Self-duality of One-dimensional Quasicrystals with Spin-Orbit Interaction
Deepak Kumar Sahu, Aruna Prasad Acharya, Debajyoti Choudhuri, Sanjoy Datta
SSelf-duality of One-dimensional Quasicrystals with Spin-Orbit Interaction
Deepak Kumar Sahu, Aruna Prasad Acharya, Debajyoti Choudhuri, and Sanjoy Datta
1, 3, ∗ Department of Physics and Astronomy, National Institute of Technology, Rourkela, Odisha, India Department of Mathematics, National Institute of Technology, Rourkela, Odisha, India Center for Nanomaterials, National Institute of Technology, Rourkela, Odisha, India (Dated: February 11, 2021)Non-interacting spinless electrons in one-dimensional quasicrystals, described by the Aubry-André-Harper (AAH) Hamiltonian with nearest neighbour hopping, undergoes metal to insulatortransition (MIT) at a critical strength of the quasi-periodic potential. This transition is related tothe self-duality of the AAH Hamiltonian. Interestingly, at the critical point, which is also known asthe self-dual point, all the single particle wave functions are multifractal or non-ergodic in nature,while they are ergodic and delocalized (localized) below (above) the critical point. In this work,we have studied the one dimensional quasi-periodic AAH Hamiltonian in the presence of spin-orbit(SO) coupling of Rashba type, which introduces an additional spin conserving complex hopping anda spin-flip hopping. We have found that, although the self-dual nature of AAH Hamiltonian remainsunaltered, the self-dual point gets affected significantly. Moreover, the effect of the complex andspin-flip hoppings are identical in nature. We have extended the idea of Kohn’s localization tensorcalculations for quasi-particles and detected the critical point very accurately. These calculationsare followed by detailed multifractal analysis along with the computation of inverse participationratio and von Neumann entropy, which clearly demonstrate that the quasi-particle eigenstates areindeed multifractal and non-ergodic at the critical point. Finally, we mapped out the phase diagramin the parameter space of quasi-periodic potential and SO coupling strength.
I. INTRODUCTION
In one dimensional (1D) lattice with random disor-ders, all the electronic single-particle states (SPS) ofthe non-interacting Anderson Hamiltonian localize ex-ponentially even if the strength of the disorder is ar-bitrarily small [1, 2]. In pure Anderson Hamiltonian,metal (extended SPS) to insulator (localized SPS) tran-sition exists only in three dimension (3D), while in twodimension (2D) there are no extended states, but forweak disorder SPS are marginally localized[2]. The ex-tended states are also ergodic, that is, in the thermo-dynamic limit the real space average of the q -th mo-ment of | ψ ( i ) | converges to its ensemble average value[3]. In contrast, a quasiperiodic lattice, described bythe Aubry-André-Harper (AAH) Hamiltonian [4–6] withnearest-neighbour hopping, undergoes metal to insula-tor (MIT) transition at a critical disorder strength, evenin 1D. The fundamental, spinless AAH Hamiltonian re-ferred above can be represented by the eigenvalue equa-tion, t ( ψ n +1 + ψ n − ) + W cos (2 πbn + φ ) ψ n = Eψ n ,where W is the strength of the disorder, t is the near-est neighbour hopping amplitude, and ψ n is the ampli-tude of the electronic wave function at the lattice site n . When b is irrational we have the quasi-periodic lat-tice. The AAH Hamiltonian in reciprocal space is givenby, W ( ψ k +1 + ψ k − ) + 2 t cos (2 πbn + φ ) ψ k = Eψ k . TheFourier transformed AAH Hamiltonian becomes same asthe original Hamiltonian when W/t = 2 and their rolesare interchanged. This unique feature of AAH Hamil-tonian makes it self-dual , as states below and above this ∗ [email protected] critical value W c /t = 2 are related by Fourier transforma-tion. All the SPS are extended for W c /t < while theyare localized for W c /t > . The self-dual point W c /t = 2 is particularly special, as all the states show multifractalcharacter, that is, they are extended but non-ergodic .Disordered electronic systems have continued to re-ceive wide attention, even after more than five decadesof the publication of the seminal paper by Anderson [1].A major driving force behind this is due to the impor-tance of disorders, which are unavoidable in real mate-rials. Also, the debate about whether MIT exists in di-mension lower than three in such lattices is not settledcompletely. The search for MIT in lower dimensionaldisordered systems is not just an academic interest, butit is an extremely important problem from the point ofview of device applications as well. Moreover, success-ful experimental simulations of these disordered Hamil-tonians in optical lattice set-up [7–10] have opened thepossibility of verification of the theoretical predictions.More recently, the critical behaviour of the spin-less ver-sion of the AAH Hamiltonian has been experimentallyobserved in polaritonic 1D wires, created with the helpof cavity-polariton devices[11]. Consistent effort towardsfinding MIT in lower dimension have led researchers toconsider Anderson Hamiltonian with SO coupling quiteextensively [12–20]. However, the effect of SO coupling inquasi-periodic systems has not received much attention.In this present work, we have explored the effect ofSO coupling on MIT in quasi-periodic 1D lattice. Morespecifically, we have studied the AAH model with near-est neighbour hopping in the presence of Rashba typespin-orbit (RSO) coupling [21]. RSO coupled electronicsystems receive wide attention because of its potential invarious device applications, especially in spintronic de- a r X i v : . [ c ond - m a t . d i s - nn ] F e b vices [22]. The main advantage of RSO coupled systemsis the possibility of easier external control of the cou-pling strength. In order to find a comprehensive andclear answer, we have systematically used three differentapproaches. First, to determine the effect of RSO on thenature of MIT in 1D quasi-periodic lattice, we have usedthe idea of many-body localization tensor [23, 24], whichis based on Kohn’s idea of localization in disordered sys-tems [25]. In the process, we