Majorana edge states in two atomic wires coupled by pair-hopping
Christina V. Kraus, Marcello Dalmonte, Mikhail A. Baranov, Andreas M. Laeuchli, P. Zoller
MMajorana edge states in two atomic wires coupled by pair-hopping
Christina V. Kraus,
1, 2
Marcello Dalmonte, Mikhail A. Baranov,
1, 2, 3
Andreas M. L¨auchli, and P. Zoller
1, 2 Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria Institute for Theoretical Physics, Innsbruck University, A-6020 Innsbruck, Austria RRC ”Kurchatov Institute”, Kurchatov Square 1, 123182, Moscow, Russia (Dated: November 8, 2018)We present evidence for the existence of Majorana edge states in a number conserving theorydescribing a system of spinless fermions on two wires that are coupled by a pair hopping. Ouranalysis is based on the combination of a qualitative low energy approach and numerical techniquesusing the Density Matrix Renormalization Group. We also discuss an experimental realization ofpair-hopping interactions in cold atom gases confined in optical lattices, and its possible alternativeapplications to quantum simulation.
PACS numbers: 37.10.Jk, 71.10.Pm, 05.10.Cc
At present there is significant interest in identify-ing physical setups where Majorana fermions (MFs) [1]emerge as a collective phenomenon in many-body quan-tum systems [2]. The motivation behind this search istwo-fold: First, the existence of MFs is intimately linkedto the concept of topological phases and their explo-ration. Second, MFs provide due to their topological na-ture a promising platform for topological quantum com-puting and quantum memory [3–5]. In a seminal paperKitaev pointed out a route towards the realization of MFsin a simple many-body system [6]: A 1D wire of spin-less fermions with a p -wave pairing can exhibit a topo-logically ordered phase with zero-energy Majorana edgemodes. The key ingredient here is the coupling of thewire to a superconducting reservoir in a grand canonicalsetting, which is induced in complex solid state struc-tures via the so called proximity effect. Building on thisresult, a remarkable theoretical and experimental efforthas been devoted in search of alternative settings sup-porting topological superconductivity in 1D condensedmatter systems, such as the combination of spin-orbitcoupling, magnetic fields and s-wave interactions [7–16].Alternatively, Majorana physics can be observed with1D quantum gases coupled to a particle reservoir rep-resented by molecular condensates, taking advantage ofthe unique tools for control and measurements in atomicsystems [17–19].In contrast, we propose and investigate in the presentLetter an alternative approach to create Majorana edgestates in a purely number-conserving setting [20–22]. Weconsider the conceptually remarkably simple system ofspinless fermions in two wires with single-particle in-trawire hopping, which are coupled via an interwire pairhopping (c.f. Fig. 1a). On an intuitive level the relationto Kitaev’s model and existence of Majorana edge statesis apparent, when we consider one of the wires as an effec-tive particle reservoir for the second wire. The essentialelement in our system is pair hopping between the wires,which breaks the U (1) symmetry associated with the con-servation of the particle-number difference between the t ... ... ... W ... a) (2,b)(C)(1,a) (2,a)(1,b) (C) (C)(2,a) (1,a)(1,b) (2,b) b)c) d) e) U ΩΩ t am t bm
10 1000 Ω f) FIG. 1. a) Ladder Hamiltonian: Atoms in the a - and b - wirescan tunnel individually along the x-directions, and can hopin pairs between the wires. b-e) Implementation of the pairhopping: b) The single wire is realized as a bipartite latticeof ↑ and ↓ fermions with Raman assisted tunneling (Rabifrequency Ω). c) Ladder scheme as a combination of twowires with opposite energy off-sets. The dashed box denotesa single plaquette, with site indices indicated in parenthesis( ... ); t am , t bm are the tunneling amplitudes from the a - resp.and b -wire to the central sites. Atoms in the center (C) ofthe plaquette interact with strength U (shaded areas). d-e) Energy off-sets along the diagonal of the plaquette in c) forthe ↓ resp. ↑ species, and corresponding virtual processesindicating pair tunneling (see text). f) Time evolution ofthe state a † ,a ↑ a † ,a ↓ (cid:105) according to the microscopic dynamicsin units t/t am (see text): the blue (red) curve indicate thepair population p a,b ( t ) in the upper ( a )/ lower ( b ) wire as afunction of time. two wires, down to the Z parity symmetry, an ingredi-ent known to be crucial for the emergence of Majoranamodes in the grand-canonical scenario. The purpose ofthis work is two-fold. First, we provide evidence for Ma-jorana edge states and related topological order usingboth field-theoretical arguments and detailed a Density-Matrix-Renormalization Group (DMRG) study [23, 24].Second, we show that the present setup with pair hop-ping has a natural implementation with cold atoms instate-dependent optical lattices [25] combined with Ra- a r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b man assisted tunneling processes.The emergence of Majorana edge states in the super-fluid phase of the system is demonstrated on the ba-sis of the following criteria: (i) two degenerate groundstates with different parities for the individual wires inthe case of open boundary conditions (OBC), (ii) non-local fermionic correlations between the edges, comingalong with (iii) topological order indicated by a degen-erate entanglement spectrum, and (iv) robustness of theabove properties against static disorder. We also showthat properties (i)-(iii) survive in the presence of a weaksingle-particle hopping between the wires, also support-ing the topological origin of the state. For experimentalrealizations, in particular with atoms, the last propertycould be crucial, as it shows the robustness against themost probable major imperfection. Model.
We consider the following Hamiltonian: H = − (cid:80) j [( t a a † j a j +1 + t b b † j b j +1 ) + h.c.]+ W (cid:80) j ( a † j a † j +1 b j b j +1 + h.c.) , (1)where a j ( a † j ) , b j ( b † j ) are fermionic annihilation (creation)operators defined on two distinct wires a and b , respec-tively, the first line describes intrawire single-particlehopping with the corresponding amplitudes t a,b (in thefollowing we consider t a = t b = t as a weak asymmetry of t a,b does not affect the results qualitatively), and the lastterm is the interwire pair hopping with the amplitude W .The choice of the Hamiltonian (1), motivated by previ-ous considerations of the number-conserving setting [20],stems from global symmetries and corresponding con-served quantities: Beside the total number of particles, N = N a + N b = (cid:80) j a † j a j + b † j b j , associated with the U (1)symmetry, there is another conserved charge – the parity P of one of the wires [say, the wire a , P = p a = ( − N a ]associated with a Z symmetry [44]. The conservation isguaranteed by the last term in H allowing only hoppingof particles between the wires in pairs, and is the keyrequirement to access a topological phase with MFs.Before presenting the analytical and numerical analy-sis of the Hamiltonian (1), let us give an intuitive picturebased on the simplest system supporting fermionic Majo-rana edge states – the 1D Kitaev quantum wire [6] with p -wave pairing described by a mean-field BCS-like Hamilto-nian resulting from the coupling to a reservoir of Cooperpairs (see Ref. [6] for details). In our case, one couldview one wire as a reservoir of pairs for the other wireand vice versa, and decompose the pair-hopping termin a mean-field manner as W (cid:80) i ( a † i a † i +1 b i b i +1 + h . c . ) → (cid:80) i (∆ b a † i a † i +1 − ∆ ∗ a b i b i +1 + h . c . ), where ∆ a = W (cid:104) a i a i +1 (cid:105) and ∆ b = W (cid:104) b i b i +1 (cid:105) are non-zero pairing amplitudeswhich can be found by applying the standard Bogolyubovprocedure. With this decomposition, the Hamiltonian(1) describes two Kitaev wires [6], each of them havingdoubly-degenerate ground states with different fermionicparities p a,b = ± for the a - and b -wire, respectively, and carrying two Majorana operators corresponding to theedge-modes. Therefore, the ground state (GS) of thedouble-wire system (1) with a fixed parity P = p a anda total parity P = ( − N is doubly degenerate. The twoground states can be connected by the product of two Majorana operators – one from each wire. Strictly speak-ing, long-wavelength fluctuations destroy long-range or-der in 1D breaking the mean-field description even atzero temperature. In the considered case, however, thisdoes not change the picture qualitatively (see Ref.[21]).
