Generating and detecting topological phases with higher Chern number
GGenerating and detecting topological phases with higher Chern number
Abhijeet Alase ∗ and David L. Feder Institute for Quantum Science and Technology, and Department of Physics and Astronomy,University of Calgary, Calgary, Alberta, Canada T2N 1N4
Topological phases with broken time-reversal symmetry and Chern number |C| ≥ |C| = 2 , |C| = 2 for ultracold atomic gases on a triangular lattice subject to spin-orbit coupling. The Chernnumber can be directly measured using Zeeman spectroscopy; for fermions the spin amplitudes canbe measured directly via time of flight, while for bosons this is preceded by a short Bloch oscillation.Our results provide a pathway to the realization and detection of novel topological phases withhigher Chern number in ultracold atomic gases. I. INTRODUCTION
Since the breakthrough provided by the quantum Halleffect, the systems that display topological features inthe absence of an external magnetic field have havebeen at the center stage in research in condensed matterphysics. The landmark discovery of topological insula-tors as well as quantum anomalous Hall systems estab-lished spin-orbit (SO) coupling as the dominant mecha-nism behind topological band inversion [1]. In contrast totime-reversal invariant topological systems including 2Dtopological insulators, protected phases characterized byhigher Chern numbers |C| ≥ |C| ≥ C = 2 phase requires d ∗ Corresponding author: [email protected] orbitals [25] or six bands [19]. Identification of solid-statematerial candidates with |C| ≥ C = ± a r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b of the proposed methods can distinguish a topologicalphase with Chern number C = ± C = 0. Moreover, it appears that topological phases withChern number C = ± |C| ≥ |C| = 2.The first result is the derivation of a relation betweenthe Chern number (mod 2 n , n ∈ Z ) and the spin-polarization at the high-symmetry points in the Brillouinzone (BZ) of a SO coupled Hamiltonian supported ona Bravais lattice with 2 n -fold rotation symmetry C n .The key insight is a relation between the Chern numberand the Berry-Zak phase of a closed loop that connectshigh-symmetry points of the BZ, Eq. (14). The relationcan be extended to systems with any number of inter-nal states, and to other non-Bravais lattices (i.e. Bravaislattices with attached basis) in a straightforward way.The second result leverages the Chern-spin polariza-tion relation to construct, among others, a minimal two-band nn tight-binding Hamiltonian on a triangular lat-tice that exhibits a topological phase with C = ± |C| ≥ |C| = 2. The results are summarized in Section IV). II. SYMMETRY PRINCIPLE YIELDINGHIGHER CHERN NUMBERA. Spin-Chern polarization relation
In this section, we derive an important relationshipbetween the Chern number and the spin polarization athigh-symmetry points in the Brillouin zone. Assume thatthe system is described by a gapped 2 × H ( k ), with symmetry property H ( S k ) = U H ( k ) U † , U := e − iπmσ z / n , (1)where σ z is the Pauli- z matrix, and m ∈ { , , } . Thesymmetry group is generated by counter-clockwise ro-tations around the z -axis by angle π/n ; we denote therotation in the real space by R := R z ( π/n ) and in thereciprocal space by S := R z ( π/n ). The symmetry condi-tion for m = 1 is satisfied by Rashba SO coupling, whichfor small | k | has the form k × (cid:126)σ · ˆ z . The symmetry for m = 2 ( m = 3) requires that the spin wavefunction ofany one of the bands on the Bloch sphere rotates twice(thrice) as rapidly as the momentum vectors in the xy -plane of the BZ.The Chern number C of the lower band wavefunction | u ( k ) (cid:105) of the Hamiltonian H ( k ) can be related to thespin wavefunctions at the high-symmetry momenta Γ and M in two steps using an approach similar to that dis-cussed in Ref. [65]. For any integer (cid:96) , Eq. (1) dictatesthat the wavefunctions at crystal momenta S (cid:96) k and k satisfy | u ( S (cid:96) k ) (cid:105) = e iθ (cid:96) ( k ) U | u ( k ) (cid:105) for some real function θ (cid:96) on the BZ. Then A i ( S (cid:96) k ) = (cid:104) u ( q ) | ∂ q i | u ( q ) (cid:105) (cid:12)(cid:12) q = S (cid:96) k = (cid:104) u ( S (cid:96) k ) | (cid:88) j S − (cid:96)ji ∂ k j | u ( S (cid:96) k ) (cid:105) (cid:12)(cid:12) k = k = (cid:104) u ( k ) | ( U † ) (cid:96) e − iθ (cid:96) ( k ) (cid:88) j S − (cid:96)ji ∂ k j e iθ (cid:96) ( k ) U (cid:96) | u ( k ) (cid:105) = (cid:88) j S (cid:96)ij (cid:104) u ( k ) | ( U † ) (cid:96) e − iθ (cid:96) ( k ) ∂ k j e iθ (cid:96) ( k ) U (cid:96) | u ( k ) (cid:105) = (cid:88) j S (cid:96)ij (cid:0) A j ( k ) + i∂ k j θ (cid:96) ( k ) (cid:1) . (2)where ∂ k i ≡ ∂/∂k i is defined for brevity. Therefore, (cid:126) A ( S (cid:96) k ) = S (cid:96) (cid:16) (cid:126) A ( k ) + i(cid:126) ∇ k θ (cid:96) ( k ) (cid:17) . (3)Because the unitary operator U is independent of k , thegauge-independence of the Berry curvature B ( k ) := (cid:126) ∇ k × (cid:126) A ( k ) · ˆ z leads to B ( S (cid:96) k ) = (cid:126) ∇ k × (cid:126) A ( S (cid:96) k ) · ˆ z = (cid:126) ∇ k × S (cid:96) (cid:16) (cid:126) A ( k ) + i(cid:126) ∇ k θ (cid:96) ( k ) (cid:17) · ˆ z = (cid:126) ∇ k × (cid:16) (cid:126) A ( k ) + i(cid:126) ∇ k θ (cid:96) ( k ) (cid:17) · ˆ z = (cid:126) ∇ k × (cid:126) A ( k ) · ˆ z = B ( k ) , (4) Γ X M MX Γ XX M M FIG. 1. (Color online) Loop γ in the BZ of a two-dimensionalBravais lattice with C (left) and C (right) symmetry. Thearea A ( γ ) enclosed by γ is shaded in each case. where we have made use of the fact that S (cid:96) is an orthog-onal transformation, and that the curl of a gradient iszero. The result is B ( S k ) = B ( k ), where ˆ z denotes theunit vector in the z -direction. The loop encloses an area A ( γ ), which covers exactly 1 / n of the total BZ and sat-isfies (cid:80) nj =1 S j A ( γ ) = BZ. By invoking Stokes’ theorem,one obtains C = 2 n πi (cid:90) A ( γ ) B ( k )d k = nπi (cid:90) γ (cid:126) A ( k ) · d k (mod 2 n ) , (5)where (mod 2 n ) accounts for the definition of the Berry-Zak phase acquired over a closed loop modulo 2 π .In the next step, the Berry phase acquired over theloop γ is related to the spin wavefunction at the pointsΓ , M and X , as shown in Fig. 1. The loop γ may bedecomposed into four parts, γ = ( X → X (cid:48) ) ∪ ( X (cid:48) → M (cid:48) ) ∪ ( M (cid:48) → Γ → M ) ∪ ( M → X ) . (6)The integral of the Berry connection over the second andthe fourth segments on the right hand side of Eq. (6)cancel each other, as the two segments correspond to thesame paths in the Brillouin zone traversed in opposingdirections. It is therefore only necessary to calculate thephases acquired over X → X (cid:48) and M (cid:48) → Γ → M .Recall that on the segment M (cid:48) → Γ → M , the wave-functions satisfy | u ( S n − k ) (cid:105) = e iθ ( k ) U n − | u ( k ) (cid:105) , where θ ( k ) := θ n − ( k ) is used for brevity. Because M and M (cid:48) represent the same points in the BZ, | u ( M ) (cid:105) ∝ | u ( M (cid:48) ) (cid:105) ,so that θ ( K ) = − Arg (cid:104) u ( K ) | U n − | u ( K ) (cid:105) , K ∈ { Γ , M (cid:48) } . (7)For (cid:96) = n −
1, Eq. (3) reduces to (cid:126) A ( S n − k ) = S n − (cid:16) (cid:126) A ( k ) + i(cid:126) ∇ k θ ( k ) (cid:17) . (8)This relation can be used to compute the Berry phasealong the path M (cid:48) → Γ → M , which may be expressedas (cid:90) M (cid:48) → Γ → M (cid:126) A ( k ) · d k = (cid:90) Γ → M (cid:126) A ( k ) · d k − (cid:90) Γ → M (cid:48) (cid:126) A ( k ) · d k (9) Using the symmetry properties, the first term on the righthand side of Eq. (9) can be expressed as (cid:90) Γ → M (cid:126) A ( q ) · d q = (cid:90) Γ → M (cid:48) (cid:126) A ( S n − k ) · S n − d k = (cid:90) Γ → M (cid:48) S n − ( (cid:126) A ( k ) + i(cid:126) ∇ k θ ( k )) · S n − d k = (cid:90) Γ → M (cid:48) ( (cid:126) A ( k ) + i(cid:126) ∇ k θ ( k )) · d k = (cid:90) Γ → M (cid:48) (cid:126) A ( k ) · d k + i ( θ ( M (cid:48) ) − θ (Γ)) , (10)which yields (cid:90) M (cid:48) → Γ → M (cid:126) A ( k ) · d k = i ( θ ( M (cid:48) ) − θ (Γ)) . (11)The symmetry of the points S n − Γ = Γ and S n − M (cid:48) = M enforce the condition that | u (Γ) (cid:105) and | u ( M ) (cid:105) areeigenstates of the generator of U , namely σ z . We canuse this fact to simplify Eq. (11) to (cid:90) M (cid:48) → Γ → M (cid:126) A ( k ) · d k = m ( n − n ( (cid:104) σ z (cid:105) M − (cid:104) σ z (cid:105) Γ ) , (12)where (cid:104) σ z (cid:105) K = (cid:104) u ( K ) | σ z | u ( K ) (cid:105) is the spin polarization.It follows that (cid:90) X → Γ → X (cid:48) (cid:126) A ( k ) · d k = m (cid:104) σ z (cid:105) Γ − (cid:104) σ z (cid:105) X ) . (13)Adding all contributions and substituting in Eq. (5)yields the Chern-spin polarization relation C = m (cid:104) σ z (cid:105) Γ + ( n − (cid:104) σ z (cid:105) M − n (cid:104) σ z (cid:105) X ] (mod 2 n ) . (14)This is the first main result of the present work.The Chern-spin polarization relation (14) provides in-sight into the mechanisms responsible for the emergenceof energy bands characterized by |C| = 2 ,
