the classical equations are as close as possible to the quantum results. Using this method we compare the dynamics of such a spin–spin system with that of a spin–boson system, in which the bath is modeled as a collection of harmonic oscillators. We verify numerically that when the system-bath coupling is spread over many bath spins 共the Brownian motion limit兲, the spin–spin model can be mapped on the spin–boson model 共although with a temperature dependent spectral density兲. We also demonstrate that the two dissipative models are qualitatively very different in a non-Brownian motion regime. |
| Ref | On The Critical Coupling For Kuramoto oscillators |
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| 摘要 | We investigate the transport of electrons in disordered and pristine graphene devices. Fano shot noise, a standard metric to assess the mechanism for electronic transport in mesoscopic devices, has been shown to produce almost the same magnitude (≈1/3) in ballistic and diffusive graphene devices and is therefore of limited applicability. We consider a two-terminal geometry where the graphene flake is contacted by narrow metallic leads. We propose that the dependence of the conductance on the position of one of the leads, a conductance profile, can give us insight into the charge flow, which can in turn be used to analyze the transport mechanism. Moreover, we simulate scanning probe microscopy (SPM) measurements for the same devices, which can visualize the flow of charge inside the device, thus complementing the transport calculations. From our simulations, we find that both the conductance profile and SPM measurements are excellent tools to assess the transport mechanism differentiating ballistic and diffusive graphene systems. |
| Ref | Dynamics of a two-level system coupled to a bath of spins |
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| 摘要 | The dynamics of a two-level system coupled to a spin bath is investigated via the numerically ex- act multilayer multiconfiguration time-dependent Hartree (ML-MCTDH) theory. Consistent with the previous work on linear response approximation [N. Makri, J. Phys. Chem. B 103, 2823 (1999)], it is demonstrated numerically that this spin-spin-bath model can be mapped onto the well-known spin-boson model if the system-bath coupling strength obeys an appropriate scaling behavior. This linear response mapping, however, may require many bath spin degrees of freedom to represent the practical continuum limit. To clarify the discrepancies resulted from different approximate treat- ments of this model, the population dynamics of the central two-level system has been investigated near the transition boundary between the coherent and incoherent motions via the ML-MCTDH method. It is found that increasing temperature favors quantum coherence in the nonadiabatic limit of this model, which corroborates the prediction in the previous work [J. Shao and P. Hanggi, Phys. Rev. Lett. 81, 5710 (1998)] based on the non-interacting blip approximation (NIBA). However, the coherent-incoherent boundary obtained by the exact ML-MCTDH simulation is slightly different from the approximate NIBA results. Quantum dynamics in other physical regimes are also discussed. |
| Ref | Kicked-Harper model versus on-resonance double-kicked rotor model: From spectral difference to topological equivalence |
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| 摘要 | Recent studies have established that, in addition to the well-known kicked-Harper model (KHM), an on- resonance double-kicked rotor (ORDKR) model also has Hofstadter’s butterfly Floquet spectrum, with strong resemblance to the standard Hofstadter spectrum that is a paradigm in studies of the integer quantum Hall effect. Earlier it was shown that the quasienergy spectra of these two dynamical models (i) can exactly overlap with each other if an effective Planck constant takes irrational multiples of 2π and (ii) will be different if the same parameter takes rational multiples of 2π . This work makes detailed comparisons between these two models, with an effective Planck constant given by 2π M/N , where M and N are coprime and odd integers. It is found that the ORDKR spectrum (with two periodic kicking sequences having the same kick strength) has one flat band and N − 1 nonflat bands with the largest bandwidth decaying in a power law as \( \sim K^{N+2} \) , where K is a kick strength parameter. The existence of a flat band is strictly proven and the power-law scaling, numerically checked for a number of cases, is also analytically proven for a three-band case. By contrast, the KHM does not have any flat band and its bandwidths scale linearly with K. This is shown to result in dramatic differences in dynamical behavior, such as transient (but extremely long) dynamical localization in ORDKR, which is absent in the KHM. Finally, we show that despite these differences, there exist simple extensions of the KHM and ORDKR model (upon introducing an additional periodic phase parameter) such that the resulting extended KHM and ORDKR model are actually topologically equivalent, i.e., they yield exactly the same Floquet-band Chern numbers and display topological phase transitions at the same kick strengths. A theoretical derivation of this topological equivalence is provided. These results are also of interest to our current understanding of quantum-classical correspondence considering that the KHM and ORDKR model have exactly the same classical limit after a simple canonical transformation. |
