One dimensional quasiperiodic mosaic lattice with exact mobility edges
Yucheng Wang, Xu Xia, Long Zhang, Hepeng Yao, Shu Chen, Jiangong You, Qi Zhou, Xiong-Jun Liu
OOne dimensional quasiperiodic mosaic lattice with exact mobility edges
Yucheng Wang,
1, 2, 3, ∗ Xu Xia, ∗ Long Zhang,
2, 3
Hepeng Yao, ShuChen,
6, 7, 8
Jiangong You, † Qi Zhou, ‡ and Xiong-Jun Liu
2, 3, 9, 10, 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 CPHT, CNRS, Institut Polytechnique de Paris, Route de Saclay 91128 Palaiseau, France Beijing National Laboratory for Condensed Matter Physics,Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China School of Physical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China Yangtze River Delta Physics Research Center, Liyang, Jiangsu 213300, China Beijing Academy of Quantum Information Science, Xibeiwang East Rd, Beijing 100193, China CAS Center for Excellence in Topological Quantum Computation,University of Chinese Academy of Sciences, Beijing 100190, China
The mobility edges (MEs) in energy which separate extended and localized states are a centralconcept in understanding the localization physics. In one-dimensional (1D) quasiperiodic systems,while MEs may exist for certain cases, the analytic results which allow for an exact understandingare rare. Here we uncover a class of exactly solvable 1D models with MEs in the spectra, wherequasiperiodic on-site potentials are inlaid in the lattice with equally spaced sites. The analyticalsolutions provide the exact results not only for the MEs, but also for the localization and extendedfeatures of all states in the spectra, as derived through computing the Lyapunov exponents, and alsonumerically verified by calculating the fractal dimension and performing the multifractal analysis.We further propose a novel scheme with high feasibility to realize our model based on an opticalRaman lattice, which paves the way for experimental exploration of the predicted exact ME physics.
Introduction.–
Anderson localization (AL) is a funda-mental and extensively studied quantum phenomenon,in which the disorder induces exponential localizationof electronic wave-functions, resulting in the absence ofdiffusion [1]. The phase transition between AL and ex-tended phases in disorder system occurs only in dimen-sion higher than two [2]. The noninteracting mobilityedge (ME), a most important concept in the extended-AL transition marking a critical energy E c separatingextended states from localized states [3], may exist inthree-dimension, but not for one-dimensional (1D) sys-tems with random disorder where all states are localizedfor arbitrarily weak disorder strengths.On the other hand, the quasiperiodic potential in-duced localization has also generated considerable in-terests in recent years. The AL, many body localiza-tion, and Bose glass have been observed in experimentin ultracold atomic gases trapped in quasiperiodic lat-tices [4–10], and the many-body critical phases havebeen predicted [11, 12]. Moreover, the presences ofextended-AL transitions and mobility edges (MEs) arealso predicted [13–27] in the 1D quasiperiodic systems.The Aubry-André-Harper (AAH) model is a simplestnontrivial example with 1D quasiperiodic potential [28],described by t ( ψ j +1 + ψ j − ) + 2 λ cos(2 πωj + θ ) ψ j = Eψ j , where ψ j , t, λ, θ denote the wavefunction amplitudeat site j , the nearest-neighbor hopping coefficient, thequasiperiodic potential amplitude, and the phase offset,respectively, and ω is an irrational number. The model exhibits a self duality for the transformation betweenlattice and momentum spaces at λ = t , leading to theextended-localization transition with all the eigenstatesof the model being extended (localized) for λ < t ( λ > t ).Thus no ME exists for the AAH model. While the tran-sition was predicted in 1980’s, its rigorous mathematicalproof only came recently [29–31]. By introducing a long-range hopping term [16, 32, 33], or breaking the self du-ality of the AAH Hamiltonian, e.g. superposing anotherquasiperiodic optical lattice [19–22, 34] or introducing thespin-orbit coupling [35, 36], one can obtain MEs in the en-ergy spectra of the system. In very few cases [16, 23] theself duality may be recovered on certain analytically de-termined energy, across which the extended-localizationtransition occurs, rendering the ME in the spectra, whilethe whole model is not exactly solvable. That is, the ex-tended and localized states in the spectra cannot be an-alytically obtained to rigorously illustrate