have demonstrated that thismany-body localization tensor, also known as Kohn’s lo-calization tensor(KLT), is a truly powerful method tostudy MIT for spinfull electronic systems, where spinstates mix . After obtaining precise answer on the effectof RSO on MIT, we studied the nature of the eigenstatesacross the entire energy spectrum for different disorderstrengths. For this we have used the inverse participationratio (IPR) and von Neumann entropy (vNE) to get aquick overall idea about the nature of the quasi-particleeigenstates. Finally, we have carried out detailed mul-tifractal analysis to prove that the self-duality of AAHHamiltonian remains preserved in the presence of RSO,although the self-dual point shifts towards higher disor-der strength with the increase of RSO coupling. Whilecarrying out the multifractral analysis, we have used bothperiodic and open boundary conditions (OBC) to showthat OBC can be used for the quasi-periodic latticesas well, effectively removing the constraint on the sys-tem sizes for numerical computation of the multifractalspectrum. To summarize our conclusions, at the end,we present the phase diagram in the parameter spacespanned by the disorder strength and RSO couplings.It is important to note that in an earlier study [26], a1D quasi-periodic AAH Hamiltonian with RSO couplingwas obtained starting from a tight-binding square lat-tice in presence of uniform magnetic field. However, theRSO Hamiltonian used in the above mentioned referenceis not same as ours. Although, it had also been con-cluded that the self-dual nature remains intact, but itwas reported that the self-dual point remains unaffectedby RSO coupling strength, while there exists two newphases, self-dual of each other, in the parameter spaceof disorder strength and SO coupling. These phases arecharacterized by coexistence of delocalized and localizedstates. Naturally, we have not found any evidence ofthese phases.The paper is organized as follows: in Sec.III, we intro-duce briefly the idea of Kohn’s localization tensor (KLT)for non-interacting electrons. From the KLT calculations,we can immediately identify the critical point for MIT.In Sec. III C, we have explored the evolution of the crit-ical point with the RSO coupling strengths. However,these results do not provide clear idea about the natureof the SPS at the critical point. To study the nature ofthe SPS, we have computed the single particle IPR spec-trum in Sec.IV and the von Neumann entropy (vNE) inSec.V. To demonstrate that all the SPS are non-ergodic at the critical point, in Sec.VI A, we have calculated themultifractal spectrum for the three regions: below, above, and at the critical point. Finally, in Sec.VII, we presentthe phase diagram with respect to the parameters thatcontrol the disorder strength and the strengths of RSOcoupling. II. AUBRY-ANDRÈ MODEL WITH RSOCOUPLING
The Hamiltonian considered in this work consists oftwo parts, H (cid:48) = H + H R , (1)where H is the usual AAH Hamiltonian given by, H = − t L − (cid:88) i =1 ,σ ( c † i +1 ,σ c i,σ + h.c ) + W L (cid:88) i =1 ,σ cos (2 πbi + φ ) c † i,σ c i.σ (2)Here, t is the hopping amplitude from site i to site i + 1 and L = N a is the length of the lattice, where N is thenumber of lattice sites and a = 1 (arbitrary unit) is thelattice spacing. c † i,σ and c i,σ are the fermionic creationand annihilation operators respectively for spin operator σ = ↑ , ↓ particle at site i . W is the strength of quasi-periodic potential. φ is an arbitrary phase varying from (0 , π ) . The choice of the phase φ does not affect ourconclusion, and henceforth we shall set it to zero. Wehave used b = ( √ / . Please note that, sometimesin the literature b = ( √ − / is also used, but ourconclusions are independent of the particular choice of b .The RSO Hamiltonian H R is given by [27], H R = − α z L − (cid:88) i =1 ,σ,σ (cid:48) ( c † i +1 ,σ ( iσ y ) σ,σ (cid:48) c i,σ (cid:48) + h.c. ) − α y L − (cid:88) i =1 ,σ,σ (cid:48) ( c † i +1 ,σ ( iσ z ) σ,σ (cid:48) c i,σ (cid:48) + h.c. ) , (3)where σ y and σ z are Pauli spin matrices in y - and z -direction respectively. α y is a complex spin-conservinghopping due to the confinement in y -direction and α z isa spin-flip hopping due to the confinement in z -directions.The hopping amplitude α y and α z could be different ingeneral and they could also be site dependent. The pureRSO Hamiltonian H R , that is studied in this work, hasalso been studied in the context of transport propertiesin quantum nanowires [28, 29]. Also, recently localiza-tion properties of attractive fermions have been stud-ied in presence of the spin-flip component of the RSOHamiltonian[30]. Since α y is a spin preserving hoppingprocess, it is expected that it will not change the self-dual nature of AAH Hamiltonian. However, it is notclear how W c /t is going to be affected by α y alone. Onthe other hand, the effect of spin-flip hopping, α z , on theself-duality as well as on W c /t is not evident. Hence, weare going to focus mainly on the problem where α z (cid:54) = 0 and α y = 0 . However, for completeness, we also presentKLT results for α y (cid:54) = 0 and α z = 0 . III. KOHN’S LOCALIZATION TENSOR ANDMETAL TO INSULATOR TRANSITION
For a comprehensive analysis of the interplay of disor-der and RSO coupling we have systematically used differ-ent approaches to capture the full picture. In this section,we discuss the idea of Kohn’s localization tensor. It is areliable way to characterize metallic and insulating state.Based on the idea first proposed by W. Kohn[25], Restaand Sorella [23, 24] formulated a localization tensor, al-ternatively called as Kohn’s localization tensor (KLT).In the thermodynamic limit, it is independent of systemsize for insulating states, while for metallic states it di-verges. For 1D AAH model with nearest-neighbour hop-ping and no RSO coupling, a metal to insulator transitiontakes place at the critical disorder strength W c /t = 2 . .Recently, it has been shown [31] that the localizationtensor λ αβ , where α and β are the space co-ordinates,can capture this transition accurately. Furthermore, ithas been shown that computation