| Ref | The large deviation approach to statistical mechanics |
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| 摘要 | The theory of large deviations is concerned with the exponential decay of probabilities of large fluctuations in random systems. These probabilities are important in many fields of study, including statistics, finance, and engineering, as they often yield valuable information about the large fluctuations of a random system around its most probable state or trajectory. In the context of equilibrium statistical mechanics, the theory of large deviations provides exponential-order estimates of probabilities that refine and generalize Einstein’s theory of fluctuations. This review explores this and other connections between large deviation theory and statistical mechanics, in an effort to show that the mathematical language of statistical mechanics is the language of large deviation theory. The first part of the review presents the basics of large deviation theory, and works out many of its classical applications related to sums of random variables and Markov processes. The second part goes through many problems and results of statistical mechanics, and shows how these can be formulated and derived within the context of large deviation theory. The problems and results treated cover a wide range of physical systems, including equilibrium many-particle systems, noise-perturbed dynamics, nonequilibrium systems, as well as multifractals, disordered systems, and chaotic systems. This review also covers many fundamental aspects of statistical mechanics, such as the derivation of variational principles characterizing equilibrium and nonequilibrium states, the breaking of the Legendre transform for nonconcave entropies, and the characterization of nonequilibrium fluctuations through fluctuation relations. |
| Ref | Quantum properties of double kicked systems with classical translational invariance in momentum |
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| 摘要 | Double kicked rotors (DKRs) appear to be the simplest nonintegrable Hamiltonian systems featuring classical translational symmetry in phase space (i.e., in angular momentum) for an infinite set of values (the rational ones) of a parameter η. The experimental realization of quantum DKRs by atom-optics methods motivates the study of the double kicked particle (DKP). The latter reduces, at any fixed value of the conserved quasimomentum β\(\hbar\), to a generalized DKR, the “β-DKR.” We determine general quantum properties of β-DKRs and DKPs for arbitrary rational η. The quasienergy problem of β-DKRs is shown to be equivalent to the energy eigenvalue problem of a finite strip of coupled lattice chains. Exact connections are then obtained between quasienergy spectra of β-DKRs for all β in a generically infinite set. The general conditions of quantum resonance for β-DKRs are shown to be the simultaneous rationality of η, β, and a scaled Planck constant \( \hbar_{s} \). For rational \( \hbar_{s} \) and generic values of β, the quasienergy spectrum is found to have a staggered-ladder structure. Other spectral structures, resembling Hofstadter butterflies, are also found. Finally, we show the existence of particular DKP wave-packets whose quantum dynamics is free, i.e., the evolution frequencies of expectation values in these wave-packets are independent of the nonintegrability. All the results for rational \( \hbar_{s} \) exhibit unique number-theoretical features involving η,\( \hbar_{s} \), and β. |
| Ref | Many-Body Localization Transition, Temporal Fluctuations of the Loschmidt Echo, and Scrambling |
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| 摘要 | We show that the transition between a ETH phase and a many-body localized phase is marked by the different finite size scaling behaviour of the decay of the Loschmidt Echo and its temporal fluctuations - after a quantum quench - in the infinite time limit, despite the fact that the finite time behaviour of such quantities is dramatically different approach the MBL phase, so that temporal fluctuations cannot be inferred from the infinite time average of the Loschmidt Echo. We also show the different scrambling powers of ETH and MBL Hamiltonians as a probe to the different approaches to equilibrium. |
| Ref | Brownian Motion in a Field of Force and the Diffusion Model of Chemical Reactions |
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| 摘要 | A particle which is caught in a potential hole and which, through the shuttling action of Brownian motion, can escape over a potential barrier yields a suitable model for elucidating the applicability of the transition state method for calculating the rate of chemical reactions. |
| Ref | Dynamics of axially localized states in Taylor-Couette flows |
