TOPOLOGICAL MATTER

Observation of the topologicalAnderson insulator in disorderedatomic wiresEric J. Meier1, Fangzhao Alex An1, Alexandre Dauphin2, Maria Maffei2,3,Pietro Massignan2,4*, Taylor L. Hughes1*, Bryce Gadway1*

Topology and disorder have a rich combined influence on quantum transport. To probetheir interplay, we synthesized one-dimensional chiral symmetric wires with controllabledisorder via spectroscopic Hamiltonian engineering, based on the laser-driven couplingof discrete momentum states of ultracold atoms. Measuring the bulk evolution ofa topological indicator after a sudden quench, we observed the topological Andersoninsulator phase, in which added disorder drives the band structure of a wire fromtopologically trivial to nontrivial. In addition, we observed the robustness of topologicallynontrivial wires to weak disorder and measured the transition to a trivial phase in thepresence of strong disorder. Atomic interactions in this quantum simulation platform mayenable realizations of strongly interacting topological fluids.

Topology and disorder share many surpris-ing connections, from the formal similarityof one-dimensional (1D) pseudodisorderedlattices and 2D integer quantum HallHofstadter lattices (1, 2) to the deep con-

nection between the symmetry classes of randommatrices (3) and the classification of symmetry-protected topological phases (4). Recently, therehas been great interest in exploring both disor-der (5) and topology (6) through quantum sim-ulation, stemming from the dramatic influencesthat these ingredients can have, separately, onthe localization properties of quantum particles(7, 8). When combined, disorder and topologycan have a rich and varied influence on quan-tum transport (9). Indeed, one of the hallmarkfeatures of topological insulators (TIs) is thetopologically protected boundary states that areimmune to certain types of disorder up to somecharacteristic strength (10). The robust conduct-ance of such boundary states, such as the 1D edgestates of integer quantumHall systems (8) or the2D surface states of 3D TIs (11), serves as animportant counterexample to the inevitabilityof localization in low-dimensional disorderedsystems (7, 12). However, topological features caneventually disappear when the disorder strengthbecomes too large, and unusual critical phenome-na related to the unwinding of the topology canaccompany such transitions (13, 14).

Conversely, static disorder can induce non-trivial topology when added to a trivial bandstructure. This disorder-driven topological phase,known as the topological Anderson insulator(TAI), was first predicted to occur in metallic2D HgTe/CdTe quantum wells (15). There hasbeen much interest in the TAI phase over thepast decade (15–17), andmany theoretical studieshave shown the TAI phenomenon to be quitegeneral, emerging across a range of disorderedsystems (18–21). However, owing to the lack ofprecise control over disorder in real materialsand the difficulty in engineering both topology

and disorder in most quantum simulators, theTAI has so far evaded experimental realization.In this work, we reach this goal by engineeringsynthetic 1D chiral symmetric wires with pre-cisely controllable disorder using simultaneous,coherent control overmany transitions betweendiscrete quantum states of ultracold atoms.The topological band structures we consider

are 1D TIs based on the Su-Schrieffer-Heegermodel with a chiral, or sublattice, symmetry(4, 19, 22, 23). We describe this system in termsof a tight-bindingmodel with a two-site unit cell,consisting of sublattice sites A andB (depicted inFig. 1A). We consider the Hamiltonian

H ¼ Xn

mnc†nScn þ tn c†nþ1

ðs1 � is2Þ2

cn þ h:c:

� �� �

ð1Þ

where c†n ¼ ðc†n;A; c†n;BÞ creates a particle at unitcell n in sublattice site A or B, cn is the cor-responding annihilation operator, h.c. denotesthe Hermitian conjugate, and si are the Paulimatrices related to the sublattice degree of freedom(23). The mn and tn characterize the intra- andintercell tunneling energies, respectively. Thismodel can describe chiral wires of the AIII orBDI symmetry classes, by choosing the intracellhopping term to be S = s1 (BDI) or S = s2 (AIII).Both the AIII (chiral unitary) and the BDI (chiralorthogonal) classmodels respect chiral symmetry—that is, they obey GHG = −H with the chiraloperator G ¼ s3 � I—whereas the BDI classalso obeys particle-hole and time-reversal sym-metries (4). In this work we chose to study bothBDI and AIII class systems because they rep-resent all possible distinguishable chiral classesfor which the ℤ topological invariant in onedimension, the winding number, is defined.We experimentally implement effective tight-

