Topologically quantized current in quasiperiodic Thouless pumps

Pasquale Marra1, 2, ∗ and Muneto Nitta2

1Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo, 153-8914, Japan2Department of Physics, and Research and Education Center for Natural Sciences,

Keio University, 4-1-1 Hiyoshi, Yokohama, Kanagawa, 223-8521, Japan

Thouless pumps are topologically nontrivial states of matter with quantized charge transport, which can berealized in atomic gases loaded into an optical lattice. This topological state is analogous to the quantum Hallstate. However, contrarily to the exact and extremely precise and robust quantization of the Hall conductance,the pumped charge is strictly quantized only when the pumping time is a multiple of a characteristic time scale,i.e., the pumping cycle duration. Here, we show instead that the pumped current becomes exactly quantized,independently from the pumping time, if the system is led into a quasiperiodic, incommensurate regime. Inthis quasiperiodic and topologically nontrivial state, the Bloch bands and the Berry curvature become flat, thepumped charge becomes linear in time, while the current becomes steady, topologically quantized, and pro-portional to the Chern number. The quantization of the current is exact up to exponentially small corrections.This has to be contrasted with the case of the commensurate (nonquasiperiodic) regime, where the current is notconstant, and the pumped charge is quantized only at integer multiples of the pumping cycle.

The hallmark of topological states of matter is the exactquantization of a physical observable in terms of a conservedquantity, the topological invariant[1, 2]. A paradigmatic ex-ample is that of the quantum Hall conductance, which is quan-tized as integer (or fractional) multiples of e2/h with a preci-sion which exceeds one part in a billion[1, 3, 4]. Moreover,this quantization is robust against perturbations, i.e., it per-sists in the presence of disorder, defects, impurities, or im-perfections of the experimental sample. This led to extremelyprecise definition of the electrical resistance standard and ex-perimental determination of the finite-structure constant[5].

A topologically equivalent state of matter is the Thoulesspump[2, 6–14], which can be engineered, for instance, withultracold atoms[15–18] in a superlattice created by the super-position of two optical lattices with different wavelengths[19–22]. When the superlattice is adiabatically and periodicallyvaried in time t, the charge pumped through the atomic cloudis quantized in terms of the topological invariant, i.e., theChern number of the Bloch bands[2]. However, the chargeis quantized only when the duration of the pumping processis an integer multiple nT of a characteristic time T , whichcorrespond to a full adiabatic cycle, and deviations from thequantized value are linear in time ∝ t− nT . In this sense, thequantization of the pumped charge is not exact: This consti-tutes a fundamental hindrance to the realization of metrologi-cal standards.

Here, we will show that the exact quantization of thepumped current can be indeed realized by a Thoulesspump in the quasiperiodic regime. In optical lattices,quasiperiodicity[23–28] is realized using a superposition oftwo standing waves with incommensurate wavelengths, i.e.,their ratio α is an irrational number. In this regime, the trans-lational symmetry is completely broken, the familiar conceptof Brillouin zone becomes ill-defined, and the usual defini-tion of the Chern number as integral of the Berry curvaturebreaks down. In order to illustrate the exact quantization ofthe current in a realistic experimental setup, we will derivean effective tight-binding model describing an atomic gas in a

bichromatic potential[22], which coincides with a generalizedHarper-Hofstadter Hamiltonian[29–34] with an extra spatial-dependent tunneling term. Secondly, we will operatively de-fine the Chern number in the quasiperiodic regime, by tak-ing the limit of an ensemble of periodic and topologically-equivalent (homotopic) states which progressively approxi-mate the quasiperiodic system. In this limit, the Bloch bandsand the Berry curvature become asymptotically flat. Finally,we describe the experimental fingerprint of the quasiperiodicnontrivial topological state, which can be observed in thecharge transport and in the adiabatic evolution of the cen-tre of mass of the atomic gas. Whereas in the commensu-rate (nonquasiperiodic) case, the current is not constant andthe pumped charge is quantized only at exact multiples ofthe pumping cycle, we find that the quasiperiodic nontriv-ial state is characterized by a steady and topologically quan-tized pumping current, independently from the duration of thepumping process. The exact quantization of the current is adirect consequence of quasiperiodicity, and may contribute tothe definition of a more accurate and direct definition of cur-rent standards[35].

