Asymmetric wave transmission through one dimensional lattices with cubic-quintic nonlinearity
AAsymmetric wave transmission through one dimensional lattices with cubic-quinticnonlinearity
Muhammad Abdul Wasay , , ∗ Department of Physics, University of Agriculture, Faisalabad 38040, Pakistan Center for Photon Information Processing, School of Electrical Engineering and Computer Science,Gwangju Institute of Science and Technology, Gwangju 61005, South Korea Center for Theoretical Physics of Complex Systems,Institute for Basic Science (IBS), Daejeon 34051, Republic of Korea
One dimensional lattice with an on-site cubic-quintic nonlinear response described by a cubic-quintic discrete nonlinear Schr¨odinger equation is tested for asymmetric wave propagation. Thelattice is connected to linear side chains. Asymmetry is introduced by breaking the mirror symmetryof the lattice with respect to the center of the nonlinear region. Three cases corresponding to dimer,trimer and quadrimer are discussed with focus on the corresponding diode-like effect. Transmissioncoefficients are analytically calculated for left and right moving waves via backward transfer map.The different transmission coefficients for the left and right moving waves impinging the lattice giverise to a diode-like effect which is tested for different variations in asymmetry and site dependentcoefficients. We show that there is a higher transmission for incoming waves with lower wavenumbersas compared to the waves with comparatively larger wavenumbers and a diode-like effect improvesby increasing the nonlinear layers. We also show that in the context of transport through suchlattices, the cooperation between cubic and quintic nonlinear response is not ”additive”. Finally,we numerically analyse Gaussian wave packet dynamics impinging on the CQDNLS lattice for allthree cases.
PACS numbers:
I. INTRODUCTION
When wave propagates through a medium which isquite sensitive to the incoming wave intensity, one mustincorporate some nonlinear corrections to accommodatethis when describing wave propagation through suchmedium. The prospect to model devices (with nonlin-ear refractive index) that could serve for controlled en-ergy or mass flow is a compelling challenge both from atechnological and scientific perspective.Diode is a device which allows a unidirectional trans-port and due to this property it can serve as one such de-vice. Search for the diode feature of various phenomenonhave been explored in the literature, for example, inacoustics [1–3], heat flow [4–6] and electromagnetic waves[7–10]. When dealing with nonlinear media, the simplestpossibility for controlled wave propagation is to devisea ’wave diode’ i.e., having asymmetric transmission ofwaves along two opposite directions of propagation. Thehypothesis of reciprocity theorem forbids this possibilityin a linear system [11–13].In a linear system, to break the time-reversal symme-try one needs to introduce an external field(electric ormagnetic), as is the case in an optical diode. However,much effort can be avoided if we instead have a nonlin-ear media. This approach [14, 15], seems to be morenatural as one can use the nonlinear properties of themedia (material) itself to break the parity symmetry and ∗ Electronic address: [email protected] it thus provides a variety of new features for controlledwave propagation.Nonlinearity leading to asymmetric propagation isstudied in many domains. To the best of our knowledge,the first work of this kind in the literature is related to theasymmetric transmission of phonons through a nonlinearlayer between two different crystals [16]. The idea hasbeen popular in the field of nonlinear optics, for exam-ple in [17, 18], a so called all-optical diode was proposed.In [19], a thermal diode was proposed which is capableof transmitting heat asymmetrically between two differ-ent sources by means of nonlinearity, a similar study in[20] was done for the rectification of heat conduction bymeans of the asymmetry and non-harmonic nature of thesystem. The propagation of sound waves is another im-portant issue and a so-called acoustic wave diode wasproposed in [21].In a discrete nonlinear setting, wave propagation hasbeen studied in different physical contexts [22–25]. Thediscreteness in such systems is attributed to a weak in-teraction between different elements of the system, forinstance, BEC trapped in optical lattices, and coupledoptical waveguides.In the context of asymmetric wave transmissionthrough a 1D layered photonic crystal lattice, it will beinteresting to see how transmission is effected by a higherorder (quintic) on-site nonlinear response. We will use aset of discrete nonlinear Schr¨odinger (DNLS) equationswith a local (on-site) cubic-quintic nonlinearity. Thiscubic-quintic DNLS model has been studied for mobilityregimes of solitons in 1D lattices [26] and in 2D lattices[27]. DNLS models have also been used to study variousrelated phenomenon [28–30]. a r X i v : . [ n li n . PS ] O c t In this paper we will work with a DNLS equation hav-ing on-site focusing cubic-quintic nonlinearity. We willinvestigate the scattering phenomenon for two, three andfour nonlinear layers. The DNLS equation is equippedwith variable site dependant coefficients in order to de-scribe the nonlinear features of different layers. The sys-tem is such that these nonlinear sites are embedded into alinear lattice, and are connected to linear side chains. Asa physically relevant case we will also analyse the dynam-ics of an incident Gaussian wave packet on this CQDNLSlattice system for all three cases.
