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Extracting in the same manner the critical exponent [MATH] of the [MATH] -scaling of the mean [MATH] value, we compare the results for [MATH] and [MATH] in the inset of Fig. (b). From the inset one can see that |
[MATH] has a change of [MATH] for available system sizes, while [MATH] changes only by [MATH] Nevertheless, [MATH] increases with [MATH] and we cannot in principle exclude that its asymptotic value may be [MATH] |
Figure (c) shows [MATH] for several [MATH] as a function of [MATH] . For [MATH] [MATH] gives the evidence that in this regime even at our system sizes, the eigenstates are ergodic (but possibly not fully ergodic as at [MATH] . For [MATH] [MATH] drops to a smaller value confirming that for available systems sizes the sy... |
[MATH] towards unity with increasing [MATH] is visible at least for [MATH] , while for [MATH] the data does not change with system size. For large disorder strength one should also keep in mind that the convergence could be induced by the finite system sizes, which are small compared to the correlation length [MATH] |
VII Conclusion We have studied the quantum dynamics of particle initially prepared in a narrow wave-packet form in three different ensembles of disordered systems, giving a characterization of multifractal phases based on the statistics of the return probability. In particular, we have studied the return probability [M... |
First, we have benchmarked these ideas in the power law random banded matrix ensemble, where one observes both ergodic and multifractal phases . We have shown that the long time limit of the mean and typical value of [MATH] scale to zero in the same way in its ergodic regime, while at criticality, where all the states ... |
Second, we have pointed out, analyzing the Rosenzweig-Porter random matrix model, that this difference in the scaling disappears in the case of fractal (but not multifractal) states. |
Finally, we have used this idea to tackle the Anderson model on random regular graph, in which the existence of an extended multifractal phase is under debate. We present the results for the return probability |
[MATH] for system sizes , where the convergence of [MATH] is ensured and provide the numerical evidence of the difference of the mean and typical values of the return probability, giving a signature of non-ergodic behavior of eigenstates, in the range [MATH] Furthermore, we have shown that in this range [MATH] decays l... |
and provides predictions of the wavepacket evolution with time, which are worth to further verify For small disorder strengths, [MATH] [MATH] shows oscillations which survive in the thermodynamic limit, confirming the existence of the fully ergodic phase consistent with the standard Wigner-Dyson behavior |
Our analysis based on dynamical properties allow us to conclude that the RRG is in the fully- ergodic phase at least for small disorder [MATH] |
and in non-ergodic extended phase at least for the [MATH] What implies that a transition between ergodic and non-ergodic phases should exist in the range [MATH] in agreement with Altshuler et al. 2016 |
The subdiffusion results for the wavepacket spreading ( ) guessed from the classical model need more analysis and we left them for further investigations. |
Acknowledgements. The authors would like to thank A. L. Burin, F. Evers, M. Heyl, V. E. Kravtsov, G. Lemarié, F. Pollmann and K. S. Tikhonov for many interesting discussions. A.S. is partially supported by a Google Faculty Award. S.B. acknowledges support from DST, India, through Ramanujan Fellowship Grant No. SB/S2/RJ... |
Note added. During the consideration of the paper in the journal, authors have become aware of the work Tikhonov and Mirlin ( 2018 considering, in particular, the return probability [MATH] in RRG and having similar results of the stretched-exponential decay of [MATH] |
with time in the corresponding range of system sizes and time intervals. Appendix A Power-law banded random matrix In this section we provide additional data for the power-law banded random matrix (PLBM). |
A.1 Return Probability In this section we study the decay of [MATH] with time for the PLBM. We perform the time evolution using exact diagonalization method as mentioned in the main text. The decay of the return probability at the critical point is already shown in the main text, here we show a similar data in the ergo... |