Low-energy theory.
Effective field theories based onbosonization [29, 30] represent a remarkable tool to in-vestigate the emergence of topological states and MFsin strongly correlated systems [11, 26–28], and has beenapplied recently to number conserving settings [20, 21].Here, we employ this formalism to qualitatively ana-lyze the low-energy properties of the Hamiltonian (1).We start with applying standard bosonization formu-las to introduce effective low-energy phase and densityfluctuation fields ϕ γ , ϑ γ , respectively, for each species γ = a, b [45]. After introducing symmetric and anti-symmetric combinations, ϕ S/A = ( ϕ a ± ϕ b ) / √
2, and ne-glecting contributions with high scaling dimensions, thebosonized Hamiltonian decouples into symmetric and an-tisymmetric sectors. The symmetric sector describes col-lective density-wave excitations, and is well-captured bya Tomonaga-Luttinger liquid Hamiltonian: H S = v S (cid:90) (cid:20) ( ∂ x ϕ S ) K S + K S ( ∂ x ϑ S ) (cid:21) dx, (2)whilst the antisymmetric one is described by a sine-Gordon Hamiltonian [29]: H A = v A (cid:90) (cid:20) ( ∂ x ϕ A ) K A + K A ( ∂ x ϑ A ) + w cos[ √ πϑ A ] (cid:21) dx, (3)where K α and v α are the Luttinger parameter and thesound velocity, respectively, for each sector α = ( A, S ),and w ∝ W results from the pair hopping. It can beshown that the parity symmetry Z and the number con-servation are exactly retained at low energies in the an-tisymmetric and symmetric sector, respectively [20, 29].We now discuss the qualitative phase diagram of the sys-tem by using standard Renormalization Group (RG) scal-ing arguments [29–31]. Away from the strong couplinglimit W (cid:29) t (where terms with higher scaling dimensionsmay become relevant), the two sectors remain decoupled,so that one can analyze them separately. While the sym-metric sector is simply a theory of free bosons, the anti-symmetric sector displays richer physics, as it undergoesa phase transition from a gapless phase at W = 0 to agapped, superconducting phase for W >
0. In analogywith the continuum model of Ref. [20], Eq. (3) can beexactly mapped to the continuum version of the Kitaevwire [20] at the Luther-Emery point K A = 2. As a result,the system with OBC displays a two-fold ground state de-generacy, where the two states have opposite parities P ,and support MFs at the boundaries [45]. Moreover, thesingle-particle correlation functions show exponential de-cay (cid:104) a † i a i + x (cid:105) (cid:39) e − ξ | x | in the bulk, signaling the presenceof a finite superconducting gap. Away from the Luther-Emery point, the MF wave function overlap increases de-pending on ( K A − e − κL , even-tually turning to power-law in the presence of certainkinds of perturbations [20–22]. On the other hand, inthe strong coupling limit | W | (cid:29) t , the presence of addi-tional terms (with higher scaling dimension) of the form w cos[ √ πϑ A ]( ∂ x ϕ S ) (cid:39) − w ( ∂ x ϕ S ) leads to a reductionof the sound velocity v S , resulting in phase separation. Numerical results
Employing this low-energy pictureas a guide, we now present a quantitative numerical in-vestigation of the Hamiltonian Eq. (1). We start with abrief description of the phase diagram of the system, andthen discuss the criteria (i)-(iv) relevant for the existenceof MFs. In the following, we set t = 1 as the energy scale.The phase diagram of the model can be divided intothree regions: a superconducting phase, an insulatingphase, and a region of phase separation. The supercon-ducting phase is characterized by a homogeneous density,leading superconducting correlations, and nonzero single-particle gap ∆ = | E ( N ) − ( E ( N + 1) + E ( N − | for periodic boundary conditions (PBC). Here E ( N ) isthe ground state energy for N particles. We find thisphase for small and moderate values of the pair hop-ping | W | (cid:38) n = 1 /
2. At ex-actly half-filling, an incompressible insulating phase isformed with exponentially decaying superconducting cor-relations. For large values of the pair hopping | W | (cid:29) W = − . n = 1 / L = 12, 24and L = 36 with even number of particles.(i) The ground state degeneracy can be studied bylooking at the energy gap ∆ E n ( N ) = E n ( N ) − E ( N )between the ground state and the n -th excited state. Asshown in Fig. 2a, in the case of OBC, the gap betweenthe ground and the first excited state ∆ E , OBC closes ex-ponentially in the system size (left panel) indicating thedegeneracy of the ground state in the thermodynamiclimit. This is in contrast to the case of PBC that is de-picted in the right panel of Fig. 2a (blue open triangles).Here we find that ∆ E , PBC closes linearly in the systemsize, and ∆ E , PBC = ∆ E , PBC , i.e. the first and secondexcited state are degenerate (blue open and closed tri-angles). For OBC, ∆ E , OBC also closes linearly in thesystem size (red diamonds). We find that the two de-generate ground states in the case of OBC differ by theparities of the individual wires. Note that for OBC we
14 13 12 11 ∆ E ∆ E ∆ E ∆ E site j G ij G G
14 13 12 11 −10−5 l og ∆ E log ∆ E a) b) − l og ( λ ) − l og ( λ ) c) d) OBC PBC non-local correlation
FIG. 2. a) Closing of the energy gaps with the system size( L = 12 , ,
36) for W = − . n = 1 /
3. For OBC,the gap ∆ E closes exponentially (left panel), in contrast tothe polynomial closing in the case of PBC (right panel, openblue triangles). The energy gap ∆ E closes polynomially in-dependent of the boundary conditions (right panel, red di-amonds for OBC and closed triangles for PBC). Note that∆ E , PBC = ∆ E , PBC . b) Non-local fermionic correlations G lj on the upper wire for L = 24. c) and d) Entanglementspectrum for the system of the size L = 24 shows doubledegeneracy for both OBC (c) and PBC (d). also have ∆ = 0.(ii) The intrawire single-particle correlation function G lj = (cid:104) a † l a j (cid:105) for the system of the length L = 24 is shownin Fig. 2b for the case where l = 1 , j ∈ [ l, L ]. We see that G lj , being exponen-tially small inside the wire, attains a finite value at theright edge showing the existence of non-local fermioniccorrelations typical for a system with MF edge states.(iii) Topological order (TO) manifests itself in the de-generacy of the entanglement spectrum (ES) [32–34]: Let ρ A = (cid:80) Nj λ ( N ) j ρ ( N ) j be the reduced density matrix of thesystem with respect to some bipartition with support onboth wires, where ρ ( N ) j describes a pure state of N parti-cles with the corresponding eigenvalues λ ( N ) j . In a topo-logical phase, the low-lying eigenvalues λ ( N ) j are expectedto be doubly degenerate for each N , for both OBC andPBC, as it is demonstrated in Figs. 2c (OBC) and 2d(PBC) for a system of the size L = 24. Moreover, thedistributions of the low-lying eigenvalues as a function of N share the same patter in the two cases.(iv) The robustness of the above properties againststatic disorder is one of the key manifestations of a non-local topological order. We model the disorder by addingthe term H V r = (cid:80) j V ( a ) j a † j a j + V ( b ) j b † j b j to the Hamilto-nian, where V ( γ ) j with γ = a, b are random local potentialsequally distributed in the interval [ − V r , V r ]. We find thateven for moderate disorder V r = 0 . t , the ground stateremains doubly degenerate, and the system still exhibitsthe non-local correlations (Fig. 3b) as well as the degen- −3 −2 −1 0−6−20 log t y l og ∆ E − l og ( λ ) site j G ij G G a) b)c) − l og ( λ ) d) OBC OBC
FIG. 3. Effects of imperfections on the topological order ( L =24 , n = 1 / H ⊥ = (cid:80) i t y a † i b i + h.c. .b)-d) Effects of static disorder: The non-local correlations(b) and the degeneracy of the ES (c) indicated the topologicalstate in the presence of disorder with V r = 0 . t . d) Breakingof the topological phase by a strong disorder with V r = 1 . t . erate ES (Fig. 3a), indicating the presence of topologicalorder. For strong local disorder, however, the topologicaleffects disappear, as exemplified by the non-degenerateES for V r = 1 . t in Fig. 3d.Remarkably, the observed topological order and itsconsequences are also stable against a single-particle hop-ping H ⊥ = (cid:80) i t y a † i b i + h.c. between the two wires, whichbreaks the parity of the wires and related Z symme-try. As an example, in Fig. 3a we show the energy gap∆ , OBC as a function of t y : The ground state of the sys-tem remains quasi-degenerate (∆ E (cid:39) − ) up to values t y of the order of 0 . t , in agreement with the predictionof Refs. [20, 21]. Note, however, that the dependenceof ∆ , OBC on L changes from exponential to power law[22]. This stability could be very important for exper-imental realizations of the model because the interwiresingle-particle hopping is one of the most probable im-perfections. Pair hopping with cold fermionic atoms.