II. THE MODEL
The set up assumes an on-site (local) cubic-quinticnonlinear response, which can be modelled by a set of dis-crete nonlinear Schr¨odinger equations with local cubic-quintic nonlinearity.The time dependent DNLS with focusing cubic-quinicnonlinearity is given by[26][27] i dψ n dt = V n ψ n − ( ψ n +1 + ψ n − )+ γ n | ψ n | ψ n + ν n | ψ n | ψ n (1)Here V n is the potential on site n , γ n and ν n representthe on-site cubic and quintic nonlinearities respectively,and γ n , ν n > ψ n is the amplitude of the field at the n -th lattice site. The Hamiltonian is H = − (cid:88) n [( ψ ∗ n ψ n +1 + ψ n ψ ∗ n +1 ) + V n | ψ n | + γ n | ψ n | + ν n | ψ n | ] (2)with this Hamiltonian one can derive the equation ofmotion (1) by using i dψ n dt = ∂H∂ψ ∗ n (3)The dynamical equations (1) have solutions of the form ψ ( t ) = φe − iωt , where φ is independent of t , substitutingthis in (1) leads to ωφ n = V n φ n − φ n +1 − φ n − + γ n | φ n | φ + ν n | φ n | φ n (4)where ω is the spatial frequency and φ n is the complexamplitude on site n with potential V n . The nonlinearsites are embedded in a linear lattice and are connectedto linear side chains where the wave can propagate freely,therefore we can say that γ n , ν n and V n are non-vanishingonly for 1 ≤ n ≤ N , here n represents nonlinearity ata particular site and N represents the total number ofnonlinear sites. Let us consider plane wave solutions ofthe form φ n = (cid:18) R e ikn + Re − ikn n ≤ T e ikn n ≥ N (cid:19) (5)where R , R and T are the amplitudes of incident,reflected and transmitted wave respectively. As men-tioned above, in the linear region with n > N or n < ω = − k ).For n = 0, we have φ = R + R (6)and for n = 1 φ = R e ik + Re − ik (7)With φ and φ we can get R and R in terms of φ and φ R = φ e ik − φ e ik − e − ik (8)and R = φ e − ik − φ e − ik − e ik (9)The extent to which an incident wave is transmittedis calculated by the transmission coefficient t ( k, | T | ) = | T | | R | . We will calculate these coefficients by a backwardtransfer map [31–33], obtained by rearranging Eq.(4), φ n − = − φ n +1 + ( V n − ω + γ n | φ n | + ν n | φ n | ) φ n (10)It is useful to introduce the following notation for laterconvenience, we define δ j = V j − ω + γ j | φ j | + ν j | φ j | (11) III. RESULTS
With the backward iterative map at hand, we willnow compute the transmission coefficients for the caseof dimer (two nonlinear sites), trimer (three nonlinearsites) and quadrimer (four nonlinear sites) in the follow-ing subsections.
