scholarly journals Survival probability in Generalized Rosenzweig-Porter random matrix ensemble

2019 ◽  
Vol 6 (1) ◽  
Author(s):  
Giuseppe De Tomasi ◽  
Mohsen Amini ◽  
Soumya Bera ◽  
Ivan Khaymovich ◽  
Vladimir Kravtsov
Keyword(s):  
Wave Packet ◽  
Thermodynamic Limit ◽  
Survival Probability ◽  
Wave Functions ◽  
Initial State ◽  
Extended Phase ◽  
Periodic Oscillations ◽  
Power Law Decay ◽  
Matrix Ensemble ◽  
Random Matrix Ensemble

We study analytically and numerically the dynamics of the generalized Rosenzweig-Porter model, which is known to possess three distinct phases: ergodic, multifractal and localized phases. Our focus is on the survival probability \bm{R(t)}𝐑(𝐭), the probability of finding the initial state after time \bm{t}𝐭. In particular, if the system is initially prepared in a highly-excited non-stationary state (wave packet) confined in space and containing a fixed fraction of all eigenstates, we show that \bm{R(t)}𝐑(𝐭) can be used as a dynamical indicator to distinguish these three phases. Three main aspects are identified in different phases. The ergodic phase is characterized by the standard power-law decay of \bm{R(t)}𝐑(𝐭) with periodic oscillations in time, surviving in the thermodynamic limit, with frequency equals to the energy bandwidth of the wave packet. In multifractal extended phase the survival probability shows an exponential decay but the decay rate vanishes in the thermodynamic limit in a non-trivial manner determined by the fractal dimension of wave functions. Localized phase is characterized by the saturation value of \bm{R(t\to\infty)=k}𝐑(𝐭→∞)=𝐤, finite in the thermodynamic limit \bm{N\rightarrow\infty}𝐍→∞, which approaches \bm{k=R(t\to 0)}𝐤=𝐑(𝐭→0) in this limit.

scholarly journals Dynamical phases in a ``multifractal'' Rosenzweig-Porter model

2021 ◽  
Vol 11 (2) ◽  
Author(s):  
Ivan Khaymovich ◽  
Vladimir Kravtsov
Keyword(s):  
Random Matrix ◽  
Survival Probability ◽  
Large Deviation ◽  
Finite Size ◽  
Critical Length ◽  
Random Matrix Model ◽  
Localization Model ◽  
Random Regular Graph ◽  
Matrix Ensemble ◽  
Random Matrix Ensemble

We consider the static and the dynamical phases in a Rosenzweig-Porter (RP) random matrix ensemble with a distribution of off-diagonal matrix elements of the form of the large-deviation ansatz. We present a general theory of survival probability in such a random-matrix model and show that the averaged survival probability may decay with time as a simple exponent, as a stretch-exponent and as a power-law or slower. Correspondingly, we identify the exponential, the stretch-exponential and the frozen-dynamics phases. As an example, we consider the mapping of the Anderson localization model on Random Regular Graph onto the RP model and find exact values of the stretch-exponent \kappaκ in the thermodynamic limit. As another example we consider the logarithmically-normal RP random matrix ensemble and find analytically its phase diagram and the exponent \kappaκ. Our theory allows to describe analytically the finite-size multifractality and to compute the critical length with the exponent \nu_{MF}=1νMF=1 associated with it.

scholarly journals Analysis of the buildup of spatiotemporal correlations and their bounds outside of the light cone

2018 ◽  
Vol 5 (5) ◽  
Author(s):  
Nils O. Abeling ◽  
Lorenzo Cevolani ◽  
Stefan Kehrein
Keyword(s):  
Correlation Function ◽  
Power Law ◽  
Light Cone ◽  
Relativistic Quantum ◽  
Initial State ◽  
Quantum Theories ◽  
Luttinger Model ◽  
Power Law Decay ◽  
Exactly Solvable ◽  
Exponentially Small

In non-relativistic quantum theories the Lieb-Robinson bound defines an effective light cone with exponentially small tails outside of it. In this work we use it to derive a bound for the correlation function of two local disjoint observables at different times if the initial state has a power-law decay. We show that the exponent of the power-law of the bound is identical to the initial (equilibrium) decay. We explicitly verify this result by studying the full dynamics of the susceptibilities and correlations in the exactly solvable Luttinger model after a sudden quench from the non-interacting to the interacting model.

A sufficient criterion for particles performing a diffusive motion with state-dependent death rate to die with probability one

1979 ◽  
Vol 11 (03) ◽  
pp. 510-526 ◽  
Author(s):  
Wolfgang Mergenthaler
Keyword(s):  
Integral Equation ◽  
State Space ◽  
Survival Probability ◽  
Death Rate ◽  
Sufficient Criterion ◽  
Initial State ◽  
Diffusive Motion ◽  
State Dependent

We consider an individual which performs a diffusive motion in a certain state space and dies according to a state-dependent death rate. An integral equation for the survival probability is derived, and finally a sufficient criterion for the existence of an initial state is given, for which the corresponding individual dies with probability one.

Oscillatory Circuits Built on Physical SBT Memristor

2019 ◽  
Vol 29 (07) ◽  
pp. 1950097 ◽  
Author(s):  
Yuman Zhang ◽  
Mei Guo ◽  
Gang Dou ◽  
Yuxia Li ◽  
Guanrong Chen
Keyword(s):  
Complex Dynamics ◽  
Circuit Parameter ◽  
Chaotic Oscillations ◽  
Initial State ◽  
Oscillatory Circuit ◽  
Periodic Oscillations ◽  
Dynamical Behaviors ◽  
In Series ◽  
A Charge ◽  
Line Equilibrium

The [Formula: see text] (SBT) nanometer film can be used as a physical memristive component. Three oscillatory circuits built on the physical SBT memristor are proposed in this paper, one is self-excited oscillatory circuit and two are forced oscillatory circuits. These three oscillatory circuits have simple structures with complex dynamics. The self-excited oscillatory circuit can generate steady periodic oscillations; the first forced oscillatory circuit can generate relatively complex quasi-periodic oscillations, while the second can generate more complex dynamics such as chaotic oscillations. The impacts of the circuit parameter and initial state values of the SBT memristor on the dynamical behaviors of the three oscillatory circuits are investigated via numerical simulations. It is found that the SBT memristor can be used to design various memristor-based circuits. Specifically, in a flux-controlled memristor-based circuit, if an inductor is in parallel with the memristor, the order of the circuit is one less than the number of energy storage elements in the circuit. The equilibrium point of the circuit is different from the typical line equilibrium for autonomous circuits. The initial state value of the memristor has no impact on the steady state of the circuit. The same phenomena are observed for a charge-controlled memristor-based circuit, when a capacitor is in series with the memristor.

scholarly journals Random Matrix Ensemble for the Level Statistics of Many-Body Localization

2019 ◽  
Vol 122 (18) ◽  
Author(s):  
Wouter Buijsman ◽  
Vadim Cheianov ◽  
Vladimir Gritsev
Keyword(s):  
Random Matrix ◽  
Many Body ◽  
Level Statistics ◽  
Matrix Ensemble ◽  
Random Matrix Ensemble

scholarly journals Energy level statistics of a critical random matrix ensemble

2003 ◽  
Vol 36 (12) ◽  
pp. 3639-3645 ◽  
Author(s):  
Macleans L Ndawana ◽  
Vladimir E Kravtsov
Keyword(s):  
Energy Level ◽  
Random Matrix ◽  
Level Statistics ◽  
Matrix Ensemble ◽  
Random Matrix Ensemble
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