The key in-gredient of the Hamiltonian (1) is the interwire pair hop-ping with coupling W in the absence of (parity violat-ing) single particle tunneling. The basic idea behind anatomic implementation is to introduce offsets in opticallattices, which suppress single particle hopping by energyconstraints, while an energy conserving pair hopping isallowed and mediated by interactions.An atomic setup illustrating these ideas is given inFig. 1, while technical details and variants of the schemecan be found in the SI. We implement the two wiresof spinless fermions as a bipartite lattice for spinfulfermions. Odd and even lattice sites j trap the spin ↑ and ↓ components of the fermions with energies (cid:15) and − (cid:15) ,respectively, and transitions between the adjacent wellsare induced by an external RF field or a Raman assisted hopping (c.f Fig. 1b). This realizes the first line of H in Eq. (1). To understand the pair hopping mechanism,consider the plaquette indicated in Fig. 1c by the dashedline. We assume an auxiliary molecular site in the cen-ter of the plaquette (indicated as (C) in Fig. 1c), whichtraps both ↑ and ↓ atoms, and is connected to the latticesites on the wire by a spin-preserving tunneling couplingwith amplitudes t am and t bm . Pairs of atoms occupyingthe molecular site are assumed to interact via an onsiteinteraction U . In addition, we introduce spin-dependentlattice offsets, which are indicated by the − (cid:15) , (cid:15) for thelattice sites on the two wires and for ↑ and ↓ species, re-spectively (Fig. 1d-e). Such offsets can be generated asZeeman shifts of the spin states, if a gradient magneticfield is applied perpendicular to the wire.Single particle hopping between the wires is suppressedin this setup: consider an atom, say in the upper wire a in lattice site 1 with spin ↑ . Spin-preserving tunnel-ing is possible via the molecular site along the diago-nal of the plaquette (virtual processes are indicated inFig. 1d-e). It corresponds to the process ↑ a →↑ m →↑ b ,which is suppressed by the corresponding energy offsets+ (cid:15) , , − (cid:15) . In a similar way also the tunneling of the ↓ atom along ↓ a →↓ m →↓ b is suppressed by energy conser-vation. However, for pair hopping ↑ a ↓ a →↑↓ m →↑ b ↓ b the overall energy will be conserved, since the two atomscan exchange energy via the interaction U . After adia-batic elimination of the intermediate sites when U, (cid:15) / (cid:29) t am,bm , the resulting amplitude for the pair-hopping termis W (cid:39) t am t bm (1 /(cid:15) − /(cid:15) ) /U (see SI). Note that thepair-hopping process ↑ a ↓ b →↓ a ↑ b will also be allowed,but does not change the number of particles on the wires,and thus preserves atom number parity on the wires. Adetailed description of this pair hopping dynamics includ-ing possible imperfections, e.g. induced by the Ramancouplings, can be found in the SI. In Fig. 1f we present anumerical analysis of the pair hopping dynamics, where(in units of t am = t bm = 1) (cid:15) = 2 (cid:15) = 2, U = −
20 (seeSI). Finally, note that the engineering of pair hopping hasfurther applications in cold atom systems. For example,the pair hopping can be used as an entangling quantumgate, where the hopping of one particle (control) triggersthe tunneling of a second atom (target). Further, it hasapplications in the context of quantum simulation, e.g.for lattice gauge theories emulation include ring-exchangeand rishon determinant interactions [38, 39].
Detection.
Finally, we address the problem of detect-ing the emerging Majorana states in our AMO setup.Following the proposals of Ref. [40], this could be done,e.g., by using standard AMO detection tools like time-of-flight imaging and spectroscopic techniques to probethe ground state degeneracy and the inherent non-localfermionic correlations. Demonstration of a non-Abelianstatistic of the MFs, on the other hand, requires some dy-namical protocols resulting in the motion of MFs aroundeach other. In our setup, one could think of a general-ization of the ideas of Ref. [41] relying on single-site ad-dressing available in current experiments with ultra-coldatoms [42, 43]. Another possibility would be an atomicanalog of the fractional Josephson effect [16] using a prop-erly shaped external potential along the x -direction. Conclusions.
In summary, we have shown that topo-logical states of matter with Majorana fermion edgestates can be created in fermionic atomic ladders with-out any additional reservoir or p-wave interaction, butwith only interwire pair hopping, which could provide aneasier, complementary way for experimental realizations.
Acknowledgments.
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We present here some details of the low-energy field theoryin the lattice pair-hopping model presented in the maintext. The Hamiltonian is (for compactness, we adopt herea different notation c a,j = a j , c b,j = b j with respect to themain text) H = − (cid:88) j ; γ = a,b t γ ( c † γ,j c γ,j +1 + h.c. )++ W (cid:88) j ( c † a,j c † a,j +1 c b,j c b,j +1 + h.c. ) (4)and has a U (1) ⊗ Z symmetry conserving both the totalnumber of particles and the parity of each wire. We applyhere the standard bosonization lattice procedure in orderto extract the low-energy field theory [1, 2]. Within thisframework, we take first the continuum limit ( a being thelattice spacing) and introduce c † j,γ = ψ † R,γ ( x = ja ) + ψ † L,γ ( x = ja ) . (5)The bosonization identities have the following form: ψ r,γ ( x ) = η r,γ √ πa e irk F,γ x e − i [ rϕ γ − ϑ γ ] == η r,γ √ πa e irk F,γ x e − i √ [ rϕ S − ϑ S +¯ γ ( rϕ A − ϑ A )] , (6)where R/L implies r = ± γ ( a/b ) = ±
1, and we havedefined ϑ A = ϑ a − ϑ b √ , ϑ S = ϑ a + ϑ b √ , and the Klein operators { η R,γ , η
L,γ } = 0 , [ η r,a , η s,b ] = 0 , (7) { η † r,γ , η s,β } = 2 δ sr δ γβ , η s,γ η † s,γ = 1 (8)in order to preserve fermionic commutation relations. Thefree part (the first line) of the lattice Hamiltonian is then H A + H S , where (∆ = A/S ) H ∆ = v ∆ (cid:90) (cid:20) ( ∂ x ϕ ∆ ) K ∆ + K ∆ ( ∂ x ϑ ∆ ) (cid:21) , (9)with Tomonaga-Luttinger parameters K ∆ = 1 and soundvelocities v ∆ = 2 ta − sin( k F, ∆ a ) with k F,S = πn .The bosonized pair-hopping operator [the second line inEq. (4)] reads H ( A ) W = 4 W (2 πa ) (cid:90) dxa (cid:104) γ ( n ) cos[ √ πϑ A ( x )++ a − cos[ √ πϑ A ( x )] cos[ a k S ]( ∂ x ϕ S ) (cid:105) , (10)where γ ( n ) is a density-dependent coefficient and we omitthe Klein factors. Away from the strong coupling limit W/t (cid:29)
1, where interactions between symmetric and(gapped) antisymmetric sector drive the system towardsphase separation, the bosonized Hamiltonian H can beeffectively split into two distinct sectors. The symmetricone is described by a Tomonaga-Luttinger liquid, whereasthe antisymmetric one by a sine-Gordon Hamiltonian with the mass-term for the phase field ϑ A . In particular, at theLuther-Emery point K = 2, it is possible to map the an-tisymmetric sector to the so called massive Dirac Hamil-tonian [4]˜ H = − iv A ( ξ † R ∂ x ξ R − ξ † L ∂ x ξ L ) + im ( ξ † R ξ † L − ξ L ξ R ) , (11)by employing standard re-fermionization techniques [1, 5].Here ξ s ( s = L/R ) are Dirac fermions with mass m ∼ W ,which are related to the bosonic operators via ξ s = η † R,b η L,a e i √ πr s ϕ s,A , where r s = R/L = ± ϕ s = R/L,A = ( ϕ A / √ ± √ ϑ A ). The mapping toEq. (11) is possible due to the specific form of the interac-tion in the sine-Gordon Hamiltonian involving phase fluc-tuations ϑ A . The effective Hamiltonian (11) correspondsto the continuum limit of the Kitaev chain [4], and, there-fore, supports Majorana edge states. Since the antisym-metric and symmetric sectors are decoupled, the energysplitting between the two degenerate ground states is ex-ponentially small in the system size [3, 4]. However, cer-tain external perturbations, such as interchain backscat-tering (in the presence of interactions between the wires)or impurities, may change the dependence of the energysplitting on the system size from exponential to algebraic(although with an extremely small prefactor in the vicin-ity of the Luther-Emery point, see the detailed instantondiscussion in Ref. (3). SI: EFFECTIVE PAIR-HOPPINGHAMILTONIAN - MODEL 1
In this and the following Section, we describe two pos-sible implementation schemes realizing the pair hoppingHamiltonian discussed in the main text.
