aa r X i v : . [ c ond - m a t . d i s - nn ] J a n Invariable mobility edge in a quasiperiodic lattice
Tong Liu, ∗ Yufei Zhu, Shujie Cheng, † Feng Li, Hao Guo, ‡ and Yong Pu § School of Science, Nanjing University of Posts and Telecommunications, Nanjing 210003, China Department of Physics, University of Otago, P.O. Box 56, Dunedin 9054, New Zealand Department of Physics, Zhejiang Normal University, Jinhua 321004, China Department of Physics, Southeast University, Nanjing 211189, China (Dated: January 5, 2021)In this paper, we study a one-dimensional tight-binding model with tunable incommensuratepotentials. Through the analysis of the inverse participation rate, we uncover that the wave functionscorresponding to the energies of the system exhibit different properties. There exists a critical energyunder which the wave functions corresponding to all energies are extended. On the contrary, thewave functions corresponding to all energies above the critical energy are localized. However, weare surprised to find that the critical energy is a constant independent of the potentials. We usethe self-dual relation to solve the critical energy, namely the mobility edge, and then we verify theanalytical results again by analyzing the spatial distributions of the wave functions. Finally, we givea brief discussion on the possible experimental observation of the invariable mobility edge in thesystem of ultracold atoms in optical lattices.
I. INTRODUCTION
Anderson localization [1], a universal phenomenon ofthe wave propagation in disordered mediums, is an ac-tive research subject in condensed matter physics. Withregard to the disordered systems, the scaling theory [2]predicts that the influence of the spatial degrees of free-dom on Anderson localization is significant. In one- andtwo-dimensional systems, uncorrelated random distur-bances account for why all the wave functions are ex-ponentially localized. Accordingly, in low-dimensionaldisordered systems, there is always a hot topic, namelythe delocalization-localization transition. By contrast, inthree-dimensional (3D) Anderson system [1, 3], there ex-ists a radically different phenomenon that neither all thewave functions are localized nor are delocalized. Thereis an energy level threshold which separate the delocal-ized functions from those are localized. This critical en-ergy is known as the mobility edge. However, it is stilla challenge to observe the the mobility edge in 3D sys-tems with the mobility edge in experiment. Therefore,to acquire more insights into the mobility edge and un-cover its essence, low-dimensional systems, such as theone-dimensional (1D) systems, are a better choice. Afterall, there has a long history to search the systems withmobility edge and plenty of theoretical models are pro-posed and studied in the decades [2–17]. Moreover, themobility edge has been observed in recent experiments bymanipulating the cold atoms trapped in 1D quasiperiodicoptical lattice [18, 19], although there are experimentalchallenges.The Aubry-Andr´e (AA) model [20] plays an impor-tant role to understand the Anderson localization anduncover the mobility edge. As a paradigmatic 1D tight-binding model, the AA model has a self-dual symmetry,and the increasingly incommensurate potential will makethe system undergo a localization transition, from the the extended phase to the localized phase. Moreover, themobility edge always appears in the extended AA mod-els. For instance, Das Sarma et al . revealed that therewas mobility edge in a class systems with slow-varyingincommensurate on-site potentials [7–9]. Biddle et al .found the exactly solvable mobility edge in the extendedAA model with long-range tunnelings [10, 11]. Ganeshan et al . verified the existence of the mobility edge whichwas protected by the self-dual symmetry [12]. Liu etal . obtained the exactly solvable mobility edge in the off-diagonal incommensurate models [13, 14]. Nowadays, theresearch about the mobility edge has been extended tothe non-Hermitian case [21–23], where the mobility edgeswere located in the real energy spectra. However, by com-prehensively analyzing the above research results on mo-bility edge, we find that the mobility edges change withthe extra parameters which are commonly the strengthof the incommensurate on-site potentials. We can’t helpbut ask, is it a universal phenomenon that the mobilityedge changes with the extra parameters? Is there a spe-cial case where the mobility edge is a constant that is abit easier to measure experimentally?In this work, we are motivated to study a one-dimensional tight-binding model with tunable incom-mensurate potentials. We make attempt to find outwhether there is a mobility edge which keeps a constantin this model and then extract it analytically, if it exists.We further verify the analytical result by the inverse par-ticipation rate and the spatial distributions of the wavefunctions.The rest of this paper are arranged as follows: In Sec. IIwe present the model and derive the mobility edge by theself-dual relations. In Sec. III we verify the analyticallyobtained mobility edge by the inverse participation rateand the distributions of the wave functions. Finally, wemake a summary in Sec. IV.
