Topological bands in the continuum using Rydberg states
TTopological bands in the continuum using Rydberg states
Sebastian Weber, ∗ Przemyslaw Bienias, ∗ and Hans Peter B¨uchler Institute for Theoretical Physics III and Center for Integrated Quantum Science and Technology,University of Stuttgart, 70550 Stuttgart, Germany Joint Quantum Institute and Joint Center for Quantum Information and Computer Science,NIST/University of Maryland, College Park, MD, 20742, USA (Dated: January 22, 2021)The quest to realize topological band structures in artificial matter is strongly focused on latticesystems, and only quantum Hall physics is known to appear naturally also in the continuum. Inthis letter, we present a proposal based on a two-dimensional cloud of atoms dressed to Rydbergstates, where excitations propagate by dipolar exchange interaction, while the Rydberg blockadephenomenon naturally gives rise to a characteristic length scale, suppressing the hopping on shortdistances. Then, the system becomes independent of the atoms’ spatial arrangement and can bedescribed by a continuum model. We demonstrate the appearance of a topological band structurein the continuum characterized by a Chern number C = 2 and show that edge states appear atinterfaces tunable by the atomic density. PACS numbers: 67.85.-d, 05.30.Jp, 73.43.Cd
Band structures characterized by topological invari-ants are at the heart of a plethora of phenomena robustto local perturbations because the topological invariantscan only change in case of band closings [1, 2]. In fi-nite systems or at interfaces, topologically protected edgestates emerge, whose robustness makes them promisingfor coherent transport of quantum information [3–5]. Incombination with strong interactions, topological bandsare candidates for realizing topologically ordered statesexhibiting anyonic excitations [6]. Remarkably, such phe-nomena occur in both continuum and lattice models.Especially, the renowned integer [7] and fractional [8]quantum Hall effect arise naturally in continuous two-dimensional electron systems subject to strong magneticfields. In contrast, the efforts to realize a wide varietyof topological band structures and topological states ofmatter mainly focus on lattice models. In this letter,we bridge this gap and demonstrate a topological bandstructure characterized by a Chern number C = 2 in thecontinuum based on the intrinsic spin-orbit coupling withdipolar exchange interaction.The quest to realize topological band structure in ar-tificial matter was pioneered by the theoretical proposal[9] to realize the Hofstadter butterfly [10, 11] with coldatomic gases in optical lattices. Since then, severalpromising experimental platforms have emerged such asclassical phononic setups [12, 13], and photonic nanos-tructures [14–16], solid-state devices [17–19], and coldatomic gases [20–25]. While all these platforms workon lattices, fewer experiments pursue the challenge toachieve topological states in the continuum, relying onimplementing the physics of magnetic fields either byartificial magnetic fields [26–28] of fast rotating traps[29]. Rydberg atoms and polar molecules have recentlyemerged as a complementary platform to generate topo-logical band structures, using the dipolar exchange in- teraction [30–32]. A fundamental advantage of this ap-proach is that the time-reversal symmetry is broken by ahomogeneous magnetic field [31], therefore avoiding po-tentially problematic heating present in approaches basedon Floquet engineering and experimentally challengingstrong spatially-inhomogeneous light fields. The under-lying principle has recently been experimentally verifiedusing Rydberg atoms [33] and the potential of Rydbergplatforms to realize topological states of matter has beendemonstrated by observing a symmetry-protected topo-logical phase [34]. xyz (a)(b) (c) | | + | µ | | + | R/s R . . . . . . . we iφ R we iφ R f ( R ) s f ( R ) t α t α FIG. 1. Setup. (a) Two-dimensional homogeneous cloud ofatoms with quantization axis z . (b) Each atom can be eitherin the ground state | (cid:105) with the magnetic quantum number m or a dipole-coupled excited state | + (cid:105) or |−(cid:105) , having m ± = m ± µ = E + − E − . We consider the case ofa single excitation in the