of λ αβ becomes par-ticularly simpler for non-interacting electrons as well asfor interacting electrons, if interaction is treated withinmean-field approximation. However, in these two abovementioned scenarios spin states were completely decou-pled. As mentioned earlier, in the presence of spin-fliphopping induced by RSO coupling, spin states mix togiving rise to quasi-particle eigenstates. Here, in thiswork we extended the computational approach of λ αβ for quasi-particle states and simultaneously capture theMIT, if it exists, accurately. We have presented our re-sults for periodic and open boundary conditions. How-ever, as we are going to see, we have found that it isnot always possible to calculate λ αβ with open boundarycondition.Here, we briefly discuss the key aspects of KLT andmethods to calculate it for different boundary conditions[31–33]. A more elaborate discussion on KLT and waysto compute it can be found in Ref.[32] and in Ref.[34].For periodic boundary condition (PBC), the localizationtensor, λ αβ ( here α = x, β = x ) can be expressed as, λ xx = − L π N ln | z xN || z xN || z xxN | , (4)where the quantity z xN is given by, z xN = (cid:104) Ψ | e i πL ˆ R x | Ψ (cid:105) , (5)whereas, z αβN can be obtained by replacing ˆ R α with ˆ R α − ˆ R β . Hence, in our case | z xxN | = 1 . In the above equation, | Ψ (cid:105) is the many-body ground state wave function, ˆ R = (cid:80) Ni =1 ˆ r i is the the many-body position operator with ˆ R x being the x component. N is the number of lattice sites,while L = N a , with a = 1 (in arbitrary unit) being thelattice constant, is the size of the system. For a half-filledsystem ( L = N ) in 1D, the localization tensor reduces to λ xx = λ = − L π ln {| z N |} . (6)In absence of electron-electron interaction, z ( x ) N canbe simplified further [32, 33] and represented as z ( x ) N = det [ S xjj (cid:48) ] , where S xjj (cid:48) is a matrix whose elements aregiven by, S xj,j (cid:48) = (cid:90) dr ψ ∗ j ( r ) e i πL ˆ r x ψ j (cid:48) ( r ) . (7)In the above equation ψ j ( r ) represents the amplitude ofsingle-particle wave function at position r for a spin-up orspin-down electron arranged in ascending order in ener-gies. The indices j and j (cid:48) indicate the energy level. Since,in absence of RSO spin-up and spin-down electrons arecompletely decoupled, it is sufficient to consider only onetype of spin and j, j (cid:48) = 1 , , · · · , N/ to compute λ xx athalf-filling. But, in presence of RSO, more specifically be-cause of the spin-flip hopping process, in Eq. 7 φ j ( r ) nowrepresents the amplitude of a single quasi-particle wavefunction at position r corresponding to the j th eigen-energy. Hence, in our case j, j (cid:48) = 1 , , · · · , N. In case of open boundary conditions (OBC), squaredlocalization length ( λ ) , in units of the nearest-neighbourdistance, can be expressed as follows [34, 35]: λ = 1 νN N (cid:88) i,i (cid:48) =1 ρ ii (cid:48) ( ν )( i − i (cid:48) ) , (8)where i, i (cid:48) = 1 , · · · , N represent the lattice site, ν is thefilling factor and ρ ii (cid:48) ( ν ) is the one-body density matrix,defined as ρ ii (cid:48) = N (cid:88) j =1 φ j ( i ) φ ∗ j ( i (cid:48) ) , (9)where φ j ( i ) is the amplitude of single quasi-particle wavefunction at lattice site i corresponding to j th eigenvalue.Since we are interested to compute λ at half-filling, weset ν = 1 in Eq. 8 . A. Localization tensor without RSO
Before presenting the localization tensor results for ourHamiltonian, in this section we first benchmark our λ calculations for pure AA model without the RSO cou-pling at half-filling. We also highlight the behaviour of λ with system sizes at the critical point. In Fig.1(a)and Fig.2(b), we have shown these calculation for openand periodic boundary conditions respectively. In case ofPBC, system sizes are restricted to lattice sizes given by -1
0 0.5 1 1.5 2 2.5 3 3.5 4(a) λ W/tL= 610:OBCL=1000:OBCL=1597:OBCL=2584:OBC 0 0.001 0.002 0.003 3.6 3.9 4.2 4.505101520 (b) / λ ( x - ) Figure 1. (a) Squared localization length λ for half-filled 1D pure AA model with respect to disorder strength W/t for someselected lattices. (b) Scaling of inverse localization length /λ with inverse chain length (1 /L ) . For finite size scaling we haveused /λ = a + b /L . (c) Plot of /λ with /L at the critical point. It is evident that at the critical point, /λ does notfollow the scaling pattern of either metallic or insulating states. All the results for open boundary condition. -1
0 0.5 1 1.5 2 2.5 3 3.5 4(a) λ W/tL= 377:PBCL= 610:PBCL=1597:PBC 0 0.001 0.002 0.003 1.2 2.2 3.20246 (b) / λ ( x - ) Figure 2. Localization tensor λ for half-filled 1D pure AAH model with PBC. (a) λ vs. W/t for a few selected chain lengths.(b) Finite size scaling of /λ vs. (1 /L ) . (c) Once again, similar to the OBC case, /λ lacks simple scaling behaviour with /L at the critical point. the Fibonacci sequence, while for OBC there is no suchrestriction. One of the main objectives to compare re-sults of different boundary conditions is to show that thelocalization tensor is a robust indicator to differentiatemetallic and insulating state irrespective of the bound-ary conditions. This observation is going to be useful,as we are going to show in Sec.VI, while computing themultifractal spectrum for quasi-periodic lattices, as wedo not need to restrict ourselves to very specific systemsizes dictated by the Fibonacci sequence in order to ex-trapolate the results to thermodynamic limit.In Fig.1(a)(OBC) and Fig.2(a)(PBC), we have shownthe variation of squared localization length λ with in-creasing disorder strength ( W/t ) for different 1D latticesat half-filling. As expected, the transition from delocal-ized to localized phase occurs at W c /t (cid:39) for both theboundary conditions. λ values are finite and indepen-dent of system sizes above W/t > whereas for W/t < ,it increases with the increase of system size. These ob-servation are similar to Ref. [31]. In Sec.III it has beenmentioned that in case of metallic phase λ diverges inthe thermodynamic limit, while it is independent of sys-tem sizes in the insulating phase. From the plot of /λ vs. /L of Fig. 1(b) and Fig.2(b), it is clear that, irrespec- tive of the boundary condition, λ diverges for L → ∞ in the metallic phase, while it is nearly constant and con-verges to a finite value in the insulating phase. For pureAA Hamiltonian, we have found that /λ scales linearlywith the inverse of the system size. For finite size scaling,in this case, we have used /λ = a + b /L , where a and b are two adjustable parameters.At the critical point W c /t = 2 , scaling behaviour of λ with increasing system size is expected to be erraticas all the single particle eigenstates are mutifractral incase of pure AA model. However, it can be an use-ful and quick indicator to detect the existence of mul-tifractal eigenstates within a spectrum. Since there isno characteristic length-scale for multifractal states, wecan expect an anomalous behaviour of /λ with respectto /L compared to pure metallic and insulating phases.In Fig. 1(c) and in Fig. 2(c) we have plotted /λ ver-sus /L for W c /t = 2 . In case of OBC, it is clear that λ neither converges to a finite value nor does it go tozero in the thermodynamic limit. With PBC as well, λ oscillates with increasing L without any indication ofconvergence. In this work, however, we have used a wellestablished and computationally less costly approach forthe multifractal analysis of the eigenstates. -1
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5(a) λ W/t α z /t=0.8L= 610:OBCL=1000:OBCL=1597:OBCL=2584:OBC
0 0.001 0.002 0.003 2.7 3.1 3.5(b)0369121520 / λ ( x - ) Figure 3. Localization tensor λ for half-filled 1D AAH model with RSO coupling. (a) λ vs. W/t for selected lattices withOBC. (b) Scaling of inverse localization length /λ vs. /L . Here /λ scales linearly with /L similar to Fig. 1(b). (c) Plotof /λ vs. /L at the critical point W c /t = 2 . . Lack of proper scaling of /λ indicates that our estimation of critical pointis quite correct. -1
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5(a) λ W/t α z /t=0.8L= 377:PBCL= 987:PBCL=2584:PBCL=4181:PBC
0 0.001 0.002 0.003 1.2 1.7 2.2(b)0369 / l ( x − )
0 0.001 0.002 0.003(c)0102030 1/LW/t=2.55:PBC
Figure 4. Localization tensor λ for half-filled 1D AAH model with RSO coupling. (a) λ vs. W/t for selected latticeswith PBC. (b) Finite size scaling of /λ with inverse chain length /L below and above the critical point. We have used /λ = a + b /L + c /L for the purpose of scaling in the metallic phase. (c) With PBC as well we find anomalous behaviourof /λ with increasing system size at W/t = 2 . . B. Localization tensor in presence RSO
We now present our localization tensor results for one-dimensional AA model in the presence of RSO coupling.Since the effect of the spin-flip hopping process, inducedby RSO, on the self-duality is much harder to antici-pate, we primarily focus on the α y = 0 case. We havechosen α z /t = 0 . . There is nothing special about thisvalue, as all our conclusions are independent of it, exceptthe location of self-dual point. In Fig.3(a)(OBC) and4(a)(PBC) the variation of squared localization length, λ , with increasing value of disorder strength W/t havebeen shown. It is evident that inclusion of RSO cou-pling enhances the critical point towards larger disorderstrength, in this case W c /t (cid:39) . , as compared to pureAA model ( W c /t (cid:39) ). Furthermore, this conclusion isindependent of the applied boundary conditions on thesystem.To confirm our observation in the thermodynamic limitwe have done finite size scaling of /λ with differentsystem sizes below and above the critical point. Theseresults are presented in Fig. 3(b) and in Fig. 4(b). Inthe case of OBC, it is evident that /λ converges to zero as L → ∞ for W/t < . , that is, for delocal-ized phase, while it is independent of system size for W/t > . , indicating an insulating phase. In case ofPBC also, these fundamental conclusions remain same.Interestingly, however, with PBC the dependence of /λ on /L deviates significantly in the presence of RSO inthe metallic phase. For finite size scaling, specially for themetallic phase, we have used /λ = a + b /L + c /L ,where a , b and c are adjustable parameters.It is easy to pinpoint the critical disorder strength fromthe λ vs W/t data almost exactly. We are going tosee that this information can reliably be extracted fromthe multifractal analysis (Sec. VI A) as well. From thelocalization tensor result we estimate W c /t (cid:39) . for α z /t = 0 . . To cross-check this estimation, we haveplotted /λ versus /L separately at W c /t (cid:39) . inFig. 3(c) and in Fig. 4(c). It is quite clear that /λ does not follow any discernible pattern with /L . This,as we have pointed out in Sec. III A, is the hallmark ofcritical/multifractal eigenstates. -2 -1
0 0.5 1 1.5 2 2.5 3 3.5 4(a) λ W/t α y /t=0.0 α z /t=0.0 α z /t=0.2 α z /t=0.4 α z /t=0.6 α z /t=0.8 α z /t=1.0 10 -2 -1
0 0.5 1 1.5 2 2.5 3 3.5 4(b) W/t α z /t=0.0 α y /t=0.0 α y /t=0.2 α y /t=0.4 α y /t=0.6 α y /t=0.8 α y /t=1.0 10 -2 -1
0 0.5 1 1.5 2 2.5 3 3.5 4(c) W/t α y /t=0.6 α z /t=0.2 α z /t=0.4 α z /t=0.6 α z /t=0.8 α z /t=1.0 Figure 5. (a) Evolution of the self-dual point in AAH with only spin-flip hopping induced by RSO coupling. The results are fora lattice L = 1597 with OBC. (b) The self dual point moves towards higher strength of the quasi-periodic potential when pureAAH Hamiltonian is considered with only the complex hopping induced by the RSO coupling. (c) Evolution of the self-dualpoint when the AAH Hamiltonian is considered with the full RSO Hamiltonian. Here we have only shown the change with thespin-flip hopping while the complex hopping has been fixed at a value . . C. Evolution of the critical point with RSOinteraction