3. As each (cid:104) σ z (cid:105) K take values in { , − } , Eq. (14) implies that C mustbe an even integer for m = 2 and any value of n . There-fore, a Hamiltonian satisfying the m = 2 symmetry afterundergoing a topological band inversion typically entersa |C| = 2 phase. Similarly, for a triangular lattice Hamil-tonian obeying the m = 3 symmetry, a topological bandinversion typically leads to a |C| = 3 phase. B. Engineering topological Hamiltonians
The Chern-spin polarization relation is a powerful toolfor engineering topological Hamiltonians, as we demon-strate next by constructing a tight-binding Hamiltonianwith a |C| = 2 band. Let a and a be the lattice vectors,with angular separation 2 π/
3, of a triangular Bravais lat-tice, and a := − a − a . The expression for a generaltwo-band spin-orbit coupled Hamiltonian is H ( k ) = h x σ x + h y σ y + h z σ z , (15)where h x , h y and h z are k -dependent coefficients, and aspin-independent term proportional to identity is omit-ted as it has no effect on the Chern number. Thanksto Eq. (14), it suffices to construct a gapped Hamiltonianthat obeys the m = 2 symmetry such that sgn( h z ( M )) = − sgn( h z (Γ)). To restrict the Hamiltonian to nn terms,each of the coefficients h x , h y and h z must be linear com-binations of { cos k q , sin k q , q = 1 , , } , with k q = k · a q .As σ z satisfies U σ z U † = σ z , the constraint on h z is h z ( k ) = h z ( S k ), which restricts possible values to h z = M z + 2 t z (cid:80) q cos( k q ) , M z , t z ∈ R . (16)Similarly, the constraints on the coefficients of σ x and σ y imposed by the symmetry lead to h x = 2 t so (cid:80) q cos( k q ) cos(2 qπ/ ,h y = − t so (cid:80) q cos( k q ) sin(2 qπ/ , t so ∈ R , (17)which completes the construction of the Hamiltonian.After simplification using trigonometric identities, theterms reduce to h x = t so (2 cos( k ) − cos( k ) − cos( k )) ,h y = −√ t so (cos( k ) − cos( k )) ,h z = M z + 2 t z (cos( k ) + cos( k ) + cos( k )) . (18)For t so (cid:54) = 0 and t z >
0, this Hamiltonian is gappedexcept for M z /t z ∈ {− , , } . Further restricting to M z /t z > − k q = 0 and k q = − π/ M points respectively for q ∈ { , , } , we obtain | u (Γ) (cid:105) = |↓(cid:105) , and | u ( M ) (cid:105) = (cid:26) |↓(cid:105) if M z /t z > , |↑(cid:105) if M z /t z < . (19)Therefore, Eq. (14) dictates that the lower band hasChern number C = − − < M z /t z < M z /t z = 2 at which the gapcloses), and 0 otherwise, which we confirm numerically.Fig. 2 shows the variation of the spin polarization of thelower energy band in the BZ. The topological phase, thespin polarization at the corners of the BZ including atthe M point, is the negative of the polarization at thecenter, Γ.One can now construct other Hamiltonians that host |C| = 2 , C = 2 is obtained by leveraging theChern-spin polarization relation (14) for the m = 1 sym-metry. The symmetry condition places the exact sameconstraints on the h z coefficient as in Eq. (16), while theconstraints on the σ x and σ y coefficients lead to h x = 2 t so (cid:88) q sin( k q ) cos(2 qπ/ h y = 2 t so (cid:88) q sin( k q ) sin(2 qπ/ . (20) k x k x k y k y FIG. 2. (Color online) Color map of spin polarization overthe BZ for the trivial (left) and the topological (right) phasewith C = −
2. The crystal momenta k x and k y are in unitsof π/a , with a = | a q | denoting the lattice constant. Thebright red (blue) color represents +1 ( −
1) value of the spinpolarization (cid:104) σ z (cid:105) k . Parameters used were t so = 0 . , t z = 1, M z = 4 (trivial phase) and M z = 1 (topological phase). For small values of | k | , it can be verified that h x σ x + h y σ y ∝ [ S z ( − π/ k ] × (cid:126)σ · ˆ z, (21)which is the Rashba SO coupling. The Chern numbercan again be evaluated directly by using the Chern-spinpolarization relation. For t so (cid:54) = 0 and M z , t z >
0, thespin at the Γ point is (cid:104) σ z (cid:105) Γ = −
