text stringlengths 128 2.05k |
|---|
[EQUATION] Time dependent survival probability In summary, the survival probability of the reaction and hopping events are (from Eq. ( D.66 ) and Eq. ( D.70 )) |
[EQUATION] Thus, the survival probability after one step is [EQUATION] where [MATH] As a consequence, the survival probability of a reactive pair at short time [MATH] after step [MATH] follows the Poisson distribution: |
[EQUATION] where [MATH] Rate coefficient at long times Here, we study the time-dependent kinetics of the diffusion-influenced scheme. We start with the definition of continuous rebinding-time probability density, and use it to express the time-dependent rate coefficient. |
Denoting the continuous rebinding-time probability density after [MATH] steps as [EQUATION] where [EQUATION] is the survival time probability density of a particle that started and ended at [MATH] on the [MATH] th step. The first term on the right-hand side of Eq. ( D.75 ) is the initial probability density [MATH] , wh... |
[EQUATION] for [MATH] . Intuitively, the first term describes the first-passage time distribution for single step while the second term accounts for the convolution of the probability of time required for the [MATH] steps after the first step. |
The continuous rebinding-time probability density is related to the rate coefficient of the particle-pair formalism through: [EQUATION] |
as shown in the main text. We then take the Laplace transform of [MATH] which is easier to work with: [EQUATION] Note that [MATH] is related to the rebinding-time and survival-time probability densities via: |
[EQUATION] The corresponding Laplace transform of Eq. ( D.75 ) is given as [EQUATION] where [EQUATION] The infinite sum of Eqs.( D.80 ) and ( D.81 ) are |
[EQUATION] [EQUATION] where [MATH] is the generating function, [MATH] is as defined in ( D.56 ). Hence, we have [EQUATION] where [MATH] |
Substituting Eq. ( D.84 ) into Eq. ( D.79 ) and by the final value theorem, we obtain the long-time behavior of [MATH] by taking the limit [MATH] in Eq. ( D.78 ). |
Assuming the asymptotic Laplace form of the rate coefficient on lattice ( Eq. 2.37a): [EQUATION] We then set [MATH] to obtain the effective lattice reaction rate constant: |
[EQUATION] Evaluating [MATH] by referring to Eq. ( D.79 ), we then get [EQUATION] which is consistent with the result shown in main text. |
The second order term of Eq. ( D.85 ) is evaluated by expanding [MATH] around [MATH] [EQUATION] where [MATH] and [MATH] Thus, by comparing the terms we obtain |
[EQUATION] where [EQUATION] Applying the definitions of reaction acceptance probability ( B.2 ) and voxel size ( C.1 ), we obtain |
[EQUATION] Note that the corresponding time domain form of Eq. ( D.85 ) is given as [EQUATION] Hence, the long-time behavior of the lattice rate coefficient follows the same form as in the continuum case: |
[EQUATION] Appendix E Production-degradation process In the coupled reactions [MATH] [MATH] , the survival probability of a newly produced [MATH] molecule in an equilibrated pool of [MATH] is |
[EQUATION] where [MATH] is the concentration of [MATH] and [MATH] is the irreversible rate coefficient according to the radiation boundary condition. Since [MATH] is removed from the system via the bimolecular reaction, the concentration of [MATH] will eventually reach a steady-state. The corresponding mean lifetime of... |
[EQUATION] where the hat denotes Laplace transform. For small [MATH] [MATH] is given by (Eq. 4.5 in [EQUATION] where [MATH] is the macroscopic rate constant, [MATH] is the effective radius and [MATH] is the intrinsic reaction rate constant. The equilibrium concentration of [MATH] is then [EQUATION] |
# Source: arxiv 1805.12354 # Title: Return probability for the Anderson model on the random regular graph # Sections: all # Downloaded: 2026-03-03T05:15:44.356183+00:00 |
Return probability for the Anderson model on the random regular graph Abstract We study the return probability for the Anderson model on the random regular graph and give the evidence of the existence of two distinct phases: a fully ergodic and non-ergodic one. In the ergodic phase the return probability decays polynom... |