Single Wire
Before discussing the two couple wires and pair hoppingbetween these wires we briefly describe our setup for asingle wire.We implement the single wire of spinless fermions as abipartite lattice of spinful fermions in an 1D-optical latticeas indicated in Fig. 4a. Odd and even lattice sites j trap ↑ and ↓ components of fermionic atoms, respectively, andtransitions between adjacent wells (and the associated spinflip) are induced either by an RF field or by an opticalRaman transition. The corresponding Hamiltonian is H = (cid:15) (cid:88) j odd a † j ↑ a j ↑ − (cid:15) (cid:88) j even a † j ↓ a j ↓ − Ω (cid:88) r = ± j even (cid:16) a † j + r ↓ a j ↓ e − iωt + h . c . (cid:17) . with (cid:15) σ =1 , the energies of the atomic states representingthe spins, Ω the Rabi frequency of the Raman drive and ω the corresponding frequency. After a transformation to a”rotating frame” the above Hamiltonian can be rewrittenwith the mapping a j ↓ → a j ( j even) and a j ↑ → a j ( j odd)as ˜ H = − t (cid:88) r = ± j a † j + r a j + ( − δ ) (cid:88) j odd a † j a j Here δ = ω − ( (cid:15) + (cid:15) ) is the detuning , which acts asan offset (superlattice) for the odd lattice sites. The tun-neling amplitude in the lattice can be identified with theRabi frequency of the drive, t ≡ Ω. Note that the aboveHamiltonian corresponds to an effective model of spinlessfermions hopping on a 1D lattice.
Two Coupled Wires
The above realization of a 1D wire of spinless fermionicatoms acts as a building block for two coupled wires (witha straightforward generalization to many coupled wires).The setup we have in mind is depicted in Fig. 4b withthe wires denoted by a and b , respectively. In addition weintroduce auxiliary sites between the wires, which we callcenter or molecular sites in Fig. 4b. They are the basicingredient in implementing pair hopping, while stronglysuppressing unwanted processes like single particle tun-neling (which violates parity of atoms on the two wires).We extend the single wire model of the previous subsectionassuming that the spins are placed in a spatially varyingmagnetic field with gradient perpendicular to the wires.This results in a spin dependent energy offset: the spins ↑ and ↓ on the (upper) a wire have Zeeman energies (cid:15) and − (cid:15) , and the corresponding energies on the (lower) wire b are − (cid:15) and (cid:15) , respectively. Thus we have the followingHamiltonians for the two (uncoupled) wires: H a = (cid:15) (cid:88) j odd a † j ↑ ,a a j ↑ ,a − (cid:15) (cid:88) j even a † j ↓ ,a a j ↓ ,a − (cid:88) r = ± j even (cid:16) Ω a † j + r ↑ ,a a j ↓ ,a e − iωt + h . c . (cid:17) ,H b = (cid:15) (cid:88) j odd a † j ↓ ,b a j ↓ ,b − (cid:15) (cid:88) j even a † j ↑ ,b a j ↑ ,b − (cid:88) r = ± j even (cid:16) Ω a † j + r ↓ ,b a j ↑ ,b e + iωt + h . c . (cid:17) . If we choose ω = (cid:15) + (cid:15) , our model reduces again to a 1Dtight binding model for spinless fermions hopping on thewires, as discussed in the previous subsection. The Zeemanoffsets will play a central role in the following discussionof pair hopping.As shown in Fig. 4b, atoms can hop from the two wires to(auxiliary) central sites, which can be occupied by both ↑ and ↓ atoms. Two atoms occupying a central site willinteract according to an onsite interaction U , effectivelyforming a ”molecule” m . The corresponding Hamiltonianis H = H + H = H a + H b + H c + H . Here H a and H b are Hamiltonians for the wire as dis-cussed above. The Hamiltonian for the central sites c is H c = (cid:88) c ≡ ( j,j +1) Ua †↑ c a †↓ c a ↓ c a ↑ c − (cid:88) c ≡ ( j,j +1) (cid:88) α = a,b t mα (cid:16) a †↑ c a j ↑ ,α + a †↓ c a j +1 ↓ ,α + h . c . (cid:17) where we adopt the notation c ≡ ( j, j + 1) for c on thelink j, j + 1, and we have an interaction term U betweenatoms with different spin in the center of the plaquette. (2,b)(C)(1,a) (2,a)(1,b) a) U ΩΩ t am t bm b) Ω FIG. 4. Microscopic illustration of Model I. Panel a) : the sin-gle ( a -)wire described by ˜ H is obtained by combining a speciesdependent lattice for the ↑ and ↓ species, with Raman assistedtunneling between the two, the small detuning thereof provid-ing a finite value of the δ parameter. Panel b) : two-wire setupof the Hamiltonian ˜ H (cid:3) , indicating the correspondent energyoff-set of each site. The upper and lower wires are denoted as a and b , respectively. Here, in the single plaquette includedin the dotted box, thin lines link sites connected by standardtunneling, and dashed curved lines link sites connected byRaman assisted tunneling.The last line is a spin-preserving hopping from the wiresto the central site.Finally, H accounts for RF or Raman induced spin-fliptransitions, H = − (cid:88) c ≡ ( j,j +1) (cid:16) Ω (cid:48) e − iωt a †↑ c a ↓ c + h . c . (cid:17) − (cid:88) j even Ω (cid:48)(cid:48) (cid:16) a † j ↑ ,a a ↑ c + a † j ↓ ,b a ↓ c ++ a †↓ c a j +1 ↑ ,a + a †↑ c a j +1 ↑ ,b (cid:17) e − iωt + h . c . − (cid:88) j odd Ω (cid:48)(cid:48) (cid:16) a † j ↑ ,a a ↑ c + a † j ↓ ,b a ↓ c ++ a †↓ c a j +1 ↑ ,a + a †↑ c a j +1 ↑ ,b (cid:17) e + iωt + h . c . . The first line is a spin-flip at the central site, and the lastlines account for an RF or Raman assisted hopping fromthe wire to the central site. Note that - in contrast to theRF or Raman couplings along the wires - all these termsare off-resonant because the drive frequency ω = (cid:15) + (cid:15) isdetuned from from the corresponding transition frequen-cies (cid:15) , (cid:15) . Thus these terms average to zero, providedΩ (cid:48) , Ω (cid:48)(cid:48) (cid:28) (cid:15) . We will neglect them in the following dis-cussion.To illustrate the physics of pair hopping we consider nowa single plaquette j = 1 , ω as˜ H (cid:3) = (cid:15) a † a a ,a − (cid:15) a † a a ,a − (cid:16) Ω a † a a a e − i (cid:15)t + h . c . (cid:17) + (cid:15) a † b a ,b − (cid:15) a † b a b − (cid:16) Ω a † b a b e − i (cid:15)t + h . c . (cid:17) ( − (cid:15) (cid:48) ) (cid:88) σ = ↑ , ↓ a † σc a σc + Ua †↑ c a †↓ c a ↓ c a ↑ c − (cid:88) α = a,b t ma (cid:16) a †↑ c a jα + a †↓ c a jα + h . c . (cid:17) For the sites on the two wires we have again used thenotation a ↑ ,a → a a , a ↓ ,b → a b and a ↓ ,a → a a , a ↑ ,b → a j,b . The first line is again the Raman or RFhopping between sites 2 ↔ − (cid:15) and + (cid:15) ,respectively, due to absorption (emission) of a photon ω = (cid:15) + (cid:15) , which is tuned to compensate the energydifference. The second line corresponds to the central sitewith interaction and hopping. The last two lines describea spin-flip on site c , and a Raman assisted hopping andaccompanying spin flip from the wires to the central site c .Let us now analyze the various processes on the plaquetteaccording to the above Hamiltonian. We will argue thatan adiabatic elimination of the central site will result in aneffective Hamiltonian for the two wires where single parti-cle tunneling is suppressed, while atoms can hop pairwisebetween the wires. • Suppression of single particle interwire hopping.