how the tran-sition between them occurs. Further, it is not clear if asingle system can have multiple MEs, and what deter-mines the number of the MEs. Addressing these issueswith exactly solvable models is critical to gain exact un-derstanding of the extended-localization transition.In this letter, we propose a class of analytically solvable1D models in quasiperiodic mosaic lattice, which hostmultiple MEs with the self-duality breaking. The local-ization and extended features can be exactly obtained forall states in these models which have an arbitrary evennumber of MEs depending on parameter. We also pro- a r X i v : . [ c ond - m a t . d i s - nn ] M a y pose a novel scheme with high feasibility to realize anddetect the exact MEs. Model.—
We consider a class of quasiperiodic mosaicmodels, which can be described by H = t (cid:88) j ( c † j c j +1 + H.c. ) + 2 (cid:88) j λ j n j , (1)with λ j = (cid:40) λ cos(2 π ( ωj + θ )) , j = mκ, , otherwise , (2)where c j is the annihilation operator at site j , n j = c † j c j is the local number operator and κ is an integer. We setthe hopping strength t = 1 for convenience. Since thequasiperiodic potential periodically occurs with interval κ , we can introduce a quasi-cell with the nearest κ latticesites. If the quasi-cell number is taken as N , i.e., m =1 , , · · · , N , the system size will be L = κN .It is obvious that this model reduces to the AAH modelwhen κ = 1 . If κ (cid:54) = 1 , the duality symmetry of thesemodels is broken, which motivate us to show the existenceof MEs. In this letter, we prove that these models with κ (cid:54) = 1 do have energy dependent MEs, which are givenby the following expression, | λa κ | = 1 , for E = E c , (3)with a κ = 1 √ E − (cid:32) ( E + √ E −
42 ) κ − ( E − √ E −
42 ) κ (cid:33) (4)In addition, all the localized and extended states can beexactly studied. This is our central result which we proveby computing the exact Lyapunov exponent (LE). Beforeshowing the analytic derivatives, we display the numeri-cal results of the κ = 2 , which is benefit to visually un-derstand this condition (Eq. (3)) representing it as a ME.Without loss of generality, we set θ = 0 and ω = √ − ,which can be approached by using the Fibonacci num-bers F n [37, 38]: ω = lim n →∞ F n − F n , where F n is definedrecursively by F n +1 = F n − + F n , with F = F = 1 .We take the quasi-cell number N = F n and the ratio-nal approximation ω = F n − /F n to ensure a periodicboundary condition when numerically diagonalizing thetight binding model defined in Eq. (1). The κ = 2 case.— For the minimal nontrivial case with κ = 2 , we obtain from Eq. (3) [39] the MEs as E c = ± λ . (5)The numerical results are obtained from the inverse par-ticipation ratio (IPR) IPR( m ) = (cid:80) Lj =1 | ψ m,j | [3], where ψ m is the m -th eigenstate. To characterize the ME, weinvestigate the fractal dimension of the wave function, Figure 1: (a) Fractal dimension Γ of different eigenstates asa function of the corresponding eigenvalues and quasiperiodicpotential strength λ with size N = F = 610 . The twodashed lines represent the MEs E c = λ and E c = − λ . Herethe hopping strength t has been set as the unit of energy.Spatial distributions of two eigenstates correspond to (b) E = − . and (c) E = − . , which respectively correspondto the nearest-neighbor eigenvalue below and above the MEof the system with fixed λ = 0 . and N = 610 . which is given by Γ = − lim L →∞ ln(IPR)ln L . It is known that Γ → for extended states and Γ → for localized states.We plot energy eigenvalues and the fractal dimension Γ of the corresponding eigenstates as a function of poten-tial strength λ in Fig. 1 (a). The two dashed lines in thefigure represent the MEs ( E c = ± λ ). As expected fromthe analytical results, Γ approximately changes from zeroto one for energies across the dashed lines. This is furtherconfirmed by the distributions of the wave functions, asshown in Fig. 1 (b) and (c). We see that the wave func-tions are localized and extended when their eigenvaluessatisfy | E | > λ and | E | < λ , respectively.It appears that the localization starts from the edgesof the spectrum, as the coupling constant λ is increased,then we have MEs, and MEs moves towards the cen-ter of the spectrum. This behavior is similar to MEsin the three-dimensional disordered systems. However,the present model has a new fundamental feature that inthe arbitrarily strong quasiperiodic potential regime, theMEs always take place, i.e, the extended states