1. For 2 < M z /t z < M and the X points are (cid:104) σ z (cid:105) M = − (cid:104) σ z (cid:105) X = 1, which leads to a C = 2 phase.Note that this Hamiltonian exhibits a |C| = 2 phase fora smaller parameter range than was the case above when m = 2. This could be attributed to the fact that incontrast to the m = 2 symmetric Hamiltonian, the m = 1symmetric Rashba SO term does not enforce the Chernnumber to be even in general, and therefore a finer tuningof parameters is required.The construction of tight-binding Hamiltonians with |C| = 3 follows a similar approach. Eq. (14) for m = 1with the additional condition (cid:104) σ z (cid:105) Γ = (cid:104) σ z (cid:105) M reduces to C = n (cid:2) (cid:104) σ z (cid:105) Γ − (cid:104) σ z (cid:105) X (cid:3) (mod 2 n ) . (22)The additional condition is satisfied for h z = M z + 2 t z [cos( k − k ) + cos( k − k )+ cos( k − k )] . (23)To satisfy the m = 1 symmetry, the coefficients of σ x and σ y are taken to have the same form as in Eq. (20). In theparameter range − < M z /t z < t so (cid:54) = 0 , t z > (cid:104) σ z (cid:105) Γ = − (cid:104) σ z (cid:105) X = −
1, and therefore thisHamiltonian displays a C = − m = 3 symmetry case. Choosing h z as in Eq. (16),any two-band nearest-neighbor Hamiltonian is necessar-ily gapped, as h x , h y ∝ (cid:80) q sin k q . One way to con-struct h x and h y coefficients satisfying the symmetry isto add next-nearest neighbor spin-orbit coupling. One FIG. 3. Proposed experimental setup for the realization andthe detection of a |C| = 2 topological phase. In the figure,LA1 and LA2 denote two lasers with the same frequency ω polarized out-of-plane (along z -axis) and in-plane (in x - y plane), respectively. BS denotes a 50 −
50 beam-splitter,M1-M4 are polarization-preserving mirrors, and AOM1 andAOM2 are acousto-optic modulators. An external magneticfield of strength B is applied out-of-plane. such choice is h x = t so (cid:88) q sin k q ; h y = t so [sin( k − k ) + sin( k − k ) + sin( k − k )] . (24)For any t so (cid:54) = 0, the total Hamiltonian displays a |C| = 3phase for the parameter range − < M z /t z < t so , which is advantageous for experimental re-alization. Most of these Hamiltonians host |C| = 2 or |C| = 3 phases for M z = 0, which could be importantfor discovering materials that show quantum anomalousHall effect in the total absence of external magnetic field. III. REALIZING AND DETECTING A |C| = 2
PHASE WITH ULTRACOLD ATOMSA. Overview of the proposed experimental setup
We now describe how a topological Hamiltonian on atriangular lattice, described by Eqs. (16) and (17), couldbe realized using ultracold Rb atoms. The scheme wepresent here only requires isolation of, and control over,three hyperfine levels forming a Λ configuration for therealization of the SO coupling via Raman coherence, andtherefore can be extended to all bosonic and fermionicatoms used in ultracold atomic experiments.By applying a red-detuned laser sheet that is tightlyfocused along the z -axis, the atoms are confined tothe xy -plane (Fig. 3). The atoms are then loadedonto a spin-independent triangular optical lattice poten-tial [60], formed by three red-detuned z -polarized (linear ∆ δ F = 1 , S / F = 1 , P / − m F | ↑ i | ↓ i ω ω FIG. 4. The Λ-configuration, responsible for the Raman tran-sition, formed by |↑(cid:105) and |↓(cid:105) states with the excited state | F = 1 , m F = − (cid:105) in the 2 P / multiplet. F = 2 levels of the2 P / multiplet are not shown for simplicity. π -polarized) ‘lattice’ laser beams. The scheme for in-ducing SO coupling, inspired by the experimental setupdescribed in Ref. [52], makes use of two hyperfine levels ofthe ground state manifold as the pseudospin-1 / |↑(cid:105) )and down ( |↓(cid:105) ) states. To simulate SO coupling betweenthese two states, a sufficiently strong external magneticfield needs to be applied in the z -direction, so that thehyperfine energy levels shift according to the quadraticZeeman effect. Three Raman laser beams with linearin-plane polarization induce Raman transitions betweenthe |↑(cid:105) and |↓(cid:105) levels via excited states. The Raman laserbeams also generate an additional spin-dependent peri-odic potential.For the detection of the Chern number, we designa scheme based on Bloch oscillations and TOF imag-ing that leverages the Chern-spin polarization relation.The spin polarization of the Bose-Einstein condensateof bosonic ultracold atoms can be obtained by releas-ing the atoms from the trap followed by a Stern-Gerlachmeasurement. The condensate can be moved from onepoint to the Brillouin zone to another by short Blochoscillations. Such oscillations are induced by slowly ac-celerating the optical lattice, which is achieved by slowlyand simultaneously chirping the frequencies of the latticeand Raman lasers. If the experiment is performed byfermionic atoms, then the Bloch oscillations are not re-quired, as a direct spin-resolved TOF imaging suffices toobtain the spin polarization. Finally, the Chern numberis calculated using the Chern-spin polarization relation.