Consider a single atom on the plaquette whichoccupies initially, say, site 1 a , i.e. has spin ↑ .The particle hopping to and from the central sitepreserves spin, and thus it can only tunnel alongthe diagonal 1 ↑ , a → c ↑→ ↑ , b . In view of theenergy mismatch + (cid:15) , 0, − (cid:15) this tunneling processis not energy conserving and thus will occur onlyas virtual process, which renormalizes the (singleparticle) tunneling parameters and onsite shifts ofthe wire sites.[Note: The above argument ignores the effect ofthe drive on the central site. We argued abovethat this coupling will be small for Ω (cid:48) (cid:28) (cid:15) , butcan result in a (weak) energy conserving transition1 ↑ , a → ↓ , b . We can suppress such terms bytilting the plaquette, so that the energies of the fourlattice 1 a, a, b, b sites are + (cid:15) (cid:48) , − (cid:15) (cid:48) , + (cid:15) (cid:48) , − (cid:15) (cid:48) with (cid:15) (cid:48) − (cid:15) (cid:48) (cid:29) t ma , Ω. This makes the spin-flip interwiretunneling an energy non-conserving process, i.e thetunneling terms of the form a † ja a jb will be absent.] • Interwire pair hopping.
While the single particle in-terwire hops 1 ↑ , a → ↑ , b and 2 ↓ , a → ↓ , b areindividually forbidden, the joint hopping is energet-ically allowed. For this pair hopping to happen thetwo particles must occupy simultaneously the centralsite to be able to exchange energy, i.e. to interact.An adiabatic elimination of the central site thereforegives a term W a † b a † b a a a a +h . c . , where in the limitof large UW = − (cid:18) (cid:15) − (cid:15) (cid:19) t am t bm U .
Particle assisted tunneling, where, for example, aparticle hops 1 ↑ , a → ↑ , b while a second particle2 ↓ , a → ↓ , a remains on wire a , is suppressed byenergy conservation. A process 1 ↑ , a → ↑ , b while2 ↓ , a → ↑ , a requires a spinflip on the central sitewhich we argued to be small; in addition this processcan be suppressed by tilting the plaquette. • Parity-preserving perturbations.
Imperfections whichpreserve the parity symmetry are also present. Local off-sets of the form: H off = (cid:88) j odd ξ j ↑ ,a a † j ↑ ,a a j ↑ ,a + (cid:88) j even ξ j ↓ ,a a † j ↓ ,a a j ↓ ,a ++ (cid:88) j even ξ j ↑ ,b a † j ↑ ,b a j ↑ ,b + (cid:88) j odd ξ j ↓ ,b a † j ↓ ,b a j ↓ ,b (12)are also generated within second order perturbationtheory, the corresponding coefficients being: ξ j ↑ ,a = t am (cid:15) , ξ j ↑ ,b = − t bm (cid:15) ,ξ j ↓ ,a = t am (cid:15) , ξ j ↓ ,b = t bm (cid:15) (13)Moreover, in fourth order perturbation theory, addi-tional diagonal interactions emerge, induced by vir-tual processes where two particles from the site j, j (cid:48) belonging to the same plaquette hop into the inter-mediate sites, and subsequently hop back to j, j (cid:48) .The corresponding terms read: H diag = K (cid:88) j,j (cid:48) (cid:88) σ,σ (cid:48) = ↑ , ↓ (cid:88) α,α (cid:48) = a,b n jσ,α n j (cid:48) σ (cid:48) ,α (cid:48) with coefficients: K (cid:39) (cid:18) (cid:15) − (cid:15) (cid:19) t am t bm U , (14)are also present. Regarding the relevance of such ad-ditional terms in the effective Hamiltonian, we referthe reader to the discussion of the imperfections inthe following section.
Numerical analysis of a single plaquette
In the following we investigate a single plaquette andcarry out a real-time evolution under the Hamiltonian˜ H (cid:3) starting from a state with two particles on the up-per wire, | Ψ (cid:105) = a † a † | (cid:105) . We plot, as a function oftime, the double occupancy p a ( t ) = (cid:104) n a, ( t ) n a, ( t ) (cid:105) and p b ( t ) = (cid:104) n b, ( t ) n b, ( t ) (cid:105) in the upper resp. lower wire. Alarge value of p b ( t ) is an indication of pair hopping. Sin-gle particle hopping, which is indicated via a finite valueof S ( t ) = (cid:80) i,j =1 , (cid:104) n a,i ( t ) n b,j ( t ) (cid:105) should be suppressed.Taking t am = t bm = 1 as the unit of energy, we depict,in Fig. 5a), numerical results for p a,b ( t ) and S ( t ), where (cid:15) = 2 (cid:15) = 2, U = − (cid:15) (cid:48) = 1 / U − ( (cid:15) − (cid:15) )) + δ , and δ = 1. We find a large and finite value for the pair hop-ping, while the single particle hopping is of the order of10 − . The occupation of the central sites (Fig. 5b) is verysmall. Relation to Model 2
As a final remark we will relate the present model to theone described in the following subsection, where the role oftunneling and Raman beams is interchanged. We can makea time-dependent transformation to rewrite the above pla-quette Hamiltonian as a) b)
FIG. 5. Real time evolution for the parameters in the maintext. a) Time evolution of the expectation values p a ( t ) = (cid:104) n a, ( t ) n a, ( t ) (cid:105) and p b ( t ) = (cid:104) n b, ( t ) n b, ( t ) (cid:105) that indicate thepair hopping. Single particle hopping indicated by S ( t ) = (cid:80) i,j =1 , (cid:104) n a,i ( t ) n b,j ( t ) (cid:105) is of the order of 10 − . b) The oc-cupation of the intermediate sites (cid:104) n a c (cid:105) , (cid:104) n b c (cid:105) , is very small˜ H (cid:3) = − (cid:16) Ω a † a a a + h . c . (cid:17) − (cid:16) Ω a † b a b + h . c . (cid:17) + Ua †↑ c a †↓ c a ↓ c a ↑ c − t ma (cid:16) a †↑ c a a e − i ( (cid:15) + (cid:15) (cid:48) ) t + a †↓ c a a e i ( (cid:15) − (cid:15) (cid:48) ) t + h . c . (cid:17) − t mb (cid:16) a †↓ c a b e + i ( (cid:15) − (cid:15) (cid:48) ) t + a †↑ c a b e − i ( (cid:15) + (cid:15) (cid:48) ) t + h . c . (cid:17) . The first line now looks like a tunneling Hamiltonian, whilethe last two lines correspond to time-dependent hoppingsto the central site with frequencies (cid:15) ± (cid:15) (cid:48) . In this rewrit-ing of the model the lattice offsets have been convertedto effective time-dependent RF or Raman couplings. Inthe following section we investigate a model, which is ageneralization of this scheme. SI: EFFECTIVE PAIR-HOPPINGHAMILTONIAN - MODEL 2Building block: Single wire with resonant couplingto intermediate states