cloud of atoms. The excitation canhop from one atom to another by dipolar exchange interactionthat keeps the excitation’s internal degree of freedom constantor changes it. In the latter case, the excitation picks up anon-trivial phase (spin-orbit coupling). (c) We design theinteraction potentials to have the distance dependence givenby the function f ( R ) with short-distance cut-off s , chosen tobe much larger than the average distance between the atoms. a r X i v : . [ c ond - m a t . qu a n t - g a s ] J a n Here, we show that the intrinsic spin-orbit coupling bythe dipolar exchange interaction of Rydberg atoms canlead to topological bands in the continuum. The basicsetup consists of cold atoms in the frozen regime, dressedto Rydberg states; a system studied extensively in thepast [35–47]. The combination of two ideas underliesour proposal: (i) The possibility to engineer non-trivialtopological band structures using the intrinsic spin-orbitcoupling of dipolar exchange interaction [31, 33], and (ii)the Rydberg blockade phenomenon, which suppresses thepresence of two Rydberg excitation on short distances[48], and introduces a natural length scale by the block-ade radius, see Fig. 1. Then, the admixture of Rydbergstates by the dressing lasers allows for the hopping of ex-citations by the dipolar exchange interaction. Still, thishopping becomes quenched on short distances due to theRydberg blockade phenomenon. Remarkably, the shortdistance cut-off by the Rydberg blockade renders the sys-tem independent of the precise atomic distribution of theatoms. We find that the emerging band structure in thetopological regime is characterized by the Chern number C = 2, and the system exhibits edge states at boundariesor interfaces, which can be tuned by the atomic density.We start with the description of our systems. A cloudof atoms is confined to two dimensions with the quan-tization axis and a magnetic field perpendicular to theplane, see Fig. 1. Each atom possesses a V-level struc-ture formed by the ground state | (cid:105) with magnetic quan-tum number m and excited states | + (cid:105) and |−(cid:105) . Theexcited states differ by their magnetic quantum number m ± = m ±
1, constituting an internal degree of freedom.The states of the V-level structure are dressed with Ry-dberg states, giving rise to an effective dipolar exchangeinteraction, leading to a single excitation hopping at largeinteratomic distances R in analogy to studies on a lattice[30–32]. However, in our case, the conventional Rydbergblockade suppresses the Rydberg states’ admixture ondistances shorter than a characteristic length scale s . Ittherefore quenches the dipolar exchange interaction andthe hopping of excitations. In the following, we focuson the dynamics of such a single excitation. If the dis-tance between the atoms in the cloud is much smallerthan the cut-off distance, i.e. ns (cid:29) n of atoms in the ground state, we canexpress the excitations in terms of field operators α † ( r )and β † ( r ) ( α ( r ) and β ( r )), which create (annihilate) a | + (cid:105) and |−(cid:105) excitation at the position r , respectively. Then,the Hamiltonian describing the dynamics of excitationsunder the dipolar exchange interaction takes the form H = n (cid:90) (cid:90) d r d r (cid:48) ψ † ( r ) (cid:18) − T α ( R ) W ( R ) e − iφ R W ( R ) e iφ R − T β ( R ) (cid:19) ψ ( r (cid:48) )+ (cid:90) d r ψ † ( r ) (cid:18) µ/ − µ/ (cid:19) ψ ( r ) . (1)Here, ψ † ( r ) = (cid:0) α † ( r ) , β † ( r ) (cid:1) is the spinor field operator, . . . . . − − (a) (b) (c)(d) −
101 0 3 6 −
101 0 3 6 − k / t k / t k / t µ / t | k | s | k | s | k | s C h e r nnu m b e r bandclosing a/s FIG. 2. (a-c) Isotropic band structure of the continuummodel for different µ/t calculated for an exemplary value of w/t = 3 and t = 0. Between the band closings at µ/t = − µ/t = 0, the Chern number is C = 2. (d) The phasediagram of the dicretized model on a square lattice approachesthe continuum model (dashed lines) when the lattice constant a gets smaller than the cut-off distance s . while R = r (cid:48) − r is the relative distance and R = | r (cid:48) − r | .The last term takes into account the energy difference be-tween the two excitations µ = E + − E − , tunable by thestrength