II. MODEL AND THE MOBILITY EDGE
In this section, we consider a one-dimensional tight-binding model with nearest-neighbor hoppings and in-commensurate on-site potentials, and we are devotedto extract the mobility edge analytically. The staticSchr¨odinger equation is written as Eψ n = ψ n +1 + ψ n − + V − b cos(2 παn ) ψ n , (1)where V is the potential strength with 0 < b < α is irrational number,which makes the Hamiltonian incommensurate. ψ n is theamplitude of wave function at the n th lattice site. Thismodel provides a extension of the quasi-periodic latticepreviously introduced in Ref. [12] and displays a gener-alized self-duality. To derive the mobility edge, namelythe critical energy E c , which separates the extended wavefunctions from whose are localized, we first multiply bothsides of the Eq. (1) by e i παnm and sum it over n . Aftersetting φ m = P n e i παnm ψ n , we obtain[ E − παm )] φ m = X n e i παnm V − b cos(2 παn ) ψ n . (2)Next, we introduce some self-dual relations which arenecessary in the following derivationcosh( s ) = 1 b , E = 2 cosh( s ) , Ω m ( s ) = cosh( s ) − cos(2 παm )sinh( s ) , (3)where, s and s are real constants. Therefore, the Eq. (2)can be rewritten as2 sinh( s )Ω m ( s ) φ m = V coth( s ) X r ′ e −| m − r ′ | s φ r ′ . (4)Then, we multiply both sides of the Eq. (4) byΩ k ( s ) e i παmk , and sum it over m . After defining therelationship ϕ k = P m e i παmk Ω m ( s ) φ m , the Eq. (4) isfurther expressed as2 sinh( s )Ω k ( s ) ϕ k = V coth( s ) X k ′ e −| k − k ′ | s ϕ k ′ . (5)We continue multiplying both sides of Eq. (5) by e i παkq , and summing it over k . By defining the rela-tionship µ q = P k e i παkq ϕ k , the Eq. (5) is transformedinto2 sinh( s )sinh( s ) cosh( s ) µ q = µ q − + µ q +1 + V coth( s )Ω q ( s ) µ q . (6)Note that when s = s , the Eq. (3) is reduced to theEq. (1). From the self-dual relations cosh( s ) = 1 /b and E = 2 cosh( s ), we obtain the mobility edge E c = 2 /b. (7) Intuitively, for a fixed value of the tuning parameter b , E c is independent of the potential strength V . The propertyof ”being constant” of the mobility edge is a intriguingresult. In the following, we will check the predictions of E c in the theoretical analysis by direct numerical simu-lations, so as to make the theoretical analysis convictive. III. VALIDATIONS AND DISCUSSIONS V -3-1135791113 E Figure 1: (Color online) The eigenvalues of Eq. (1) andIPR as a function of V with the parameter b = 0 . L = 500. Dif-ferent colors of the eigenvalue curves indicate differentmagnitudes of the IPR of the corresponding wave func-tions. The black region where the IPR approaches zerodenote the extended states, and the orange region with awhitish hue where the IPR tends to 1 denotes the local-ized states. The blue solid line represent the boundarybetween spatially localized and extended states, i.e., themobility edge E c = 2 /b = 5.In this section, we verify the accuracy of the analyti-cally obtained mobility edge E c by exhaustively numer-ical analysis. In order to acquire the eigenenergies E ofthe system and the associated wave functions ψ , we nu-merically diagonalize the model’s Hamiltonian which isdefined in terms of the Schr¨odinger equation in Eq. (1).Then, it is convenient to calculate other typically physicalquantities which are needed in our numerical validation.For example, the inverse participation rate and the spa-tial distributions of the wave functions will be used todistinguish the extended and localized states. Althoughour numerical simulations are carried out at the systemwith finite size, our results are exactly consistent withthe theoretical analysis under the thermodynamic limit.To begin with, we calculate the inverse participationrate (IPR). As it was studied in Ref. [24–27], the IPR ofa given wave function ψ is defined asIPR j = L X n =1 | ψ jn | ; (8)where ψ has been normalized, L is the total number ofthe lattice sites, and j denotes the energy level index. Itis widely known that for the extended wave functions, thecorresponding IPRs scale like L − , which tend to 0 when L → ∞ , whereas they keep finite values for localizedstates.Figure 1 is the energy spectrum which exhibits vari-ation of eigenenergies E as a function of the parameter V , where we have considered the size of the system is L = 500 and the tuning parameter b = 0 .
