Duality between two generalized Aubry-Andre models with exact mobility edges
Yucheng Wang, Xu Xia, Yongjian Wang, Zuohuan Zheng, Xiong-jun Liu
DDuality between two generalized Aubry-Andr´e models with exact mobility edges
Yucheng Wang,
1, 2, 3, ∗ Xu Xia, ∗ Yongjian Wang,
5, 6, ∗ Zuohuan Zheng,
5, 6, 7 and Xiong-Jun Liu
2, 3, 8, 1 Shenzhen Institute for Quantum Science and Engineering, and Department of Physics,Southern University of Science and Technology, Shenzhen 518055, China International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China Collaborative Innovation Center of Quantum Matter, Beijing 100871, China Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, China Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China University of Chinese Academy of Sciences, Beijing 100049 , China College of Mathematics and Statistics, Hainan Normal University, Haikou, Hainan 571158, China CAS Center for Excellence in Topological Quantum Computation,University of Chinese Academy of Sciences, Beijing 100190, China
A mobility edge (ME) in energy separating extended from localized states is a central conceptin understanding various fundamental phenomena like the metal-insulator transition in disorderedsystems. In one-dimensional quasiperiodic systems, there exist a few models with exact MEs, andthese models are beneficial to provide exact understanding of ME physics. Here we investigate twowidely studied models including exact MEs, one with an exponential hopping and one with a specialform of incommensurate on-site potential. We analytically prove that the two models are mutuallydual, and further give the numerical verification by calculating the inverse participation ratio andHusimi function. The exact MEs of the two models are also obtained by calculating the localizationlengths and using the duality relations. Our result may provide insight into realizing and observingexact MEs in both theory and experiment.
I. INTRODUCTION
Anderson localization (AL) , a fundamental quantumphenomenon in nature, reveals that the single-particlestates can become localized due to disorder effect. Thequantum phase transition from extended (metal) phaseto localized (insulator) phase can occur by increasingthe disorder strength in three dimensional (3D) systems.Near the transition point, the mobility edges (MEs) canoccur and separate the extended and localized states .ME lie at the heart in understanding various fundamen-tal phenomena such as the metal-insulator transition in-duced by varying disorder strength or particle numberdensity. Moreover, a system with ME has strong ther-moelectric response , and can be applied to thermo-electric devices. MEs exist widely in 3D systems withrandom disorder, but for one and two dimensions, thescaling theory shows that all states are localized for ar-bitrarily small disorder strengths, so no MEs exist.Unlike random disorder, the quasiperiodic potentialcan induce the extended-AL transition at a finite strengthof the potential even in the 1D systems, which bringabout rich interesting physics, e.g., the existence ofMEs even in 1D systems and non-ergodic criti-cal phases . The most celebrated example with1D quasiperiodic potential is the Aubry-Andr´e (AA)model , described by t ( ψ j +1 + ψ j − ) + V cos(2 πβj + δ ) ψ j = Eψ j , where ψ j , t, V, δ denote the wavefunc-tion amplitude at site j , the nearest-neighbor hoppingstrength, the strength of quasiperiodic potential, and thephase parameter, respectively, and β is an irrational num-ber. The model exhibits a self duality for the transforma-tion between real and momentum spaces at V = 2 t , lead-ing to the extended-localization transition with all the eigenstates of the model being extended (localized) for V < t ( V > t ). Thus no ME exists for the AA model.This model has been realized in ultracold atomic gasestrapped in incommensurate optical lattices, and the lo-calization transition has been observed . The existenceof the many body localization phase in the quasiperi-odic AA model in the presence of weak interactions hasalso been well established in both theory and exper-iment .By introducing the short-range or long-range term ,or breaking the self duality of the AA model , onecan obtain the MEs in the system. However, very few ofthem can provide the accurate expression of MEs ,and undoubtedly, these models with exact MEs are ben-eficial to provide exact understanding of the ME physicsfor both the non-interacting and interacting systems. Inthis work, we will focus on the two most commonly usedmodels with exact MEs. One is E a n = (cid:88) n (cid:48) (cid:54) = n t e − p | n − n (cid:48) | a n (cid:48) + V cos(2 πβn + δ ) a n , (1)where p > t e − p | n − n (cid:48) | is the hopping rate between thesites n and n (cid:48) and V is the strength of the quasiperi-odic potential. There exists an exponential rather thannearest-neighbor hopping, and this model will becomethe AA model in the limit p → ∞ . The exact expressionof ME is E c = V cosh( p ) − t . (2)The other widely studied model with exact MEs we con-sidered is E b n = t ( b n − + b n +1 ) + 2 λ cos(2 πβn + δ )1 − α cos(2 πβn + δ ) b n , (3) a r X i v : . [ c ond - m a t . d i s - nn ] D ec where t represents the hopping strength between neigh-boring sites, λ and α ( α ∈ ( − , α = 0, this model reduces to theAA model. The exact expression of the ME is E c = 2 sgn ( λ )( | t | − | λ | ) /α. (4)For convenience, we call the above-mentioned first (sec-ond) model Model I (II). The two models have widelybeen used to study the ME physics, e.g., the dynami-cal behavior of a system with MEs , fate of MEs in thepresence of interactions or non-Hermitian term .In recent years, MEs have been observed in disorderedsystems and quasiperiodic systems in experi-ments based on ultracold atoms. In particular, the re-cent work has accurately realized the Model II (3) byusing synthetic lattices of laser-coupled atomic momen-tum modes, and accurately detected the location of MEsin the absence and presence of interactions.This study is motivated by two nontrivial questionsraised here. First, is there any profound relation betweenthe above two models even they seem to be quite differ-ent, and whether the Model I can be accurately realizedin experiment? Secondly, can the localization lengths ofthe states in two models can be exactly computed, whichclearly necessitates to go beyond the dual transformationapplied to determine the ME in the previous studies. An-swering these questions is important to unveil the funda-mental properties of the two important models. In thiswork, we prove that the above two generalized AA mod-els (Eq. (1) and Eq. (3)) have the mutually dual relation,and further provide exact study of the localization prop-erties of the states. In particular, the Hamiltonian of theModel II can be written as a tri-diagonal matrix, whose ME expression can be obtained by using a self-consistenttheory or by calculating the localization length numer-ically by using the recursive methods and analyticallyby using Avila’s global theory . By using the du-ality relation between Model I and Model II, we furtherdetermine the exact ME of Model I, whose localizationlength is difficult to be directly computed and the self-consistent theory can not also be used due to the expo-nential hopping. With the dual relation proved in thiswork, the recent experimental work that realized theModel II (3) in momentum space could be regarded tohave also effectively realized the Model I in real space. II. ANALYTICAL AND NUMERICAL RESULTSA. analytical derivation for dual relations