binding models of the form of Eq. 1 using thecontrolled, parametric coupling of many discretemomentum states of ultracold atoms (24). Westart with a weakly trapped Bose-Einstein con-densate (BEC) of 87Rb atoms and apply a pair ofcounter-propagating laser fields with nominalwavelength l and wave vector k = 2p/l. Theselasers are far-detuned from any atomic transi-tion; however, their interference pattern couplesto the atoms through the ac Stark effect. Thespatial periodicity of the laser interference pat-tern, p/k, defines the set of momentum stateswhose momenta are separated by integer valuesof 2ħk (where h = 2pħ is the Planck constant).Atoms initially in the BEC, which is a source ofatoms with essentially zero momentum, mayundergo transitions between thesemany discretemomentum states, which represent the sites ofour synthetic lattice. The effective tunneling ofatoms between these sites is precisely controlledby simultaneously driving many two-photonBragg transitions with the applied laser fields.The individual, spectroscopically resolved con-trol over many such transitions is allowed forby the Doppler shifts experienced by the atoms,which are specific to the various Bragg transitions

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1Department of Physics, University of Illinois at Urbana–Champaign, Urbana, IL 61801-3080, USA. 2ICFO–Institutde Ciencies Fotoniques, The Barcelona Institute of Scienceand Technology, 08860 Castelldefels, Barcelona, Spain.3Dipartimento di Fisica, Università di Napoli Federico II,Complesso Universitario di Monte Sant’Angelo, Via Cintia,80126 Napoli, Italy. 4Departament de Física, UniversitatPolitècnica de Catalunya, Campus Nord B4-B5, 08034Barcelona, Spain.*Corresponding author. Email: bgadway@illinois.edu (B.G.);hughest@illinois.edu (T.L.H.); pietro.massignan@upc.edu (P.M.)

Fig. 1. Synthetic chiral symmetric wiresengineered with atomic momentum states.(A) Schematic lattice of the nearest-neighbor-coupled chiral symmetric wire. Site-to-sitelinks within the unit cell (solid) and thoseconnecting different unit cells (dashed) haveindependent tunneling energies mn and tn,respectively. (B) Schematic of the experimentalimplementation of the tight-binding modeldepicted in (A), with tunnelings based on two-photon Bragg transitions between discreteatomic motional states with momentum p.

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(Fig. 1B). This provides local (inmomentumspace)control of the intra- and intercell tunneling am-plitudes and phases, directly through the ampli-tudes and phases of the corresponding Bragglaser fields (24). This control gives us access toboth BDI and AIII class wires, in contrast toprevious studies based on real-space superlatticesthat were restricted to exploring BDI wires, be-cause quantum tunneling between stationary lat-tice sites is real-valued (25–27).The ability to create precisely defined disorder

in the off-diagonal tunneling terms is crucial forthis study. Unlike the site-potential disorder thatismore naturally realized in real-space cold atomexperiments—for example, through optical speckle(28) or quasiperiodic lattice potentials (29)—puretunneling disorder is important for preservingthe chiral symmetry of our wires (19, 23). In par-ticular, we let

tn = t(1 + W1wn) (2)

mn = t(m + W2w′n) (3)

define the controlled fluctuations of our hop-ping terms, where t is the characteristic intercelltunneling energy, m is the ratio of intra- tointercell tunneling in the clean limit, wn and w′nare independent random real numbers chosenuniformly from the range [−0.5, 0.5], and W1

and W2 are the dimensionless disorder strengthsapplied to inter- and intracell tunneling.We begin by considering the influence of dis-