of the magnetic field applied along the quantiza-tion axis. The first term in Eq. (1) describes two differenthopping processes: First, the | + (cid:105) and |−(cid:105) excitations canhop mediated by the amplitudes T α,β ( R ) while conserv-ing the internal angular momentum. Second, the exci-tations can change their internal angular momentum bya hopping process with amplitude W ( R ), i.e., a |±(cid:105) ex-citation becomes a |∓(cid:105) . The phase factor accompaniesthe change of internal angular momentum of the exci-tation e − iφ R , where φ R is the polar angle of R . Notethat for an excitation hopping on a closed loop, the totalcollected phase is independent of the chosen coordinatesystem and cannot be gauged away. This phase accountsfor a spin-orbit coupling and guarantees that the totalangular momentum is conserved, i.e., the change in in-ternal angular momentum of the excitation is transferredinto orbital angular momentum.The effective hopping Hamiltonian (1) gives rise to aband structure with two bands. In the following, we fo-cus on characterizing the topological properties of thisband structure. A discussion on optimal parameters foran experimental realization and the detailed microscopicinteraction potentials is presented at the end. As theprecise shape of the hopping amplitudes is not essentialfor the topological properties of the band structure, weassume that each hopping amplitude obeys the same dis-tance dependency given by the cut-off function f ( R ) = 9 s π R ( R + s ) . (2)with the characteristic cut-off distance s . The nor-malization is chosen such that (cid:82) d R f ( R ) = 1. Wewrite T α ( R ) n = t α f ( R ) with the interaction energy t α = (cid:82) d R T α ( R ) n . Analogously, we introduce the in-teraction energies t β and w . By Fourier transforming thefield operators as ψ † k = (cid:82) d r e i kr ψ † ( r ) / √ V with quantiza-tion volume V , we obtain the Hamiltonian in momentumspace H = (cid:88) k ψ † k (cid:0) t(cid:15) k + n ( k ) · σ (cid:1) ψ k , (3)Here, σ is the vector of Pauli matrices, and we have in-troduced the vector n ( k ) = w (cid:0) (cid:15) ( k ) + (cid:15) − ( k ) (cid:1) / w (cid:0) (cid:15) ( k ) − (cid:15) − ( k ) (cid:1) / iµ/ t(cid:15) k (4)with the average hopping strength t = ( t β + t α ) / t = ( t β − t α ) /
2. Furthermore, we have made use of theFourier transformation for the hopping amplitudes (cid:15) m k = (cid:90) d r e i kr + imφ r f ( | r | ) = 2 π ( − m e i mφ k F m ( | k | ) , (5)where F m is the Hankel transform of order m of the cut-off function (2). In our continuum model, momenta canhave arbitrary high values in contrast to lattice models.However, interesting physics happen only at momentacomparable to the characteristic momentum 2 π/s definedby the cut-off distance s . As the Hamiltonian is block-diagonal in momentum space with block-size 2 ×
2, theband structure consists of two bands whose Chern num-ber can be calculated as a winding number of the three-dimensional unit vector ˆ n ( k ) = n ( k ) / | n ( k ) | [30]. Note,that ˆ n ( k ) is pointing into the same direction ˆ z sign( µ )for all | k | → ∞ , and therefore we can compactify the in-finite Brillouin zone to the sphere S . We obtain for aninfinite quantization volume the Chern number for thelower band as C = 14 π (cid:90) d k (cid:0) ∂ k x ˆ n ( k ) × ∂ k y ˆ n ( k ) (cid:1) · ˆ n ( k ) . (6)For − < µ/t <
0, the system is in its topological phasewith Chern number C = 2, see Fig. 2(b). This condi-tion implies that µ (cid:54) = 0 and t (cid:54) = 0. Outside this pa-rameter regime, the system is in the topologically trivialphase. At the phase boundaries, the bandgap closes, seeFig. 