We firstly analytically establish the duality betweenthe Model I and Model II. Since the phase offset δ isredundant in the context of localization , without loss ofgenerality, we set δ = 0. We start from the Model I (1),and introduce the transformation a j = 1 √ L (cid:88) m b m e − i πmβj , (5)where b m = 1 √ L (cid:88) j a j e i πmβj , (6)By using the transformation (5), Eq. (1) becomes E √ L (cid:88) m b m e − i πmβn = (cid:88) n (cid:48) (cid:54) = n t e − p | n − n (cid:48) | √ L (cid:88) m b m e − i πmβn (cid:48) + V cos(2 πβn ) 1 √ L (cid:88) m b m e − i πmβn . (7)Then we rewrite the first term on the right sideof the equation (cid:80) n (cid:48) (cid:54) = n t e − p | n − n (cid:48) | √ L (cid:80) m b m e − i πmβn (cid:48) as √ L (cid:80) m b m e − i πmβn (cid:80) n (cid:48) (cid:54) = n t e − p | n − n (cid:48) | e − i πmβ ( n (cid:48) − n ) ,where (cid:80) n (cid:48) (cid:54) = n t e − p | n − n (cid:48) | e − i πmβ ( n (cid:48) − n ) is the summation of a geometric sequence, and one can obtain that it equalsto t ( − e − p + e − p cos(2 πmβ ))1+ e − p − e − p cos(2 πmβ ) . Then Eq. (7) can be writtenas E √ L (cid:88) m b m e − i πmβn = 1 √ L (cid:88) m b m e − i πmβn t ( − e − p + e − p cos(2 πmβ ))1 + e − p − e − p cos(2 πmβ ) + V √ L (cid:88) m ( b m − + b m +1 ) e − i πmβn , Utilizing the above formula, one can directly obtain E b m = 2 t ( − e − p + e − p cos(2 πmβ ))1 + e − p − e − p cos(2 πmβ ) b m + V b m − + b m +1 ) . (8) Let t = V , α = 2 e − p e − p , (9a) E = E + 2 t e − p e − p , λ = t ( − e − p + e − p )(1 + e − p ) , (9b)then Eq. (8) is equivalent to Eq. (3). Therefore, Model I(Eq. (1)) and Model II (Eq. (3)) are mutually dual. B. Localization lengths and mobility edges
Since the two models (Model I and Model II) are mu-tually dual, we can obtain some properties of one modelfrom the other model. Model I is not exactly solvable dueto the existence of the exponential hopping, but Model IIcan be written as a tri-diagonal matrix, whose all states’extended and localized properties can be analytically ob-tained by using the Avila’s global theory . Wefirstly represent the Eq. (3) in the transfer matrix form, (cid:18) b n +1 b n (cid:19) = T n (cid:18) b n b n − (cid:19) where the transfer matrix T n is given by T n = (cid:32) E t − λt cos(2 πβn + δ )1 − α cos(2 πβn + δ ) −
11 0 (cid:33) (10)Using the transfer matrix, one can define and computethe Lyapunov exponent (LE), γ ( E ) = lim n →∞ πL (cid:90) ln (cid:107) T L ( δ ) (cid:107) dδ, where T L = (cid:81) Ln =1 T n and (cid:107) T L (cid:107) denotes the norm of thematrix T L . The LE can be exactly obtained by usingAvila’s global theory , and the details for the calcula-tion are put in the Appendix. By the LE, we can obtainthe localization length ξ , which is the reciprocal of theLE, i.e., ξ ( E ) = 1 γ ( E ) = 1ln | | αE +2 λ | + √ ( αE +2 λ ) − α t + √ t − α ) | . (11)When | | αE +2 λ | + √ ( αE +2 λ ) − α t + √ t − α ) | > ξ is a finite(infinite) value and the corresponding state is localized(decolized). Thus the critical points and MEs are de-termined by | | αE c +2 λ | + √ ( αE c +2 λ ) − α t + √ t − α ) | = 1, which cangive the ME expression (Eq. (4)) (see the Appendix).The ME expression of Model II can also be analyticalobtained by using a self-consistent theory .Naturally, combining Eqs. ((9a),(9b)) and the expres-sion Eq. (4) of Model II’s ME, one can obtain the expres-sion Eq. (2) of Model I’s ME. C. numerical results