order added to a BDI-class wire. The wire is

strongly dimerized, as characterized by a smallintracell-to-intercell tunneling ratio of m=0.100(5)(with t/ħ ≈ 2p × 1.2 kHz), and hence is in thetopological regime in the clean limit. We fix thedisorder amplitudes to be W ≡W2 ¼ 2W1 andshow in Fig. 2A the disorder-averaged topologi-cal phase diagram of this model as a function ofW and m, as determined numerically by a real-space calculation of the winding number n for asystem with 200 unit cells, together with thecritical phase boundary predicted for an infinitesystem based on the divergence of the localiza-tion length L (23, 30).The strong dimerization produces a large (in

units of the bandwidth) energy gap in the bandstructure. Such large bandgaps are typically fa-vorable for experimentally observing the topo-logical nature of disorder-free nontrivial wiresvia adiabatic charge pumping (25, 26) or theadiabatic preparation of boundary states (31).However, it is expected that in disordered chiralsymmetric wires, the bulk energy gap will essen-tially vanish at moderate disorder strengths, wellbelow those required to induce a change in topol-ogy (23). The energy gap is replaced by amobilitygap, and the band insulator of the clean system isreplaced by an Anderson insulator that remainstopological, with topology carried by localizedstates in the spectrum (23). Thus, without thespectral gap, experimental probes relying onadiabaticity are expected to fail in evidencingthe topology of disordered wires.We instead characterize the topology of our

wires bymonitoring the bulk dynamical response

of atoms to a sudden quench. Specifically, wemeasure the mean chiral displacement of ouratoms. This observable was recently introducedin the context of discrete-time photonic quantumwalks (32); here we measure it for continuous-time dynamics. We define the expectation valueof the chiral displacement operator as

C ¼ 2hGXi ð4Þ

given in terms of the chiral operator G and theunit cell operator X (32). The dynamics of C , ingeneral, display a transient, oscillatory behavior,and their time- and disorder-average h�Ci con-verges to the winding number n, or equivalentlyto the Zak phase ϕZak divided by p, in both theclean and the disordered cases. Moreover, attopological critical points, h�Ci converges to theaverage of the invariants computed in the twoneighboring phases (30, 32, 33).For our experiment, we begin with all tunnel

couplings turned off, and the entire atomic popu-lation localized at a single central bulk lattice site(site A of unit cell n = 0, for a systemwith 20 unitcells). We then quench on the tunnel couplingsin a stepwise fashion. The projection of the local-ized initial state onto the quenched system’seigenstates leads to rich dynamics, as depictedin Fig. 2B for both weak (W = 0.5) and strong(W = 5) disorder. Such site-resolved dynamics ofthe atomic population distribution are directlymeasured by a series of absorption images takenafter dynamical evolution under the Hamilto-nian of Eq. 1 for a variable time t (given in units

Meier et al., Science 362, 929–933 (2018) 23 November 2018 2 of 4

Fig. 2. Disorder-driven transition from topological to trivial wires.(A) Calculated topological phase diagram of the BDI wire model describedin Eq. 1, showing the winding number n (inset color scale) as a function ofdisorder strength W and tunneling ratio m with tunneling disorder strengthsW ≡ W2 ¼ 2W1. The striped black and white line at m = 0.1 indicates theregion explored experimentally in (B) to (D). The solid red curve indicatesthe critical phase boundary (i.e., the set of points where the localizationlength L diverges for an infinite chain) (23, 30). (B) Integrated absorptionimages of the bulk dynamics after a sudden quench of the tunnel couplings,for both weak disorder (W = 0.5) and strong disorder (W = 5), each for asingle disorder configuration. (C) Dynamics of C calculated from the datashown in (B). The solid red curves are numerical simulations according to

Eq. 1 with no free parameters (30). The dashed gray horizontal lines denote �Cfor each dataset. (D) h�Ci as a function of W for m = 0.100(5). The data are

averaged over 20 independent disorder configurations and times in the range0.5 ħ/t to 8 ħ/t in steps of 0.5 ħ/t. The solid gold line represents a numericalsimulation according to Eq. 1, with no free parameters for 200 disorderconfigurations but with the same finite-time sampling as the data (30). Thedashed gold line is based on the same simulation as the solid gold line butsampled to much longer times (t = 1000 ħ/t) in a wire with 250 unit cells(30). The dotted gray curve shows the topological index in the thermody-namic limit (23), which changes value at the same point as the red lineposition in (A). This topological index takes a value of 0.5 at the critical point,as indicated by the horizontal dashed line. The inset shows C for W = 3 as afunction of time for all 20 disorder configurations (distinguished by color)with the number of disorder configurations corresponding to various values

of �C shown in the histogram. All error bars in (C) and (D) denote one standard

error of the mean.