In a typical experiment[19, 20] a topologically nontrivialThouless pump is realized by an ultracold Fermi gas loadedinto a dynamically controlled bichromatic lattice, which is asuperposition of a stationary lattice (short lattice) and a dy-namical sliding lattice (long lattice) in the form

V (x, φ) = VS cos2

(πx

dS

)+ VL cos2

(πx

dL+φ

2

), (1)

where VL,S and dL,S are the depths and the wavelenghts re-spectively for the long and short lattices, and φ is the phasedifference between the two lattices, which we assume to varylinearly in time φ = 2πt/T with a period T . The com-mensuration α = dS/dL between the two lattices can be inprinciple controlled by tilting the long lattice beam with re-spect to the short lattice[14]. We assume a deep lattice regimeVS > Er (here Er = h2/(8Md2

S) is the recoil energy ofthe short lattice[16]. If VS > VL the continuum Hamilto-

arX

iv:2

001.

1102

2v1

[co

nd-m

at.q

uant

-gas

] 2

9 Ja

n 20

20

2

Figure 1. Energy spectra of the tight-binding Hamiltonian in Eq. (3)calculated for V = 2J and K = J (a) and J/2 (b) respectively. Thespectra are asymmetric with respect to the transformationsE → −Eand α → 1 − α, and they are gapped around E = 0 for α = p/qwith q even. The central gap at α = 1/2 has Chern numberC = −1.The symmetries are restored and the central gaps at α = p/q with qeven close in the limit case K → 0 (not shown).

nian H = p2/2M + V (x, φ) can be discretized using lo-calized states at the minima of the short lattice and treat-ing the long lattice as a perturbation[9]. This gives rise toa site-dependent tunneling Jn,n+1 = 〈φn|H |φn+1〉 and on-site potential Vn = 〈φn|H |φn〉, where |φn〉 are the Wan-nier functions localized around the minima of the short latticexn = dS(n+ 1/2). We thus obtain an effective tight-bindingHamiltonian corresponding to a generalized Harper equationwhich reads

−[J+Kα sin (πα) cos(2πα(n+ 1)+φ)](ψn−1+ψn+1) +

+V cos(2πα(n+ 1/2) + φ)ψn = Eψn. (2)

This is a generalization of the Harper-Hofstadter model,which includes an extra site-dependent tunneling term propor-tional to K ∝ VL. On the other hand, one can verify that forα = 1/2, Eq. (2) reduces to the Rice-Mele model[36]. In thecommensurate case, i.e., α = p/q with p, q integer coprimes,

one can directly verify that Eqs. (1) and (2) are invariant up totranslations n→ n+q, and consequently the superlattice unitcell has lenght qdS. In momentum space one has

H = −∑k

2J cos k c†kck + ei(φ+πα)×

×[

12V +Kα sin (πα) cos (k+πα)

]c†kck+2πα+h. c., (3)

where the momentum can be restricted in the first Brillouinzone k ∈ [0, 2π/q]. Figure 1 shows the energy spectraof the tight-binding model calculated for different values ofK. These spectra are a deformed version of the Hofstadterbutterfly[30]: Indeed, whereas the Hofstadter butterfly (K =0) is symmetric with respect to the transformations α→ 1−αandE → −E (which correspond to k → k+π), the spatially-dependent tunneling term breaks these symmetries. Theseasymmetries are inherited by the properties of the continu-ous potential in Eq. (1) (cf. Ref. [22]). For small K, one canassume that the intraband gaps remain open for K → 0 andare thus homeomorphic to the gaps of the Hofstadter butter-fly. This mandates that the intraband gaps of the model aretopologically nontrivial and characterized by a nonzero Chernnumber C which satisfies the diophantine equation −pC ≡ jmod q (analogously to the Hofstadter butterfly K = 0). Wenotice that, unlike the case of the Hofstadter butterfly K = 0,the energy spectra aroundE = 0 are gapped for α = p/q withq even. These central gaps correspond to a nontrivial Chernnumber satisfying the diophantine equation −pC ≡ (q/2)mod q.