Introduction The problem of Anderson localization on locally tree-like structures Abou-Chacra et al. 1973 ); Mirlin and Fyodorov ( 1991 , or Bethe lattices, which are limits of families of random regular graphs (RRG), has been at the center of a recent spur of research activity De Luca et al. 2014 ); Altshuler et al. 2... |
The MBL phase Nandkishore and Huse ( 2015 , which now looks like the prototypical dynamical behavior of an interacting quantum system with strong disorder, has been characterized completely in terms of emergent, local integrals of motion Huse et al. 2014 ); Serbyn et al. 2013 ); Ros et al. 2015 ); Chandran et al. 2016 ... |
Other works have found a diffusive to subdiffusive phase transition in the delocalized region Luitz and Bar Lev ( 2016 ); Žnidarič et al. 2016 ); Luitz et al. 2016 |
(but Karahalios et al. 2009 ); Steinigeweg et al. 2016 ); Bera et al. 2017 questioned these findings). Subdiffusion has been interpreted sometimes in terms of rare-region effects Luitz et al. 2016 ); Luitz and Bar Lev ( 2016 ); Bar Lev et al. 2015 2017 ); Khait et al. 2016 ); Agarwal et al. 2015 ); Vosk et al. 2015 ); ... |
In light of these findings, and if the mapping of the MBL problem to the Anderson model on the RRG has to be taken to its extreme consequences, one is led to wonder if different |
flavors of the delocalized phase should be present there too (this is at some level conjectured in Altshuler et al. 1997 ). This is an intriguing possibility, and an interesting question per se Since numerical analysis of the Anderson model on [MATH] lattices for small [MATH] (mainly up to [MATH] |
Evers and Mirlin ( 2008 ); Tarquini et al. 2017 ) found no such phase, this possibility is clearly linked to the nature of the RRG, or to mean field approximations valid when [MATH] |
However, recently in long-range random matrix models such behavior has been found in several models Kravtsov et al. 2015a ); Nosov et al. 2018 ); Khaymovich More or less simultaneously, it has been proposed that the Anderson model on the RRG might have a new phase within the extended phase (where states span the entire... |
In this work, we focus on the characterization of the delocalized phase, basing on time evolution of observables, which more sensitive to non-ergodicity and converge at available system sizes, unlike eigenfunction statistics. Studying the return probability of a particle initially localized in a small region of the sys... |
First, we benchmarked this characterization on two known models that possess critical states: The power-law random banded matrix (PLRBM) Mirlin et al. 1996 ); Evers and Mirlin ( 2008 , and the Rosenzweig-Porter random matrix (RPRM) models Rosenzweig and Porter ( 1960 ); Kravtsov et al. 2015b |
The former model mimics the Anderson localization transition at finite [MATH] , showing only ergodic and localized phases and giving the access to multifractal states at the Anderson transition point. The latter exhibits an entire fractal phase Kravtsov et al. 2015b in a range of parameters along with the standard ergo... |
Then, we use the same concept to study the Anderson model on the RRG, showing similarities and differences with the previous two models , PLRBM and RPRM. |
We find that the ratio of logarithms of the mean and typical values of the return probability is, to a good approximation, a constant. While this constant equals to unity |
in the ergodic phase of the PLRBM and for the RPRM model, it is smaller than unity for the multifractal phase of the PLRBM and for the AM on the RRG. |
In the ergodic phase in all models, both mean and typical values of the return probability show an universal algebraic decay with time with oscillations, due to the rigidity of their spectrum de Tomasi et al. 2018 ); Távora et al. 2017 ); Torres-Herrera et al. 2018 ); Santos and Torres-Herrera ( 2018 |
This kind of ergodicity is usually referred to as full ergodicity and characterized by standard Gaussian ensembles with ergodic fully correlated wavefunctions and Wigner-Dyson level statistics Mehta. ( 2004 |