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of the tunneling time ħ/t ≈ 130 ms) and after thediscrete momentum states separate accordingto their momenta during a time-of-flight period(24). From the data shown in Fig. 2B, we cal-culate C as a function of t as shown in Fig. 2C,along with the time average �C. We additionallyobtain the disorder-averaged topological char-acterization of the system h�Ci by averaging �Cover many independent disorder configurations.The dependence of h�Ci on the strength of

applied disorderW is summarized in Fig. 2D.The inset of Fig. 2D depicts the determinationof h�Ci (shown for the caseW = 3), first from thetime average of C over 16 values of t evenly spacedbetween 0.5ħ/t and 8ħ/t, followed by an averageover 20 unique realizations of disorder. We ob-serve that h�Ci is robust to weak disorder, main-taining a nearly quantized value close to 1. Forstrong disorder, W ≳ 2, we observe a relativelysteep drop in h�Ci, with it falling below h�Ci ¼ 0:5for W ≳3. Our observed decrease of h�Ci with in-creasing disorder is in reasonably good agreementwith a numerical simulation (solid gold line inFig. 2D) of the Hamiltonian in Eq. 1 for experi-mental time scales. The observed decay of h�Ci isassociated with a disorder-driven transition be-tween topological (W ≲4) and trivial wires (W ≳4).On an infinitely long chain, we would expect to

observe a sharp phase transition in the infinite-time limit of our h�Ci measurement, yieldingquantized values of the invariant for all disorders,and half-integer values at the critical phaseboundary (30, 34). However, we instead observea smooth crossover owing to broadening fromour finite period of quench dynamics and thecorresponding finite number of sites. The ob-

servation of a moderately sharper transition,such as that of the dashed-line numerical simu-lation in Fig. 2D, would require that we measureat extremely long time scales (shown for 1000tunneling times) and for very large systems(shown for 250 unit cells), which at the momentis beyond the capabilities of our experimentaltechnique (30). The slow convergence of thistransition with increasing measurement timeand system size is a characteristic feature ofrandom-singlet transitions (14), such as thosefound in chiral symmetricwires at strong disorder.Having demonstrated a disorder-driven change

of topology in BDI-class wires, we now turn ourattention to AIII-class wires, for which we inves-tigate the surprising feature that an initially clean,trivial system can be driven topological throughthe addition of disorder. This phenomenon ismanifest in the calculated phase diagram ofAIII-class wires shown in Fig. 3A for m justexceeding 1. The value jmj ¼ 1 is the criticalpoint between the topological and trivial phasein the clean limit, and values of jmj > 1are in thetrivial phase in the absence of disorder. However,we see that random tunneling disorder inducesthe TAI phase over a broad range of weak tomoderateW values, eventually giving way to atrivial Anderson insulator phase again for verylarge disorder. Beyond numerics, a mechanismfor the formation of a TAI phase was first elab-orated in (17) for 2D systems. In that work, dis-order is taken into account perturbatively usingthe self-consistent Born approximation and wasshown to effectively renormalize the parametersin the Hamiltonian [including the parameter(s)that tune between the topological and trivial

phases]. The TAI phase arises because, as dis-order is added to the trivial phase tuned near theclean critical point, the effective Hamiltonian isrenormalized through the critical point and intothe topological phase. This type of reasoning wasadapted, with strong support from numericalevidence, and extended to describe the TAI phasein 1D systems, including both the BDI- and AIII-class wires that we consider here (19, 23, 34).Here we probe the influence of tunneling dis-

