Realistic Time-Reversal Invariant Topological Insulators With Neutral Atoms

N. Goldman *,1 I. Satija,2, 3 P. Nikolic,2, 3 A. Bermudez,4 M. A. Martin-Delgado,4 M. Lewenstein,5, 6 and I. B. Spielman7

1Center for Nonlinear Phenomena and Complex Systems - Universite Libre de Bruxelles (U.L.B.),Code Postal 231, Campus Plaine, B-1050 Brussels, Belgium

2 Department of Physics, George Mason University, Fairfax, Virginia 22030, USA3National Institute of Standards and Technology (NIST), Gaithersburg, Maryland 20899, USA

4 Departamento de Fısica Teorica I, Universidad Complutense, 28040 Madrid, Spain5 ICFO-Institut de Ciencies Fotoniques, Parc Mediterrani de la Tecnologia, E-08860 Castelldefels (Barcelona), Spain

6ICREA - Instituciio Catalana de Recerca i Estudis Avancats, 08010 Barcelona, Spain7Joint Quantum Institute, NIST and University of Maryland, Gaithersburg, Maryland, 20899, USA

We lay out an experiment to realize time-reversal invariant topological insulators in alkali atomic gases sub-jected to synthetic gauge fields in the near-field of an atom-chip. We introduce a feasible scheme to engineersharp boundaries where the hallmark edge states are localized. Our multi-band system has a large parameterspace exhibiting a variety of quantum phase transitions between topological and normal insulating phases. Dueto their remarkable versitility, cold-atom systems are ideally suited to realize topological states of matter anddrive the development of topological quantum computing.

Topological insulators are a broad class of unconventionalmaterials that are insulating in the interior but conduct alongthe edges. This edge transport is topologically protected anddissipationless. Until recently, the only known topological in-sulators – quantum Hall (QH) states – violated time-reversalsymmetry. However, the discovery of the quantum spin Hall(QSH) effect demonstrated the existence of novel topologi-cal states not rooted in time-reversal violations [1–3] and hasopened the possibility to design new spintronic devices ex-ploiting the edge states’ dissipationless, spin-dependent, cur-rent.

Realizing topological insulators with cold atoms is partic-ularly attractive [4, 5]. In this Letter, we propose a concretesetup, using fermionic 6Li subjected to a synthetic gauge field,which provides a simple system for investigating quantumspin-Hall physics. There are numerous proposals for engi-neering gauge fields [6], which generally depend on laser-induced Raman coupling between internal atomic states. Sucha method was recently implemented [7] for bosonic 87Rbatoms, but would lead to to large spontaneous emission ratesfor the alkali fermions. Here we describe a setup that com-bines state-independent optical potentials with micron-scalestate-dependent magnetic potentials in an atom chip. Bycompletely eliminating spontaneous emission, this approachmakes practical the realization of gauge fields for all alkaliatoms. In particular, it allows us to reach the purest realiza-tion of the QSH effect and to explore striking aspects of thistopological state of matter. To go beyond the non-interactinglimit – the QSH is an essentially non-interacting effect – thestability of the topological phases can be explored by meansof Feschbach resonances and engineered disorder [8].

In condensed-matter systems, the QSH effect originatesfrom a material’s intrinsic spin-orbit coupling [1, 2], that isanalogous a gauge field A = Aσz, where σx,y,z are the Paulimatrices. This observation emphasizes that the QSH effectconsists of spin-1/2 fermions where the two spin componentsare described as IQH states at equal but opposite “magneticfields.” We hereby demonstrate how to synthesize such a

gauge field in an square lattice and show how it leads to QSHphysics.

Our proposal, an extension of Ref. [6], for realizing afermionic model with a SU(2) gauge structure requires fouratomic states |g1〉 = |F = 1/2,mF = 1/2〉, |g2〉 =|3/2,−1/2〉, |e1〉 = |3/2, 1/2〉, and |e2〉 = |1/2,−1/2〉, in asquare lattice described by the Hamiltonian

H =− t∑m,n

c†m+1,neiθxcm,n + c†m,n+1e

iθycm,n + h.c.