In the multifractal phase of the PLRBM , mean and typical values of return probability are power-law decaying but with different powers, while in the RPRM the decay is exponential. In the multifractal phase of the AM on the RRG, the mean and typical values are stretched exponentials [MATH] , with the same power [MATH] ... |
Our analysis gives a characterization of the delocalized region of the AM on the RRG, [MATH] Indeed, at small enough disorder strengths, [MATH] a fully ergodic phase is established, while for [MATH] from [MATH] to [MATH] |
a non-ergodic extended phase appears, which is somehow intermediate between the PLRBM one and the fractal region of the RPRM. Moreover, within the range of values of disorder in which the non-ergodic phase has been found ( [MATH] ), |
the parameters of the stretched exponential [MATH] and [MATH] evolve smoothly with [MATH] , where [MATH] as [MATH] approaching the critical value [MATH] |
In the last section we provide a classical random walk model in which the particle jumps in random directions but at random times [MATH] which are distributed in a power-law way [MATH] . The exponent [MATH] is the exponent of the stretched exponential. |
II Model and methods We study the Hamiltonian [EQUATION] represented in the basis of the site states [MATH] , where [MATH] is the number of sites in the system. We consider three different models that have a metal-insulator transition (MIT) with wave-functions changing properties from ergodic to localized via multifrac... |
First, we consider the power-law random banded matrix ensemble (PLRBM) Mirlin et al. 1996 ); Evers and Mirlin ( 2008 , which is obtained from [MATH] ) with [MATH] Here and further [MATH] are independent uniformly distributed random variables taken from [MATH] This ensemble of matrices parameterized by [MATH] and [MATH] |
has an MIT at [MATH] , for any [MATH] For [MATH] , the model shows an ergodic phase and at [MATH] the eigenstates are power-law localized. At the critical point ( [MATH] ) all the states are multifractal and the parameter [MATH] tunes the multifractal properties of eigenstates from strong ( [MATH] ) to weak ( [MATH] ) ... |
Second, we discuss the Rosenzweig-Porter random matrix model (RPRM) Rosenzweig and Porter ( 1960 ); Kravtsov et al. 2015a , which is obtained choosing [MATH] for [MATH] , while for [MATH] [MATH] Like the PLRBM the RPRM has no mobility edge, but it has three distinct phases. For [MATH] all the states are fully ergodic w... |
take a simple linear form [MATH] [MATH] Third, we examine the random regular graph (RRG) with the uncorrelated diagonal disorder [MATH] uniformly distributed in the interval |
[MATH] The hopping amplitudes are deterministic and equal to [MATH] if the sites [MATH] and [MATH] are linked in RRG with fixed local connectivity [MATH] and [MATH] otherwise. The local connectivity is taken to be three (i.e., [MATH] like in many previous studies. This model is believed to have the Anderson transition ... |
(this number is the most recent one in Kravtsov et al. 2018 and Parisi ). Moreover, the matter of discussion is the possibility of the existence of an non-ergodic (multifractal) phase constituted by extended states at [MATH] |
De Luca et al. 2014 and thus a transition at even smaller disorder strength between these multifractal states and ergodic states Altshuler et al. 2016 This putative transition has been estimate to be around [MATH] (EMT, ergodic to multifractal) Altshuler et al. 2016 It implies existence of an entire phase ( [MATH] ) co... |
In this work we focus on the study of these different extended phases (ergodic, non-ergodic multifractal, fractal) by investigating their dynamical properties. In particular, we study the return probability starting from a projected state [MATH] |
de Tomasi et al. 2018 ); Bera et al. 2017 ); De Tomasi et al. 2016 , defined as: [EQUATION] where for RRG [MATH] is the projector to eigenstates of [MATH] with energy [MATH] |
which belongs to a small energy shell [MATH] around the middle of the spectrum of [MATH] [MATH] is considered to be a fraction of the whole bandwidth [MATH] for the Anderson model on the RRG. For PLRBM and RPRM (where there is no mobility edge) the projector is taken to be |