order on atomic wires of the AIII class. Becausewe are interested in the TAI phase, we start witha slight dimerization [m = 1.12(2)] that places thesystem in a trivial phase in the clean limit. Wenote that being so near the critical point atm = 1causes the bandgap in the clean limit to be muchsmaller than in the previous experimental setup.The choice of disorder we consider here differsfrom the previous case: We add disorder onlyto the intracell hopping terms, that is, settingW1 = 0 andW ≡W2 . The difference in the topo-logical phase diagrams for the BDI case (Fig.2A) and the AIII case (Fig. 3A) comes entirelyfrom this change in disorder configuration. Inone dimension, the topological phase diagramsfor BDI and AIII systems are identical whenexposed to equivalent disorder configurations(W1 andW2 values) andwith onlynearest-neighbortunnelings present. From (17, 19, 34), we expectthat, for weak disorder of this form, the intracellhopping m should be renormalized toward thetopological phase, resulting in a TAI. Thanks tothe smaller bandgap in this case of reduced di-merization, the effects of off-resonant driving(24, 30) in our system becomemore pronounced.Tomitigate these effects, we reduce our tunneling

Meier et al., Science 362, 929–933 (2018) 23 November 2018 3 of 4

Fig. 3. Observation of the TAI phase. (A) Calculated topologicalphase diagram of the AIII wire model described in Eq. 1, showing thecomputed winding number (color scale at right) as a function ofdisorder strength W and tunneling ratio m with tunneling disorderstrengths W ≡W2 (W1 = 0). The striped black and white line at m = 1.12indicates the region explored experimentally in (B). The solid redcurve indicates the critical boundary (i.e., the set of points where thelocalization length L diverges for an infinite chain) (23, 30). (B) h�Cias a function of W for m = 1.12(2). The data are averaged over 50independent disorder configurations and are averaged in time over therange 1.5 ħ/t to 4.5 ħ/t in steps of 0.5 ħ/t. The solid gold line shows

a numerical simulation according to Eq. 1 for 200 disorder configurationsbut with the same finite-time sampling as the data (30). The dashedgold line is based on the same simulation as the solid gold line butsampled to much longer times (t = 1000 ħ/t) in a 250–unit cell system (30).The dotted gray curve shows the topological index in the thermodynamiclimit (23), which changes value at the same point as the red line positionin (A). This topological index takes a value of 0.5 at the critical points,as indicated by the horizontal dashed line. C as a function of time for all50 disorder realizations (distinguished by color) is shown at the rightfor W = 2.5 and 6; histograms of �C are shown to the right of each plot.All error bars in (B) denote one standard error of the mean.

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energy to t/ħ ≈ 2p × 600 Hz, resulting in a cor-respondingly lessened experimental time rangeof t = 1.5 ħ/t to 4.5 ħ/t.Figure 3B shows the dependence of h�Ci on the

strength of added disorder in the AIII-class wire.Themeasured h�Ci values are obtained, as in Fig. 2,through the nonequilibrium bulk dynamics ofthe atoms after a quench of the tunneling. Be-cause of the restricted range of t, we includemany more disorder configurations (50) to allowfor stable measures of h�Ci. For weak disorder, h�Cirises and reaches a pronounced maximum atW ≈ 2.5. This is consistent with the expectedchange in the renormalized m parameter, thatis, given the negative sign of the lowest-ordercorrection to m, for weak disorder (17, 19, 34).h�Ci then decays for very strong applied disor-der. This observation of an initial increase ofh�Ci followed by a decrease is indicative of twophase transitions, first from trivial wires tothe TAI phase and then to a trivial Andersoninsulator at strong disorder, broadened by ourfinite interrogation time.Despite the effects of finite-time broadening,

we see that our measured h�Ci rises to a valuegreater than 0.5 (the infinite-time h�Ci valueassociated with the critical point) forW ≈ 2.5,lending further evidence to our observationof the TAI phase. Based on the measurementsof h�Ci for W = (2.0, 2.5, and 3.0) and their sta-tistical errors, there is only a 0.3% chance for allthree measurements to fall below 0.5, the criticalvalue that indicates a change in topology (30).Further, the excellent agreement of our experi-mental h�Ci data with a short-time sampled nu-merical simulation (solid gold line in Fig. 3B),combined with the sharper transitions expectedfor long-time measurements based on the samesimulations (dashed gold line in Fig. 3B for 1000tunneling times in a 250–unit cell system), providestrong evidence for the observation of disorder-driven topology in an otherwise trivial bandstructure (30).Unlike condensed-matter systems and pho-

tonic simulators, where carrier mobility or lat-tice parametersmay vary from sample to sample,the spectroscopic control of our atomic physicsplatform has allowed us to engineer many dif-ferent, precisely tuned realizations of disorder.