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| 摘要 | We present numerical simulations of the flow confined in a wide gap Taylor-Couette system, with a rotating inner cylinder and variable length-to-gap aspect ratio. A complex experimental bifurcation scenario differing from the classical Ruelle-Takens route to chaos has been experimentally reported in this geometry. The wavy vortex flow becomes quasiperiodic due to an axisymmetric very low frequency mode. This mode plays a key role in the dynamics of the system, leading to the occurrence of chaos via a period-doubling scenario. Further increasing the rotation of the inner cylinder results in the appearance of a new flow pattern which is characterized by large amplitude oscillations localized in some of the vortex pairs. The purpose of this paper is to study numerically the dynamics of these axially localized states, paying special attention to the transition to chaos. Frequency analysis from time series simultaneously recorded at several points has been applied in order to identify the flow transitions taking place. It has been found that the very low frequency mode is essential to explain the behavior associated with the different transitions towards chaos including localized states. |
| Ref | Analogue studies of nonlinear systems |
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| 摘要 | The design of analogue electronic experiments to investigate phenomena in nonlinear dynamics, especially stochastic phenomena, is described in practical terms. The advantages and disadvantages of this approach, in comparison to more conventional digital methods, are discussed. It is pointed out that analogue simulation provides a simple, inexpensive, technique that is easily applied in any laboratory to facilitate the design and implementation of complicated and expensive experimental projects; and that there are some important problems for which analogue methods have so far provided the only experimental approach. Applications to several topical problems are reviewed. Large rare fluctuations are studied through measurements of the prehistory probability distribution, thereby testing for the first time some fundamental tenets of fluctuation theory. It has thus been shown for example that, whereas the fluctuations of equilibrium systems obey time-reversal symmetry, those under non-equilibrium conditions are temporally asymmetric. Stochastic resonance, in which the signal-to-noise ratio for a weak periodic signal in a nonlinear system can be enhanced by added noise, has been widely studied by analogue methods, and the main results are reviewed; the closely related phenomena of noise-enhanced heterodyning and noise- induced linearization are also described. Selected examples of the use of analogue methods for the study of transient phenomena in time-evolving systems are reviewed. Analogue experiments with quasimonochromatic noise, whose power spectral density is peaked at some characteristic frequency, have led to the discovery of a range of interesting and often counter-intuitive effects. These are reviewed and related to large fluctuation phenomena. Analogue studies of two examples of deterministic nonlinear effects, modulation-induced negative differential resistance (MINDR) and zero-dispersion nonlinear resonance (ZDNR) are described. Finally, some speculative remarks about possible future directions and applications of analogue experiments are discussed. |
| Ref | Experiments on Critical Phenomena in a Noisy Exit Problem |
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| 摘要 | We consider a noise-driven exit from a domain of attraction in a two-dimensional bistable system lacking detailed balance. Through analog and digital stochastic simulations, we find a theoretically predicted bifurcation of the most probable exit path as the parameters of the system are changed, and a corresponding nonanalyticity of the generalized activation energy. We also investigate the extent to which the bifurcation is related to the local breaking of time-reversal invariance. |
| Ref | Transfer matrix study of the Anderson transition in non-Hermitian systems |
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| 摘要 | The Anderson transition driven by non-Hermitian (NH) disorder has been extensively studied in recent years. In this paper, we present in-depth transfer matrix analyses of the Anderson transition in three NH systems, NH Anderson, U(1), and Peierls models in three-dimensional systems. The first model belongs to NH class \(AI^{+}\) , whereas the second and the third ones to NH class A. We first argue a general validity of the transfer matrix analysis in NH systems, and clarify the symmetry properties of the Lyapunov exponents, scattering (S) matrix and two-terminal conductance in these NH models. The unitarity of the S matrix is violated in NH systems, where the two-terminal conductance can take arbitrarily large values. Nonetheless, we show that the transposition symmetry of a Hamiltonian leads to the symmetric nature of the S matrix as well as the reciprocal symmetries of the Lyapunov exponents and conductance in certain ways in these NH models. Using the transfer matrix method, we construct a phase diagram of the NH Anderson model for various complex single-particle energy E. At E = 0, the phase diagram as well as critical properties become completely symmetric with respect to an exchange of real and imaginary parts of on-site NH random potentials. We show that the symmetric nature at E = 0 is a general feature for any NH bipartite-lattice models with the on- site NH random potentials. Finite size scaling data are fitted by polynomial functions, from which we determine the critical exponent ν at different single-particle energies and system parameters of the NH models. We conclude that the critical exponents of the NH class \(AI^{+}\) and the NH class A are ν = 1.19 ± 0.01 and ν = 1.00 ± 0.04, respectively. In the NH models, a distribution of the two-terminal conductance is not Gaussian. Instead, it contains small fractions of huge conductance values, which come from rare-event states with huge transmissions amplified by on-site NH disorders. Nonetheless, a geometric mean of the conductance enables the finite-size scaling analysis. We show that the critical exponents obtained from the conductance analysis are consistent with those from the localization length in these three NH models |