and [MATH] decay algebraically, but with different power laws. Instead, in the ergodic phase ( [MATH] [MATH] and [MATH] decay asymptotically at the same rate, as shown in Fig. As a consequence of a different rate of decay between [MATH] and [MATH] , in the multifractal phase the saturation values [MATH] and [MATH] |
can have different scaling to zero as a function of [MATH] [MATH] and [MATH] [MATH] ), while in an ergodic phase [MATH] Figure shows [MATH] and [MATH] in the ergodic phase for the PLBM for one system size [MATH] in this case we projected onto eigenstates with energies within [MATH] with [MATH] , where [MATH] is the ene... |
A.2 Probability distribution of IPR The difference in scaling between the mean and typical value originates from the tails of the probability distribution of the long time limit of the return probability, [MATH] , which is the inverse participation ratio ( [MATH] ). In the multifractal region the probability distributi... |
While in the ergodic phase, Fig. (a) the probability distribution of [MATH] has exponentially decaying tails and it shrinks with increasing system size with a well defined mean close to 1. This validates again our previous observation in Fig. which is that at log time the two exponents [MATH] , i.e., [MATH] |
Appendix B Random Regular Graph In this section we provide additional data of return probability and the probability distribution of inverse participation ratio for Random Regular Graph (RRG). |
B.1 Return probability Figure (Figure 10 ) shows [MATH] [MATH] ) for several values of [MATH] and for the three largest system sizes ( [MATH] ) and for several [MATH] For [MATH] and for all [MATH] oscillations are present and the decay of [MATH] is consistent with [MATH] , giving the evidence of the existence of an erg... |
Figure 11 shows mean and typical values of [MATH] in the ergodic phase [MATH] for [MATH] . As described in the main text in the ergodic phase there is no distinction between the way of tacking the disorder average (mean or typical value). Moreover, in the main text, we claim that [MATH] for [MATH] . In what follows, we... |
The idea is to consider separately [MATH] and [MATH] . If a stretched exponent is assumed for the decay of [MATH] [EQUATION] [EQUATION] |
In the long time limit [MATH] , nevertheless this regime requires [EQUATION] Figure 12 shows [MATH] for [MATH] , underlining the plateau ( [MATH] ). Furthermore, [MATH] , what also implies that [MATH] |
B.2 Probability distribution of [MATH] Like in the previous section for PLBM, here we also analyze the full probability distribution of [MATH] at different disorder strength for the RRG. Figure 13 shows the probability distribution [MATH] of the rescaled variable [MATH] for disorder values [MATH] In the so-called ergod... |
# Source: arxiv 1805.12478 # Title: Snakes and ghosts in a parity-time-symmetric chain of dimers # Sections: all # Downloaded: 2026-03-02T08:56:05.481774+00:00 |
Snakes and ghosts in a parity-time-symmetric chain of dimers Abstract We consider linearly coupled discrete nonlinear Schrödinger equations with gain and loss terms and with a cubic-quintic nonlinearity. The system models a parity-time ( [MATH] )-symmetric coupler composed by a chain of dimers. Particularly we study si... |
pacs: 47.54.-r, 46.32.+x, 47.20.Ky, 47.55.P- Introduction Many nonlinear dynamical systems, such as spatially extended nonlinear dissipative systems Purwins2010 , vertical-cavity semiconductor optical amplifiers Barbay2008 , nematic liquid crystal layers with spatially modulated input beam Haudin2011 , and magnetic flu... |
In most of previous works devoted to localized states and snaking in continuous systems, the Swift-Hohenberg equation has been widely used as a model for pattern formation since it is the simplest model equation that illustrates the pinning effect Sakaguchi1996 Woods2006 Burke2006 Burke2007 Burke2007a . In general, the... |
Like spatially continuous systems, several discrete systems can display the snaking behavior with the locking effect, however, being attributed to the imposed lattice. Examples include the discrete bistable nonlinear Schrödinger equation Carretero-Gonzalez2006 Chong2009 Chong2011 , which leads to a subcritical Allen-Ca... |