Now we display the numerical evidence for the dualrelation. The numerical results are obtained by calcu-lating the inverse participation ratio (IPR) IP R ( κ ) = IPR ' FIG. 1: (a) IPR of different eigenstates as a function ofthe corresponding eigenvalues E and quasiperiodic potentialstrength V with fixed p = 1 . t = 1 in Model I, (b) IPRas a function of E (cid:48) and 2 t with fixed α = 0 . λ =0 . α and λ are obtainedfrom Eq. (9a) and Eq. (9b) with fixed p = 1 . t = 1. E (cid:48) = E − t e − p e − p . The blue and green dotted line in (a) and(b) are obtained from Eq. (2). Here we fix β = ( √ − / L = 500. (cid:80) Lj =1 | ψ κ,j | , where ψ κ is the κ -th eigenstate. It is knownthat tends to zero in the thermodynamic limit for ex-tended states, but approaches to a finite value of O (1) fora localized state. Fig. 1 (a) shows the energy eigenvaluesand the IPR of the corresponding eigenstates for ModelI as a function of V under open boundary conditions.The dotted line represents the ME given in Eq. (2). Wesee that IPR values are approximately zero for energiesabove the ME and are finite for energies below the ME.In fig. 1 (a), we take p = 1 . t = 1, which can give α = 0 . λ = 0 . t , we can diagonalize the Model II and ob-tain its eigenvalues E and the IPR of the correspondingeigenstates. Fig. 1 (b) shows the IPR as a function of E (cid:48) and 2 t , where E (cid:48) = E − t e − p e − p . Due to V = 2 t ,we take the horizontal axis being 2 t to compare withfig. 1 (a). One can see that the two energy spectrum infig. 1 (a) and fig. 1 (b) are exactly the same, but the IPRvalues in fig. 1 (b) are finite for energies above the MEand are approximately zero for energies below the ME,which is contrary to fig. 1 (a), indicating that the ModelI and Model II are mutually dual.Besides the IPR, we further introduce the Husimi func-tion to gain a better intuition for the localizationbehavior in both the real space and momentum space. Itis given by ρ ( j , k ) = |(cid:104) j , k | ψ (cid:105)| . (12)Here the Husimi function is the probability density func-tion for finding the system with state | ψ (cid:105) in a minimum-uncertainty state centered at j in coordinate space andat k in momentum space. Note that while the momen-tum is not a good quantum number here, the projection j j FIG. 2: The Husimi function ρ ( j, k ) for the eigenstate cor-responding the lowest energy in (a) and (c) and highest en-ergy in (b) and (d). (a) and (b) correspond to Model I with p = 1 . t = 1 and V = 0 .
5, (c) and (d) correspond to ModelII with α = 0 . λ = 0 . t = 0 .
25. Here we fix β = ( √ − / L = 500. on the selected k can be done. Using the minimal un-certainty state in real space (cid:104) j | j , k (cid:105) = ( 12 πσ ) / exp ( − ( j − j ) σ + ik ( j + j / , where σ is taken as σ = (cid:113) L π , and inserting (cid:80) j | j (cid:105)(cid:104) j | inEq. (12), one can obtain the Husimi function, as shownin fig. 2. From fig. 2 (a), we see that there exists onevertical stripe for the lowest state of Model I indicatingthat the considered state is localized in real space andextended in momentum space. By contrast, fig. 2 (b)shows three horizontal stripes symmetrically placed withrespect to k = 0 for the highest state of Model I indicat-ing that the state is localized in momentum space andextended in real space. Comparing fig. 2 (a) with fig. 2(b), we see that there exists a ME for the Model I with p = 1 . , t = 1 and V = 0 .
5, and the eigenstates are spa-tially localized and extended below and above the ME.The same analysis applies to the Model II (see fig. 2 (c)and (d)), whose eigenstates are spatially extended and lo-calized below and above the ME, and it is consistent withour conclusion that Model I and Model II are mutuallydual.