[MATH] The reason to use the projector [MATH] in the Anderson model on the RRG is dual. On one hand one wants to avoid the mixing of states with different dynamical properties, and in general some [MATH] have overlap with both localized and delocalized states. So, for the RRG, [MATH] has been chosen small enough so tha... |
[MATH] and [MATH] such that the uncertainty principle [MATH] is satisfied (here [MATH] is some velocity [MATH] ). The average over matrix ensemble and initial states [MATH] is indicated with a bar over the quantity considered. In particular, we focus on mean and typical values of [MATH] , defined as [MATH] and |
[MATH] , respectively The scaling of [MATH] to zero with the system size [MATH] in the long time limit is also in our main focus (both typical and mean averages) |
[EQUATION] [EQUATION] These quantities will give information on the properties (ergodicity or multifractality) of the eigenstate belonging in the energy shell [MATH] as the mean [MATH] can be expressed in terms of the inverse participation ratio (IPR) of wavefunctions [MATH] of [MATH] |
[EQUATION] The typical value [MATH] of [MATH] is not equal to [MATH] in general. This difference possibly originates from the time fluctuations of [MATH] Nevertheless for long times (of the order of the saturation time of [MATH] in a finite system) the time fluctuations of [MATH] scale to zero as a function of [MATH] ,... |
Nevertheless, the scaling of [MATH] and [MATH] can, in principle, be different depending on the phase. Indeed, in the ergodic phase the envelope of the wavefunctions [MATH] is in the first approximation uniformly distributed over the entire system [MATH] thus it does not reveal strong spatial fluctuations. In this case... |
emerged due to a fractal spatial support set of wavefunctions, forming subbands in the entire energy spectrum from eigenstates living in the same fractal set and fully correlated to each other Kravtsov et al. 2015a ); de Tomasi et al. 2018 . Thus, in this case we expect a situation similar to that of the ergodic phase.... |
III PLRBM [MATH] RPRM In this section we study [MATH] and its long time saturation value for the PLRBM and RPRM. We perform the time evolution using exact full diagonalization. At the critical point of the PLRBM, [MATH] , where all states are multifractal, both [MATH] and |
[MATH] decay algebraically, [MATH] and [MATH] in full agreement with the previous analytical investigations for [MATH] Kravtsov et al. 2010 2011 2012 Figure (a) shows the algebraic decay of [MATH] |
and [MATH] at criticality (multifractal phase). As observed, the two decay rates ( [MATH] ) are different from each other, and due to the inequality between arithmetic and geometric mean [MATH] |
Instead, in the ergodic phase ( [MATH] ) the asymptotic decay rates of [MATH] and [MATH] are the same and [MATH] demonstrates power-law decay with oscillations, being an attribute of Wigner-Dyson fully ergodic behavior (see Appendix). As a consequence of the difference in decay rates of |
[MATH] and [MATH] in the multifractal phase the saturation values [MATH] ) and [MATH] ) may have different scaling to zero as functions of [MATH] [MATH] |
and [MATH] [MATH] ), while in the ergodic phase the exponents are the same and equal to unity [MATH] To emphasize the difference in the behavior of the typical and mean [MATH] in different phases of PLRBM, in Fig. (c) we show |
[MATH] as a function of [MATH] in a log-log plot for two different set of values of [MATH] : one in the ergodic phase and another in the multifractal phase. In the ergodic phase [MATH] and [MATH] scale in the same way as a function of system size [MATH] ). In the multifractal phase [MATH] and [MATH] scale in a differen... |
[MATH] with [MATH] . This difference is also possible to observe in the probability distribution of [MATH] (for the scaling with [MATH] of the probability distribution of [MATH] see Appendix), which in the multifractal region becomes long-tailed giving the discrepancy in the scaling between mean and typical values. In ... |
[MATH] ) in the latter one obtains that [MATH] with the same [MATH] as in [MATH] In the RPRM both [MATH] and [MATH] decay exponentially in time in the non-ergodic phase, [MATH] [MATH] |