This paper is devoted to a detailed numerical and analytical study of homoclinic snaking in a parity-time ( [MATH] ) system. The physical problem is a chain of dimers that has two arms with each arm described by a discrete nonlinear Schrödinger equation with gain or loss and with cubic-quintic nonlinearity. While the c... |
A system of equations is [MATH] symmetric when it is invariant with respect to combined parity ( [MATH] ) and time-reversal ( [MATH] ) transformation bend98 bend99 bend07 . In the context of Schrödinger Hamiltonians with a complex potential [MATH] [MATH] symmetry requires the potential to satisfy the condition [MATH] ,... |
The continuum limit of the set-up studied herein was considered in burl16 burl13 . In optical media, such nonlinearity can be obtained from a saturation of the Kerr response, which with the increase of the intensity will introduce a self-defocusing quintic term in the expansion of the refractive index cout91 smir06 . I... |
The report is outlined as follows. The [MATH] -symmetric chain of dimers with cubic-quintic nonlinearity is discussed in Section II . In Section III , we study spatially uniform solutions and their stability. We obtain that symmetric states can become unstable due to pitchfork bifurcations. The emanating solutions are ... |
II Mathematical model and stability of solutions The governing equations describing [MATH] -symmetric chains of dimers are of the form |
[EQUATION] The derivative with respect to the evolution variable (i.e., the propagation distance, if we consider their application in fiber optics) is denoted by the overdot, [MATH] [MATH] are complex-valued wave function at site [MATH] with the propagation constant [MATH] [MATH] is the constant coefficient of the hori... |
System ( ) is [MATH] -symmetric because it is invariant with respect to the action of the parity [MATH] and time-reversal [MATH] operators given by |
[EQUATION] Next, we consider the equations for standing wave solutions of Eqs. ( ), obtained from setting [MATH] and substituting [MATH] into ( ), |
[EQUATION] Here, [MATH] . We can assume that [MATH] is real-valued because of the phase invariance of the governing equations ( ). Splitting the real and imaginary parts of the equations and simplifying them will yield |
[EQUATION] which is also known as the discrete Allen-Cahn equation, where [MATH] [MATH] with [MATH] for the minus sign and [MATH] for the plus sign, which corresponds to the so-called symmetric and antisymmetric configuration between the arms, respectively. Note that ( ) will have no real solution when [MATH] . This is... |
The linear stability of a standing wave solution is determined as follows. Introducing the ansatz [MATH] [MATH] [MATH] , and substituting it into Eq. ( ) will yield from the terms at [MATH] ) the linear eigenvalue problem |
[EQUATION] where [MATH] Generally the spectrum will consist of two types, i.e. continuous and discrete spectrum or eigenvalue. A solution is unstable when there exists [MATH] with Re [MATH] . However, if [MATH] is a spectrum, so are [MATH] and [MATH] |
kiri16 . A solution is therefore (linearly) stable only when Re [MATH] for all [MATH] , i.e. it is neutral stability. Nonlinear stability may be obtained numerically by evolving a perturbed solution in Eqs. ( ) for a long while, which analytically is still an open problem due to the absence of a Hamiltonian structure o... |
Numerically we solve the steady-state equations of ( ) using a Newton-Raphson method in Matlab . A pseudo-arclength continuation scheme is implemented to do numerical continuation past a turning point. To model the infinite domain, we use a periodic boundary condition with a large number of lattices. The typical value ... |
III Uniform solutions Equation ( ) has uniform solutions [MATH] that are given by [EQUATION] Besides [MATH] must be less than [MATH] , the uniform solution ( ) also requires [MATH] to exist. Under competing cubic-quintic nonlinearities, i.e., [MATH] , we will have two branches of non-zero uniform solutions. |
The stability of uniform solutions ( ) can be determined by computing their continuous spectrum. Introducing the plane-wave ansatz |
[MATH] [MATH] , and substituting it into ( ) will yield a dispersion relation. Continuous spectrum of the equilibrium is then obtained by setting [MATH] and [MATH] in the dispersion equation. |
The dispersion relation of the trivial equilibrium [MATH] is [EQUATION] with [MATH] , from which we obtain the continuous spectrum |