| Ref | Variational dynamics of the sub‐Ohmic spin‐boson model on the basis of multiple Davydov D1 states |
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| 摘要 | Dynamics of the sub-Ohmic spin-boson model is investigated by employing a multitude of the Davy- dov \(D_{1}\) trial states, also known as the multi-\(D_{1}\) Ansatz. Accuracy in dynamics simulations is improved significantly over the single \(D_{1}\) Ansatz, especially in the weak system-bath coupling regime. The reliability of the multi-\(D_{1}\) Ansatz for various coupling strengths and initial conditions is also system- atically examined, with results compared closely with those of the hierarchy equations of motion and the path integral Monte Carlo approaches. In addition, a coherent-incoherent phase crossover in the nonequilibrium dynamics is studied through the multi-\( D_{1} \) Ansatz. The phase diagram is obtained with a critical point \( s_{c} \) = 0.4. For \(s_{c}\) < s < 1, the coherent-to-incoherent crossover occurs at a certain coupling strength, while the coherent state recurs at a much larger coupling strength. For s < \(s_{c}\) , only the coherent phase exists. |
| Ref | Aperiodically Driven Integrable Systems and Their Emergent Steady States |
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| 摘要 | Does a closed quantum many-body system that is continually driven with a time-dependent Hamiltonian finally reach a steady state? This question has only recently been answered for driving protocols that are periodic in time, where the long-time behavior of the local properties synchronizes with the drive and can be described by an appropriate periodic ensemble. Here, we explore the consequences of breaking the time- periodic structure of the drive with additional aperiodic noise in a class of integrable systems. We show that the resulting unitary dynamics leads to new emergent steady states in at least two cases. While any typical realization of random noise causes eventual heating to an infinite-temperature ensemble for all local properties in spite of the system being integrable, noise that is self-similar in time leads to an entirely different steady state (which we dub the “geometric generalized Gibbs ensemble”) that emerges only after an astronomically large time scale. To understand the approach to the steady state, we study the temporal behavior of certain coarse-grained quantities in momentum space that fully determine the reduced density matrix for a subsystem with size much smaller than the total system. Such quantities provide a concise description for any drive protocol in integrable systems that are reducible to a free-fermion representation. |
| Ref | Scarring of Dirac fermions in chaotic billiards |
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| 摘要 | Scarring in quantum systems with classical chaotic dynamics is one of the most remarkable phenomena in modern physics. Previous works were concerned mostly with nonrelativistic quantum systems described by the Schrödinger equation. The question remains outstanding of whether truly relativistic quantum particles that obey the Dirac equation can scar. A significant challenge is the lack of a general method for solving the Dirac equation in closed domains of arbitrary shape. In this paper, we develop a numerical framework for obtaining complete eigensolutions of massless fermions in general two-dimensional confining geometries. The key ingredients of our method are the proper handling of the boundary conditions and an efficient discretization scheme that casts the original equation in a matrix representation. The method is validated by (1) comparing the numerical solutions to analytic results for a geometrically simple confinement and (2) verifying that the calculated energy level-spacing statistics of integrable and chaotic geometries agree with the known results. Solutions of the Dirac equation in a number of representative chaotic geometries establish firmly the existence of scarring of Dirac fermions. |
| Ref | State selection in the noisy stabilized Kuramoto-Sivashinsky equation |
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| 摘要 | In this work, we study the one-dimensional stabilized Kuramoto Sivashinsky equation with additive uncor- related stochastic noise. The Eckhaus stable band of the deterministic equation collapses to a narrow region near the center of the band. This is consistent with the behavior of the phase diffusion constants of these states. Some connections to the phenomenon of state selection in driven out of equilibrium systems are made. |
| Ref | A critical strange metal from fluctuating gauge fields in a solvable random model |