In the commensurate case, where α is a rational numberα ∈ Q, assuming homogeneously populated bands below theFermi level EF, and at zero temperature T = 0, the totalcharge pumped during an adiabatic evolution φ → φ + 2πis quantized and equal to the Chern number C of the filledBloch bands[2] Q = C = (1/2π)

∫ φ+2π

φdφ∫ 2π/q

0dkΩ.

Here Ω =∑i Θ(EF − Ei)Ωi is the total Berry curva-

ture at the Fermi level EF, with Θ(E) the Heaviside stepfunction and Ωi = 2= 〈∂φui|∂kui〉 the Berry curvature ofthe ith band, defined in terms of the Bloch wavefunctions|ψi(k, x)〉 = eikx |ui(k, x)〉. On the other hand, the currentI = ∂tQ = (1/T )

∫ 2π/q

0dkΩ is not quantized and it is not

constant during the pumping process, and oscillates around anaverage value 〈I〉 = 2π〈Ω〉/(qT ) with a maximum variationδI ≤ 2πδΩ/(qT ) where δΩ = max Ω−min Ω.

Due to translational invariance, the Hamiltonian in Eq. (3)is periodic in the momentum k → k + 2π/q. However, itis not periodic in the phase since H(φ + 2π/q) 6= H(φ).By direct substitution into Eqs. (1) and (2), one can showthat a phase shift φ → φ + 2πm/q is equivalent to a trans-lation n → n + c where c satisfies the diophantine equa-tion pc ≡ m mod q. Thus, the Hamiltonian is “unitarily”periodic[12, 13] in the phase φ up to lattice translations, in thesense that it is periodic up to unitary transformations (transla-tions)H(φ+2πm/q) = T−cH(φ)T c, where T translates thelattice by one site. This is a general property of the continuousmodel in Eq. (1) as well as the discrete tight-binding model in

3

Figure 2. Pumped charge (a) and current (b) for a bichromatic optical lattice and confined by a harmonic trap for the central gap withChern number C = −1 (see Fig. 1), calculated in the adiabatic limit Different curves correspond to successive rational approximationsαn = 1/2, 1/3, 2/5, 3/8, 5/13 of the commensuration α = 1/Φ2. We use VS = 2Er , VL = 0.5Er . (c) and (d) same as above, but calculatedusing the effective tight-binding model in Eq. (3) with V = 2J and K = J . The pumped charge is quantized as mC/q for pumping periods∆φ = 2πm/q. In the limit αn → α, the charge has a linear dependence Q = ∆φCα/2π. The current shows large fluctuations but becomessteady for large denominators q by reaching its quantized value Iα = C/T . We use VS = 2Er , VL = 0.5Er . (e) The current approaches itsquantized value exponentially as δI = |I − Iα| ∝ exp(−q/ξ). Different data sets correspond to V = 2J, 2.5J and K = J/2, 3J/2, J .

Eq. (2). As a consequence, the energies and Berry curvaturesare periodic in the phase φ → φ + 2π/q. By considering theperiodicities in the momentum and phase, one can concludethat the energies and Berry curvatures of Bloch bands satisfy

Ei(k + n2π/q, φ+m2π/q) = Ei(k, φ), (4a)Ωi(k + n2π/q, φ+m2π/q) = Ωi(k, φ). (4b)

Hence, the energy bands and the Berry curvatures be-come flat in the limit of large denominators q. Conse-quently, the pumped charge at well defined fractions ofthe pumping cycle ∆φ = 2πm/q is quantized as frac-tions of the Chern number[12, 13] Q = mC/q =

(1/2π)∫ φ+2πm/q

φdφ∫ 2π/q

0dkΩ.