[MATH] and [MATH] with the spectrum boundaries [EQUATION] The equilibrium is therefore stable for [MATH] and unstable otherwise. Continuous spectra of the non-zero solution can be obtained similarly. |
When [MATH] , bifurcation diagrams of the nonzero solutions are shown in Fig. (a,b) for two values of [MATH] . In this case, the chain is uncoupled and one obtains the dimer, which was studied in pick13 (see also references therein) for [MATH] and in li17 for nonzero [MATH] |
Consider antisymmetric solutions along branch ’a’. We obtain that the low intensity solution, i.e. the lower branch, is stable, while the high one is not. Branch ’s’ generally corresponds to stable symmetric solutions, but there is a small portion of unstable branch due to pitchfork (i.e. spontaneous symmetry breaking)... |
In panel (b), we consider [MATH] . As the gain/loss parameter increases towards the critical value [MATH] , branches ’s’ and ’a’ become closer to each other. At the critical value, the two branches coincide, i.e. we obtain a turning point. This is due to the fact that when studying time-independent solutions, the gover... |
Panel (c) shows the effect of coupling constant that clearly only affects the stability of the equilibrium. Now we obtain that branch ’a’ and the lower part of branch ’s’ have become unstable. |
IV Asymmetric solutions as ghost states In the classical cubic dimer, i.e., Eqs. ( ) with [MATH] , symmetric solutions are known to be unstable for [MATH] due to a pitchfork (symmetry-breaking) bifurcation (see, e.g., smer97 ragh99 kirr08 kirr11 jian14 rodr13 and references therein). At the bifurcation point, an asymme... |
When [MATH] , symmetric solutions still can become unstable, but the bifurcating asymmetric ones will no longer exist pick13 rodr13 . This observation was first reported in hill06 . Cartarius et al. cart12 provide an analytic continuation of the asymmetric solutions that emerge as ghost states, namely, a solution of th... |
To obtain asymmetric states of our problem, consider again time-independent equations of Eqs. ( ) and their conjugates, where the propagation constant [MATH] is now complex-valued, i.e., [MATH] |
[EQUATION] The imaginary part [MATH] needs to be determined from a consistency equation below. Multiplying Eqs. ( 10a )-( 10d ) with [MATH] [MATH] [MATH] , and [MATH] , respectively, summing up the infinite-dimensional vectors over [MATH] , and adding the resulting equations will lead to the equation for [MATH] |
[EQUATION] It is clear that [MATH] will vanish either when [MATH] , i.e., symmetric and antisymmetric solutions, or when [MATH] In Fig. (b,c) the branch of asymmetric solutions is obtained from time-independent equations of ( ) with ( 11 ). We also have determined the states’ stability by solving the corresponding line... |
Localised solutions We consider discrete solitons of Eqs. ( ) satisfying the localisation conditions [MATH] as [MATH] . It is known that there are two fundamental localized solutions existing for any coupling constant [MATH] , i.e. an intersite (bond-centred) and onsite (or site-centred) discrete mode with an even and ... |
Fixing the coupling [MATH] and varying the propagation constant [MATH] , we depict the bifurcation diagrams of the two types of discrete modes in Fig. . For each symmetric and antisymmetric configuration between [MATH] and [MATH] , there are two branches that correspond to the site-centred and bond-centred solutions. |
In addition to symmetric solutions, there are also solutions that are asymmetric between the arms or asymmetric in the same arm. The former type of solutions corresponds to that giving ’as’ branches in Fig. , while the latter one constitutes the ’ladders’ connecting snaking branches of onsite and intersite modes in Fig... |
In Fig. we plot profiles of several localized solutions and their spectrum in the complex plane. Unstable solutions are due to spectra with nonzero real part, which belong to the red dashed segment in Fig. |
Bifurcation diagrams in Fig. form a snaking structure. Even though such structures have been reported before Carretero-Gonzalez2006 Chong2009 Chong2011 Taylor2010 Matthews2011 , the effect of the gain/loss parameter that yields different stability behaviours along the curves is novel. The region between the boundaries ... |
Up in the snaking structures (represented by, e.g., point 4 in Fig. (a)), the stability of the branches is similar to those in Fig. . This is because the corresponding localized solutions have long plateau of nonzero uniform solutions, i.e. the stability is mainly determined by the continuous spectrum of the nonzero un... |