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| 摘要 | Building upon techniques employed in the construction of the Sachdev-Ye-Kitaev (SYK) model, which is a solvable 0 + 1 dimensional model of a non-Fermi liquid, we develop a solvable, infinite- ranged random-hopping model of fermions coupled to fluctuating U(1) gauge fields. In a specific large-N limit, our model realizes a gapless non-Fermi liquid phase, which combines the effects of hopping and interaction terms. We derive the thermodynamic properties of the non-Fermi liquid phase realized by this model, and the charge transport properties of an infinite-dimensional version with spatial structure. |
| Ref | Quantum dynamics of the damped harmonic oscillator |
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| 摘要 | The quantum theory of the damped harmonic oscillator has been a subject of continual investigation since the 1930s. The obstacle to quantization created by the dissipation of energy is usually dealt with by including a discrete set of additional harmonic oscillators as a reservoir. But a discrete reservoir cannot directly yield dynamics such as Ohmic damping (proportional to velocity) of the oscillator of interest. By using a continuum of oscillators as a reservoir, we canonically quantize the harmonic oscillator with Ohmic damping and also with general damping behaviour. The dynamics of a damped oscillator is determined by an arbitrary effective susceptibility that obeys the Kramers–Kronig relations. This approach offers an alternative description of nano-mechanical oscillators and opto-mechanical systems. |
| Ref | Cold bosons in optical lattices: a tutorial for exact diagonalization |
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| 摘要 | Exact diagonalization (ED) techniques are a powerful method for studying many-body problems. Here, we apply this method to systems of few bosons in an optical lattice, and use it to demonstrate the emergence of interesting quantum phenomena such as fragmentation and coherence. Starting with a standard Bose–Hubbard Hamiltonian, we first revise the characterisation of the superfluid to Mott insulator (MI) transitions. We then consider an inhomogeneous lattice, where one potential minimum is made much deeper than the others. The MI phase due to repulsive on-site interactions then competes with the trapping of all atoms in the deep potential. Finally, we turn our attention to attractively interacting systems, and discuss the appearance of strongly correlated phases and the onset of localisation for a slightly biased lattice. The article is intended to serve as a tutorial for ED of Bose–Hubbard models. |
| Ref | Quantum ergodicity: fundamentals and applications. |
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| Ref | Exact solution of the Bose-Hubbard model on the Bethe lattice |
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| 摘要 | The exact solution of a quantum Bethe lattice model in the thermodynamic limit amounts to solve a functional self-consistent equation. In this paper we obtain this equation for the Bose-Hubbard model on the Bethe lattice, under two equivalent forms. The first one, based on a coherent-state path integral, leads in the large connectivity limit to the mean-field treatment of Fisher et al. [Phys. Rev. B 40, 546(1989)] at the leading order, and to the bosonic dynamical mean field theory as a first correction, as recently derived by Byczuk and Vollhardt [Phys. Rev. B 77, 235106(2008)]. We obtain an alternative form of the equation using the occupa- tion number representation, which can be easily solved with an arbitrary numerical precision, for any finite connectivity. We thus compute the transition line between the superfluid and Mott insulator phases of the model, along with thermodynamic observables and the space and imaginary-time dependence of correlation functions. The finite connectivity of the Bethe lattice induces a richer physical content with respect to its infinitely connected counterpart: a notion of distance between sites of the lattice is preserved, and the bosons are still weakly mobile in the Mott insulator phase. The Bethe lattice construction can be viewed as an approximation to the finite-dimensional version of the model. We show indeed a quantitatively reasonable agreement between our predictions and the results of Quantum Monte Carlo simulations in two and three dimensions. |
| Ref | Numerical study of the trapped and extended Bose-Hubbard models |
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| 摘要 | The Bose-Hubbard model describes the physics of a system of bosonic ultracold atoms in an optical lattice, in which a phase transition is present between a superfluid phase and a Mott insu- lator one. The exact solution of this Hamiltonian is only feasible to find the ground-state of small systems, while other techniques (as mean-field schemes or quantum Monte Carlo) are necessary to study systems of larger size. As a first application, we study the trapped model – relevant for the comparison with current experiments – through an in- homogeneous mean-field scheme. We describe some signatures of the phase crossover between superfluid and Mott insulator. In particular, the visibility of the quasimomentum distribution shows some kinks as a function of the lattice depth; we describe these features and we link them with the ones observed in other works in the literature. As a second application, we use quantum Monte Carlo tech- niques to study the one-dimensional Bose-Hubbard model with long-range interactions and we focus on the appearance of the Haldane insulating phase, distinguishable from the Mott one through the presence of non-local hidden order. Non-local corre- lation functions are also used to describe the difference between the superfluid phase and the Mott insulator one. |