1. Laser configuration
As per convention, the z -axis is the quantizationaxis of the atom, so that J z | J, m, F, m F (cid:105) = (cid:126) ( m + m F ) | J, m, F, m F (cid:105) , where J z = L z + S z + I z is the z -component of the total angular momentum operator.The pseudospin-1 / |↑(cid:105) := | F = 1 , m F = − (cid:105) and |↓(cid:105) := | F = 1 , m F = 0 (cid:105) , as depicted in Fig. 4. The threelattice laser beams with amplitude E propagate in the xy -plane with wavevectors k = k ( − , , k = k (cid:32) , − √ (cid:33) , k = k (cid:32) , √ (cid:33) . (25)The frequency ω of the lattice laser beams is close to,but less than, the D1 transition frequency. The threeRaman laser beams are linear polarized in the xy -planeand propagate along the same directions as the latticelaser beams with amplitude G , and frequency ω = ω +( (cid:15) ↑ − (cid:15) ↓ )+ δ , where (cid:15) ↑ − (cid:15) ↓ is the Zeeman energy shift, and δ denotes the two-photon detuning. The frequency shiftcan be implemented with the help of an acousto-opticmodulator AOM1, as shown in Fig. 3. The arm length L needs to be adjusted such that the relative phase acquiredby adjacent Raman laser beams is φ c := 3 Lδω/c = 2 π/ xy -plane can be separated into su-perpositions of σ + and σ − circular polarization. In thespherical basis e + = (cid:18) − √ , − i √ (cid:19) , e − = (cid:18) √ , − i √ (cid:19) , (26)and using the definition G ± ( r ) = e ∗± · (cid:126)G ( r ), one obtains r · (cid:126)G = − ( x + iy ) √ G + + ( x − iy ) √ G − . (27)The σ − -polarized component at frequency ω in combi-nation with the linearly polarized field at frequency ω leads to Raman coherence in the Λ system formed by |↑(cid:105) , |↓(cid:105) and | F, − (cid:105) for each F = 1 , P / multi-plet (Fig. 4). All transitions to the level | F = 1 , m F = 1 (cid:105) in the S / multiplet are suppressed due to a large two-photon detuning as a result of the quadratic Zeemanshift, and therefore can be ignored. The detuning ∆from the center of the D1 transition is large comparedto both the hyperfine splitting and Zeeman energy shiftin the P / levels.We now derive the symmetry properties of the electricfields, which will be used later to derive the optical lat-tice potential and the form of spin-orbit coupling inducedby this laser configuration. The hyperfine splitting ismuch smaller than the D1 energy gap, so that ω − ω (cid:28) ω ; therefore, | k | ≈ | k | = k is a good assumption.The E ( r , t ) is the complex-valued electric field in the z -direction, and (cid:126)G ( r , t ) := ( G x ( r , t ) , G y ( r , t )) is the in-plane electric field along the x and y direction. One cannow write E ( r , t ) = E ( r ) e − iω t , (cid:126)G ( r , t ) = (cid:126)G ( r ) e − iω t ,where E ( r ) = E (cid:88) j =1 e i k j · r , (cid:126)G ( r ) = G (cid:88) j =1 e i ( k j · r + jφ c ) ˆ z × k j . (28)Let a = 4 π k (1 , , a = 4 π k (cid:32) − , √ (cid:33) . (29) The electric fields satisfy E ( r + a q ) = e i π/ E ( r ); G ± ( r + a q ) = e i π/ G ± ( r ) , q = 1 , , . (30)Eq. (30) follows directly from the observation k j · a q =2 π/ π for all q, j ∈ { , , } . Therefore, a , a arethe lattice vectors of the triangular optical lattice formedby the lattice lasers [60]. We also define a = − a − a as before. The reciprocal lattice vectors are b = √ k (cid:32) √ , (cid:33) , b = √ k (0 , . (31)For the rotation by 2 π/ z -axis imple-mented by R = R z (2 π/
3) = (cid:20) cos(2 π/ − sin(2 π/ π/
3) cos(2 π/ (cid:21) = (cid:20) − / −√ / √ / − / (cid:21) , (32)one obtains E ( R r ) = E ( r ) , G ± ( R r ) = e i ( φ c ∓ π/ G ± ( r ) . (33)Eq. (33) follows from the rotational symmetry of the in-plane and out-of-plane field components. The out-of-plane field is invariant under rotation by 2 π/
3, whichleads to the first equality in Eq. (33). The second equal-ity can be derived as follows: first note that by symmetryof the experimental setup, the field (cid:126)G satisfies (cid:126)G ( R r ) = e iφ c R (cid:126)G ( r ) . Using e ∗± R = e ∓ i π/ e ∗± one obtains G ± ( R r ) = e ∗± · (cid:126)G ( R r )= e iφ c e ∗± · R (cid:126)G ( r )= e i ( φ c ∓ π/ e ∗± (cid:126)G ( r )= e i ( φ c ∓ π/ G ± ( r ) , as desired.
2. Optical lattice potential
We now compute the optical lattice potential due tothe lattice laser beams. The frequency ω can be chosento be close to D1 transition frequency so as to ensuresthat D2 transitions can be safely ignored. This assump-tion is made only for simplicity; the scheme works evenif both D1 and D2 transitions have comparable contribu-tion. The Rabi frequency of oscillation due to the opticallattice lasers is (cid:126) Ω ↑ , ,F ( r ) = (cid:104) F, − | ez | ↑(cid:105) E ( r ) , (cid:126) Ω ↓ , ,F ( r ) = (cid:104) F, | ez | ↓(cid:105) E ( r ) , (34)where F corresponds to the excited level in the P / space. The optical lattice potential generated by the laserwith ω frequency alone is V ( r ) = (cid:88) F =1 , (cid:126) | Ω σ, ,F ( r ) | , (35)where ∆ is the detuning. Note that since Ω σ, ,F ∝ E ( r ),then V ( r ) ∝ | E ( r ) | . The lattice potential is generatedby π -polarized light and therefore is spin-independent(see discussion above Eq. (44) in Ref. [66] for a rigorousjustification). The details of the exact potential can befound in Ref. [60]. The contribution to the continuumHamiltonian due to lattice lasers alone is H lat = − (cid:126) ∇ m + V ( r ) . (36)
3. Spin-dependent potential
In contrast to the out-of-plane polarized lattice lasers,the Raman lasers add a spin-dependent component tothe optical lattice potential. Consider first the spin-dependent change in the lattice potential caused by theRaman lasers. The dipole potential for alkali atoms inthe state (
F, m F ) due to D1 transitions is given by theformula [67] V m F ( r ) = πc Γ2 ω (cid:18) − P g F m F ∆ (cid:19) I ( r ) , (37)where the detuning ∆ is with respect to the center ofthe D1 line, g F is the Land´e factor, m F is the relevantmagnetic spin state of the atom, and P = ± P = 0represent the local polarization σ ± and π , respectively,of the light field relative to the chosen quantization axis.For the |↓(cid:105) state, m F = 0, and therefore the second termvanishes. However, for the |↑(cid:105) , we have m F = − W ( r ) = V ↑ ( r ) − V ↓ ( r ) = πc Γ g F ω ∆ (cid:88) i = σ ± P i I i ( r ) (38)as additional spin-dependent dipole potential due to Ra-man lasers. This contributes the term W ( r ) σ z / V ↑ ( r )+ V ↓ ( r ) term can be ignoredbecause it is spin-independent and can be absorbed in V ( r ).