We will now consider the incommensurate and quasiperi-odic case, that is, when α is an irrational number α ∈ R−Q.In order to do so, we will approach quasiperiodicity by takingthe limit of the commensurate case with αn = pn/qn → α,with pn/qn a succession of rational approximations of thenumber α, obtained in terms of continued fraction represen-tation. Every irrational number α can be written uniquely asan infinite continued fraction[37] as α = [a0; a1, a2, . . .] =a0+1/(a1+1/(a2+1/(a3+. . . ))) with ai integers. The suc-cessive approximations obtained by truncating the continuedfraction representation αn = [a0; a1, a2, . . . , an] are rationalnumbers, and converge to α. We thus consider the ensembleof Hamiltonians H(αn) describing the commensurate systemswith commensurations αn = pn/qn = [a0; a1, a2, . . . , an].We assume that the insulating gap at the Fermi level remainsopen, such that the Hamiltonians H(αn) are homotopic andthus topologically equivalent. Since the denominator qn in-creases for n → ∞, the Brillouin zone [0, 2π/qn] shrinksat each successive approximations and therefore becomes ill-defined in the quasiperiodic limit. Thus, the usual definition ofthe Chern number as an integral of the Berry curvature in theBrillouin zone needs to be redefined. However, it is clear that

in the quasiperiodic limit (qn → ∞), following Eq. (4), theenergy bands and the Berry curvatures become constant, i.e.,they no longer depend on the momentum and phase. Hence,the limit of the Berry integral for n → ∞ converges, and wecan define the Chern number in the quasiperiodic limit as

Cα = limαn→α

1

2π

∫ 2π

0

dφ

∫ 2π/qn

0

dkΩ(αn) = limαn→α

2π

qnΩ(αn). (5)

In this limit, the Chern number is simply proportional to theBerry curvature, which diverges asymptotically as Ω(αn) ∼qC/2π. Moreover, due to the flat Berry curvature, we obtainthat the pumped charge pumped during adiabatic transforma-tions, for any initial and final values of the phase φ→ φ+∆φbecomes

Qα = limαn→α

1

2π

∫ φ+∆φ

φ

dφ

∫ 2π/qn

0

dkΩ(αn) =∆φ

2πCα, (6)

whereas the instantaneous charge current becomes

Iα = limαn→α

1

T

∫ 2π/qn

0

dkΩ(αn) =C

T. (7)

In the quasiperiodic limit, the Berry curvature becomes flat,the pumped charge is linear in the phase difference ∆φ,whereas the current I = ∂tQ becomes constant and propor-tional to the Chern number, defined in Eq. (5).

The quantization of the pumped current is the distinctivefingerprint of the quasiperiodic topological state. In order toillustrate how to verify this experimentally, we now considerthe original continuous Hamiltonian which describes the ul-tracold atomic gas in a bichromatic potential. The atomicgas is typically confined by a shallow harmonic potentialVT(x/dS)2. The continuous system is thus described by thetime-dependent Schrödinger equation

i~ ∂tΨ(x, t) =

[−~2∂2

x

2M+ V (x, t) +

VT

d2S

x2

]Ψ(x, t). (8)

4

The pumped current I = ∂tQ can be linked to a sim-ple physical observable, i.e., the center of mass of theatomic cloud. The variation of the center of the cloud〈x(t)〉 = (1/N)

∫∞−∞ dx|Ψ(x, t)|2 is proportional to the

pumped charge[10, 12], that is Q = ρ[〈x(t + ∆t)〉 − 〈x(t)〉]where ρ = j/(qdS) is the number of atoms j per unit cellqdS. The length L of the cloud can be controlled by the trap-ping potential VT. For a given choice of the commensurationα = p/q, the length of the superlattice unit cell is equal toqdS. Therefore, assuming the number of filled bands to be j,the total length of a cloud of N atoms is given by N/j unitcells. Hence, the number of atoms N must be an integer mul-tiple of the filling factor j, and the trapping potential VT mustbe tuned such that the length of the cloud satisfies

N ≡ j L

qdS. (9)

This condition guarantees a uniform density of the atomiccloud, equal to j atoms per unit cell qdS. Moreover, in or-der to minimize thermal and nonadiabatic effects, one shouldconsider a filling factor j = p which correspond to the largecentral gap of Fig. 1 with Chern number C = −1.