In this section, we study the evolution of the criticalpoint for three different cases, (i) α y = 0 , (ii) α z = 0 and(iii) α z (cid:54) = 0 , α y = fixed . For case (i) and (iii) a latticewith N = 1597 and OBC have been used, while for case(ii) we have used PBC and N = 987 . It is important tonote that λ could not be computed with OBC for case(ii). In Fig. 5(a), we have considered only the spin-fliphopping α z in the RSO Hamiltonian. It is clear, evenfrom the results for a single finite lattice, that the criti-cal point moves to higher strength of the quasi-periodicpotential. Interestingly, when only the complex hoppingof the RSO Hamiltonian is considered along with pureAA Hamiltonian, similar variation of the critical pointis observed in Fig. 5(b). Fig 5(c), represents the gen-eral nature of the evolution of the critical point whenthe full RSO Hamiltonian is considered along with pureAA Hamiltonian. In these results, variation of the criti-cal point with spin-hopping has been studied for a fixedstrength of the spin conserving complex hopping havingthe amplitude α y /t = 0 . . For α z = 0 . , λ appears toindicate a cross-over rather than a clear transition. How-ever, this is an artifact of the finite system being used forthese calculations. We have checked that as the systemsize is increased, a clear transition from delocalized tolocalized transition emerges. IV. INVERSE PARTICIPATION RATIO
Existence of MIT in presence of RSO is evident fromthe results of the localization tensor calculations. Al-though at the critical point, anomalous behaviour of thelocalization tensor with the inverse of the system size doindicate the absence of a characteristic length scale, thetrue nature of the quasi-particle eigenstates across the en-tire energy spectrum is not clear. For example, if an en- ergy spectrum contains predominantly delocalized statesthen the localization tensor can suppress the contributionof localized states while calculating it upto certain fillingfraction. To distinguish localized and delocalized states,Inverse Participation Ratio (IPR) is regularly used as afirst indicator. IPR can also provide some hint, althoughonly qualitatively, of multifractal eigenstates, if it exists.Generally, IPR is used to quickly identify the existenceof mobility-edge. Typically the mobility edge is definedas the energy which separates localized and delocalizedeigenstates in the energy spectrum. If a mobility edgeexists, IPR jumps from system size independent (insu-lating/localized states) higher value to a lower value (er-godic metallic/extended states) that scales inversely withthe system size. In our case, we have observed that awayfrom the critical point, the states are either delocalized ( W < W c ) or localized ( W > W c ) . The other impor-tant question that remains to be answered is, what arethe nature of the eigenstates at the critical point? InAA model without RSO, all the eigenstates are extendedbut non-ergodic at the critical point. IPR data qualita-tively indicates that RSO does not alter this behaviour.At this point we would like to mention that, studyingthe single particle energy spectra E n and the distribu-tion of the level-spacing δ n = E n +1 − E n can also reveala great deal about the nature of the eigenstates [36–38];for example, in AAH model the energy level-spacing fol-lows Poissonian distribution, while at the critical pointthis distribution follows an inverse power law. Typically,the energy spectrum of AAH Hamiltonian consists of sub-bands and many gaps between these subbands. Recently,it has been reported that there are some special statesinside these subband gaps [39], which are localized evenin the metallic phase of pure AAH Hamiltonian. Wehave checked that these states are mostly concentratedat the lattice edges. Appearance of these states dependson many factors; for example, they can appear for systemwith OBC, with PBC, but the system size is not sameas certain (not all) Fibonacci number. These states donot influence the critical behaviour in any way, as can beseen from the results of the previous section.Formally, for ν -th quasiparticle eigenstate the usualdefinition of IPR [40] can be generalized as follows; P ν ( q ) = (cid:88) σ N (cid:88) i =1 | ψ ν,σ ( i ) | q , q = 2 . (10)Here ν th single quasiparticle eigenstate is given by | Ψ ν (cid:105) = (cid:80) σ (cid:80) Ni =1 ψ ν,σ ( i ) | , σ (cid:105) i , where | , σ (cid:105) i = | , , · · · , σ i , · · · , , (cid:105) represents the localized basis statehaving one particle with spin σ at site i . Since, this isa non-interacting problem, usual scaling properties withrespect to system size is also expected to hold for thequasiparticle states, i.e. for a perfectly extended metal-lic state P ν ( q ) = 1 /N , and for a completely localizedstate P ν ( q ) = 1 . For a localized state, the IPR value issupposed to be system size independent. These distinctscaling properties of IPR enables it to identify delocalizedand localized states quickly. We have numerically verifiedthat the spin-up and spin-down contributions are identi-cal towards the IPR of the quasiparticle they constitute.In Fig. 6 we have plotted the IPR spectrum for (a) W/t = 2 . , (b) W/t = 2 . , and (c) W/t = 2 . . Theseresults are for α z /t = 0 . , α y /t = 0 . It is evident that for
W/t < . , all the eigenstates are delocalized as the IPRvalue in this region depends inversely on the system sizeacross the entire energy spectrum, while all the statesare localized for W/t > . . On the other hand, at thecritical point the IPR spectrum behaves differently, it isneither independent of system size nor does it scale in-versely with L like the extended states. This behaviour issimilar to the IPR spectrum of AA model without RSOat the critical point, i.e. these are multifractal or in otherwords extended yet non-ergodic . Furthermore, the corre-lation between the energy level-spacing spectra and thescaling behaviour of IPR with L for all three differenttypes of electronic states are similar to the AA modelwithout RSO [39]. In the delocalized phase, IPR typi-cally behaves inversely with L across the whole energy-spectrum except at the special positions where the level-spacing jumps abruptly. At the critical point, both ofthem show anomalous behaviour with the system size,while in the localized phase IPR spectrum behaves op-posite compared to the delocalized phase at these specialpoint of level-spacing spectra apart from being systemsize independent. All of these results are presented forPBC. OBC does not change the fundamental conclusions. V. VON NEUMANN ENTROPY