4. Spin-orbit coupling
The Raman laser beams, together with the lattice laserbeams, induce Raman coupling between the | (cid:105) and | (cid:105) states. Both F = 1 and F = 2 levels in the P / space contribute to Raman resonance. The effective two-photon Rabi frequency isΩ R ( r ) = 12∆ (cid:88) F =1 , Ω ∗↓ , − ,F ( r )Ω ↑ , ,F ( r ) , (39) where (cid:126) Ω ↓ , − ,F ( r ) = 1 √ (cid:104) F, − | e ( x − iy ) | ↓(cid:105) G − ( r ) . (40)Note that the two-photon Rabi frequency satisfiesΩ R ( r ) ∝ E ( r ) G ∗− ( r ) . It follows from Eqs. (30) and (33) thatΩ R ( r + a q ) = Ω R ( r ) , q = 1 , , , (41)Ω R ( R r ) = e − i ( φ c +2 π/ Ω R ( r ) . (42)Then, the contribution to the Hamiltonian due to Ramanresonance is [68] V R ( r ) = − (cid:126) (cid:34) δ + (cid:80) F | Ω ↓ , ,F ( r ) |
4∆ Ω ∗ R ( r )2Ω R ( r )2 − δ + (cid:80) F | Ω ↑ , − ,F ( r ) | (cid:35) , (43)where δ is the two-photon detuning. Note that the terms (cid:80) F | Ω ↓ , ,F ( r ) | in the top left and (cid:80) F | Ω ↑ , − ( r ) | in thebottom right corner have already been taken into ac-count when calculating the optical lattice potential andthe spin-dependent correction to it.After combining contributions from all factors, the to-tal effective Hamiltonian in the continuum becomes H = H lat + (cid:18) W ( r )2 − (cid:126) δ (cid:19) σ z − (cid:18) (cid:126) Ω R ( r )2 σ + + H.c. (cid:19) , (44)where H lat is the sum of the kinetic energy and the lat-tice potential generated by the lattice lasers, and the co-efficients W ( r ) and Ω R ( r ) depend on experimentally ad-justable parameters, namely the amplitudes E , G , andthe single-photon and two-photon detunings ∆ and δ re-spectively. B. Derivation of the tight-binding Hamiltonian
The effective tight-binding Hamiltonian is constructedby restricting the continuum Hamiltonian to the Wan-nier states of the lowest energy band. We show througha detailed analysis of the symmetry properties of theseWannier states that the resulting Bloch Hamiltonian hasmatrix coefficients of the form in Eqs. (16),(17). The keystep involves taking advantage of the symmetry proper-ties of the Wannier functions as well as the electric fieldconfiguration to obtain the phases acquired by atomswhile hopping in various directions accompanied by (pos-sible) spin flips.Consider the Bloch eigenstates ψ n, k of the latticeHamiltonian H lat corresponding to n th band and crys-tal momentum k . Next, construct the Wannier states φ σn,j ,j ( r ) = 1 √ N (cid:88) e − i k · r ψ σn, k ( j a + j a ) , (45)defined for all pairs of integers (cid:126)j := ( j , j ). From hereonwards, we will restrict attention to the Wannier func-tions of the lowest energy band n = 0, and assume thatthe Wannier functions φ σj ,j ( r ) := φ σ ,j ,j ( r ) are sym-metric under six-fold rotation generated by R , that is φ σ(cid:126) ( r ) = φ σ(cid:126) ( R r ) . Because the lattice Hamiltonian H lat is spin-independent,one also has φ ↑ j ,j ( r ) = φ ↓ j ,j ( r ) (46)Further, since the lattice potential V ( r ) is real, it is pos-sible to use a gauge in which ψ n, − k ( r ) = ψ ∗ n, k ( r ), whichleads to the conclusion that the Wannier functions φ σ arereal-valued.The lattice Hamiltonian term H lat leads to isotropichopping without spin-flip with strength t hop , (cid:98) H hop = − t hop (cid:88) (cid:104) (cid:126)j,(cid:126)j (cid:48) (cid:105) ˆΦ † (cid:126)j ˆΦ (cid:126)j (cid:48) , (47)where (cid:104) (cid:126)j,(cid:126)j (cid:48) (cid:105) denote the restriction that (cid:126)j,(cid:126)j (cid:48) are nearest-neighbor sites on the triangular lattice, ˆΦ † (cid:126)j = (cid:104) ˆ φ † (cid:126)j, ↑ ˆ φ † (cid:126)j, ↓ (cid:105) are arrays of creation operators corresponding to theWannier orbitals, and − t hop = (cid:90) d r ( φ ↓ (cid:126) ) ∗ ( r ) H lat ( r ) φ ↑ (cid:126)a ( r ) , (48)where (cid:126)a := (1 , (cid:126)a := (0 ,
1) and (cid:126)a := ( − , −