A. DIMER
We consider the simplest case of two nonlinear layers,i.e., N = 2: The dimer. The cubic-quintic DNLS with N = 3 and N = 4 will be considered in the followingsubsections. From Eq.(10) with n = 2, φ = T e ik ( δ − e ik ) (12)where, δ = ( V − ω + γ | T | + ν | T | )For n = 1, we have φ = T e ik [ δ ( δ − e ik ) −
1] (13)where, δ = V − ω + γ | T | | ( δ − e ik ) | + ν | T | | ( δ − e ik ) | and from Eq.(9) | R | = | T | | ( δ − e ik )( δ − e ik ) − | | e − ik − e ik | (14) ⇒ | T | = | R | | e − ik − e ik | | ( δ − e ik )( δ − e ik ) − | (15)So, the sought transmission coefficient is t ( k, | T | ) = (cid:12)(cid:12)(cid:12)(cid:12) e − ik − e ik ( δ − e ik )( δ − e ik ) − (cid:12)(cid:12)(cid:12)(cid:12) (16)We have calculated the transmission coefficient for leftmoving wave with k >
0. To calculate the transmis-sion coefficient for right moving wave we assume thatthe sample is flipped such that the encountered sites arenow labelled as,1 → N (cid:48) , → ( N − (cid:48) , ..., N → k < V n = V N − n +1 i.e to change V by V andvice versa. Let us introduce the notation ζ = V − ω + γ | T (cid:48) | + ν | T (cid:48) | (17)and ζ = V − ω + γ | T (cid:48) | | ( ζ − e ik (cid:48) ) | + ν | T (cid:48) | | ( ζ − e ik (cid:48) ) | (18)Using a backward transfer map as discussed above, forthe left moving wave with k (cid:48) = − k we get | R (cid:48) | = | T (cid:48) | | ( ζ − e ik (cid:48) )( ζ − e ik (cid:48) ) − | | e − ik (cid:48) − e ik (cid:48) | (19)and thus the transmission coefficient turns out to be t (cid:48) ( k (cid:48) , | T (cid:48) | ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e − ik (cid:48) − e ik (cid:48) ( ζ − e ik (cid:48) )( ζ − e ik (cid:48) ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (20)The lattice is still mirror symmetric with respect tothe center of the nonlinear region, we must break thissymmetry to achieve an asymmetric transmission leading to the desired diode effect. Once the symmetry is broken,we will have different transmission coefficients for left andright moving wave. This can be done in different ways.We break the symmetry by chosing different potentialson each of the two nonlinear sites, i.e., V n (cid:54) = V N − n +1 .For dimer N = 2, we take V n = V (1 ± ε ), and we chooseto have positive sign at site 1 and negative sign at site 2, ε is the extent of asymmetry and V is the depth of thepotential. FIG. 1: | T | as function of | R | . (a): CQDNLS dimer . (b):
Purely cubic DNLS dimer
Fig.1 depicts the relationship between incident ( | R | along horizontal axis) and transmitted intensity ( | T | along vertical axis). Transmission curves in Fig.1(a) cor-respond to CQDNLS dimer with V = − . , ε = 0 . , γ =1 , ν = 0 . | k | = 0 .
1, and in Fig.1(b) for a purely cu-bic DNLS dimer with V = − . , ε = 0 . , γ = 1 . | k | = 0 .
1. The dashed black line corresponds to the sym-metric case ε = 0. The asymmetric branch with ε (cid:54) = 0have two oppositely directed waves with differently de-tuned resonances responsible for different transmissioncoefficients and thus for the asymmetric transmission.The first window of maximal transmission in CQDNLSmodel is slightly broader as compared to the correspond-ing window in cubic DNLS model [14, 35] with same pa-rameter values, however, this trend is reversed for the sec-ond window in both models. Moreover, the first windowsoccur roughly at the same incoming intensities in bothmodels, while the second window is displaced to slightlylower intensities in the CQDNLS model as compared tothe cubic DNLS model. Further, please note that thebistable behavior persists in the CQDNLS dimer model,which is apparent from the corresponding (bistability)windows in Fig.1(a).The purpose of plotting curves for the cubic DNLSdimer stems from the idea that one could possibly mimicsimilar transmission curves (as for the CQDNLS dimer)by increasing the purely cubic response in a cubic DNLSdimer. However, as one can see in Fig.1, the transmissionpattern is somewhat different in Fig.1(a) and Fig.1(b),with same parameters except we chose γ = 1 . γ = 1 since we enhanced the cubic response in the hopethat this could lead to the same transmission curves asfor the CQDNLS model. Thus the cooperation betweencubic and quintic response in a CQDNLS model doesnot seem to be of an ”additive” type which one usuallypresumes.Further also note that in the CQDNLS dimer case,the windows of maximal transmission keep broadeningwith the corresponding increase in asymmetry level until ε ∼ .
1, where the first window starts to diminish.
FIG. 2: Transmission coefficient t as a function of transmittedintensity | T | : For Dimer
Fig.2 shows the transmission coefficient t ( k, | T | ) alongvertical axis as a function of transmitted intensity | T | along horizontal axis with V , = V (1 ± ε ) , N = 2 , V = − . , γ = 1 , ν = 0 . , ε = 0 . , | k | = 0 .