From the results of the previous sections, it is evi-dent that at half-filling there is a transition from metal-lic phase to an insulating phase at a critical disorderstrength W c /t > . , which increases as the strength ofRSO is increased. Furthermore, these results also hint that at the critical point the states are multifractal, apreliminary observation that we are going to establishfirmly in Sec. VI. In this section, we present the resultsof von-Neumann entropy (vNE), an alternative indicatorof single particle properties, which can also be used toqualititavely identify the nature of the eigenstates.In case of non-interacting spin-1/2 fermions, and inpresence of RSO coupling, individual eigenstates are oc-cupied by quasiparticles. The quasiparticle eigenstatehaving energy E ν can be written as, | Ψ ν (cid:105) = N (cid:88) i =1 (cid:2) ψ νi, ↑ | , ↑(cid:105) i + ψ νi, ↓ | , ↓(cid:105) i (cid:3) , (11)where | , ↑(cid:105) i = | (cid:105) i ⊗ |↑(cid:105) and | , ↓(cid:105) i = | (cid:105) i ⊗ |↓(cid:105) . | , ↑(cid:105) i = c † i, ↑ | (cid:105) and | , ↓(cid:105) i = c † i, ↓ | (cid:105) . Here | (cid:105) represents the vac-uum state for the lattice in real space basis. c † i, ↑ , c † i, ↓ arethe creation operators for spin up and spin down parti-cles respectively at the lattice site i . The average num-ber of spin up and down particles at site i are given by | ψ νi, ↑ | = (cid:104) Ψ ν | c † i, ↑ c i, ↑ | Ψ ν (cid:105) and | ψ νi, ↓ | = (cid:104) Ψ ν | c † i, ↓ c i, ↓ | Ψ ν (cid:105) respectively. Then, the local density matrix ρ νj can be ob-tained from the total density matrix ρ ν by tracing overall the lattice sites except site j and can then be writtenas, ρ νj = (cid:12)(cid:12) ψ νj, ↑ (cid:12)(cid:12) | , ↑(cid:105) j (cid:104) , ↑| j + (1 − | ψ νj, ↑ | ) | , ↑(cid:105) j (cid:104) , ↑| j + (cid:12)(cid:12) ψ νj, ↓ (cid:12)(cid:12) | , ↓(cid:105) j (cid:104) , ↓| j + (1 − | ψ νj, ↓ | ) | , ↓(cid:105) j (cid:104) , ↓| j (12)It is important to note that | , ↑(cid:105) j and | , ↑(cid:105) j represent lo-cal vaccum states for j -th site. A similar interpretationapplies to the states | , ↑(cid:105) j and | , ↓(cid:105) j . The von Neu-mann entropy for spin-1/2 quasiparticles follows easilyfrom Eq. 12 as, S nisj,ν = − (cid:0) | ψ νj, ↑ | log | ψ νj, ↑ | + (1 − | ψ νj, ↑ | )ln(1 − | ψ νj, ↑ | ) (cid:1) + − (cid:0) | ψ νj, ↓ | ln | ψ νj, ↓ | + (1 − | ψ νj, ↓ | )ln(1 − | ψ νj, ↓ | ) (cid:1) (13)Finally summing over all the lattice sites, the von-Neumann entropy for a quasiparticle eigenstate is definedas, S nisν = N (cid:88) j =1 S nisj,ν . (14)Similar to the spinless AA model, for a purely extendedquasi-particle state S nisν ≈ (log N +1) and for completelylocalized state it is ≈ .In Fig. 7(a)-(c) results of our von Neumann entropycalculations are presented for α z /t = 0 . and α y /t = 0 . .To show the dramatic change in vNE as we move slightlyaway from the critical point ( W c /t (cid:39) . ), in Fig. 7(a)and in Fig. 7(c) we have plotted our results for W/t = 2 . and W/t = 2 . respectively. It is clear that for W/t = 2 . the von Neumann entropy increases with system size androughly scales as expected, while it is independent of the -3 -2 -1
0 0.2 0.4 0.6 0.8 1(a) I n IndexL= 610:PBCL= 987:PBCL=1597:PBC 10 -3 -2 -1
0 0.2 0.4 0.6 0.8 1(b) IndexL= 610:PBCL= 987:PBCL=1597:PBC 10 -3 -2 -1
0 0.2 0.4 0.6 0.8 1(c) IndexL= 610:PBCL= 987:PBCL=1597:PBC
Figure 6. Evolution of IPR I n with increasing disorder strength for a fixed RSO coupling α z /t = 0 . and α y = 0 . (a) W/t =2 . ,(b) W/t = 2 . , and (c) W/t = 2 . . Here index represents the ratio of serial number of eigenstate to total number ofeigenstates ( i / (2 L ) ), where i = 1 , , · · · , L . S n IndexL= 610:PBCL= 987:PBCL=1597:PBC 4 8 12 16 0 0.2 0.4 0.6 0.8 1(b) IndexL= 610:PBCL= 987:PBCL=1597:PBC 4 8 12 16 0 0.2 0.4 0.6 0.8 1(c) IndexL= 610:PBCL= 987:PBCL=1597:PBC
Figure 7. Evolution of von Neumann Entropy S n with increasing disorder strength for a fixed RSO coupling α z /t = 0 . and α y = 0 . (a) W/t = 2 . ,(b) W/t = 2 . , and (c) W/t = 2 . . system sizes and the value is close to zero for W/t = 2 . .It is clear from the results of Fig. 7(b), at the criticalpoint these results do not completely follow the expectedpattern of purely extended or localized states. These re-sults along with the localization tensor calculations andIPR data qualitatively capture the nature of the eigen-states at the critical point. However, it is necessary tohave a more rigorous analysis to quantify the degree ofmultifractality of the eigenstates at the critical point. Toaddress this, in the next section, we present a detail andcareful analysis of the multifractal spectrum around thecritical point. VI. MULTIFRACTAL SPECTRUM AND THEQUASI-PARTICLE EIGENSTATES “Absence of length scale” at the critical point of a phasetransition has led to the understanding that localization-delocalization (LD) transition can also be viewed as aclass of a much broader set of critical phenomena. Typi-cally, the critical phenomena are characterized by criticalexponents. In case of LD transition, multifractal analy-sis of the eigenstates plays a similar role like the criticalexponents. Following the arguments of multifractal anal-ysis [41] in our case, we start by identifying the q-th mo- ment of the probability of finding a quasiparticle withina linear box of length L ( L = N a, a = 1 in arb. unit ) is P ( q ) = (cid:80) Ni =1 | ψ n ( i ) | q ∝ N − τ ( q ) , where ψ n is the quasi-particle wave function corresponding to n-th eigenvalueand i = 1 , , · · · , N . The exponent τ ( q ) is alternativelyexpressed in terms of D ( q ) as τ ( q ) = D ( q )( q − , (15)where D ( q ) is called the generalized dimension. In caseof ergodic extended (EE) eigenstates [41] τ ( q ) = q − This conclusion follows from the argument that the realspace average P ( q ) /N converges to the ensemble average (cid:104) P q (cid:105) /N = (cid:104)| ψ n ( i ) | q (cid:105) in the limit N → ∞ . Effectively itmeans that for EE states, D ( q ) = 1 , whereas for a com-pletely localized eigenstate D ( q ) = 0 . For multifractalstates, τ ( q ) deviates from these two limiting cases lead-ing to q dependence of the generalized dimension D ( q ) .These states are extended yet non-ergodic . Out of thepossible set of generalized dimesnions, D (2) is frequentlyused to characterize different states. In practice, how-ever, instead of computing τ ( q ) directly, an equivalentquantity f ( α ) is evaluated. It characterizes the multi-fractal property of the eigenstates. f ( α ) and τ ( q ) areconnected by the Legendre transformation, f ( α ( q )) = qα ( q ) − τ ( q ) , (16) f( α ) α L= 610L=2584L=6765Thermodynamic Limit α L= 610L=2584L=6765Thermodynamic limit α L= 610L=2584L=6765Thermodynamic Limit f( α ) α L=1000L=2000L=4000L=8000Thermodynamic limit α L=1000L=2000L=4000L=8000Thermodynamic limit α L=1000L=2000L=4000L=8000Thermodynamic limit