1) = − (cid:126)a − (cid:126)a are the integer coordinates of lattice vectors a , a and a in the units of a and a , and (cid:126) , φ ↓ , ) ∗ was replaced by φ ↓ , in the integrand.The optical lattice formed by a z -polarized laser hasa spin-independent correction due to the in-plane polar-ized lasers. We claim without proof that the correctionto the hopping strength due to this contribution is alsoisotropic. Whether this claim is true or false does notaffect further analysis, because the Wannier basis of theoptical lattice potential is governed by z -polarized lasers,and the spin-independent correction only adds a factorproportional to the identity to the Hamiltonian in mo-mentum space that does not affect the eigenvectors ortheir topology.Next consider the contribution to the tight-bindingHamiltonian due to the term W ( r ) σ z / σ z type hopping between all nearest neighbors, (cid:98) H zhop , = t z (cid:88) (cid:104) (cid:126)j,(cid:126)j (cid:48) (cid:105) ˆΦ † (cid:126)j σ z ˆΦ (cid:126)j (cid:48) , (49)where t z := 12 (cid:90) d r ( φ ↓ (cid:126) ) ∗ ( r ) W ( r ) φ ↑ (cid:126)a ( r ) (50) Apart from a σ z -hopping term, W ( r ) σ z / σ z mass term, given by (cid:98) H zhop , = m z (cid:88) (cid:126)j ˆΦ † (cid:126)j σ z ˆΦ (cid:126)j , (51)where m z := 12 (cid:90) d r ( φ ↓ (cid:126) ) ∗ ( r ) W ( r ) φ ↑ (cid:126) ( r ) (52)The Raman process is responsible for inducing hoppingwith spin-flip in the tight-binding Hamiltonian. The am-plitude of the hopping term between the Wannier centers(0 ,
0) and ( j , j ) can be computed using the formula t so ( j , j ) = (cid:126) (cid:90) d r ( φ ↓ (cid:126) ) ∗ ( r )Ω R ( r ) φ ↑ j ,j ( r ) . (53)Because φ ↑ = φ ↓ and φ σ is real-valued, one obtains t so ( − (cid:126)a q ) = (cid:126) (cid:90) d r ( φ ↓ (cid:126) ) ∗ ( r )Ω R ( r ) φ ↑− (cid:126)a q ( r )= (cid:126) (cid:90) d r ( φ ↓ (cid:126) )( r )Ω R ( r )( φ ↑− (cid:126)a q ) ∗ ( r )= (cid:126) (cid:90) d r ( φ ↑− (cid:126)a q ) ∗ ( r )Ω R ( r ) φ ↑ (cid:126) ( r )= (cid:126) (cid:90) d r ( φ ↓− (cid:126)a q ) ∗ ( r )Ω R ( r ) φ ↑ (cid:126) ( r )= (cid:126) (cid:90) d r ( φ ↓ (cid:126) ) ∗ ( r )Ω R ( r ) φ ↑ (cid:126)a q ( r )= t so ( (cid:126)a q ) . (54)Using a = R a = R a along with the translation in-variance of the Wannier functions φ j ,j ( r ) = φ (cid:126) ( r − j a − j a ), one obtains φ σ(cid:126)a ( R r ) = φ σ(cid:126) ( R r − a ) = φ σ(cid:126) ( R ( r − a )) = φ σ(cid:126) ( r − a )= φ σ(cid:126)a ( r ) (55)and φ σ(cid:126)a ( R r ) = φ σ(cid:126) ( R r − a ) = φ σ(cid:126) ( R ( r − a )) = φ σ(cid:126) ( r − a )= φ σ(cid:126)a ( r ) . (56)The hopping amplitudes t so ( (cid:126)a ) and t so ( (cid:126)a ) can now becalculated as follows: t so ( (cid:126)a ) = (cid:126) (cid:90) d r ( φ ↓ (cid:126) ) ∗ ( r )Ω R ( r ) φ ↑ (cid:126)a ( r ) (57)= (cid:126) (cid:90) d s ( φ ↓ (cid:126) ) ∗ ( R s )Ω R ( R s ) φ ↑ (cid:126)a ( R s ) (58)= (cid:126) (cid:90) d s ( φ ↓ (cid:126) ) ∗ ( s ) e − i ( φ c +2 π/ Ω R ( s ) φ ↑ (cid:126)a ( s ) (59)= e − i ( φ c +2 π/ t so ( (cid:126)a ) , (60)where the change of variable r = R s is employed in thefirst step and Eqs. (42) and (55) in the later steps. Notethat since R is a rotation, the Jacobian of the transfor-mation is 1. A similar calculation yields t so ( (cid:126)a ) = e − i ( φ c +2 π/ t so ( (cid:126)a )The on-site spin-flipping term does not survive for φ c =2 π/
3. This follows from the relation t so ( (cid:126)
0) = (cid:126) (cid:90) d r ( φ ↓ (cid:126) ) ∗ ( r )Ω R ( r ) φ ↑ (cid:126) ( r ) (61)= (cid:126) (cid:90) d s ( φ ↓ (cid:126) ) ∗ ( R s )Ω R ( R s ) φ ↑ (cid:126) ( R s ) (62)= (cid:126) (cid:90) d s ( φ ↓ (cid:126) ) ∗ ( s ) e − i ( φ c +2 π/ Ω R ( s ) φ ↑ (cid:126) ( s ) (63)= e − i ( φ c +2 π/ t so ( (cid:126) , (64)which implies that t so ( (cid:126)
0) = 0 for φ c = 2 π/ φ ↓ by a global gauge transformationsuch that t so ( (cid:126)a ) is real and positive. From here onwards,we set t so := t so ( (cid:126)a ). Now the total contribution to thetight-binding Hamiltonian due to the Raman process is (cid:98) H so = (cid:88) (cid:126)j (cid:88) q =1 t so (cid:110) ˆΦ † (cid:126)j (cid:2) cos( − qπ/ σ x + sin( − qπ/ σ y (cid:3) ˆΦ (cid:126)j + (cid:126)a q + H.c. (cid:111) . (65)Finally, the two-photon detuning leads to on-site σ z termwith strength − (cid:126) δσ z / φ c , (cid:98) H det = − (cid:126) δ (cid:88) (cid:126)j ˆΦ † (cid:126)j σ z ˆΦ (cid:126)j , (66)It is safe to assume that the two-photon detuning can beadjusted to be small enough so that any nearest-neighborhopping coefficients can be ignored.Combining all terms, one obtains the full tight-bindingHamiltonian Eqs. (47), (49), (51), (65), and (66). Be-cause the Hamiltonian is number conserving, one maywrite the single-particle Hamiltonian as H = H hop + H