alwaysexist. This is in sharp contrast to models with randomdisorder and to other quasiperiodic models, where all thestates are localized when the disorder is large enough.To further determine this point and strengthen the cal-culated MEs in Eq. (3) and Eq. (4), we further study thescaling behavior of eigenstates by performing a multifra-cal analysis [36, 40, 41]. For a normalized eigenstate, thescaling index α j is defined by n j ∼ L − α j . If this stateis extended, for all the lattice sites j , we have α j → when L → ∞ . If this state is localized, there exist thenon-vanishing probabilities only on a finite number of Figure 2: α min as a function of /n for (a) the lowest stateand (b) center state of the spectrum for different λ , where n is the subscript of the Fibonacci numbers F n . (c) α min versuseigenvalues E with fixed N = F = 6765 for λ = 0 . (blackdots) and λ = 2 (blue dots). The two red (green) dashed linesrepresent the MEs E c = ± ( ± . ). sites even when L → ∞ , i.e., α j → for these sites but α k → ∞ for remaining sites k with n k = 0 . Therefore,to identify the extended and localized eigenstates, onecan simply examine the minimal value of α , which takes α min = 1 (extended) or α min = 0 (localized) in the ther-modynamic limit. Fig. 2 (a) and (b) display the α min of two typical eigenstates corresponding to the bottomand center of the spectrum, respectively. We see thatthe α min tend to for λ = 0 . , indicating that both ofthem are extended. In contrast, α min tend to and for the ground state and the center state of this system,respectively, with λ = 0 . and λ = 10 , signifying the ex-istence of MEs. We can determine the extended-localizedtransition point of the ground state at λ c = 0 . byperforming a finite size scaling analysis (see Supplemen-tal Materials [42]), which coincides with a critical valuethat MEs start to appear. Fig. 2 (c) shows the α min ofall eigenstates with size N = F = 6765 . There existdramatic changes of α min at MEs given by Eq. (5) withincreasing eigenvalues E , suggesting that the predictedMEs well separate localized states from extended states.The above results also show a novel phenomenonthat the critical strength of quasiperiodic potential inextended-localization transition of the ground state issmaller than that in the standard AAH model. This isbecause for the mosaic lattice the particle tends to stay atthe site with the smallest potential and the potential dif- ference strongly impedes the nearest-neighbor hopping. Rigorous mathematical proof and general case.—
Nowwe provide the analytical derivation for the MEs by com-puting the LE. Denote by T n ( θ ) the transfer matrix of theSchrödinger operator [43], then LE can computed as γ (cid:15) ( E ) = lim n →∞ n ˆ ln (cid:107) T n ( θ + i(cid:15) ) (cid:107) dθ, where (cid:107) A (cid:107) denotes the norm of the matrix A . The com-plexification of the phase is important for us, since ourcomputation relies on A.Avila’s global theory of one-frequency analytical SL (2 , R ) cocycle [43], one of hisFields Medal work. First note that the transfer matrixcan be written as T κ ( θ ) = (cid:18) E − λ cos 2 π ( θ + κω ) −
11 0 (cid:19) (cid:18) E −
11 0 (cid:19) κ − , where (cid:18) E −
11 0 (cid:19) κ − = (cid:18) a κ − a κ − a κ − − a κ − (cid:19) and a κ is defined in (4). Let us then complexify thephase, and let (cid:15) goes to infinity, direct computation yields T κ ( θ + i(cid:15) ) = e π(cid:15) e i π ( θ + κω ) (cid:18) − λa κ λa κ − (cid:19) + o (1) . Thus we have κγ (cid:15) ( E ) = 2 π(cid:15) + log | λa κ | + o (1) . Avila’sglobal theory [43] shows that as a function of (cid:15), κγ (cid:15) ( E ) is a convex, piecewise linear function, and their slopesare integers multiply π . This implies that κγ (cid:15) ( E ) =max { ln | λa κ | + 2 π(cid:15), κγ ( E ) } . Moreover, by Avila’s globaltheory, if the energy does not belong to the spectrum,if and only if γ ( E ) > , and γ (cid:15) ( E ) is locally constantas function of (cid:15) . Consequently, if the energy E lies inthe spectrum, we have κγ ( E ) = max { ln | λa κ | , } . When | λa κ | > , γ ( E ) = ln | λa κ | κ , the state with the energy E is localized has the localization length ξ ( E ) = 1 γ = κ ln | λa κ | . (6)When | λa κ | < , the localization length ξ → ∞ , andthe corresponding state is delocalized. Thus the MEs aredetermined by | λa κ | = 1 (i.e., Eq. (3)). In fact, we canfurther show that the operator has purely absolutely con-tinuous energy spectrum (extended states) for | λa κ | < ,while it