Figures 2(a) and (b) show respectively the pumped chargeQ and the current I = ∂tQ pumped through the atomic cloudobtained by calculating the center of the cloud of the continu-ous system described in Eq. (8) in the adiabatic limit, for thecentral gap with Chern number C = −1. Different curvescorrespond to successive rational approximations of the com-mensuration α = 1/Φ2 ∈ R−Q, where Φ is the golden ratio.We tune the trapping potential such that the length of the cloudL satisfies Eq. (9). We notice that the presence of the har-monic trap necessarily breaks the periodicity in the phase φ.Figures 2(c) and (d) show respectively the charge and currentcalculated by numerical integrating Eqs. (6) and (7) for the ef-fective tight-binding Hamiltonian in Eq. (3). The charge andthe current calculated for the continuous lattice (in the pres-ence of the harmonic trap) and for the effective tight-bindingmodel show the same asymptotic behavior approaching thequasiperiodic limit. The pumped charge is quantized as inte-ger fractions of the Chern number (m/q)C for well-definedfractions of the pumping period ∆φ = 2πm/q. For increas-ing denominators q, the pumped charge approximates a lineardependence Q = ∆φCα/2π, whereas the current approachesits quantized value Iα = Cα/T for αn → α.

Hence, the pumped current Iα in the quasiperiodic limitis quantized and equal to the Chern number (in elemen-tary units). This result has been obtained by only using thesymmetry of the systems, in particular the unitary periodic-ity in the phase φ. We now determine the asymptotic be-havior of the current as one approaches the quasiperiodiclimit. For K = 0, Eq. (3) reduces to the Harper-HofstadterHamiltonian: In this case, it has been shown numericallyand perturbatively[38] that the Berry curvature takes the formΩ(p/q) ≈ F +Ge−q/ξ[cos (qk)+cos (qφ)]. It is reasonable toextrapolate this result also to the case K 6= 0. From Eq. (5),one has that F = qC/2π, whereas G ∝ q2 (see Ref. [38]).

Hence, the flattening of the Berry curvature is exponential,and for large q one has asymptotically δΩ(q) ≈ gq2e−q/ξ

where g > 0 is a constant. Thus, the current approaches itsquantized value exponentially as

δI = |I − Iα| .2πgq

Texp (−qn/ξ) ≈

≈ 2πg

T√D|α− αn|

exp

(− 1

ξ√D|α− αn|

), (10)

where |α−αn| ∼ 1/Dq2n withD <

√5, due to the Dirichlet’s

approximation theorem and Hurwitz’s theorem[37]. Thus,Eq. (10) describes the scaling behavior of the current in thequasiperiodic limit, in terms of the difference |α − αn| be-tween the irrational commensuration α and its successive ra-tional approximations αn = pn/qn.

In Fig. 2(e) we verify numerically the scaling behavior ofthe current, by calculating the variations δI using Eq. (7) forthe effective tight-binding Hamiltonian in Eq. (3). The currentapproaches its quantized value Iα = Cα/T in the quasiperi-odic limit αn → α exponentially as ∝ exp(−q/ξ). By lin-ear fitting the exponential decays, we determine numericallythe scaling length ξ, as a function of J and K. The numer-ical analysis suggests a functional dependence of the formξ ≈ J exp (−βK). The current is quantized and equal to theChern number (in natural units), and approaches its quantizedvalue exponentially fast in the quasiperiodic limit.

In summary, we have shown how a quasiperiodic and topo-logically nontrivial Thouless pump can be realized by anatomic gas confined in a bichromatic and quasiperiodic op-tical lattice, which is a superposition of two harmonic po-tentials with incommensurate wavelengths. This system ischaracterized by a topological invariant defined as the limit ofthe Chern numbers of an ensemble of topologically-equivalentand periodic Hamiltonians. The distinctive fingerprint of thisquasiperiodic and topologically nontrivial state is the exactquantization of the current in units of the Chern number,which is a consequence of the flattening of the Bloch bandsand of the Berry curvature in the quasiperiodic limit. Thisexact quantization is measurable in a typical experimental set-ting of ultracold atomic gases in optical lattices, and may opennew perspectives for a more accurate and direct definition ofcurrent standards.