4. As the fig-ure shows, the spectrum is not a single band but consistsof several bands. The color shows the IPR of the corre-sponding eigenenergy. Concretely, the black region wherethe IPR approaches zero means the wave functions thatthese eigrnenergies correspond to are all extended. Onthe contrary, the orange region with a whitish hue wherethe IPR tends to 1 means the wave fuctions that theseeigenenergies correspond to are all localized. Intuitively,there is a critical energy E c = 5 separating the extendedstates and the localized states. The result of this nu-merical analysis is consistent with what the theoreticalanalysis predicts.In order to have a deeper insight into the phase sepa-ration in the aspect of energy, and further to verify thereliability of the analysis results of IPR, we will plot thespatial distributions of the associated wave functions,whose corresponding eigenenergies are above or belowthe E c . Before performing the numerical calculation, wehave specified that the required parameters are b = 0 . V = 4 . L = 500. After the Schmidt decomposition,we obtain all the wave functions of the systems. Wechoose four typical wave functions whose correspondingeigenenergies are above or below but close to the mobilityedge E c (Here E c = 5). Figure 2 shows the spatial distri-butions of the four wave functions, in which Fig. 2(a) andFig. 2(b) are the extended states whose corresponding en-ergies are below but close to the E c , whereas Fig. 2(c)and Fig. 2(d) are the localized states whose correspond-ing energies are located above but close to the E c .Besides, we adopt the same strategy to obtain the wavefunctions at the parameter V = 7 .
3, and other involvedparameters are same as those used for Fig. 2. Similarly,we choose four typical wave functions whose correspond-ing energies are above or below but close to the mobilityedge E c . We plot the spatial distributions of the setof wave functions and they are depicted in Fig. 3. ForFig. 3(a) and Fig. 3(b), the energies of the chosen wavefunctions are located below but near the E c , and it is in-tuitively seen that the two wave functions are extended.On the contrary, the distributions of the wave functionsare localized in Fig. 3(c) and Fig. 3(d), where their cor- -0.500.5-0.500.5-1010 100 200 300 400 500-0.500.5 Figure 2: (Color online) Wave functions obtained fromEq. (1) with b = 0 . V = 4 . L = 500 nearby themobility edge E c = 5. Four different eigenenergies: (a)and (b) extended states with energies below themobility edge, (c) and (d) localized states with energiesabove the mobility edge.responding energies are located above but close to the E c .Finally, we would like to point out that this invariablemobility edge can be realized inthe ultracold-atom exper-iment. Using synthetic lattices of laser-coupled atomicmomentum modes, Ref. [28] experimentally realize a re-cently proposed family of nearest-neighbor tight-bindingmodels having quasiperiodic site energy modulation thathost an variable mobility edge protected by a dualitysymmetry. The experimental method in this paper isbased on momentum space lattice tuning. The on-sitepotential at each lattice point in momentum space isequal to the detuning of Raman Coupling between differ-ent momentum states, hence the on-site potential formrequired for the precise modulation of these detuning tomatch is obtained.Alternatively, we can use synthetic dimension to simu-late this type of on-site lattice. For example, we can useeach spin state of large spin atoms (such as Dy atoms)as