has pure point spectrum for | λa κ | > (localizedstates) [44]. This proof also shows the analytic results forthe extended and localization features of all the states.To explicitly verify the ME expression, we further con-sider the cases of κ = 3 and κ = 5 . From Eq. (3) andEq. (4), one can obtain four MEs when choosing κ = 3 and eight MEs when fixing κ = 5 , respectively given by E c = ± (cid:112) ± /λ, E c = ± (cid:113) / ± (cid:112) / ± /λ. (7) (a) (b) Figure 3: Fractal dimension Γ versus the eigenvalues andquasiperiodic potential strength λ for (a) κ = 3 and (b) κ = 5 .The red dashed lines represent the MEs given in Eq. (7). Herewe fixed the size N = 610 . In the same way as the study for κ = 2 case, Fig. 3(a) (Fig. 3 (b)) shows numerically the fractal dimension Γ of different eigenstates for the κ = 3 ( κ = 5 ) case,which confirms that the four (eight) MEs described inEq. (7) well separate the localized states from the ex-tended states. Further, for any κ , one can obtain κ − MEs well described by Eq. (3) and Eq. (4).
Experimental realization.—
We show that the Hamilto-nian (1) with κ = 2 can well describes the tight-bindinglimit of the following 1D lattice for spin-1/2 atoms H = k x m ⊗ + V p ( x ) σ z + M σ x + V s ( x ) | ↓(cid:105)(cid:104)↓ | , V p ( x ) = V p k p x ) , V s ( x ) = V s k s x ) , (8)where σ x,y,z are Pauli matrices, V p ( x ) is the primary lat-tice, M -term denotes the Raman coupling between thetwo spins, and V s ( x ) is a secondary incommensurate po-tential felt only by spin-down atoms. The above Hamil-tonian is highly achievable in utracold atoms based onoptical Raman lattice scheme [45–47], and the details forthe realization are put in Supplementary Material [42].As shown in Fig. 4(a), due to the spin-dependence ofthe primary lattice, atoms with different spins are al-ternately trapped in the x direction: Spin-up and -downatoms are located at odd and even sites, respectively. Theincommensurate potential V s ( x ) then takes effect only ateven sites. We assume V p (cid:29) M , V s , such that the spin-conserved hopping t p is relatively negligible, and the spin-flipped hopping induced by the Raman coupling playsthe role of nearest-neighbor tunneling t . For example,when V p = 10 E r and M = 1 . E r with E r ≡ k p / (2 m ) ,we have t (cid:39) . t p [42]. Thus, regardless of the atomspin and taking into account only s -bands, this latticeHamiltonian (8) indeed leads to the tight-binding modeldescribed by Eq. (1) with κ = 2 . For a realistic setup, wepropose to realize this system in alkaline-metal atoms,where well-tuned Raman couplings can generate both V s /E r -101 E / E r V s /E r -2-1012 E / E r M
We have proposed a class of exactlysolvable 1D mosaic models to realize MEs in energy spec-tra, where quasiperiodic on-site potentials are inlaid inthe lattice with equally spaced sites, and proposed theexperimental realization. By calculating the Lyapunovexponents, we have analytically demonstrated the exis-tence of MEs and obtained their expressions, which arein excellent agreement with the numerical studies. Forthe integer inlay parameter κ > of our proposed mod-els, one obtains κ − MEs, which are symmetricallydistributed in energy spectra and always exist even in thestrong quasiperiodic potential regime. Our work opensa new avenue to analytically explore ME physics withexperimental feasibility.We thank Laurent Sanchez-Palencia for helpful com-ments. Y. Wang, L. Zhang and X.-J. Liu are sup-ported by National Nature Science Foundation of China(11825401, 11761161003, and 11921005), the NationalKey R&D Program of China (2016YFA0301604), Guang-dong Innovative and Entrepreneurial Research TeamProgram (No.2016ZT06D348), the Science, Technologyand Innovation Commission of Shenzhen Municipality(KYTDPT20181011104202253), and the Strategic Pri-ority Research Program of Chinese Academy of Sci-ence (Grant No. XDB28000000). S. Chen was sup-ported by the NSFC (Grant No. 11974413) and theNKRDP of China (Grants No. 2016YFA0300600 andNo. 2016YFA0302104). H. Yao acknowledges the sup-port from the Paris region DIM-SIRTEQ. X. Xia is sup-ported by NanKai Zhide Foundation. J. You was par-tially supported by NSFC grant (11871286) and NankaiZhide Foundation. Q. 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In the Supplementary Materials, we first perform a finite size scaling analysis for the ground states. Then, wegive some mathematical basis of computing the Lyapunov exponent. Finally, we give some details of experimentalrealization.