The basic building block of model II is represented by asingle plaquette, illustrated in Fig. 6a, where the uppersites belong to the a -wire and the lower sites to the b - one,while the middle sites are employed as intermediate sites(which will then be adiabatically eliminated) to generatethe pair tunneling term. Before discussing the emergenceof the full pair-tunneling Hamiltonian, we show how onecan couple one of the wires to the intermediate sites suchthat pair tunneling from the wire to the middle sites isgenerated in second order perturbation theory. This dis-cussion will then be naturally extended to the full pla-quette treatment, where the emergence of inter-wire pairtunneling will emerge as a combination of a pair tunnelingprocess from the a -wire to the central sites, and from thecentral sites to the b -wire.We start by analyzing the system composed by the lowerpart of the plaquette in Fig. 6a. Atoms in the atomic state | a (cid:105) can occupy the ’wire’ sites | a j (cid:105) , | a j +1 (cid:105) and the centralsite | a c (cid:105) ; atoms in the atomic state | b (cid:105) can occupy thecentral site | b c (cid:105) . The microscopic Hamiltonian ˜ H for the subsystem can be decomposed as a sum of three terms:˜ H a = ˜ H t,a + ˜ H C + ˜ H Ω ,a . (15)The first term describes tunneling of the fermions withinthe wire: ˜ H t,a = − ˜ t a ( a † j a j +1 + h.c.) , (16)where ˜ t a is the tunneling amplitude, and a † j ( a j ) are cre-ation (annihilation) operators of fermions in the state | a j (cid:105) .The second term˜ H C = V a a † c a c + V b b † c b c + Ua † c a c b † c b c , (17)describes the two intermediate states | a c (cid:105) and | b c (cid:105) in themiddle of the plaquette. Here, a † c ( a c ) and b † c ( b c ) are cre-ation (annihilation) operators of fermions for the interme-diate states | a c (cid:105) and | b c (cid:105) , respectively, V a and V b are thecorresponding potential off-sets, and U is the interparticleinteraction. The last term˜ H Ω ,a = J a ( a † c ( a j + a j +1 ) + h.c.)+ − (cid:126) ( b † c (Ω ,j a j + Ω ,j a j +1 ) e − iω t + h.c.) (18)describes the coupling between the wires and the states inthe middle of the plaquette: Atoms in the | a (cid:105) state cantunnel from the wire to the intermediate site | a c (cid:105) , witha tunneling coefficient J a , and can be transferred to thestate | b c (cid:105) via a Raman process characterized by (space-dependent) Rabi frequency Ω ,j and detuning (cid:126) ω . Wework here with time-dependent fields assuming a rotatingwave approximation (Ω ,j (cid:28) ω ). Note that the same re-sults can be obtained by describing Raman process as anauxiliary quantized single photon mode.We are now interested in the dynamics of such basic build-ing block in the regime where single particle occupationin the intermediate sites is suppressed as a far-off reso-nant state, while double occupancies of the form a † c b † c | (cid:105) are allowed (here, | (cid:105) is the fermionic vacuum). In order toillustrate it, we perform a quasi-degenerate perturbationtheory (within the rotating wave approximation) in thelimit | U | , (cid:126) ω , V a , V v (cid:29) ˜ t a , | (cid:126) Ω ,j/j +1 | , with the additionalquasi-resonant condition: U = − V a − V b − (cid:126) ω + δ U , | δ U | (cid:39) ˜ t a , | (cid:126) Ω | . (19)In this limit, the two states a † j a † j +1 | (cid:105) and a † c b † c | (cid:105) are theonly quasi-degenerate with total number of fermionic par-ticles equal to two. After eliminating the states with singlyoccupied intermediate sites in the second order perturba-tion theory, we obtain the following effective Hamiltonian: H eff , a = − t a ( a † j a j +1 + h.c.) + W a ( a † c b † c a j a j +1 + h.c.)++ δ U ( a † c a c b † c b c ) + ξ a,j a † j a j + ξ a,j +1 a † j +1 a j +1 , (20)with a renormalized tunneling rate t a = ˜ t a − ( J a ) /V a − Ω ,j +1 Ω ∗ ,j / ( V a − (cid:126) ω ) , (21)an effective interaction in the intermediate site δ U , and apotential off-set on the a-wire: ξ a,j = − J a V a − | Ω a,j | ( V b + (cid:126) ω ) . (22) Moreover, a pair tunneling from the wire to the centralsites emerges, with coefficient: W a = (cid:18) V a + 1 V b + (cid:126) ω (cid:19) (cid:126) J a (Ω ,j +1 − Ω ,j ) . (23)Note that in order to have a finite coefficient of the pairtunneling, space-dependent Rabi frequencies are required.This is due to the fact that, for a short-range interpar-ticle interaction, one has to use spatially inhomogeneousRabi frequencies in order to change the symmetry of thespatial part of the wave function for two particles from an-tisymmetric (for two initial particle on one of the wires) tosymmetric (for two particles in the center of a plaquette).In a similar way, one describe the system of the b -wire(upper part of the plaquette) coupled to the intermediatesites. The corresponding microscopic Hamiltonian reads:˜ H b = ˜ H t, ∆ ,b + ˜ H C + ˜ H Ω ,b , (24)where˜ H t, ∆ ,b = − ˜ t b ( b † j b j +1 + h.c.) − ∆( b † j b j + b † j +1 b j +1 ) , (25)describes tunneling in the b -wire, and an off-set ∆. Here, b † j ( b j ) are creation/annihilation operators correspondingto the states | b j (cid:105) . The term˜ H Ω ,b = − (cid:126) ( a † c (Ω ,j b j + Ω ,j +1 b j +1 ) e iω t + h.c.)++ J b ( b † c ( b j + b j +1 ) + h.c.) , (26)describes the tunneling between the b -wire and the inter-mediate site | b c (cid:105) with the amplitude J b , and the couplingbetween the b -wire and the intermediate site | a c (cid:105) via a Ra-man process with detuning (cid:126) ω and Rabi frequency Ω ,j .The energy off-set is introduced in order to suppress singleparticle tunneling between the wires (as discussed below).This off-set, however, has to match the condition2∆ = (cid:126) ( ω + ω ) (27)in order to ensure resonant pair-tunneling.After eliminating the states with singly occupied inter-mediate sites in the second order perturbation theory in | U | , (cid:126) ω , V a , V b (cid:29) t, (cid:126) Ω, we obtain the effective Hamilto-nian: H eff , b = − t b ( b † j b j +1 + h.c.) + W b ( a † c b † c b j b j +1 + h.c.)++ δ U ( a † c a c b † c b c ) + ξ b,j b † j b j + ξ b,j +1 b † j +1 b j +1 (28)with W b = (cid:18) V b + ∆ + 1 V a − (cid:126) ω + ∆ (cid:19) (cid:126) J b (Ω ,j +1 − Ω ,j ) ,t b = ˜ t b − ( J b ) / ( V b + ∆) − Ω ,j +1 Ω ∗ ,j / ( V a − (cid:126) ω + ∆) , (29) ξ b,j = − J b ( V b + ∆) − | Ω b,j | ( V a + ∆ + (cid:126) ω ) . (30)Despite its simplicity, this setup already displays the fun-damental features of the emergence of inter-wire pair tun-neling. We will now show how the two wires can be res-onantly coupled to the intermediate sites while avoidingsingle particle inter-wire tunneling. a) c) I III IV b) II I II III IV d) FIG. 6. Panel a) : Schematic Hilbert space of a singleplaquette, including states in the upper and lower wires( | a j/j +1 (cid:105) , | b j/j +1 (cid:105) ) and states in the center of the plaquette | a (cid:105) c , | b (cid:105) c . Panels b,c) : Schematic examples of perturbationtheory processes generating the pair tunneling term. Beloweach perturbation step, denoted as I, II, III and IV, the en-ergy of the intermediate state is indicated. In case of constantΩ / ,j , these two processes interfere destructively. Panel d) :Diagrams corresponding to the illustrations in panels b and c) ,where τ defined the time evolution of the perturbative steps,and dashed (thin) lines denote tunneling (Raman) processes. Coupled wires