2(a,c). Note that these findings are independent ofthe value of t and w as long as w (cid:54) = 0. However, thebandstructure has a quantitative dependence on theseparameters. Finally, we point out that µ (cid:54) = 0 or t (cid:54) = 0give rise to a broken time-reversal symmetry as requiredfor the appearance of a non-zero Chern number [49].An interesting aspect is to connect the continuummodel to a lattice model by discretizing the Hamilto-nian on a square lattice with a lattice constant a . This − − . − . . . . k x / t k x s P r o b a b ili t y d/s (a) (b) − − FIG. 3. (a) For a semi-infinite cloud of atoms, the bandstructure in the topological sector hosts two edge states perboundary, bridging the bandgap. The two edge states coloredin red (blue) are localized at the cloud’s upper (lower) bound-ary. The numerical analysis is based on the discretized modelwith lattice spacing a/s = 1 / l = 20 s (b) Theprobability of finding an excitation in an edge state decreasesexponentially with the distance d/s from the edge, as shownexemplarily for the mode marked with the red arrow. discretized model is suitable to study numerically theappearance of edge states, and for large lattice spacing a (cid:29) s , we recover the models studied previously [30, 31].We introduce the operators a † i and b † i ( a i and b i ), whichcreate (annihilate) a | + (cid:105) or |−(cid:105) excitation at the latticesite i , respectively. With R ij = r j − r i being the distancevector pointing from site i to j , the Hamiltonian reads H sq = (cid:88) i,j a f ( | R ij | ) (cid:18) a i b i (cid:19) † (cid:18) − t α we − iφ R ij we iφ R ij − t β (cid:19) (cid:18) a j b j (cid:19) + (cid:88) i (cid:18) a i b i (cid:19) † (cid:18) µ/ − µ/ (cid:19) (cid:18) a i b i (cid:19) . (7)It approaches the Hamiltonian of the continuum model(1) in the limit a/s →
0. In momentum space, theperiodic Brillouin zone of the square lattice becomesthe infinite Brillouin zone of the continuum model, andthe discrete Fourier transform of the interaction (cid:15) m sq , k = (cid:80) i a e i kr i + imφ r i f ( | r i | ) becomes the continuous Fouriertransform (5). The convergence is also visible in thephase diagram Fig. 2(d), where we calculated the Chernnumber for different values of µ/t as a function of a/s [50]. Already for a lattice constant a , which is half of thecut-off distance s , we obtain the phase boundaries of thecontinuum model in good approximation. For such a lat-tice spacing, the nearest neighbor interaction is stronglysuppressed due to the fast decay of the cut-off function(2) on short distances. Interestingly, for a/s ≈ .
8, thereis an additional phase with Chern number C = 4.The discretized version of the continuum model allowsus to numerically calculate the band structure in a semi-infinite cloud of atoms, which is infinite in the x -directionand finite in the y -direction, see Fig. 3. For the calcula-tion, we have chosen a small lattice constant comparedto the cut-off distance, a/s = 1 /
6, to ensure convergencetowards the continuum model. According to the bulk-boundary correspondence [51], the system hosts two edge (a) (b)(c)
Species A Species B | A | + A F = 23 P / S / n sA S / n pA P / |− B | B n sB S / n pB P / F = 24 S / P / F = 1 F = 1 F = 2 R ( µm ) R ( µm ) V ( π k H z ) V ( π k H z ) + A B | V | + A B A − B | V | A − B A + A | V | A + A B − B | V | B − B A A | V | A A B B | V | B B A B | V | A B F = 2 W = + A B | V | A − B T α = 0 A + A | V | + A A T β = 0 B − B | V | − B B δ A δ B δ B δ A Ω A Ω A Ω B Ω B Ω B Ω B Ω A Ω A Ω A Ω A Ω B Ω B FIG. 4. Experimental realization of the interaction potentials.(a) The hyperfine states | A (cid:105) and | + A (cid:105) of species A ( Na)are dressed by Rydberg states, likewise | B (cid:105) and |− B (cid:105) ofspecies B ( K). The arrows visualize applied laser fieldswith detunings δ and Rabi frequencies 2Ω. Dashed arrowssymbolize EIT dressing, while solid arrows depict single-photon dressing. A static magnetic field B z perpendicularto the cloud of atoms isolates the Rydberg states and bringsthe Rydberg pair states, with which the states | A − B (cid:105) and | + A B (cid:105) are dressed, energetically close. (b,c) Effective ex-change (b) and static (c) interaction between different pairsof the dressed hyperfine states for the following parameters: n sA = 61, n pA = 60, n sB = 61, n pB = 61, Ω A,B / (2 π ) = 7 MHz,Ω A,B / (2 π ) = 35 MHz, Ω (cid:48) A,B / (2 π ) = 7 MHz, δ A,B / (2 π ) =50 MHz, δ (cid:48) A,B / (2 π ) = 130 MHz, B z = −