The work of P. M. is supported by the Japan Science andTechnology Agency (JST) of the Ministry of Education, Cul-ture, Sports, Science and Technology (MEXT), JST CRESTGrant. No. JPMJCR19T2 and by the (MEXT)-SupportedProgram for the Strategic Research Foundation at PrivateUniversities “Topological Science” (Grant No. S1511006).The work of M. N. is partially supported by the Japan So-ciety for the Promotion of Science (JSPS) Grant-in-Aid forScientific Research (KAKENHI) Grant Numbers 16H03984and 18H01217 and by a Grant-in-Aid for Scientific Researchon Innovative Areas “Topological Materials Science” (KAK-ENHI Grant No. 15H05855) from MEXT of Japan.

5

∗ pmarra@ms.u-tokyo.ac.jp[1] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den

Nijs, “Quantized Hall conductance in a two-dimensional peri-odic potential,” Phys. Rev. Lett. 49, 405 (1982).

[2] D. J. Thouless, “Quantization of particle transport,” Phys. Rev.B 27, 6083 (1983).

[3] K. v. Klitzing, G. Dorda, and M. Pepper, “New methodfor high-accuracy determination of the fine-structure constantbased on quantized Hall resistance,” Phys. Rev. Lett. 45, 494(1980).

[4] R. B. Laughlin, “Quantized Hall conductivity in two dimen-sions,” Phys. Rev. B 23, 5632 (1981).

[5] K. von Klitzing, “Quantum Hall effect: Discovery and applica-tion,” Annu. Rev. Condens. Matter Phys. 8, 13 (2017).

[6] Q. Niu, “Towards a quantum pump of electric charges,” Phys.Rev. Lett. 64, 1812 (1990).

[7] R. Shindou, “Quantum spin pump in S=1/2 antiferromagneticchains – Holonomy of phase operators in sine-Gordon theory,”J. Phys. Soc. Jpn. 74, 1214 (2005).

[8] L. Fu and C. L. Kane, “Time reversal polarization and a Z2

adiabatic spin pump,” Phys. Rev. B 74, 195312 (2006).[9] G. Roux, T. Barthel, I. P. McCulloch, C. Kollath, U. Scholl-

wöck, and T. Giamarchi, “Quasiperiodic Bose-Hubbard modeland localization in one-dimensional cold atomic gases,” Phys.Rev. A 78, 023628 (2008).

[10] L. Wang, M. Troyer, and X. Dai, “Topological charge pump-ing in a one-dimensional optical lattice,” Phys. Rev. Lett. 111,026802 (2013).

[11] R. Wei and E. J. Mueller, “Anomalous charge pumping in aone-dimensional optical superlattice,” Phys. Rev. A 92, 013609(2015).

[12] P. Marra, R. Citro, and C. Ortix, “Fractional quantization of thetopological charge pumping in a one-dimensional superlattice,”Phys. Rev. B 91, 125411 (2015).

[13] P. Marra and R. Citro, “Fractional quantization of charge andspin in topological quantum pumps,” Eur. Phys. J. Spec. Top.226, 2781 (2017).

[14] F. Matsuda, M. Tezuka, and N. Kawakami, “Two-dimensionalThouless pumping of ultracold fermions in obliquely in-troduced optical superlattice,” arXiv:1907.10357 [cond-mat,physics:quant-ph] (2019).

[15] M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski,A. Sen(De), and U. Sen, “Ultracold atomic gases in optical lat-tices: Mimicking condensed matter physics and beyond,” Adv.Phys 56, 243 (2007).

[16] I. Bloch, J. Dalibard, and W. Zwerger, “Many-body physicswith ultracold gases,” Rev. Mod. Phys. 80, 885 (2008).