+ λstag

∑m,n

(−1)m c†m,ncm,n. (1)

cm,n is a 2-component field operator defined on a lattice site(x = ma, y = na), a is the lattice spacing, m, n are inte-gers, and t is the nearest-neighbor hopping. In this model,the Peierls phases θx,y result from a synthetic gauge field [6]that modifies the hopping along x and y. Here, all the statesexperience a primary lattice potential V1(x) = Vx sin2(kx)along x which gives rise to a hopping amplitude t≈0.4 kHz.A secondary much weaker lattice V2(x) = 2λstag sin2(kx/2)slightly staggers the primary lattice with λstag ≈ t. These lat-tices, with an approximate period of a= 2 µm, are producedby two pairs of λ = 1064 nm lasers, slightly detuned fromeach other and incident on the atom chip’s reflective surface[see Fig. 1(a)]. Additionally, these beams create a latticealong z with a 0.55 µm period, confining the fermions to a2D plane.

We study the SU(2) hopping operators

θx = 2πγσx, and θy = 2πxασz, (2)

in units where a = ~ = 1. The hopping operator θy corre-sponds to opposite “magnetic fluxes” ±α for each spin com-ponent, whereas θx mixes the spins. In order to engineerthese state-dependent tunnelings, the states |g〉 and |e〉 mustexperience oppositely-signed lattices along y [Fig. 1(b)]. Thiscan be directly implemented with the Zeeman shift gµB |B|of atoms provided that |g1,2〉 and |e1,2〉 have equal, but op-posite magnetic moments g. In our proposal, the magnetic

2

1 µm

4 µm

1064 nm beams

Goldsurface

Fermi gas

Substrate

“Raman”wiresState dependent

lattice wires

Bias field

θ2 θ1

Ener

gy (a

rb.)

Y Position (μm)

0

h×230 kHzh×22

8 M

Hz

h×50 kHz

-8 -4 0 4 8

1e

2e

1g

2g

+1,2 1,2

B0z

z

x

a b

Figure 1: (a) Experimental layout showing the origin of optical (stateindependent) and magnetic (state dependent) potentials and couplingfields. A state-independent, staggered, lattice along x is formed bythe separate interference of two pairs of λ = 1064 nm laser beamsslightly detuned from each other to eliminate cross interference. Therespective intersection angles are chosen so the lattice period differby a factor of two. Both beams reflect from the chip-surface and formvertically aligned lattices, trapping the degenerate Fermi gas about5 µm above the surface. The inset shows the chip geometry, fromtop to bottom: a reflective chip-surface, gold wires aligned alongy placed every 2 µm along x (producing the |g〉-|e〉 coupling), andfinally gold wires aligned along x with a 2 µm spacing (producingthe state dependent lattice). (b) Computed atomic potentials and RFdriven “Raman” transitions.

moments are correctly signed and differ by less than 1% inmagnitude at a bias field B = 0.25 G. With these states, astate-dependent lattice potential can be generated by an arrayof current-carrying wires with alternating +I and−I currents,spaced by a distance a. A modest I=5 µA current [9] in wires3 µm below the chip-surface produces a 6 EL Zeeman lattice(EL = h2/8ma is the lattice recoil energy), with a negligi-ble 3 Hz hopping matrix element. The assisted-hopping alongy, with an x-dependent phase, can be realized with an addi-tional grid of wires spaced by a=2 µm along x, with currentsIm. This provides moving Zeeman lattices with wave-vectorq, where In = I0 sin(qma−ωt), leading to effective “Ramancouplings”. The ω/2π ≈ 228 ± 0.23 MHz transitions indi-cated with arrows in Fig. 1 (b), are independently controllablein phase, amplitude, and wave-vector by commanding con-current running waves at the indicated resonant frequencies.The minimum wavelength d = 2π/q of this moving lattice isNyquist limited by d > 2a. In the frame rotating at the angularfrequency ω, and after making the rotating wave approxima-tion the coupling terms have the desired form t exp(iqma). Apotential gradient along y detunes this Raman coupling intoresonance and is produced by simply shifting the center ofthe harmonic potential. In the model Hamiltonian (1)-(2), thephase for hopping along y is α = qa/2π = a/d in terms ofphysical parameters. Our scheme also requires a contributionto the hopping along x that mixes the |e〉 and |g〉. This canbe realized using a Zeeman lattice moving along x, but tunedto drive transitions between |g1〉 → |g2〉 and |e1〉 → |e2〉.The corresponding control parameter γ, and the additionalstaggered potential which induces alternate on-site energiesε = ±λstag along the x-axis, are shown below to enrich theQSH physics in a novel manner.