1. Transmission Coefficient
In this subsection we present plots for the transmissioncoefficient as a function of transmitted intensity | T | and k . Fig.3 corresponds to dimer with varying asymmetrywhile the quintic response fixed at ν = 0 .
5, and all otherparameters as before. The transmission seems to reduceas we increase the asymmetry, specially above ε = 0 . FIG. 3: Transmission coefficient as a function of | T | and k .Varying asymmetry level:(a) ε = 0 .
05 to (d) ε = 0 . Fig.4 is produced by fixing the asymmetry to ε = 0 . ν (the quintic nonlinearity) strengthen ascompared to γ (= 1), the cubic nonlinearity. The plotin Fig.4(a) corresponds to the purely cubic case ( ν = 0),while we increase ν in the subsequent plots from ν = 0 . ν = 0 . ν is increased, theoverall transmission is reduced, however, this effect is notsignificantly large for waves with small k , as compared tothe waves with large k . Note that with asymmetry fixed at ε = 0 .
05 the region for maximal transmission with CQnonlinearity is smaller as compared to the purely cubiccase [14].
FIG. 4: Transmission coefficient as a function of | T | and k : For varying quintic nonlinear response . Finally, in both figures Fig.3 and Fig.4, one may notethat there are two transmission peaks which split intofour as k increases. This split occurs roughly around k ∼ π/ FIG. 5: Transmission coefficient as a function of | T | and k :(a): CQDNLS dimer with the nonlinear strength distributedbetween cubic and quintic response. (b): Cubic DNLS dimerwith the same nonlinear strength purely as cubic response
2. Rectifying Factor
To see where the best diode effect occurs, following[14], a rectifying factor is introduced as follows f = t ( k, | T | ) − t ( − k, | T | ) t ( k, | T | ) + t ( − k, | T | ) (21)Fig.6 is a plot for rectifying factor as a function of | T | and k , for the dimer case. FIG. 6: Rectifying factor as a function of | T | and k : Forvarying asymmetry level.
The asymmetry is increased from ε = 0 .
05 in (a) to ε = 0 .
3. Gaussian wavepacket dynamics for dimer
It is instructive to look at a Gaussian wave packet scat-tering for CQDNLS lattice system. We consider a dimerembedded inside a lattice with M = 1000 sites with openboundary conditions. The dimer is placed at 500th and501th site. The wave packet at t = 0 (initial condition)for the right incidence is of the form [14, 15] ψ n (0) = B exp (cid:34) − ( n − n − M ) w + ik ( n − M (cid:35) (22) B is the incoming amplitude, n lattice starting point, w width of wavepacket.The initial wave packet for the left incidence is givenby ( k → − k ) ψ n (0) = B exp (cid:34) − ( n − n − M − M ) w − ik ( n − M − M ) (cid:35) (23)Fig.7 depicts how the wave packet is scattered whenit hits the CQDNLS dimer in the middle of the lattice.For the same system parameters, the wave packet trans-mission coefficients are found to be t k> = 0 . t k< = 0 . FIG. 7: Gaussian wavepacket impinging on CQDNLS dimer.(a): Right incidence (b): Left incidence
B. TRIMER
We want to examine the transmission phenomenon forthe case when we have three nonlinear layers (sites), i.e.,a trimer, N = 3. Adopting the same procedure of back-ward transformer map as above, the transmission coeffi-cient for trimer is found to be t ( k, | T | ) = (cid:12)(cid:12)(cid:12)(cid:12) e ik − e − ik e ik − δ + ( e ik − δ )(1 − δ ( δ − e ik )) (cid:12)(cid:12)(cid:12)(cid:12) (24)with δ = V − ω + γ | T | + ν | T | (25) δ = V − ω + γ | T | | δ − e ik | + ν | T | | δ − e ik | (26) δ = V − ω + γ | T | | δ ( δ − e ik ) − | + ν | T | | δ ( δ − e ik ) − | (27) FIG. 8: | T | as function of | R | . (a): CQDNLS Trimer with ε = 0 .
05. (b): CQDNLS Trimer with ε = 0 . Plots in Fig.8 represent relationship between incidentintensity | R | and transmitted intensity | T | . The maxi-mal transmission occurs at two intervals. Plot in Fig.8(b)corresponds to an increased level of asymmetry ε = 0 . ε ∼ .