This level of control will also enable future studiesof quantum criticality in disordered topologicalsystems (9, 13). By simple extension to longerevolution times, the interesting physics of log-arithmic delocalization at the random-singlettransition may be studied (14). Combined withthe ability to engineer tunneling phases (24)and artificial gauge fields, our technique maybe extended to study disordered quantum Hallsystems (13). And although our present study hasbeen restricted to a regime where interactionsare relatively unimportant, the presence of stronginteractions in syntheticmomentum-space lattices(35) will enable future studies of strongly interact-ing topological fluids.Note added in proof: After completion andsubmission of this work, a related work pres-ented complementary evidence for the TAI inphotonic waveguide arrays (36).

REFERENCES AND NOTES

1. Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, O. Zilberberg,Phys. Rev. Lett. 109, 106402 (2012).

2. D. R. Hofstadter, Phys. Rev. B 14, 2239–2249 (1976).3. A. Altland, M. R. Zirnbauer, Phys. Rev. B 55, 1142–1161 (1997).4. S. Ryu, A. P. Schnyder, A. Furusaki, A. W. W. Ludwig,

New J. Phys. 12, 065010 (2010).5. L. Sanchez-Palencia, M. Lewenstein, Nat. Phys. 6, 87–95 (2010).6. N. Goldman, J. C. Budich, P. Zoller, Nat. Phys. 12, 639–645

(2016).7. P. W. Anderson, Phys. Rev. 109, 1492–1505 (1958).8. K. von Klitzing, G. Dorda, M. Pepper, Phys. Rev. Lett. 45,

494–497 (1980).9. F. Evers, A. D. Mirlin, Rev. Mod. Phys. 80, 1355–1417 (2008).10. X.-L. Qi, S.-C. Zhang, Rev. Mod. Phys. 83, 1057–1110 (2011).11. Y. L. Chen et al., Science 325, 178–181 (2009).12. E. Abrahams, P. W. Anderson, D. C. Licciardello,

T. V. Ramakrishnan, Phys. Rev. Lett. 42, 673–676 (1979).13. H. Aoki, J. Phys. C Solid State Phys. 16, L205–L208 (1983).14. D. S. Fisher, Phys. Rev. B 50, 3799–3821 (1994).15. J. Li, R.-L. Chu, J. K. Jain, S.-Q. Shen, Phys. Rev. Lett. 102,

136806 (2009).16. H. Jiang, L. Wang, Q. Sun, X. C. Xie, Phys. Rev. B 80, 165316

(2009).17. C. W. Groth, M. Wimmer, A. R. Akhmerov, J. Tworzydło,

C. W. J. Beenakker, Phys. Rev. Lett. 103, 196805 (2009).18. H.-M. Guo, G. Rosenberg, G. Refael, M. Franz, Phys. Rev. Lett.

105, 216601 (2010).19. A. Altland, D. Bagrets, L. Fritz, A. Kamenev, H. Schmiedt,

Phys. Rev. Lett. 112, 206602 (2014).20. P. Titum, N. H. Lindner, M. C. Rechtsman, G. Refael,

Phys. Rev. Lett. 114, 056801 (2015).21. C. Liu, W. Gao, B. Yang, S. Zhang, Phys. Rev. Lett. 119, 183901

(2017).22. W. P. Su, J. R. Schrieffer, A. J. Heeger, Phys. Rev. Lett. 42,

1698–1701 (1979).