and polynomially with oscillatory time-dependence in ergodic phase, [MATH] [MATH] de Tomasi et al. 2018 Here [MATH] is the Bessel function of the first kind, [MATH] is the Thouless’s energy and [MATH] coincides in this case with the energy bandwidth [MATH] (as we take [MATH] for this model). |
Some of authors of this paper have also studied in de Tomasi et al. 2018 an accurate extraction of the [MATH] -dependence of [MATH] from [MATH] nearly free from the finite size effects. |
Figure (b) shows [MATH] and [MATH] versus [MATH] in the fractal critical region. It gives an evidence that both [MATH] and [MATH] decay exponentially in time with the same rate [MATH] The same dependence with time between mean and typical implies that their saturation values scale to zero as functions of [MATH] in the ... |
Figure (d) shows [MATH] as a function of [MATH] both in ergodic and in fractal phases. In both phases typical and mean return probabilities scale in the same manner [MATH] , confirming above mentioned arguments about the fractal states. |
IV Anderson model on RRG Having shown that the difference in the behavior between the mean and the typical value of [MATH] can be used to distinguish ergodic and multifractal phases, we now study [MATH] in the RRG. In the RRG the existence of the multifractal phase is under active debate because of two issues: First, t... |
The Anderson model on the RRG has a mobility edge, thus in our study we consider only the energies in the middle of the spectrum, choosing |
[MATH] , ensuring that all the states [MATH] share the same properties for our choice of the disorder strength [MATH] We perform the time evolution using full diagonalization for small systems sizes [MATH] , and using Chebyshev integration technique Weiße et al. 2006 for larger [MATH] The projector [MATH] has been cons... |
Figure (a) demonstrates time dependence of the mean of [MATH] at rather small disorder strength [MATH] ) for several system sizes. The presence of oscillations in the return probability [MATH] surviving in the thermodynamic limit [MATH] confirms the existence of the fully ergodic phase consistent with Wigner-Dyson beha... |
de Tomasi et al. 2018 ); Távora et al. 2017 ); Torres-Herrera et al. 2018 ); Santos and Torres-Herrera ( 2018 The inset of Fig. (a) confirms the form of oscillations without any fitting parameter |
[EQUATION] which is valid for small energy shell [MATH] , approximating the local density of states with a box function and uncovering the rigidity of the spectrum However, at moderate disorder strength [MATH] the time dependence of [MATH] (shown in Fig. (b)) demonstrates absence of oscillations and a clear bending in ... |
For disorder strengths between [MATH] ergodic oscillations are also present. Nevertheless, their amplitudes reduce with the increasing system size, preserving us from giving a final conclusion on the existence of the fully ergodic or multifractal phases in this regime. Indeed, in de Tomasi et al. 2018 it has been shown... |
[MATH] for [MATH] , which develops a large plateau over more than 2 orders of magnitudes of [MATH] , increasing with increasing system size. The formation of this plateau gives an evidence that the power [MATH] is the same for mean and typical. Since value of the ratio [MATH] at the plateau is less than unity one can c... |
For disorder strengths between [MATH] , where both the residual oscillations and the proximity to AT do not matter, the stretched-exponential parameter [MATH] decays approximately linearly. The linear extrapolation of [MATH] gives reasonable values of the Anderson localization transition [MATH] , where [MATH] Although ... |
de Tomasi et al. 2018 ). he linear extrapolation to this region [MATH] is consistent with works on classical diffusion on the Bethe lattice Cassi ( 1989 ); Chinta et al. 2015 ). |
To avoid any problems with unstability of multi-parameter fit, in Fig. we show [MATH] exponentially decaying as a function of [MATH] for several values of [MATH] providing the direct indication that [MATH] In summary, our analysis for [MATH] provides the evidence that for small disorder [MATH] the RRG is in the fully e... |