III. SUMMARY AND DISCUSSION
We have analytically proven that the two widely stud-ied models (Model I (1) and Model II (3)) with exactMEs are mutually dual. By using the Avila’s global the- ory, one can give the localization length and ME expres-sions of Model II. Then Model I’s ME expressions canbe obtained by using the dual relation. We further nu-merically verified our result by calculating the IPR andHusimi function. Our conclusion will make the ME’sstudy more convenient. In theory, studying the physicalproperties of one of the models, one can deduce the cor-responding properties of the other model. On the otherhand, we provide the new approach to study the systemwith the exponential hopping, whose some properties aredifficult to investigated both numerically and analyticallybut can be obtained for its dual model. In experiment,the realization of the Model I and the detection of thelocation of ME can be replaced by detecting the locationof Model II’s ME in momentum space, which has beenrealized and detected recently . Acknowledgments
Yucheng Wang and X.-J. Liu are supported by Na-tional Nature Science Foundation of China (11825401,11761161003, and 11921005), the National Key R&DProgram of China (2016YFA0301604), Guangdong In-novative and Entrepreneurial Research Team Program(No.2016ZT06D348), the Science, Technology and In-novation Commission of Shenzhen Municipality (KYT-DPT20181011104202253), and the Strategic Priority Re-search Program of Chinese Academy of Science (GrantNo. XDB28000000). X. Xia is supported by NanKaiZhide Foundation. Yongjian Wang is supported bythe National Natural Science Foudation of China (No.12061031).Zuohuan Zheng acknowledges financial sup-ports of the NSF of China (No. 12031020, 11671382),CAS Key Project of Frontier Sciences (No. QYZDJ-SSW-JSC003), the Key Lab. of Random Complex Struc-tures and Data Sciences CAS and National Center forMathematics and Interdisciplinary Sciences CAS.
Appendix A: Details for localization length
In this appendix, we give the detail of the derivation ofLyapunov exponents. For convenience, we set the hop-ping strength t = 1. The transfer matrix (10) can bedecomposed into two parts, T n = A n B n , where A n = 11 − α cos(2 πβn + δ ) , (A1)and B n = (cid:18) B B B (cid:19) (A2)with B = E (1 − α cos(2 πβn + δ )) − λ cos(2 πβn + δ )and B = − B = 1 − α cos(2 πβn + δ ). Then γ ( E ) = γ A ( E ) + γ B ( E ) , (A3)where γ A = lim n →∞ πL (cid:82) ln (cid:107) A L ( δ ) (cid:107) dδ with A L = (cid:81) Ln =1 A n . By the ergodic theory, γ A ( E ) = π (cid:82) π ln( − α cos( δ ) ) dδ = − ln | √ − α | . InEq. (A3), γ B = lim n →∞ πL (cid:82) ln (cid:107) B L ( δ ) (cid:107) dδ with B L = (cid:81) Ln =1 B n . Below we calculate γ B relies on Avila’s global theory . We firstly complexify the phase, i.e., B = E (1 − α cos(2 πβn + δ + i(cid:15) )) − λ cos(2 πβn + δ + i(cid:15) ), B = − B = 1 − α cos(2 πβn + δ + i(cid:15) ). Then let (cid:15) tends to infinity, the matrix B n becomes B n ( δ + i(cid:15) ) = e π(cid:15) e i (2 πβn + δ ) (cid:18) − αE − λ α − α (cid:19) + o (1) (A4)Thus we have γ B ( E, (cid:15) ) = 2 π(cid:15) +ln | | αE +2 λ | + √ ( αE +2 λ ) − α | + o (1). By the global the-ory , we obtain γ B ( E ) = ln | | αE +2 λ | + √ ( αE +2 λ ) − α | .Plugging γ A ( E ) and γ B ( E ) into Eq. (A3), wehave γ ( E ) = ln | | αE +2 λ | + √ ( αE +2 λ ) − α √ − α ) | , whichgive the localization length, as shown in Eq. (11). As our discussions in the main text, MEs satisfy | | αE +2 λ | + √ ( αE +2 λ ) − α √ − α ) | = 1. Now we set P = αE + 2 λ ,then MEs satisfy | P | + √ P − α = 2(1 + √ − α ),which give | P | = 2, i.e., | αE + 2 λ | = 2, which can giveEq. (4) in the main text. ∗ These authors contribute equally to this work.Corresponding author: [email protected] (YuchengWang). P. W. Anderson, Absence of diffusion incertain randomlattices, Phys. Rev. , 1492 (1958). A. Lagendijk, B. Tiggelen, and D. S. 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