[17] D.-W. Zhang, Y.-Q. Zhu, Y. X. Zhao, H. Yan, and S.-L. Zhu,“Topological quantum matter with cold atoms,” Adv. Phys 67,253 (2018).

[18] N. Cooper, J. Dalibard, and I. Spielman, “Topological bandsfor ultracold atoms,” Rev. Mod. Phys. 91, 015005 (2019).

[19] S. Nakajima, T. Tomita, S. Taie, T. Ichinose, H. Ozawa,L. Wang, M. Troyer, and Y. Takahashi, “Topological Thoulesspumping of ultracold fermions,” Nat. Phys. 12, 296 (2016).

[20] M. Lohse, C. Schweizer, O. Zilberberg, M. Aidelsburger, andI. Bloch, “A Thouless quantum pump with ultracold bosonicatoms in an optical superlattice,” Nat. Phys. 12, 350 (2016).

[21] L. Taddia, E. Cornfeld, D. Rossini, L. Mazza, E. Sela, andR. Fazio, “Topological fractional pumping with alkaline-earth-like atoms in synthetic lattices,” Phys. Rev. Lett. 118, 230402(2017).

[22] K. K. Das and J. Christ, “Realizing the Harper model with ul-tracold atoms in a ring lattice,” Phys. Rev. A 99, 013604 (2019).

[23] Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and O. Zilber-berg, “Topological states and adiabatic pumping in quasicrys-tals,” Phys. Rev. Lett. 109, 106402 (2012).

[24] Y. E. Kraus and O. Zilberberg, “Topological equivalence be-tween the Fibonacci quasicrystal and the Harper model,” Phys.Rev. Lett. 109, 116404 (2012).

[25] Y. E. Kraus and O. Zilberberg, “Quasiperiodicity and topologytranscend dimensions,” Nat. Phys. 12, 624 (2016).

[26] T. Ozawa and H. M. Price, “Topological quantum matter in syn-thetic dimensions,” Nat. Rev. Phys. 1, 349 (2019).

[27] M. Valiente, C. W. Duncan, and N. T. Zinner, “Super Hamil-tonian in superspace for incommensurate superlattices andquasicrystals,” arXiv:1908.03214 [cond-mat, physics:quant-ph](2019).

[28] Y. Kuno, “Disorder-induced Chern insulator in the Harper-Hofstadter-Hatsugai model,” Phys. Rev. B 100, 054108 (2019).

[29] P. G. Harper, “Single band motion of conduction electrons in auniform magnetic field,” Proc. Phys. Soc. A 68, 874 (1955).

[30] D. R. Hofstadter, “Energy levels and wave functions of Blochelectrons in rational and irrational magnetic fields,” Phys. Rev.B 14, 2239 (1976).

[31] Y. Hatsugai and M. Kohmoto, “Energy spectrum and the quan-tum Hall effect on the square lattice with next-nearest-neighborhopping,” Phys. Rev. B 42, 8282 (1990).

[32] D. Osadchy and J. E. Avron, “Hofstadter butterfly as quantumphase diagram,” J. Math. Phys. 42, 5665 (2001).

[33] Y. Hatsuda, H. Katsura, and Y. Tachikawa, “Hofstadter’s but-terfly in quantum geometry,” New J. Phys 18, 103023 (2016).

[34] K. Ikeda, “Hofstadter’s butterfly and Langlands duality,” J.Math. Phys. 59, 061704 (2018).

[35] N.-H. Kaneko, S. Nakamura, and Y. Okazaki, “A review ofthe quantum current standard,” Meas. Sci. Technol. 27, 032001(2016).

[36] M. J. Rice and E. J. Mele, “Elementary excitations of a lin-early conjugated diatomic polymer,” Phys. Rev. Lett. 49, 1455(1982).

[37] G. H. Hardy and E. M. Wright, An Introduction to the Theory ofNumbers, 6th ed. (Oxford University Press, Oxford, 2008) seechapters 10 and 11.

[38] F. Harper, S. H. Simon, and R. Roy, “Perturbative approachto flat Chern bands in the Hofstadter model,” Phys. Rev. B 90,075104 (2014).