It is important to underline that the time scale [MATH] in which the decay of [MATH] can be distinguished from an algebraic decay diverges approaching the Anderson transition (see also Tikhonov and Mirlin ( 2018 i.e. for [MATH] the bending in a log-log plot is only visible for [MATH] and it requires having system size ... |
[MATH] is consistent with an algebraic dependence of the overlap of different wavefunctions [MATH] defined as [EQUATION] with a normalization constant [MATH] ensuring [MATH] |
However, using stationary phase approximation it is possible to show that for [MATH] , the overlap decays as [MATH] for moderately large [MATH] and as [MATH] for very large [MATH] As for observed values of [MATH] the difference between above mentioned exponents is less than [MATH] , a stretched-exponential behavior for... |
as well as with the power-law with logarithmic corrections Classical random walk approximation Let us now present a classical model of subdiffusion which can explain stretched-exponential behavior of the return probability on the RRG, while giving normal sub-diffusion on a regular lattice. Let us consider a random walk... |
We assume that different dwelling times [MATH] and jump directions, determining the random number [MATH] of steps the walker takes in time [MATH] , are statistically independent. Within this assumption on a line , one can easily see that the averaged square distance from the initial point is determined solely by [MATH] |
[EQUATION] where [MATH] is the lattice constant. If we have [MATH] we straightforwardly obtain a subdiffusion law [EQUATION] It is possible to see that, by choosing [MATH] for [MATH] the typical number of steps in an interval is indeed scaling as |
[MATH] . For [MATH] , instead we have [MATH] On any regular lattice, the probability distribution follows the Markov rate equation |
[EQUATION] where [MATH] is the connectivity and [MATH] the adjacency matrix of the graph. While for a typical non-expander like a square lattice [MATH] , or similar, the decay of the return probability after [MATH] steps is power-law [MATH] , for a RRG/Bethe lattice the return probability scales exponentially |
[EQUATION] irrespective of [MATH] , where [MATH] is the gap in the adjacency matrix, [MATH] If we now take into account the above mentioned assumption |
[MATH] we immediately obtain a stretched-exponential form [EQUATION] which is in accord with our numerics. Moreover, for a RRG/Bethe lattice the averaged distance from the initial point grows linearly with [MATH] |
[EQUATION] giving in our case subballistic wavepacket spreading [EQUATION] The prediction of the subdiffusive spreading of the wavepacket following from ( ) needs to be verified in further works. |
Notice moreover, that the fluctuations of [MATH] can explain the difference between [MATH] and [MATH] , even if the distribution of [MATH] does not have long tails. For example a distribution like [MATH] for [MATH] and 0 otherwise gives a ratio [MATH] |
The above mentioned assumption of the classical dynamics in RRG can be justified in our numerics due to not too small width [MATH] of the initial wavepacket [MATH] The power-law distribution of the dwelling times [MATH] is possibly related to the strength of the on-site disorder. |
VI Participation ratios In this section we analyze the system size dependence of the saturation values ( ) of mean [MATH] and typical [MATH] return probability in AM on RRG. |
We directly observe scaling of IPRs with system size [MATH] , but the scaling exponents [MATH] have not yet reached saturation. Using the time evolution algorithm it is difficult to extract the saturation values of [MATH] systematically and reliably, since very large times are needed. Therefore, we find it easier to an... |
as a function of [MATH] in a log-log scale for a fixed disorder strength [MATH] Strong finite size effects are visible for available systems sizes, which makes the extrapolation of [MATH] and [MATH] |
unreliable. Nevertheless, [MATH] and [MATH] seem to suffer from similar finite-size effects. Indeed, plotting [MATH] parametrically as a function of [MATH] drastically reduces finite-size effects. Figure (b) shows [MATH] as a function of [MATH] for several values of [MATH] , giving an indication that [MATH] As we have ... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.