Our goal is to derive an effective Hamiltonian for the fullplaquette where i) single occupancies in the intermediatesites are suppressed, and ii) single particle tunneling be-tween the wires is suppressed by energy constraints. Thefirst condition is fulfilled in the regime discussed in theprevious section; moreover, as single particle tunnelingbetween the wires is always driven by a single auxiliaryfield [7], the condition: | ∆ − (cid:126) ω | , | ∆ − (cid:126) ω | (cid:29) ˜ t a , ˜ t b , | (cid:126) Ω ,j/j +1 | , | (cid:126) Ω ,j/j +1 | (31)guarantees that inter-wire single particle tunneling are far-off resonant processes, and thus suppressed within pertur-bation theory (see Fig.8). FIG. 7. Schematic two-particle level scheme in a single plaque-tte (here, V a = V b > V a = V b < l, i run over( j, j + 1), indicating the multiplicity of the various interme-diate states (15 states are involved). The Raman detunings (cid:126) ω / are indicated by the green/red line, respectively, show-ing the two particle resonant condition in Eq. (27).In the perturbative regime | U | , (cid:126) ω , (cid:126) ω , ∆ , V a , V v (cid:29) ˜ t a , ˜ t b , | (cid:126) Ω ,j/j +1 | , | (cid:126) Ω ,j/j +1 | , after combining the resultsof the previous section, one obtains the following effectiveHamiltonian: H eff , = − t a ( a † j a j +1 + h.c.) − t b ( b † j b j +1 + h.c.)++ W a ( a † c b † c a j a j +1 + h.c.) + W b ( a † c b † c b j b j +1 + h.c.)++ δ P ( a † c a c b † c b c ) + ξ a,j a † j a j + ξ a,j +1 a † j +1 a j +1 ++ ξ b,j b † j b j + ξ b,j +1 b † j +1 b j +1 . (32)Here, the first line describes renormalized intrawire tun-neling; the second one contains terms coupling the a and b -wire to the intermediate sites, respectively. Finally, inanalogy with the previous discussion, the last two linesdescribe a residual interaction in the intermediate sitesdue to the quasi-resonant condition in Eq. (19), and lo-cal potential offsets. Beyond constituting the basis for therealization of the pair-tunneling Hamiltonian, Eq. (32) dis-plays interesting physics by itself, as its effective descrip-tion in terms of two Luttinger liquids coupled with a 1Dsuperconductor (as an effective description of the interme-diate wire composed by the center sites of each plaquette)has been also investigated in the context of emergent Ma-jorana edge states [3]. FIG. 8. Single particle level scheme, illustrating how the off-resonant condition between | a (cid:105) j and | b (cid:105) j can be engineeredby fulfilling Eq. (31). Inter-wire pair tunneling
Adiabatic elimination of the intermediate sites
The pair tunneling Hamiltonian discussed in the text isthen obtained by adiabatically eliminating the central sitesfrom Eq. (32) in the regime δU (cid:29) t, (cid:126) Ω, where the quasi-resonant condition in Eq. (19) is not met anymore. Theeffective Hamiltonian will then read: H eff , = − t a ( a † j a j +1 + h.c.) − t b ( b † j b j +1 + h.c.)++ W ( b † j b † j +1 a j a j +1 + h.c.)++ ξ a,j a † j a j + ξ a,j +1 a † j +1 a j +1 ++ ξ b,j b † j b j + ξ b,j +1 b † j +1 b j +1 + H imp , (33)where the second line describe the pair tunneling termbetween the wires (which, in perturbation theory, emergesas a sum of terms similar to the ones illustrated in Fig. 6),with coefficient: W = (cid:18) V a + 1 V b + (cid:126) ω (cid:19) (cid:18) (cid:126) J a J b δU (cid:19) ×× (cid:18)
1∆ + V a − (cid:126) ω + 1∆ + V b (cid:19) ×× (Ω ,j +1 − Ω ,j )(Ω ∗ ,j − Ω ∗ ,j +1 ) , (34)while H imp denotes additional imperfections (such asintra-wire and inter-wire interactions) emerging from ad-ditional fourth order processes. While all of them preserveparity symmetry, some of them may change the energysplitting between the two degenerate states in the topo-logical region discussed in the text from exponential topower law as a function of the system size. Additional terms
Let us now discuss the imperfections described by H imp . Intra-wire interactions.
These interactions are gener-ated via virtual processes, where, e.g., two particles onnearest-neighbor sites in the a -wire are transferred to theintermediate sites | a (cid:105) c and | b (cid:105) c via tunneling and Raman assisted process, respectively, and then move back to thesame two sites. The effective contribution in the Hamilto-nian reads: G a n a,j n a,j +1 + G b n b,j n b,j +1 (35)and the corresponding pre-factors in fourth-order pertur-bation theory are: G a = − (cid:18) V a + 1 (cid:126) ω + V b (cid:19) ( (cid:126) J a ) δU × (36) × ( | Ω ,j | + | Ω ,j +1 | − Ω ∗ ,j Ω ,j +1 − Ω ∗ ,j +1 Ω ,j ) , and G b = − (cid:18)
1∆ + V b + 1 − (cid:126) ω + V a + ∆ (cid:19) ( (cid:126) J b ) δU × (37) × ( | Ω ,j | + | Ω ,j +1 | − Ω ∗ ,j Ω ,j +1 − Ω ∗ ,j +1 Ω ,j ) . The effect of these terms is to renormalized the single wireLuttinger parameter, producing only some quantitativeshifts in the model Hamiltonian phase diagram. In case G b , G a <
0, the topological phase is expected to emergeat smaller values of W , as the value of K A becomes largerwhen an additional intra-wire attraction is introduced [2]. Inter-wire interactions.
Inter-wire terms emerge as wellin fourth order perturbation theory. They are induced byvirtual processes, where a particle hops from | a (cid:105) i to | a (cid:105) c and another one hops from | b (cid:105) l to | b (cid:105) c (or similar pro-cesses induced by the Raman couplings), and then both ofthem hop back to the original wires in | a (cid:105) k and | b (cid:105) r . Thecontribution in the effective Hamiltonian then reads: (cid:88) i,l,k,r = j,j +1 ( K i,l,k,r a † i a l b † k b r + h.c.) . (38)Here, the indices i, l, k, r belong to the same plaquette,that is, i, l, k, r ∈ { j, j + 1 } . The corresponding coefficientsare: K i,l,k,r = − (cid:34)(cid:18) V a + 1 V b + ∆ (cid:19) (cid:18) J a J b V b + V a + ∆ + U (cid:19) ++ (cid:18) V a − (cid:126) ω + ∆ + 1 V b + (cid:126) ω (cid:19) ×× (cid:32) (cid:126) Ω ,l Ω ,r Ω ∗ ,i Ω ∗ ,k V b + V a + ∆ + U + (cid:126) ω − (cid:126) ω (cid:33)(cid:35) . (39)While not breaking the parity symmetry, these terms maylift the splitting of the ground and first excited state fromexponential to algebraic as a function of the system size [3];as such, it’d be preferable to strongly reduce their effectsin order to guarantee observability of the topological fea-tures even in small size systems. In general, the rationbetween W and K i,l,k,r can be minimized by tuning theHamiltonian parameter in such a way that the sum of allprocesses involved in K i,l,k,r is annihilated via quantuminterference; this requires matching conditions involvingboth interactions, off-sets and detunings. A simpler, al-ternative way is to tune the potential offsets such that V a = − ∆ − V b , so that the first contribution is killed viainterference, and then take J (cid:39) W/K i,l,k,r will be of order (cid:39) / (4 ∗ (16)) = 1 /
64, thusnegligible with respect to other possible imperfections inthe system.