| Ref | Synchronization in the Second-order Kuramoto Model |
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| 摘要 | Synchronization phenomena are ubiquitous in the natural sciences and engi- neering, but also in social systems. Among the many models that have been proposed for a description of synchronization, the Kuramoto model is most popular. It describes self-sustained phase oscillators rotating at heterogeneous intrinsic frequencies that are coupled through the sine of their phase differences. The second-order Kuramoto model has been used to investigate power grids, Josephson junctions, and other systems. The study of Kuramoto models on networks has recently been boosted because it is simple enough to allow for a mathematical treatment and yet complex enough to exhibit rich phenom- ena. In particular, explosive synchronization emerges in scale-free networks in the presence of a correlation between the natural frequencies and the network topology. The first main part of this thesis is devoted to study the networked second- order Kuramoto model in the presence of a correlation between the oscillators’ natural frequencies and the network’s degree. The theoretical framework in the continuum limit and for uncorrelated networks is provided for the model with an asymmetrical natural frequency distribution. It is observed that clusters of nodes with the same degree join the synchronous component successively, starting with small degrees. This novel phenomenon is named cluster explosive synchronization. Moreover, this phenomenon is also influenced by the degree mixing in the network connection as shown numerically. In particular, discontinuous transitions emerge not just in disassortative but also in strong assortative networks, in contrast to the first-order model. Discontinuous phase transitions indicated by the order parameter and hystere- sis emerge due to different initial conditions. For very large perturbations, the system could move from a desirable state to an undesirable state. Basin stability was proposed to quantify the stability of a system to stay in the desirable state after being subjected to strong perturbations. In the second main part of this thesis, the basin stability of the synchronization of the second-order Kuramoto model is investigated via perturbing nodes sepa- rately. As a novel phenomenon uncovered by basin stability it is demonstrated that two first-order transitions occur successively in complex networks: an onset transition from a global instability to a local stability and a suffusing transition from a local to a global stability. This sequence is called onset and suffusing transition. Different nodes could have a different stability influence from or to other nodes. For example, nodes adjacent to dead ends have a low basin stability. To quantify the stability influence between clusters, in particular for cluster synchronization, a new concept of partial basin stability is proposed. The concept is implemented on two important real examples: neural networks and the northern European power grid. The new concept allows to identify unstable and stable clusters in neural networks and also explains how dead ends undermine the network stability of power grids. |
| Ref | Quantum Phase Transition of Light in the Jaynes-Cummings Lattice |
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| 摘要 | The main goal of the present thesis is to derive the Jaynes-Cummings-Hubbard model and to study its thermodynamic properties in the grand-canonical ensem- ble. For this purpose this thesis is structured as follows. In Chapter 2, I derive the Jaynes-Cummings (JC) model which provides the on-site potential in the considered lattice model. Subsequently, I discuss the JC eigenstates and their energy spectrum and introduce polaritons whose number is the conserved quantity in this model. In Chapter 3, I go ahead and generalize the JC model to the Jaynes-Cummings- Hubbard (JCH) model describing a lattice of cavities, each filled with a single two- level system. For a first rough analysis of this model, I subsequently consider the limits of no dynamics at all and the opposite extreme of hopping domination. This approximative treatment qualitatively shows the existence of a quantum phase tran- sition from a Mott insulator to a superfluid phase in the considered model. Af- terwards, in order to obtain a more quantitative description of this phase transi- tion, I establish a mean-field theory, which eventually leads to the mean-field phase boundary at zero temperature. The investigation of temperature effects on the JCH model is then accomplished in Chapter 4. To this end, I shortly review the Dirac interaction picture, in which the partition function of the system is expressed. Using a current approach to break the symmetry of the system, which is essential for describing a quantum phase transi- tion, the partition function is then expanded in terms of cumulants. This procedure yields a perturbative expansion of the grand-canonical free energy, which is then Legendre-transformed to an effective Ginzburg-Landau action. Finally, in Chapter 5, I derive the excitation spectra and effective masses of the po- laritons in the Mott phase for finite temperature from this Ginzburg-Landau action. A summary of this thesis and an outlook on further investigations is given in Chap- ter 6. |