I. Finite size scaling analyses for ground states
In this section, we turn to determine the transition point λ c between the extended and localization ground states,which coincides with a critical value corresponding to that the mobility edges (MEs) start to appear. Fig.1 (a) inthe main text tells us that the critical value λ c is near λ = 0 . . To exactly obtain λ c and some critical exponents,we perform a finite-size scaling analysis. We firstly introduce the participation ratio: I L = IP R and then define σ L = ( I L /L ) / , which tends to ( ) when L → ∞ for the localized (extended) state, so we can use it as an orderparameter. Near λ c , we can introduce two critical exponents [1, 2]: ξ ∼ | ∆ λ | − ν , I ∼ (∆ λ ) − γ , (S1)where I = lim L →∞ I L , ∆ λ = ( λ − λ c ) /λ c and ξ represents the correlation or localization length. Near λ c , we assumethe following finite size scaling relationship when this system is finite: σ L L − γ/ν = f ( L /ν (∆ λ )) , (S2)where f ( x ) is a scaling function. λ R [ L , L ] (89,377)(377,1597)(89,1597) -100 -50 0 50 ∆ λ L ν σ L2 L - γ / ν N=89N=377N=1597N=6765 (a) (b)
Figure S1: (a) R [ L , L ] versus λ for several pairs of ( L , L ) . (b) σ L L − γ/ν as a function of ∆ λL /ν for different N ( N = 89( F ) , F ) , F ) , F ) ). We set ν = 1 and different lines are superposed together. At the transition point λ = λ c , we have ∆ λ = 0 , and thus Eq. (S2) becomes σ L = f (0) L γ/ν − . From the definitionsof the fractal dimension Γ and σ L , one can obtain σ L ∼ L Γ − , i.e, Γ = γ/ν . Then a function of two size-variables canbe defined as: R [ L , L ] = ln( σ L /σ L )ln( L /L ) + 1 , (S3)which equals to γ/ν at the transition point for any pair ( L , L ) , where L and L represent two different systemsizes. Fig. S1 (a) displays the behaviors of R [ L , L ] versus λ for different pairs of L and L . From the crossingpoint, we can determine the transition point λ c = 0 . and the corresponding critical exponent γ/ν = 0 . .The critical exponent ν can be obtained by calculating σ L L − γ/ν versus L /ν (∆ λ ) and making them superposetogether for different sizes as shown in Fig. S1 (b). It is shown that lines corresponding to different sizes superposetogether if setting ν = 1 , which indicates ν = 1 at the transition point of the ground state and it is the same with theAubry-André-Harper model. For other states, performing the similar finite-size analysis, one will find that the MEswe proposed in the main text well marks a sharp transition between extended and localized states, with no a criticalregion. II. Global theory of one-frequency cocycle
Suppose that A is an analytic function form the circle S to the group SL (2 , C ) , an analytic cocycle ( ω, A ) is alinear skew product: ( ω, A ) : S × R → S × R ( θ, v ) (cid:55)→ ( θ + ω, A ( θ ) · v ) . If A ( θ ) admits a holomorphic extension to |(cid:61) θ | < δ , then for | (cid:15) | < δ we can define A (cid:15) ( θ ) = A ( θ + i(cid:15) ) , and define itsLyapunov exponent by γ (cid:15) ( A ) = lim n →∞ n ˆ ln (cid:107) A n ( θ + i(cid:15) ) (cid:107) dθ, where A n is the transfer matrix. The key observation of Avila’s global theory [3] is that (cid:15) → γ (cid:15) ( A ) is convex andpiecewise linear, with right-derivatives satisfying lim (cid:15) → π(cid:15) ( γ (cid:15) ( A ) − γ ( A )) ∈ Z . Note that in our case, a sequence ( u n ) n ∈ Z is a formal solution of the eigenvalue equation u n +1 + u n − + v ( n ) u n = Eu n if and only if it satisfied (cid:18) u n +1 u n (cid:19) = (cid:18) E − v ( n ) −