The engineered Hamiltonian (1)-(2) satisfies TR invari-

ance, since it commutes with the TR-operator defined asT = iσyK, whereK is the complex-conjugate operator. Thissynthetic, yet robust, TR symmetry enables the realization ofZ2-topological insulators in cold-atom laboratories. We stressthat the two components of the field operators correspond ingeneral to a pseudo-spin 1/2, but in our proposal refer specifi-cally to spin components of 6Li in its electronic ground state.

In the absence of the confining trap Vconf(x, y), the systemcan be solved on a cylinder. We first study the topologicalinsulators with this partially closed geometry and then showhow they can be detected when the trap is applied to the opengeometry. When γ = 0, Eqs. (1)-(2) describe two uncou-pled QH systems and for generic α = p/q, where p, q areintegers, the fermion band-structure splits into q subbands.Our setup thus provides a multi-gap system, where a varietyof band-insulators can be reached by varying the atomic fill-ing factor. As discussed below, some of these insulators aretopologically non-trivial and feature gapless edge states. Thelatter are localized at the boundaries of the sample, and cor-respond to gapless excitations. When the Fermi energy EFlies inside a bulk gap, the presence of these states is respon-sible for the spin transport along the edges. In this Letter, wepresent results for the case α=1/6, which exhibits extremelyrich phase diagrams with almost all possible topological phasetransitions. The topological phases discussed below are robustagainst small variations δα∼0.01 and rely on the existence ofbulk gaps, which are continuously deformed when α is varied.Other configurations of the gauge field could be experimen-tally designed and would lead to similar effects. In Fig. 2(a),illustrating the energy spectrum for γ= 0, the bulk bands areclearly differentiated from the gapless edge states within thebulk gaps. In the lowest bulk gap, the edge state channel A,and its TR-conjugate B, correspond to localized excitationswhich travel in opposite directions [cf. Fig. 2(b)]. The bound-aries are populated by a single pair of counter-propagatingstates, each corresponding to an opposite spin: this lowestbulk gap describes a topological QSH phase. Conversely, thenext gap located at E≈−1 is traversed by an even number ofKramers pairs, thus yielding a normal band insulator (NI).

An alternative approach to the above even-odd criteria re-lies on the computation of the Z2 index ν that characterizesthe bulk gaps [1, 3]: ν(QSH) = 1 and ν(NI) = 0. We verifiedthat the four gaps depicted in Fig.2(a) are indeed associated tothe sequence ν = {1, 0, 0, 1}. When γ= 0, spin is conservedand the spin conductivity is quantized [1, 2] as σs = eν/2π.Surprisingly, different topological insulators can be realized inthis multi-band scenario by simply varying the atomic fillingfactor.

In our multi-band system, the lattice-potential distortionscan drive direct transitions between normal and topologicalinsulating states. Figure 2(c) shows the bulk gaps and edgestates for successive values of the experimentally controllablestaggered potential’s strength λstag. We demonstrate that λstaginduces a quantum phase transition (QPT) from a NI to a QSHphase even in the uncoupled case γ = 0. This transition oc-curs within the bulk gaps around E = ±1 at the critical value

3

0 /2 3 /2 2

E/t

-4

-2

0

2

4

A B

A

Bx = 0 x = L

024

-4-2

0 /2 3 /2 2

E/t

ky

c

QSH

QSH

NI

NI

]

]

stag = 0.5t

QSHNI

metal

tc

0 1.50.75

0.1

0

-0.1

da b

0 /2 3 /2 2ky

QS

QSH

QSH

QS

stag = 1.5t

ky

]

]

stag /t

Figure 2: (a) Energy spectrum of the uncoupled system (γ = λstag =0 and α=1/6) computed in a cylindrical geometry: the bulk energybands (thick blue bands) are traversed by edge states (thin purplelines). (b) Schematic representation of the two edge states, A and B,that lie inside the first bulk energy gap depicted in (a). The spins trav-eling around the edges are respectively represented by red and greenarrows. (c) Energy bands E(ky) for γ = 0 and α = 1/6, with anexternal staggered potential λstag = 0.5t and 1.5t. The topologicalphases associated with the open bulk gaps are indicated. The pur-ple rectangles highlight the NI to QSH phase transition that occursaround EF = −t. (d) Phase diagram in the (γ, λstag)-plane in thevicinity of the uncoupled case γ=0 for EF = −t.