23. I. Mondragon-Shem, T. L. Hughes, J. Song, E. Prodan,Phys. Rev. Lett. 113, 046802 (2014).

24. E. J. Meier, F. A. An, B. Gadway, Phys. Rev. A 93, 051602(2016).

25. M. Lohse, C. Schweizer, O. Zilberberg, M. Aidelsburger, I. Bloch,Nat. Phys. 12, 350–354 (2016).

26. S. Nakajima et al., Nat. Phys. 12, 296–300 (2016).27. C. G. Velasco, B. Paredes, Phys. Rev. Lett. 119, 115301

(2017).28. J. Billy et al., Nature 453, 891–894 (2008).29. G. Roati et al., Nature 453, 895–898 (2008).30. Full details on the experimental and theoretical procedures are

available as supplementary materials.31. E. J. Meier, F. A. An, B. Gadway, Nat. Commun. 7, 13986 (2016).32. F. Cardano et al., Nat. Commun. 8, 15516 (2017).33. M. Maffei, A. Dauphin, F. Cardano, M. Lewenstein, P. Massignan,

New J. Phys. 20, 013023 (2018).34. A. Altland, D. Bagrets, A. Kamenev, Phys. Rev. B 91, 085429

(2015).35. F. A. An, E. J. Meier, J. Ang’ong’a, B. Gadway, Phys. Rev. Lett.

120, 040407 (2018).36. S. Stützer et al., Nature 560, 461–465 (2018).37. E. J. Meier et al., Replication data for: Observation of the

topological Anderson insulator in disordered atomic wires,Version 1, Harvard Dataverse (2018); https://doi.org/10.7910/DVN/7BMFTD.

ACKNOWLEDGMENTS

We thank N. Goldman, M. Lewenstein, H. Shapourian, andI. Mondragon-Shem for helpful discussions. Funding: Thismaterial is based upon work supported by the National ScienceFoundation under grant no. PHY1707731 (E.J.M., F.A.A., andB.G.). A.D., M.M., and P.M. acknowledge Spanish MINECO(Severo Ochoa SEV-2015-0522, FisicaTeAMO FIS2016-79508-P,and SWUQM FIS2017-84114-C2-1-P), the Generalitat deCatalunya (SGR874 and CERCA), the EU (ERC AdG OSYRIS339106 and H2020-FETProAct QUIC 641122), the FundacióPrivada Cellex, a Cellex-ICFO-MPQ fellowship, and the “Ramón yCajal” program. T.L.H. was supported by the ONR YIP AwardN00014-15-1-2383. Author contributions: E.J.M. and F.A.A.performed the experiments. E.J.M. analyzed the data. B.G.supervised the experiments and data analysis. A.D., M.M., P.M.,and T.L.H. developed the theoretical framework. All authorsdiscussed the results and contributed to the preparation of themanuscript. Competing interests: The authors declare nocompeting financial interests. Data and materials availability:All data shown in this work can be found in tables S1 to S3 inthe supplementary materials or in an online database (37).

SUPPLEMENTARY MATERIALS

www.sciencemag.org/content/362/6417/929/suppl/DC1MethodsSupplementary TextFigs. S1 to S7Tables S1 to S3References (38–44)

19 February 2018; accepted 27 September 2018Published online 11 October 201810.1126/science.aat3406

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Observation of the topological Anderson insulator in disordered atomic wiresEric J. Meier, Fangzhao Alex An, Alexandre Dauphin, Maria Maffei, Pietro Massignan, Taylor L. Hughes and Bryce Gadway

originally published online October 11, 2018DOI: 10.1126/science.aat3406 (6417), 929-933.362Science 

, this issue p. 929Sciencewires first became topologically nontrivial and then went back to trivial for sufficient disorder strengths.tunneling between the lattice sites and monitored the atomic ''wires'' as the amplitude of the fluctuations increased. Thein momentum space whose band structure was topologically trivial. The researchers then introduced fluctuations in the caused it to become topologically nontrivial. The starting point was a one-dimensional lattice of ultracold rubidium atomsdemonstrated a counterintuitive transition in the opposite direction: Controlled fluctuations in the system's parameters

et al.Adding irregularity to a system can lead to a transition from a more orderly to a less orderly phase. Meier A messy topological wire

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MATERIALSSUPPLEMENTARY http://science.sciencemag.org/content/suppl/2018/10/10/science.aat3406.DC1

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