FIG. 9. Real time evolution for the parameters in the maintext and δU = 0 kHz , δV = 0 . kHz . a) Time evolution ofthe expectation values p a ( t ) = (cid:104) n a, ( t ) n a, ( t ) (cid:105) and p b ( t ) = (cid:104) n b, ( t ) n b, ( t ) (cid:105) that indicate the pair hopping. Single particlehopping indicated by S ( t ) = (cid:80) i,j =1 , (cid:104) n a,i ( t ) n b,j ( t ) (cid:105) is of theorder of 10 − . b) Since the detuning δ = 0, the occupation ofthe intermediate sites (cid:104) n a c (cid:105) , (cid:104) n b c (cid:105) , is relatively large large. Other contributions.
Within the effective Hamiltonian,the additional potential off-sets should have the same ef-fects as the intra-wire terms, and it can be shown thatthey strength can be tuned by slightly shifting the ini-tial potential off-set ∆ from exact degeneracy accordingto Eq. (27). Moreover, it can be shown (by employing afloquet formalism) that imperfect addressing of the indi-vidual sites by the Raman beams does not qualitativelymodify the effective Hamiltonian, as the quasi-degeneratesubspace within perturbation theory is not affected.
Numerical results
In the following we present quantitative results on modelI defined on a single plaquette. Starting from a state withtwo particles on the upper wire, | Ψ (cid:105) = a † a † | (cid:105) , we carryout a real time evolution under the Hamiltonian ˜ H . Basedon our numerics, we answer the following two questions:(1) Can we suppress single particle hopping between thetwo wires? (2) Do we get pair hopping? Further, we checkthe validity of our perturbation theory assumptions. First,we investigate the effect of giving up the space dependenceof the Rabi frequencies and take Ω j = Ω j +1 . We expectthat the pair hopping disappears. Second, we check theresonance condition for pair hopping, ∆ = (cid:126) ( ω + ω ),by comparing to a protocol where ∆ = (cid:126) ω , i.e. sin-gle particle hopping is resonant. For the numerical inves-tigation, we use the experimentally realistic parameters t a = t b = 10 Hz , J a = J b = | Ω j | = 1 kHz , V b = − kHz , (cid:126) ω = 20 kHz , (cid:126) ω = 12 kHz . The resonance conditionsimply that V a = − ∆ − V b + δV, (40) U = − (cid:126) ω − V a − V b + δU, (41)where we have defined δU and ∆ V to account for smalldeviations from the resonance conditions, as the are ex-pected in a realistic experimental setup. Suppression of single particle hopping
We carry out a numerical analysis that proves the sup-pression of single particle hopping in model I using theparameters defined above. As a figure of merit, we take FIG. 10. Real time evolution for the parameters in textand δU = 0 . kHz , δV = 0 . kHz . a) Time evolution ofthe expectation values p a ( t ) = (cid:104) n a, ( t ) n a, ( t ) (cid:105) and p b ( t ) = (cid:104) n b, ( t ) n b, ( t ) (cid:105) that indicate the pair hopping. Single parti-cle hopping indicated by S ( t ) = (cid:80) i,j =1 , (cid:104) n a,i ( t ) n b,j ( t ) (cid:105) is ofthe order of 10 − . b) Occupation of the intermediate sites, (cid:104) n a c (cid:105) , (cid:104) n b c (cid:105) as a function of time. The detuning δ (cid:54) = 0 leadsto relatively small occupation of the intermediate site.the expectation value S ( t ) = (cid:80) i,j =1 , (cid:104) n a,i ( t ) n b,j ( t ) (cid:105) thatindicates if we find, for some time t , one particle on theupper and the other particle on the lower wire. We find,that for δU and δV on the order of a few hundred Hertzthe expectation of S ( t ) is of the order of 10 − . Creation of pair hopping
Next, we present a numerical proof that we can engi-neer pair hopping in our setup. As a figure of merit,we take the expectations p a ( t ) = (cid:104) n a, ( t ) n a, ( t ) (cid:105) and p b ( t ) = (cid:104) n b, ( t ) n b, ( t ) (cid:105) indicating that the two particlesare both on the same wire. If pair hopping is possible, then p a ( t ) should decrease with time, while p b ( t ) increases. InFig. 9 we present results for δU = 0, δV = 0 . kHz . Wesee that the occupation of the intermediate site is rela-tively large. This problem can be overcome by introduc-ing a detuning δU (cid:54) = 0. As shown in Fig. 10, already asmall detuning of δU = 0 . kHz , δV = 0 . kHz signif-icantly reduces the occupation of the intermediate sites.Further, from the width of p a ( t ) and p b ( t ) we can deducethat W ∼ / kHz = 10 Hz is of the order of the hop-ping t a . Check of the perturbation theory
Now, we investigate the validity of our perturbative ap-proach. First, we give up the space dependence of the Rabifrequencies and take Ω j = Ω j +1 . We expect that the pairhopping disappears, and that there is no hopping betweenthe two wires. This assumption is confirmed by our nu-merical analysis: The expectation p a ( t ) = 1 for all times,while p b ( t ) = 0. Further, S ( t ) is of the order of 10 − .Finally, we consider the case where we tune the parametersin our model such that they are resonant to single particlehopping, i.e. we take ∆ = (cid:126) ω . We find that p b ( t ) = 0for all times in agreement with a vanishing pair hopping(see Fig. 11a)). The results for the single particle hopping,indicated by a non-vanishing values of S ( t ) are depictedin Fig. 11b). b)a) FIG. 11. Real time evolution for the parameters in the maintext and single particle resonance, ∆ = (cid:126) ω . a) Time evo-lution of the expectation values p a ( t ) = (cid:104) n a, ( t ) n a, ( t ) (cid:105) and p b ( t ) = (cid:104) n b, ( t ) n b, ( t ) (cid:105) that indicate a vanishing pair hopping.b) Single particle hopping indicated by S ( t ) is large. FINAL REMARKS
Let us conclude with some final remarks on the realiza-tion of the pair hopping Hamiltonian. As discussed in theprevious sections, pair tunnelings of order
W/t (cid:39) t/h (cid:39) − (cid:39) π × W a , W b may be much larger that W , as they arederived in second order perturbation theory. The numeri-cal results confirm this enhancement (in the specific case,by a factor of (cid:39) a and b -wire are couple to an additional 1D s -wave su-perfluid with a large spin gap (described by the dynamicsin the intermediate sites).Minimal instances of the present scheme can be validatedon a single plaquette, by measuring the relative parity ofa certain state under the evolution of the effective Hamil-tonian by using band-mapping or in-situ imaging tech-niques [9]. Moreover, a generalization to multi-leg laddersand 2D setups is indeed conceivable under the same as-sumptions described in the previous sections.[1] A.O. Gogolin, A.A. Nersesyan, A.M. Tsvelik, Bosonization and strongly correlated systems , (Cam-bridge University press, Cambridge, 1998).[2] T. Giamarchi,
Quantum Physics in one dimension ,(Oxford University press, Oxford, 2003).[3] L. Fidkowski, R. M. Lutchyn, C. Nayak and M. P. A.Fisher, Phys. Rev. B , 195436 (2011).[4] M. Cheng and H.-H. Tu, Phys. Rev. B , 094503(2011).[5] J. von Delft and H. Schoeller, Annalen Phys. , 225(1998). [6] A. Messiah, Quantum Mechanics , Dover Publication,1999 (Mineola, New York).[7] Starting from, e.g., | a j (cid:105) , the particle can either: I) tun-nel to | a c (cid:105) , and then emit a photon (cid:126) ω be transferredto | b j (cid:105) , or II) emit a photon (cid:126) ω be transferred to | b c (cid:105) ,and then tunnel to | b j (cid:105) . Higher order processes can be decomposed as a product of the two.[8] S. Sugawa, K. Inaba, S. Taie, R. Yamazaki, M. Ya-mashita, and Y. Takahashi, Nat. Phys. , 642 (2011).[9] M. Aidelsburger et al. , Phys. Rev. Lett.107