λstag = t. At half-filling, the existence of Dirac points re-sist the opening of the gap for small staggered potential andeventually lead to a NI phase for λstag > 1.25t.

The Z2-index analysis provides an efficient tool to obtainthe full phase diagram in the (γ, λstag)-plane. The phase dia-gram represented in Fig. 2(d) has been obtained numericallyby evaluating ν inside the gap at E ≈ ±1 for small spin-mixing γ < 0.1. We observe three distinct phases: metallic(blue), QSH (red), and NI (cyan). These three phases coexistat a tricritical point situated at γ= 0 and λstag = t. The QSHphase occurs for a wide range of γ, indicating the robustnessof this topological phase under small spin-mixing perturba-tions.

The possibilities offered by cold-atom experiments enablesus to consider the strong coupling regime corresponding toγ = 0.25. In this limit, the previously independent QH sub-systems (γ=0) become maximally coupled, drastically mod-ifying the topological phase transitions presented above. InFig. 3(a), we illustrate the bulk bands and edge states forλstag = 0.5t and 1.5t, and the gaps are labeled according to theeven-odd number of TR pairs. In this strong-coupled regime,a radically different emerges: opposite phase transitions oc-cur successively in the neighboring gaps. First, gap-closingsaroundE ≈ ±1 occur and trigger NI to QSH phase transitions

at λstag = t. Then, for λstag = 1.25t, the opposite transitionQSH-NI occurs at half-filling (E = 0). To fully capture therichness of this phenomenon, we numerically compute the in-dex ν for a wide range of the parameters around γ = 0.25.At half-filling, the phase diagram features tri-critical points,and the QSH-NI phase transitions occur along a well-definedcurve [cf. Fig. 3(b)]. On the other hand, in the neighbor-ing gaps, the QSH to NI phase transition is separated by anintermediate metallic region [Fig. 3(c)]. Therefore, by manip-ulating λstag, EF , and the coupling between the states γ, itis possibile to explore different topological phase transitionswithin the several bulk gaps. These novel features endow thetopological phase diagram with an intrinsic richness and com-plexity, not present in other condensed-matter realizations ofthe QSH effect.

QSH

QSH

QSH

NI ]

]

]QSH

0 /2 3 /2 2ky

024

-4-2

0 /2 3 /2 2

E/t

ky

a

QSHNI

NIQSH

QSH]

]

]

stag = 0.5t

0

1.5

0.350.250.15

0.75

NI

QSH

metal

0.5

1.5

1

metal metal

NI

QSH

0.350.250.15

t1 t2

stag = 1.5t

b c EF = -1tGap aroundHalf filling ( ) EF = 0

stag/t

stag/t

Figure 3: (a) Energy bands E(ky) for γ=0.25 and α=1/6, with anexternal staggered potential λstag = 0.5t and 1.5t. The topologicalphases associated to the bulk gaps are indicated. (b)-(c) Phase dia-grams in the (γ, λstag)-plane in the vicinity of the maximally coupledcase γ=0.25 for (b) EF =0 and (c) EF =−1.

We now describe a new feasible scheme to engineer a sharpinterface where edge states can be localized. This is essentialfor detecting topological states in optical lattices, where theindispensable harmonic trap used to confine atoms stronglyinfluences the edge states when Vconf(edge) ∼ ∆, where ∆is the gap’s width [5]. The key aspect of our proposal ex-ploits the fact that the hopping along the y-direction, ty , iscontrolled by spatially periodic RF transitions and hence canbe tuned. Since the harmonic trap has a minimal effect at thecenter of the trap, we divide the chip into three regions: thecentral region is characterized by a large hopping t′y , whilethe two surrounding regions feature small hoppings ty � t′y .This can be realized by abruptly changing the current in the“Raman” wires to a much smaller value, thereby reducing thecoupling matrix element on the single lattice site scale [cf.Fig. 4(a)]. The resulting highly anisotropic hopping creates asharp interface where the edge states of the central region –