16 onlyfirst window for maximal transmission corresponding tolower incoming intensity survives, while the window cor-responding to the higher incoming intensity diminishesabove a critical value of asymmetry ε ∼ .
16. This is incontrast to the dimer case where instead the first diodewindow diminishes.
FIG. 9: Transmission coefficient t as a function of transmittedintensity | T | : For Trimer
We find that the first window for maximal transmis-sion has been displaced to higher incoming intensities andboth windows have shrunk (for fixed asymmetry) as com-pared to the dimer case. However, increasing asymmetrycan broaden the windows upto ( ε ∼ .
16) after whichonly one diode window survives. The overall transmis-sion seems to be reduced as compared to the dimer.Fig.9 shows for trimer, the transmission coefficient t ( k, | T | ) along vertical axis as a function of transmit-ted intensity | T | along horizontal axis with V , = V (1 ± ε ) , N = 3 , , V = V , V = − . , γ = 1 , ν = 0 . , ε =0 . , | k | = 0 .
1. Transmission Coefficient
As before, transmission coefficient as a function oftransmitted intensity | T | and k for increasing asymme-try level for the trimer are plotted in Fig.10 below. Thetransmission peak splitting phenomenon occurs againroughly around k = π/
2. Waves with smaller k ( ∼≤ π/ k ( ∼≥ π/ ε = 0 .
05 in (a) to ε = 0 . FIG. 10: Transmission coefficient as a function of | T | and k .For varying asymmetry level (Trimer). (a): ε = 0 .
05 to (d): ε = 0 .
2. Rectifying Factor
Using (21), the rectifying plots in Fig.11 for trimercase with varying asymmetry ε = 0 .
05 to ε = 0 . FIG. 11: Rectifying factor as a function of | T | and k : Forvarying asymmetry level (Trimer). (a): ε = 0 .
05 to (d): ε =0 .
3. Gaussian wavepacket dynamics for trimer
The scattering of an incoming Gaussian wave packetby a CQDNLS trimer in the middle of the lattice systemis depicted in Fig.12 below. We take Eqs. (22),(23) asthe initial condition. The trimer in this case correspondsto sites 500, 501 and 502 in the lattice with 1000 sites.The on site potential is chosen as V , = V (1 ± ε )and V = V , remaining parameters as before. FIG. 12: Gaussian wavepacket impinging on CQDNLSTrimer. (a): Right incidence. (b): Left incidence.
The wave packet transmission coefficients for both leftand right incidences are: t k> = 0 . t k< = 0 . C. QUADRIMER
We keep on increasing the number of nonlinear layers,and now we examine transmission phenomenon throughfour nonlinear sites, i.e., a quadrimer. For quadrimerthen N = 4. Using backward transformer map, the trans-mission coefficient for quadrimer is found to be | R | = (28) | T | | δ ( e ik − δ ) + ( δ − e ik )( δ δ δ − e ik δ δ − δ − δ + e ik ) | | e ik − e − ik | t ( k, | T | ) = (29) (cid:12)(cid:12)(cid:12)(cid:12) e ik − e − ik δ ( e ik − δ ) + ( δ − e ik )( δ δ δ − e ik δ δ − δ − δ + e ik ) (cid:12)(cid:12)(cid:12)(cid:12) with, δ = V − ω + γ | T | + ν | T | (30) δ = V − ω + γ | T | | δ − e ik | + ν | T | | δ − e ik | (31) δ = V − ω + γ | T | | δ ( δ − e ik ) − | + (32) ν | T | | δ ( δ − e ik ) − | δ = V − ω + γ | T | | ( δ − e ik )( δ δ − − δ | + (33) ν | T | | ( δ − e ik )( δ δ − − δ | FIG. 13: | T | as function of | R | : For Quadrimer
Fig.13 represents | T | as function of | R | , for N = 4.there is no diode effect for smaller incoming intensities ∼| R | ≤
1, however there are high transmission peaks forhigher incoming intensities, which become complicatedwith different merging peaks.Fig.14 is for t ( k, | T | ) vs | T | , with parameter values: V , = V (1 ± ε ) , N = 4 , V = V = V , V = − . , γ =1 , ν = 0 . , ε = 0 .