Figure 8. Multifractal spectrum of AAH Hamiltonian with RSO coupling for PBC (top row) and OBC (bottom row). Here α y /t = 0 and α z /t = 0 . . From left to right: (a, d) W/t = 2 . , (b, e) W/t = 2 . and (c, f) W/t = 2 . . D (2) is the generalizeddimension D ( q ) for q = 2 . For extended ergodic states D (2) = 1 and D (2) = 0 for insulating states, while for multifractal/non-ergodic extended states < D (2) < . where α ( q ) = dτ ( q ) /dq . In general, f ( α ) is a smoothnon-monotonic positive valued function having negativecurvature and a global maxima but no local minima ormaxima. In fact, f max = f ( α ( q = 0)) = d , where d isthe Euclidean dimension of the system [42]. From theanalysis of the function f ( α ) , one can easily identify thenature of the eigenstates. For EE states, in the thermo-dynamic limit f ( α = 1) = 1 , while f ( α (cid:54) = 1) = −∞ .For non-ergodic (NE) eigenstates f ( α ( q )) → , for <α min < α ( q ) < α max , while f α ( q =0) = f max appears for α ( q = 0) > . In contrast to EE and NE states, for in-sulating states f ( α ( q )) → as α ( q ) → , while α ( q = 0) ,i.e., the position of the maxima of f ( α ) spectrum shiftstowards larger value than 1 as the disorder strength isincreased. It is quite evident that to identify the na-ture of the quasi-particle eigenstates it is sufficient tohave an estimation of α min ( f ( α min ) → and α ( q = 0) ( f max = f ( α ( q = 0)) = d ). This allows us to use a well-established method [42, 43] to compute the multifractalspectrum for our Hamiltonian. This spectrum is com-puted and compared for PBC and OBC. Both of theseboundary conditions lead to identical conclusions. A. Calulcation of Multifractal Spectrum
Before discussing the results, we briefly summarize thekey steps of to compute the multifractal spectrum. Ini-tially the lattice is divided into small boxes of linear size l < L
The first step is to find the normalized box-probability, P k ( l, q ) = P qk ( l ) (cid:80) N b j =1 P qj ( l ) , (17)where ≤ k ≤ N b = L/l represents the k -th box and P k ( l, q ) = (cid:80) i ∈ l k | ψ n ( i ) | q , l k = l ∀ k, is the probabilityof the n -th eigenstate. Then α ( q, L ) and f ( α ( q, L )) areobtained from the following relations; α ( q, L ) = lim δ → (cid:80) N b k =1 P k ( l, q ) ln ( P k ( l, ln δ (18) f ( α ( q, L )) = lim δ → (cid:80) N b k =1 P k ( l, q ) ln ( P k ( l, q )) ln δ , (19)where δ = l/L . It is important to note that thismethod of computing the multifractal spectrum is validfor a (cid:28) l < L . For different system sizes L , we have cho-sen l in a way that the above condition is satisfied and . ≤ δ ≤ . . α ( q, L ) and f ( α ( q, L )) have been com-puted for system size upto L = 8 × and averagedover the entire energy window till half filling. Finally,the thermodynamic limit value of α ( q ) = lim L →∞ α ( q, L ) and f ( α ( q )) = lim L →∞ f ( α ( q, L )) have been estimated us-ing finite size scaling. For finite size scaling, we proposethe following set of functions for α q ( L ) and f ( α q ( L )) , α q ( L ) = α q + a q L − + b q L − ,f ( α q ( L )) = f ( α q ) + c q L − + e q L − , (20)0 α α z /t=0.8,W/t=2.5q=0.0q=0.5q=1.0q=1.5q=2.0q=5.0 0 1 2 3 0 0.0002 0.0004 0.0006 0.0008 0.001(b)1/L α z /t=0.8,W/t=2.55q=0.0q=0.5q=1.0q=1.5q=2.0q=5.0 0 0.2 0.4 0.6 0.8 1 0 0.0002 0.0004 0.0006 0.0008 0.001 1 5 9 13 17(c)1/L α z /t=0.8,W/t=2.6q=0.0q=0.5 0 0.5 1 1.5 2 0 0.0002 0.0004 0.0006 0.0008 0.001(d) f( α ) α z /t=0.8,W/t=2.5q=0.0q=0.5q=1.0q=1.5q=2.0q=5.0 0 0.5 1 1.5 2 0 0.0002 0.0004 0.0006 0.0008 0.001(e)1/L α z /t=0.8,W/t=2.55q=0.0q=0.5q=1.0q=1.5q=2.0q=5.0 0 0.5 1 1.5 0 0.0002 0.0004 0.0006 0.0008 0.001(f)1/L α z /t=0.8,W/t=2.6q=0.0q=0.5 Figure 9. Top row: finite size scaling of α ( q ) with /L and bottom row: finite size scaling of f ( α ( q )) with /L for some limitingvalues of the moment q. The scaling functions are given in Eq.20. The complex hopping amplitude α y /t = 0 . From left toright,
W/t = 2 . , . , and . . All of these results are presented for open boundary condition. where α q , f ( α q ) , a q , b q , c q , and e q are adjustable parame-ters. In Fig. 9 we present results of these scaling of α q ( L ) and f ( α q ( L )) for three different region, at the criticalpoint and below and above it. Before discussing the scal-ing results, we first discuss the f ( α ( q )) vs α ( q ) spectrum,presented in Fig. 8. In Fig. 8(a)-(c) we present the re-sults for PBC, while the results with OBC are presentedin Fig. 8(d)-(f). All these results are for α z /t = 0 . . Asmentioned in earlier, one of the main purpose of present-ing the results with two different boundary conditions isto demonstrate that it does not affect our fundamentalconclusion about the nature of the states. Furthermore,these results indicate that one can use OBC to calculatethe multifractal spectrum quite accurately using reason-ably large system sizes, at least in 1D.In Fig 8(a) and (d), α z /t = 0 . and W/t = 2 . . Fromthe results of localization tensor, the quasi-particle statesare expected to be extended and ergodic. It is clearfrom the results of multifractal spectrum that, in thethermodynamic limit we have f ( α ( q = 0)) → , while α ( q ) → . These two values are not exactly as onewould expect for ideal extended ergodic states, but theyare very close to the ideal value. Using Eq. 15 andEq. 16 we have computed D (2) . As expected for metal-lic states, our estimated value of D (2) is .