We show in Fig. the typical time evolution of unstable solutions in Fig. . While Fig. indicates a clear blow up of the wave field with gain, which is common in [MATH] -systems pick13 , Fig. shows intensity oscillations. The fact that the oscillations persist for quite a while is interesting by itself as [MATH] -symmetr... |
In the spatially uniform case, the branches of symmetric and antisymmetric solutions between the arms move towards each other as [MATH] increases and merge at [MATH] . It is also the same with the case of localized solutions, i.e., the two snaking bifurcation diagrams in Fig. become closer with the increase of [MATH] a... |
VI Ghost states in the [MATH] broken phase In the broken [MATH] symmetric region ( [MATH] ), the trivial state [MATH] is unstable. The typical time-evolution is that [MATH] as the field with gain will blow-up, while [MATH] that experiences loss decays. |
The [MATH] -phase transition ( [MATH] ) is characterized by the merger of symmetric and antisymmetric solutions in a fold bifurcation. A follow-up question is what becomes of them past the critical point. It was also due to cart12 that it is possible to provide an analytic continuation for the original model in a nontr... |
[EQUATION] where [MATH] . The parameter [MATH] is again complex valued where the imaginary part must satisfy a self-consistency equation. Doing the same calculation, we also obtain Eq. ( 11 ). Because of complex [MATH] , Eqs. ( 12 ) are not [MATH] -symmetric and their solutions are also ghost states. The relation betwe... |
We have computed the continuation of branches in Fig. past the [MATH] phase transition point. We present bifurcation diagrams of the ghost states in Fig. . We have also computed their stability from the corresponding linear eigenvalue problem of the dual system ( 12 ). |
There are two uniform states that are mirror images of each other. Solutions with high intensity in [MATH] are stable (in the sense of ( 12 )), while the other ones with low [MATH] are unstable, i.e., stable solutions correspond to [MATH] |
In the sense of the original system ( ), the stable solutions will lead to growth in time as the parameter [MATH] is positive. On the other hand, the ones with negative [MATH] decay in time and shall not be observed in direct numerical simulations. This thus means that ghost states of ( 12 ) may be interpreted as self-... |
In Figs. and we plot bifurcation diagrams of localized ghost states and their profiles and stability computed through the ’dual’ equations ( 12 ) and ( 11 ). We observe that the homoclinic snaking persists and that solutions with larger [MATH] are stable (in the sense of the dual equations ( 12 )). It is important to n... |
VII Analytical approximations In this section, we will study the width of the snaking region in Fig. as a function of, e.g., the coupling constant [MATH] . We will derive an asymptotic approximation of the width. The approach is distinguished in two different regions, i.e. small and large coupling. Because the width of... |
VII.1 Small coupling case When [MATH] is small, as we follow the snaking structure upward (see Fig. ), at the leading order there is only one site that is ‘active’, with the remaining sites being either at [MATH] or at the plateau of a nonzero uniform solution. Such behaviours were observed and exploited in many ways b... |
From ( ), we assume that up in the snaking diagram only the following nodes are involved in the dynamics, i.e. [EQUATION] Note that we only use the ‘ [MATH] ’ sign for the uniform solution forming the plateau, which is the upper branch in Fig. . Substituting ( 13 ) into the time-independent discrete equation ( ) will y... |
[EQUATION] In general ( 14 ) will have five roots. The roots relevant to our study are the positive ones. As [MATH] varies, two of the roots will collide in a saddle-centre bifurcation. This condition corresponds to the boundaries of the snaking region. The condition for the collision is when a local maximum or minimum... |
[EQUATION] i.e. [EQUATION] Substituting ( 16 ) into ( 14 ) and solving the resulting equation for [MATH] asymptotically give us [EQUATION] |
The snaking width [MATH] is then given approximately by the difference between the two functions. VII.2 Large coupling case Following flac96 sagd88 , Eq. ( ) is identical to the equation |