05 and | k | = 0 . N increases, the transmission pat-tern becomes complicated and it becomes difficult toidentify the peak shift which occurs due to asymmetry.As compared to the simplest case of two sites, the trans-mission pattern for increased sites is rather complicatedand regions with maximal transmission are scarce, see[34] for a related discussion on a cubic DNLS model. FIG. 14: Transmission coefficient t as a function of transmit-ted intensity | T | : For Quadrimer1. Transmission Coefficient
The transmission coefficient is plotted in Fig.15 for thequadrimer. There seems to be very little transmission forlarger wavenumbers k , as compared to a slightly bettertransmission for smaller wavenumbers. The transmissiontends to reduce as we increase the asymmetry. The over-all transmission for quadrimer is smaller than both thedimer and trimer case. FIG. 15: Transmission coefficient as a function of | T | and k : For varying asymmetry level (Quadrimer) .
2. Rectifying Factor
Rectifying factor for N = 4 with increasing asymme-try level is presented in Fig.16 below. The diode effectimproves but with overall transmission smaller than thepreceding cases (dimer and trimer). FIG. 16: Rectifying factor as a function of | T | and k : Forvarying asymmetry level (Quadrimer) .
3. Gaussian wavepacket dynamics for quadrimer
How an incoming Gaussian wave packet specified bythe initial conditions given in Eqs. (22) and (23) is scat-tered by a CQDNLS quadrimer placed at the center ofthe lattice, is shown in Fig.17. The quadrimer is chosento be embedded at sites 500,501,502 and 503. Along thelines of our previous discussion on quadrimer, we chose onsite potentials as: V , = V (1 − ε ) and V , = V ,other parameters as above. FIG. 17: Gaussian wavepacket impinging on CQDNLSQuadrimer. (a): Right incidence. (b): Left incidence.
Transmission coefficients for the wave packet in thiscase are: t k> = 0 . t k< = 0 . IV. SUMMARY AND CONCLUSION
We have investigated wave propagation through a non-linear system having an on-site cubic-quintic nonlinearresponse, using a set of cubic-quintic discrete nonlinearSchr¨odinger equations with site dependent coefficients asa model of the system. The purpose of this paper was toexamine this model for scattering phenomenon and as awave diode candidate. We introduced an asymmetry in the form of different on-site potentials to break the mirrorsymmetry of the system which resulted in an asymmet-ric transmission of the incoming waves. Bistable behaviorleading to an asymmetric transmission also persists in allthree cases, i.e., dimer, trimer and quadrimer.Based on the presumption that a slightly lower trans-mission in CQDNLS as compared to its purely cubicDNLS counterpart [14] could be due to an ”additive co-operation” between cubic and quintic response, we testedthe dimer and it turned out that this cooperation is notof an ”additive” type. A more exhaustive investigationinto the ”type of this cooperation” is left for the future.We probed for parameter regimes where this CQDNLSmodel can be used as a wave diode. We examined thecases of a dimer, trimer and a quadrimer and foundthat when we increase the nonlinear sites from two, thetransmission pattern becomes complicated while retain-ing some diode behavior which improves with increasingasymmetry but smaller overall transmission, see [14, 34]and references therein.Another important aspect of this study is that oneof the windows of maximal transmission diminish afterreaching a critical value of asymmetry, for dimer andtrimer. After ε ∼ .
1, first window disappears for thedimer, whereas in trimer case the second window disap-pears after ε ∼ . ν = 0 . k ∼ π in the plots for trans-mission coefficient for the dimer, trimer and quadrimer.Finally, to put the theoretical work in context withphysically relevant phenomenon, we presented results ofour numerical considerations of how a Gaussian wavepacket impinging on the CQDNLS lattice system is scat-tered by a dimer, trimer and quadrimer with their corre-sponding transmission coefficients.As a future prospect, it will be interesting to see whathappens when we saturate the nonlinear response [36],then we have an extra parameter to play with and per-haps we can fine tune the model and test it for bettertransmission. The work in this direction is in progressand we will soon report our results on this. Acknowledgments
M.A. Wasay would like to thank the IBS Centerfor Theoretical Physics of Complex system in Daejeon, South Korea for hospitality and financial support whereinitial part of this work was carried out.
Competing Financial Interests
The author declares no competing interests. [1] Li, X. F. et al.
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