11 0 (cid:19) (cid:18) u n u n − (cid:19) , while the operator can be seen as a cocycle, however,the cocycle is not analytic since the potential is not a smooth function. The useful observation is that its iterate T κ can be seen as an analytic cocycle ( κω, T κ ( · )) . Thus by the general theory, the Lyapunov exponent of the cocycle γ (cid:15) ( T κ ) = κγ (cid:15) ( E ) is a convex, piecewise linear function, their slopes are integers multiply π . III. Experimental Realization
In this section, we illustrate how to to realize the lattice model (8) in the main text. Our basic idea is to usewell-tuned Raman couplings to generate both the primary and secondary lattice potentials. We shall first realize theHamiltonian H = k x m ⊗ + V p k p x ) σ x − M σ z + V s k s x )( − + σ x ) , (S4)which can be transformed into the form (8) under the spin rotation σ x → σ z , σ z → − σ x . (S5)Our proposed experimental setup is sketched in Fig. S2(a). In the following we shall take K atoms as an examplewhile all our results are applicable to other alkali atoms. For K, the spin- / system can be constructed by | ↑(cid:105) = | F = 9 / , m F = +9 / (cid:105) and | ↓(cid:105) = | / , +7 / (cid:105) . The lattice and Raman coupling potentials are contributed fromboth the D ( S / → P / ) and D ( S / → P / ) lines [Fig. S2(b-d)]. A. The primary lattice
The primary lattice is generated by a Raman coupling via a standing wave field E = 2 E ˆ ze i ( φ + φ e / cos( k x − φ e / of frequency ω and a plane wave E = ˆ xE e i ( k y + φ ) of frequency ω , where φ , denote the initial phases,and φ e is the phase acquired by E for an additional optical path back to the atom cloud. As shown in Fig. S2(b),the standing-wave field E creates a lattice V ( x ) , which is given by V σ ( x ) = (cid:88) F (cid:12)(cid:12)(cid:12) Ω (3 / σF, z (cid:12)(cid:12)(cid:12) ∆ (p)3 / + (cid:88) F (cid:12)(cid:12)(cid:12) Ω (1 / σF, z (cid:12)(cid:12)(cid:12) ∆ (p)1 / , (S6) | F, +9 / i | F, +7 / i | / , +9 / i | / , +7 / i S / P / P / (p)3 /
F σ , J (cid:105) ˆ z · E ( J = 1 / , / ). From the dipole matrix elements of K [4], we obtain V ( z ) = V cos ( k z − φ (cid:48) / ,V = 4 t / | ∆ (p)3 / | − | ∆ (p)1 / | E , (S7)with the transition matrix elements t / ≡ (cid:104) J = 1 / || e r || J (cid:48) = 1 / (cid:105) , t / ≡ (cid:104) J = 1 / || e r || J (cid:48) = 3 / (cid:105) and t / ≈ √ t / .The Raman coupling potential via E , is M ( x ) = (cid:88) F Ω (3 / ∗↑ F, z Ω (3 / ↓ F, ∆ (p)3 / + (cid:88) F Ω (1 / ∗↑ F, z Ω (1 / ↓ F, ∆ (p)1 / , (S8)with Ω ( J ) σF, = (cid:104) σ | er | F, m