[EQUATION] where [MATH] and [MATH] . The proof is as follows. First, from ( 18 ), we obtain that [MATH] or upon integration [MATH] . Thus, |
[EQUATION] Next, integrate ( 18 ) from [MATH] to [MATH] to obtain [EQUATION] Using Eq. ( 19 ), Eq. ( 20 ) becomes the lattice equation ( ). |
Using Fourier series, we can then write the summation [MATH] , which converges to the Dirac comb non-uniformly. Taking only the first harmonic, ( 18 ) then becomes |
[EQUATION] which can be expected to approximate ( ) in the large coupling case for [MATH] flac96 Without the periodic potential [MATH] , Eq. ( 21 ) has a front solution given by |
[EQUATION] when [EQUATION] Following Susanto2011 Matthews2011 , we will approximate the solutions along the snaking structure by |
[EQUATION] where [MATH] is the phase-shift distinguishing the two branches, i.e. [MATH] for the on-site and intersite solutions, respectively. [MATH] is the length of the plateau, which is presently an unknown variable. |
Using the standard variational argument, requiring ( 24 ) to be an optimal solution of ( 21 ) implies that [MATH] must satisfy the equation (see, e.g., dawe13 |
[EQUATION] where [MATH] is set to be near the Maxwell point ( 23 ), i.e. [MATH] Equation ( 25 ) can be simplified at the leading order for [MATH] to |
[EQUATION] The width of the snaking region is then simply given by [EQUATION] which is exponentially small. As pointed out by one of the referees, the exponential factor in the approximation ( 27 )-( 28 ) is correct, which can be justified in the following way, explained in details in kozy13 . The continuous problem ( ... |
We show in Fig. (a) the width of the snaking region computed numerically and our approximations ( 17 ) and ( 27 ). One can see good agreement between them. |
Note that Fig. (a) does not allow one to check ( 27 ) because of the very fast exponential decay as [MATH] increases. We depict in Fig. (b) the comparison in a log plot, where it is clear that the approximation deviates from the numeric as [MATH] increases. Using the function |
[EQUATION] we curve fit the numerical result where obtain that [MATH] and [MATH] . There is a slight difference in the algebraic scale, which may be attributed to the step of taking the first harmonic only in Eqs. ( 21 ). |
VIII Conclusion Spatially uniform and localized solutions (site-centered and bond-centered modes) and their bifurcation diagrams that form a snaking structure in a parity-time ( [MATH] )-symmetric coupler composed by a chain of dimers have been discussed. It has been shown that the gain/loss coefficient does not influe... |
In the broken [MATH] symmetry region [MATH] , we have also analysed the continuations of the time-independent solutions, that are called ghost states. Interestingly localized ghost states have also been observed to exhibit a homoclinic snaking in their bifurcation diagrams, with the same width of pinning region as that... |
Asymptotic approximations of the width of the snaking region have been derived in two different limits, i.e. strong and weak coupling between the dimers. The approximations have been compared with numerical results where good agreement is obtained. |
Acknowledgement We acknowledge the two referees for their detailed and valuable suggestions. The H.S. and N.L. acknowledge financial support from the UK Engineering and Physical Sciences Research Council (Grant No. EP/M024237/1). R.K. gratefully acknowledges financial support from Lembaga Pengelolaan Dana Pendidikan (I... |
# Source: arxiv 1805.12556 # Title: Renormalization of Sparse Disorder in the Ising Model # Sections: all # Downloaded: 2026-03-03T05:15:41.304553+00:00 |
Renormalization of Sparse Disorder in the Ising Model Abstract We consider the renormalization of quenched bond disorder in the Ising model in the limit that it is sparse – highly localized and vanishing in the thermodynamic limit. We begin in 1D with arbitrary disorder assigned to a finite number of bonds and study ho... |
Introductory remarks The study of structural variations or quenched disorder in spin systems is of intrinsic interest due to the wide ranging relevance of spin models in representing collective behavior across diverse physical systems spin glass . For example, in complex networks one often finds the existence of pre-sp... |
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