| Ref | Ultra Large-scale Exact-diagonalization for Confined Fermion- Hubbard Model on the Earth Simulator: Exploration of Superfluidity in Confined Strongly-Correlated Systems |
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| 摘要 | In order to explore a possibility of superfluidity in confined strongly-correlated fermion systems, e.g., nano-scale cuprate High-Tc superconductors and atomic Fermi gases loaded on optical lattice, we imple- ment an exact diagonalization code for their mathematical model, i.e., a trapped Hubbard model on the Earth Simulator. We compare two diagonalization algorithms, the traditional Lanczos method and a new algorithm, the preconditioned conjugate gradient (PCG) method, and find that when using the PCG the total CPU time can be reduced to 1/3 ~ 1/5 compared to the former one since the convergence can be dramatically improved by choosing a good preconditioner and the communication overhead is much more efficiently concealed in the PCG method. Consequently, such a performance improvement enables us to do systematic studies for sev- eral parameters. Numerical simulation results reveal that an unconventional type of pairing specific to the confined system, which may cause superfluidity, develops under a strong repulsive interaction. |
| Ref | Biologically inspired ant colony simulation |
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| 摘要 | We present a unified biologically inspired approach to simulate ant colonies inspired by the key observation of collective behaviors of ants in nature. To determine the motion states and the paths of virtual ants, considering dynamic internal and external interactions. The motion controller computes a target posi- tion for each ant at every time step according to its motion states. The motion states include four states: basic movement, the stop state, and two dynamic interactions (i.e., internal and external, respectively referring to interaction with neighbors for necessary information transfer about the destination, and interac- tion with surroundings such as food sources, nests, and obstacles) to represent basic exploration, casual or intentional stop, and purposeful movement, respec- tively. Based on the motion states, the motion controller plans an optimal path for each virtual ant. Through many simulation experiments, we demonstrate that our method is controllable, scalable, and flexible to simulate hybrid colonies with a large number of ants. |
| Ref | General Theory of Spin-Wave Interactions |
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| 摘要 | An ideal model of a ferromagnet is studied, consisting of a lattice of identical spins with cubic symmetry and with isotropic exchange coupling between nearest neighbors. The aim is to obtain a complete description of the thermodynamic properties of the system at low temperatures, far below the Curie point. In this temperature region the natural description of the states of the system is in terms of Bloch spin waves. The nonorthogonality of spin-wave states raises basic difBculties which are examined and overcome. The following new results are obtained: a practical method for calculating thermodynamic quantities in terms of a nonorthogonal set of basic states; a proof that in 3 dimensions there do not exist states (shown by Bethe to exist in a one-dimensional chain of spina) in which two spina are bound together into a stable complex and travel together through the lattice; a calculation of the scattering cross section of two spin waves, giving a mean free path for spin-spin collisions. proportional to \( T^{-7/2} \) at low temperatures; and an exact formula for the free energy of the system, showing explicitly the effects of spin-wave interactions. Quantitative results based on this theory will be published in a second paper. |
| Ref | Zero-temperature phases of the two-dimensional Hubbard-Holstein model: A non-Gaussian exact diagonalization study |
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| 摘要 | We propose a numerical method which embeds the variational non-Gaussian wave-function approach within exact diagonalization, allowing for efficient treatment of correlated systems with both electron-electron and electron-phonon interactions. Using a generalized polaron transformation, we construct a variational wave func- tion that absorbs entanglement between electrons and phonons into a variational non-Gaussian transformation; exact diagonalization is then used to treat the electronic part of the wave function exactly, thus taking into account high-order correlation effects beyond the Gaussian level. Keeping the full electronic Hilbert space, the complexity is increased only by a polynomial scaling factor relative to the exact diagonalization calculation for pure electrons. As an example, we use this method to study ground-state properties of the two-dimensional Hubbard-Holstein model, providing evidence for the existence of intervening phases between the spin and charge-ordered states. In particular, we find one of the intervening phases has strong charge susceptibility and binding energy, but is distinct from a charge-density-wave ordered state, while the other intervening phase displays superconductivity at weak couplings. This method, as a general framework, can be extended to treat excited states and dynamics, as well as a wide range of systems with both electron-electron and electron-boson interactions. |