223 G. states on each of the two boundaries of the semi-infinitecloud in the topological sector. We have chosen the semi-infinite cloud’s width l/s = 20, and indeed find two local-ized edge states within the band gap on each edge. Wefind that the edge states are exponentially localized at theedge for the studied system sizes; due to the slow decayof the dipolar exchange interaction, one can not excludea transition to a power-law behavior at larger distances[52, 53]. Note that within the continuum model, the hop-ping strength t ∼ n depends linearly on the atomic den-sity. Therefore, the density in an inhomogeneous cloudcan exhibit edge states as the ratio µ/t ∝ /n becomesposition-dependent with a topologically trivial phase inareas with low density and a topological phase in high-density regions.Next, we discuss possible experimental realizations ofthe continuum model using Rydberg dressing. We mustuse Rydberg states for the dressing, which are energeti-cally isolated from other Rydberg states, especially thosewithin the same Zeeman manifold. We apply a strongstatic magnetic field perpendicular to the atomic cloudto achieve the isolation, also for interatomic distancessmaller than the cut-off distance. On the other hand,the Rydberg states used for dressing the states | (cid:105) and |− (cid:105) should be of similar energy to obtain a strong, ef-fective interaction W ( R ). In principle, a static electricfield can compensate for the Zeeman splitting and shiftthe Rydberg pair states close to resonance, as proposedin Ref. [31]. However, if the Zeeman splitting is too large,this cannot be achieved without the electric field heav-ily mixing the Rydberg states. Thus, we propose a dif-ferent approach. We suggest to realize the model witha homogeneous atomic cloud composed of two differentspecies rather than a single species as in Ref. [31], op-tically pumped into the hyperfine states | A (cid:105) and | B (cid:105) of species A and B. The excitations | + (cid:105) and |−(cid:105) cor-respond to the hyperfine states | + A (cid:105) and |− B (cid:105) , respec-tively. Rydberg states weekly dress these four hyper-fine states, and we use Na as species A and K asspecies B. We dress | A (cid:105) with | s A (cid:105) = | n sA S / , m j = 1 / (cid:105) , | + A (cid:105) with | p A (cid:105) = | n pA P / , m j = 3 / (cid:105) , | B (cid:105) with | s B (cid:105) = | n sB S / , m j = − / (cid:105) , and |− B (cid:105) with | p B (cid:105) = | n pB P / , m j = − / (cid:105) . For details on the dressing andsuitable parameters, see Fig. 4. The magnetic field ap-plied to isolate the states is tuned, such that | s A p B (cid:105) and | p A s B (cid:105) are energetically close. We calculate the effectiveinteraction by restricting the two-atom Hamiltonian tothe manifold of the chosen hyperfine states within per-turbation theory via the Schrieffer-Wolff transformation[54–56], generalizing the numerical method in Ref. [36].The hopping amplitudes are shown in Fig. 4(b) for op-timal parameters. Furthermore, we show in the supple-ment [57], that the cut-off function in Eq. (2) can be ob-tained analytically in fourth-order perturbation theoryin case of energetically well separated Rydberg states.In addition to the effective dipolar exchange interaction(2), such an experimental realization also yields staticinteractions, see Fig. 4(c). The latter leads to a density-dependent shift of the chemical potential µ . For a homo-geneous system, this shift is everywhere in the bulk thesame. By tuning the frequencies of the dressing lasersor the relative energy using the magnetic field, we cancompensate for it and ensure that we are in the topolog-ical regime. At short interatomic distances, the Rydberginteraction unavoidably shifts some Rydberg states intoresonance with the dressing lasers. However, the result-ing divergencies in the potential curves are very sharp asthey appear on short interatomic distances so that theymight be ignored, particularly in the presence of dissipa-tion (leading effectively to the resonance suppression), orcould be avoided altogether by placing the atoms in anoptical lattice with carefully chosen lattice spacing [58].Besides, the level structure might be tuned further byapplying an electric field.In summary, we demonstrated that the combination ofbroken time-reversal symmetry and spin-orbit coupling inthe spatial continuum can give rise to topological bands.Our continuum model features topological bands withChern number C = 2 and exponentially localized edgestates in a semi-infinite system. We proposed to usea two-dimensional system of