98 (0 . forperiodic (open) boundary conditions respectively. As weincrease W/t to . (Fig. 8 (b) and (e)), we can seethe dramatic change in the multifractal spectrum. In thethermodynamic limit, f ( α q ) → for α ( q ) < , while f max = f ( α ( q = 0)) = 1 for α ( q = 0) > . It clearly in- dicates that all the states are extended, yet non-ergodic .This observation is also supported by our estimation of D (2) . At the critical point D (2) = 0 . for PBC, while D (2) = 0 . for OBC according to our estimate.It is interesting to have a closer look at the behaviourof f ( α q ( L )) vs. α q ( L ) spectrum with system size L . Fordifferent system sizes the spectrum cross each other atsome point. How these spectrum move with increas-ing system size on either side of the crossing indicatesthe nature of the eigenstates. For EE and NE states,the spectrum move towards α = 1 with increasing sys-tem size. This evolution of the multifractal spectrumwith system size gets reversed quite dramatically as weincrease the disorder strength just a little to the value W/t = 2 . . In this case, with increasing system size thespectrum moves towards α = 0 on the the left of thecrossing point, while on the right of the crossing pointit moves further away from α = 1 . The results are pre-sented in Fig. 8(c) and (f). It is evident that in thethermodynamic limit, f ( α q ) → for α q (cid:39) . , while f max = f ( α ( q = 0)) = 1 for α ( q = 0) >> , indicat-ing that all the states are localized. From the numericaldata, we find that D (2) = 0 . . for PBC (OBC), asexpected. These results are also consistent with our esti-mation of the critical point from the localization tensorcalculations, as well as with the IPR and vNE results.In Fig. 9, we have presented the scaling data for for α y /t = 0 , α z /t = 0 . and few limiting values of q . Hereresults are presented for OBC only. For PBC the re-sults are nearly identical. To demonstrate the dramatic1change in the multifractal spectrum we have chosen theself dual point W c /t = 2 . and two different potentialstrengths just below and above it. In Fig. 9(a)-(c), α q ( L ) has been plotted with /L , while Fig. 9(d)-(f) are for f ( α q ( L )) . From Fig. 9(a) it is clear that just below thecritical point α q ( L ) → with increasing system size forall q . Same pattern can be observed for f ( α q ( L )) as well,although for higher q the convergence is not perfect. Theconvergence of f ( α q ( L )) to 1 becomes perfect as disor-der strength is lowered slightly from W/t = 2 . . FromFig. 9(b) we can see that at the critical point α q ( L ) and f ( α q ( L )) do not converge to a single value for different q in the thermodynamic limit. α q =0 ( L ) converges to avalue greater than 1, while it converges to a single valuemuch less than 1 as q increases beyond 2.0. At the sametime, lim L →∞ f ( α q ( L )) → as q is increased. This indicatesthat the eigenstates are extended but non-ergodic. Quitedramatically, as W/t is increased just by a small amountto . , lim L →∞ α q =0 ( L ) converges to a value much largerthan 1, while in the thermodynamic limit α q (cid:54) =0 ( L ) and f ( α q ( L )) converge rapidly to the origin at the same timewith higher moment q. This indicates that the states arelocalized across the entire spectrum. W /t α z /t α y /t=0.0 α y /t=0.3 α y /t=0.6 α y /t=0.9 Figure 10. Phase diagram in the parameter space spanned bythe strength of quasi-periodic potential
W/t and the spin-fliphopping amplitude α z /t . α y /t represents the spin conservingcomplex hopping amplitude induced by RSO coupling. Thedotted lines indicate the phase boundary. VII. PHASE DIAGRAM
Finally, we present the phase diagram in the parameterspace spanned by
W/t and α z /t . The phase boundarieshas been obtained for increasing strength of the complexhopping of the RSO Hamiltonian. Each of these phaseboundaries are indicated schematically by dotted line inFig. 10. The phase is metallic below it, while it is aninsulating phase above the boundary. On each of theseboundaries, all the states are extended and non-ergodic. VIII. CONCLUSIONS
In conclusion, we have studied the effect of RSO cou-pling on the critical behaviour of one dimensional quasi-periodic lattice, described by AAH Hamiltonian. Wehave found that RSO coupling does not affect the self-dual nature of AAH model, however the self-dual pointmoves towards higher strength of the quasi-periodic po-tential. Moreover, individual influences of the two differ-ent kinds of hopping induced by Rasha spin-orbit cou-pling, that is the spin conserving complex hopping andspin-flip real hopping, are identical in nature. We havefound no evidence of coexistence of extended and local-ized states in the entire parameter space. Furthermore,the phases are insensitive towards the existence of anysub-gap states that might exist in the energy spectrumdepending on the lattice size and boundary conditions.In the process of studying the MIT in this system, wehave also demonstrated that can be also used to studythe transition from delocalized to localized phase in sys-tem where the spin states mix and eigenstates are quasi-particles rather pure spin states. For a rigorous analysisof the eigenstates, we have performed the multifractalanalysis, and it has been demonstrated that the algo-rithm works equally well for PBC as well OBC. This ef-fectively removes the constraint on the system sizes thatcan be used to obtain the results in the thermodynamiclimit for quasi-periodic systems.
IX. ACKNOWLEDGEMENT
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