4

-5 -4 -3 -2 -1

012345

E/t

0 40 0

40

x/ay/a

0 40 0

40

x/ay/a

0 40 0

40

x/ay/a

2

2

2

-60 -40 -20 0 20 40 60Position in units of lattice period a

-400

-200

0

200

400

Cou

plin

g St

reng

th (H

z)10%-90% change

over a length of 1.1 a

: Outer regime

: inner regime

: edge-state regime

a

b

Figure 4: (a) Position dependent coupling strength. The shaded blueregion displays the position dependent coupling strength averagedover one period of oscillation of the RF fields. The black trace depictsthe Zeeman shift from the computed magnetic field along x whichproduces the RF coupling (at a representative time). (b) Discrete en-ergy spectrum (blue dots) and typical amplitudes |ψ↑(x, y)|2 in thepresence of a harmonic potential and anisotropic hopping. The openlattice has 42× 42 sites, γ = λstag = 0 and α=1/6. The harmonicpotential Vconf(x, y) ∝ x2 + y2 is such that Vconf(42, 42) = 0.5tand the hopping parameters of the inner and outer regions are respec-tively t′y = 2t and ty = 0.1t. The amplitudes |ψ↑(x, y)|2 mark threedistinct regimes in the associated spectrum: the outer (light green),the inner (dark blue) and the edge-state (magenta) regimes.

a topological insulator – localize. By controlling the strengthof the Raman coupling, we squeeze the energy bands describ-ing the outer parts so that it does not interfere with the bulkgaps of the central topological phase, where the edge statesreside. We show that the helical edge states are robust andexactly localized at the interface between the inner and outerregions. Since the topological phases are confined in the cen-ter of the trap, one verifies that the phase diagrams discussedin the previous sections are valid for a much wider range ofthe harmonic potential’s strength. Fig. 4(b) illustrates the dis-crete energy spectrum and the typical wave-functions com-puted within this new scheme. When the Fermi energy liesin the spectrum’s “edge-state” regimes, the edge states are ro-bustly localized within the interface. Obviously, probing thesetopological edge states remains a fundamental challenge. Re-cently, a very promising method exploiting light Bragg scat-tering has been proposed in Refs. [5, 10]: whenever the Fermienergy crosses a gapless edge-state, a delta peak is distinc-tively observed in the dynamical structure factor. The presentproposal is particularly suited for testing the latter detectionmethod, since the light scattering can be focused on the in-terfaces where the edge-states have been shown to be robustin the presence of a confining harmonic potential. Besides,the QSH effect could be detected through spin-resolved den-

sity measurements: a synthetic electric field should separatethe different spin-components, thus creating spin-currents inthe transverse direction [11]. Finally, when γ = 0, the quan-tized spin Hall conductivity is related to a Chern number, sinceν = N↑mod2, and thus can be evaluated through the Stredaformula applied to the spin-up density [12]. The anisotropyt′y > tx leads to bulk gaps of the order ∆ ∼ 1tx, requiringcold, though realistic, temperatures T ∼ 10 nK to detect theQSH phase.

We described a concrete and promising proposal of syn-thetic gauge fields in optical lattices that overcomes the se-vere drawbacks affecting earlier schemes. We showed howsuch gauge fields are ideally suited to experimentally realizethe most transparent QSH phase and hence allow to explorethe validity of the Z2 classification against interaction anddisorder [3]. In our multi-band system, a staggered poten-tial is shown to drive gap-dependent QPT’s which constitutea unique and rich feature. The cold-atom realization of topo-logical band-insulators and helical metals proposed in this pa-per will pave the way for engineering correlated topologicalsuperfluids and insulators. Considering atoms with more in-ternal states, it is possible to envisage situations where the he-lical edge states present a richer spin structure, and thus offerthe opportunity to explore new avenues and exotic topologicalphases.

N.G. thanks the F.R.S-F.N.R.S for financial support. M.L.thanks the Grants of Spanish MINCIN (FIS2008-00784 andQOIT), EU (NAMEQUAM), ERC (QUAGATUA) and ofHumboldt Foundation. A. B. and M.-A. M.-D. thank theSpanish MICINN grant FIS2009-10061, CAM research con-sortium QUITEMAD, European FET-7 grant PICC, UCM-BS grant GICC-910758 and FPU MEC grant. I.S. and P.N.are supported by ONR grant N00014-09-1-1025A, and grant70NANB7H6138, Am 001 by NIST. I.B.S. is supported bythe ARO with funds from the DARPA OLE Program, and theNSF through the PFC at JQI.

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