lattice points, and use Raman Coupling to couple ad-jacent spins (i.e., the coupling of adjacent lattice points),then realize the on-site potential trough controlling thedetuning property of Raman coupling. -0.500.5-0.500.5-1010 100 200 300 400 500-101 Figure 3: (Color online) Wave functions obtained fromEq. (1) with b = 0 . V = 7 . L = 500 nearby themobility edge E c = 5. Four different eigenenergies: (a)and (b) extended states with energies below themobility edge, (c) and (d) localized states with energiesabove the mobility edge. IV. SUMMARY
In conclusion, we have studied a 1D tight-bindingmodel with tunable quasi-periodic potential. We uncov-ered there exists mobility edge in this model, and ob-tained its analytical expression by the self-dual relations.Surprisingly, unlike the previous researches, the mobilityedge in our model keeps invariable with the strength ofthe potential, and which is only determined by the con-trol parameter b . This finding may contribute to a moreconvenient observation of the mobility edge in the nearfuture experiment. Subsequent detailed analyses, suchas the inverse participation rate and the spatial distri-butions of the wave functions, show that our results arereliable.T. Liu acknowledges X.-J. Liu for fruitful discussions.This work is supported by the National Natural Sci-ence Foundation of China (Grants No. 61874060, No.11674051, and No. U1932159), Natural Science Foun-dation of Jiangsu Province (Grant No. BK20200737 andNo. BK20181388), and NUPTSF (Grant No. NY220090,No. NY220208 and No. NY217118). ∗ [email protected] † ‡ [email protected] § [email protected][1] P. W. Anderson, Phys. Rev. , 1492 (1958).[2] E. Abrahams, P. W. Anderson, D. C. Licciardello, andT. V. Ramakrishnan, Phys. Rev. Lett. , 673 (1979).[3] N. Mott, J. Phys. C , 3075 (1987).[4] C. M. Soukoulis and E. N. Economou, Phys. Rev. Lett. , 1043 (1982).[5] P. A. Lee and T. V. Ramakrishnan, Rev. Mod. Phys. ,287 (1985).[6] S. Das Sarma, A. Kobayashi, and R. E. Prange, Phys.Rev. Lett. , 1280 (1986).[7] S. Das Sarma, S. He, and X. C. Xie, Phys. Rev. Lett. ,2144 (1988).[8] S. Das Sarma, S. He, and X. C. Xie, Phys. Rev. B ,5544 (1990).[9] T. Liu, H.-Y. Yan, and H. Guo, Phys. Rev. B , 174207(2017).[10] J. Biddle and S. Das Sarma, Phys. Rev. Lett. , 070601(2010).[11] J. Biddle, D. J. Priour, B. Wang, and S. Das Sarma,Phys. Rev. B , 075105 (2011).[12] S. Ganeshan, J. H. Pixley, and S. Das Sarma, Phys. Rev.Lett. , 146601 (2015).[13] T. Liu, G. Xianlong, S. Chen, and H. Guo, Phys. Lett.A , 3683 (2017).[14] T. Liu and H. Guo, Phys. Rev. B , 104201 (2018).[15] H. Yin, J. Hu, A.-C. Ji, G. Juzeliunas, X.-J. Liu, and Q.Sun, Phys. Rev. Lett. , 113601 (2020).[16] Y. Wang, L. Zhang, S. Niu, D. Yu, and X.-J. Liu, Phys.Rev. Lett. , 073204 (2020).[17] Y. Wang, X. Xia, L. Zhang, H. Yao, S. Chen, J. You,Q. Zhou, and X.-J. Liu, Phys. Rev. Lett. , 196604(2020).[18] H. P. L¨uschen, S. Scherg, T. Kohlert, M. Schreiber, P.Bordia, X. Li, S. Das Sarma, and I. Bloch, Phys. Rev.Lett. , 160404 (2018).[19] F. A. An, E. J. Meier, and B. Gadway, Phys. Rev. X ,031045 (2018).[20] S. Aubry and G. Andr´e, Ann. Isr. Phys. Soc. , 18 (1980).[21] Y. Liu, X.-P. Jiang, J. Cao, S. Chen, Phy. Rev. B ,174205 (2020).[22] T. Liu, H. Guo, Y. Pu, S. Longhi, Phys. Rev. B ,024205 (2020).[23] Q.-Z. Zeng, Y.-B. Yang, and Y. Xu, Phys. Rev. B ,020201 (2020).[24] D. J. Thouless, Phys. Rep. , 93 (1974).[25] M. Kohmoto, Phys. Rev. Lett. , 1198 (1983).[26] M. Schreiber, J. Phys. C , 2493 (1985).[27] Y. Hashimoto, K. Niizeki, and Y. Okabe, J. Phys. A25