Rydberg-dressed atoms forexperimental realization. The dressing gives rise to aneffective interaction that vanishes on short interatomicdistances, allowing us to neglect the atoms’ spatial ar-rangement and treat it as a continuum. Besides thepractical benefit, the continuum model is homogeneous,similarly to the quantum Hall effect. Our work leadsto the open question of whether topologically non-trivialmany-body states could be realized — complementary tolattice-based fractional Chern insulators — if we combineour continuum model with strong interactions. Becauseof the hard-core constraint of excitations and inevitablevan der Waals interactions, strong interactions are natu-rally present in experimental realizations.We thank S. Hofferberth and K. Jachymski for in-teresting discussions. This work received funding fromthe European Union’s Horizon 2020 research and innova-tion program under the ERC consolidator grant SIRPOL(Grant No. 681208) as well as the French-German collab-oration for joint projects in NLE Sciences funded by theDeutsche Forschungsgemeinschaft (DFG) and the AgenceNational de la Recherche (ANR, project RYBOTIN).P.B., acknowledge support by ARL CDQI, AFOSR, AROMURI, DoE ASCR Quantum Testbed Pathfinder pro-gram (award No. DE-SC0019040), U.S. Department ofEnergy Award No. DE-SC0019449, DoE ASCR Acceler-ated Research in Quantum Computing program (awardNo. DE-SC0020312), NSF PFCQC program, and NSFPFC at JQI. ∗ These authors contributed equally.[1] M. Hasan and C. Kane, Colloquium: Topological insula-tors, Rev. Mod. Phys. , 3045 (2010).[2] X.-L. Qi and S.-C. Zhang, Topological insulators and su-perconductors, Rev. Mod. Phys. , 1057 (2011).[3] B. 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I. SIMPLIFIED MODEL ILLUSTRATING THE CUT-OFF POTENTIAL
To illustrate the interplay ofblockade phenomena and dipolar exchange interaction, we present a simplified toymodel. We consider two ground states | g (cid:105) , | e − (cid:105) dressed with two Rydberg states | s (cid:105) , | p − (cid:105) . staticstaticexchangeexchange-- FIG. S1. (a) The setup: Two identical atoms i and j are dressed with Rydberg states | p − (cid:105) and | s (cid:105) .(b) Effective interaction infourth order between dressed ground states | ˜ g (cid:105) and | ˜ e − (cid:105) in the units characteristic for this problem (details in the text). The single atom Hamiltonian H A and the laser drive Hamiltonian ˆ h read H ( i ) = δ s | s (cid:105)(cid:104) s | i + δ − | p − (cid:105)(cid:104) p − | i , ˆ h ( i ) = Ω s | g (cid:105)(cid:104) s | i + Ω − | e − (cid:105)(cid:104) p − | i + H.c. (S1)Dipolar interaction between Rydberg states is given byˆ V ( i,j ) = W ij e − iφ ij | s (cid:105)(cid:104) p − | i | p − (cid:105)(cid:104) s | j + V ij | s (cid:105)(cid:104) s | i | p − (cid:105)(cid:104) p − | j + h.c. , (S2)where W ij = C /R , V ij = C /R , R = | r i − r j | . The total Hamiltonian reads H = (cid:88) i H ( i ) + ˆ h ( i ) + (cid:88) i>j V ( i,j ) (S3)We are interested in the interaction between weakly dressed ground states, which are denoted by | ˜ g (cid:105) and | ˜ e − (cid:105) .In thelimit of weak laser fields, we obtain effective interactions between ground states by treating the laser couplings ˆ h asa perturbation [41, 59]. The static and exchange interactions between dressed ground states take the form˜ V ij = ω c Ω s δ s Ω − δ − V ij ( ω c + V ij ) − W ij ( V ij + ω c ) − W ij , (S4)˜ W ij = ω c Ω s δ s Ω − δ − W ij ω c ( V ij + ω c ) − W ij , (S5)where ω c = δ − + δ s . We express the interactions using ξ = ( C /ω c ) / , κ = C ξ /C and x = R/ξ , giving rise to˜ V ij ( R ) = − ω c Ω s δ s Ω − δ − (cid:0) κ − (cid:1) x − x + 1) − κ x , (S6)˜ W ij ( R ) = ω c Ω s δ s Ω − δ − κx ( x + 1) − κ x . (S7)Intuitively, the off-diagonal interactions ˜ W ij can be thought of asmicrowave-assisted multi-photon processes leadingto the hopping of the excitations — which for the toy model involves four optical photons and an exchange of twomicrowave photons.We see that in order to have resonance-free effective potentials, ω c < κ should be smallenough so that for any separations (cid:0) x + 1 (cid:1) − κ x >
0, and hence κ <
2. The exchange potential height ismax[ ˜ W ij ] ≈ κω c Ω s Ω − / ( δ − δ s ) (S8)for κ (cid:28) κκ