F σ + 1 , J (cid:105) ˆ e + · E , and takes the form M ( x ) = M cos( k x − φ e / e i ( k y + φ − φ − φ e / − π ) ,where M = 2 t / | ∆ (p)1 / | + 1 | ∆ (p)3 / | E E . (S9)We assume that the wavelength of E is λ = 769 nm, which satisfy ∆ (p)3 / = − (p)1 / = 2∆ FS / where ∆ FS denotesthe fine structure splitting. We then have V = 0 , and M = t / E E / ∆ FS . We further set φ − φ − φ e / π +2 nπ ( n = 0 , , , · · · ). Under the spin rotation (S5), we have the primary lattice V p ( x ) = V p cos(2 k p x − φ e / , with V p = 2 t / E E / ∆ FS , k p = k / . (S10)0 V s /E r t/t p t p / E r Figure S3: Spin-conserved and -flipped hoppings versus the lattice depth V s . Here we set M = 1 . E r . Moreover, we set a non-zero two-photon detuning δ = ∆ ω z − ω + ω , where ∆ ω z denotes the energy differencebetween the two spin states, leads to an effective Raman coupling M σ x with M = − δ/ . B. The incommensurate lattice
We apply three standing wave fields together to generate an incommensurate lattice only for spin-down atoms, whichare E = 2 E ˆ ze i ( φ + φ e / cos( k x − φ e / of frequency ω , E = 2 E ˆ ye i ( φ + φ e / cos( k x − φ e / of frequency ω , E = 2 E ˆ ze i ( φ + φ e / cos( k x − φ e / of frequency ω ∼ ω , where ∆ ω z − ω + ω = δ . These standing waves createthree spin-independent lattices V j ( x ) = V j cos ( k x − φ je / ( j = 3 , , ) [see Fig. S2(d)] with V j = 4 t / | ∆ (s)3 / | + 1 | ∆ (s)1 / | E j . (S11)If φ e − φ e = (2 n + 1) π ( n is an integer) and E = E , we have V ( x ) + V ( x ) = const . . The Raman coupling potentialvia E , is M ( x ) = − M sin(2 k x − φ e ) (assuming φ − φ = ( n + 1 + 2 m ) π with m being an integer), where M = 2 t / | ∆ (s)1 / | − | ∆ (s)3 / | E E . (S12)We further assume φ e − φ e = π/ nπ and V = 2 M ; the latter can be achieved by tuning ∆ (s)1 / , / and E .Hence, under the rotation (S5), we have the secondary incommensurate lattice V s ( x ) = V s sin(2 k s x − φ e + π ) feltonly by the spin-down state, with V s = 2 t / | ∆ (s)1 / | − | ∆ (s)3 / | E , k s = k . (S13) C. Tight-binding model
In the tight-binding limit and only considering s -bands, the spin-conserved hopping induced by the primary latticeis t p = − ´ dxφ s ( x ) (cid:104) k x m + V p ( x ) (cid:105) φ s ( x − a ) , where a = π/k is the lattice period of the primary lattice and φ s ( x ) denotes the Wannier function. When t p is negligible, the Hamiltonian (8) can take the form of Eq. (2) with thespin-flipped hopping playing the role of nearest-site tunneling, i.e. t = M ˆ dxφ s ( x ) φ s ( x − a/ , (S14)1and λ j = 12 ˆ dxV s ( x ) | φ s ( x − ja ) | = λ cos(2 πωj − φ e − π/ ,λ ≡ V s ˆ dx cos(2 k x ) | φ s ( x ) | , (S15)with ω = { k /k } . Here {·} denotes the fractional part. We define the recoil energy E r ≡ k p / (2 m ) = k / (8 m ) , andcalculate both spin-conserved and -flipped hoppings as a function of V s . The results are shown in Fig. S3. We findthat when setting M = 1 . E r , V s = 10 E r is deep enough to meet our need. ∗ These authors contribute equally to this work. † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected][1] Y. Hashimoto, K. Niizeki, and Y. Okabe, A finite-size scaling analysis of the localization properties of one-dimensionalquasiperiodic systems, J. Phys. A , 5211 (1992).[2] Y. Wang, Y. Wang, and S. Chen, Spectral statistics, finite-size scaling and multifractal analysis of quasiperiodic chain withp-wave pairing, Eur. Phys. J. B , 254 (2016).[3] A. Avila, Global theory of one-frequency Schrödinger operators, Acta. Math.1