zhop + H so + H det , (67)where H hop = − t (cid:88) (cid:126)j,q (cid:16) | (cid:126)j (cid:105) (cid:104) (cid:126)j + (cid:126)a q | + H.c. (cid:17) ,H zhop = t z (cid:88) (cid:126)j,q (cid:16) | (cid:126)j (cid:105) (cid:104) (cid:126)j + (cid:126)a q | σ z + H.c. (cid:17) ,H so = t so (cid:88) (cid:126)j,q (cid:110) | (cid:126)j (cid:105) (cid:104) (cid:126)j + (cid:126)a q | (cid:2) cos( − qπ/ σ x + sin( − qπ/ σ y (cid:3) + H.c. (cid:111) ,H det = (cid:18) m z − (cid:126) δ (cid:19) (cid:88) (cid:126)j | (cid:126)j (cid:105) (cid:104) (cid:126)j | σ z (68) To obtain the tight-binding Hamiltonian in momentumspace, define V q = (cid:88) (cid:126)j | (cid:126)j (cid:105) (cid:104) (cid:126)j + (cid:126)a q | , q = 1 , , , (69)and the momentum states | k (cid:105) = 1 √ N (cid:88) (cid:126)j e i k · j | (cid:126)j (cid:105) (70)where again (cid:126)j = ( j , j ) and j = j a + j a . It is easyto verify that V q | k (cid:105) = e ik q | k (cid:105) , V † q | k (cid:105) = e − ik q | k (cid:105) . (71)After omitting the term proportional to identity in spinspace, and M z := m z − (cid:126) δ , one obtains the spin-orbitHamiltonian in momentum space H ( k ) = H ( k ) · σ = h x σ x + h y σ y + h z σ z , with the coefficients h x , h y , h z as inEq. (16) and (17). This is the second main result of thepresent work.We conclude the description of the experimentalscheme for the realization of SO coupling and the desiredtopological state on a triangular lattice by a qualitativecomparison to the scheme used in Ref. [52] for the squarelattice. Despite the apparent similarity, our scheme dif-fers substantially from the scheme used in Ref. [52] forthe square lattice. The lattice potential in our schemeis generated by π -polarized lasers, which are truly spin-independent. More important, the Raman potential inour scheme has the same periodicity as the optical lat-tice in the standard gauge. In contrast, the scheme inRef. [52] leads to a Raman potential that has twice theperiodicity of the optical lattice before a particular gaugetransformation is implemented. Such a gauge transfor-mation is not possible in a triangular lattice due to thelack of sublattice symmetry. We turn to the detection ofChern number in the next section. C. Chern number via Zeeman Spectroscopy andBloch oscillations
We now show how the Chern-spin polarization rela-tion, (14), can be leveraged to obtain the Chern number(mod 6) of the lower band wavefunction The triangu-lar SO-coupled lattice requires n = 3 and m = 2, andthe relation can be further simplified to C = 2( (cid:104) σ z (cid:105) M −(cid:104) σ z (cid:105) Γ ) (mod 6). Typically, the spin-independent hoppingterm − t hop (cos( k ) + cos( k ) + cos( k )) of the Hamil-tonian, which was omitted in Eq. (18), dominates asfar as the energy eigenvalues are concerned. Thereforethe Rb atoms condense to form a Bose-Einstein con-densate (BEC) at the Γ point in the center of the BZthat minimizes this term, as shown in Fig. 1. The ratioof the populations in the |↑(cid:105) , |↓(cid:105) levels obtained by Stern-Gerlach imaging (i.e. Zeeman spectroscopy) can be usedto infer the spin polarization at the Γ point.0The spin polarization at the point M in the BZ can beobtained by first performing a short Bloch oscillation tomove the BEC adiabatically [61] from the Γ point to the M point. For a triangular lattice, the coordinates of the M point, shown in Fig. 1, are given by M = ( − b + 2 b ) / √ k (cid:34) −
13 (0 ,
1) + 23 (cid:32) √ , (cid:33)(cid:35) = ( k, . (72)To map the condensate from the Γ to the M point, it suf-fices to accelerate the lattice along the x -direction withsome magnitude a , which is accomplished by varying thefrequency of the laser beams travelling in the − x direc-tion. Recall that when all three ω beams meet at theorigin (0 ,
0) in phase, one of the lattice sites coincideswith the origin, taken to be the center of the lattice. Howdoes the center of the lattice shift when the third beam(propagating along − x ) reaches the origin with a phasedifference ϕ added by AOM2 with respect to the initialconfiguration? Due to the symmetry of the system, thecenter must shift to a point r c = ( r,
0) along the x -axis,satisfying ϕ + k · r c = k · r c = k · r c . (73)The second equality is automatically satisfied for anypoint r c along the x -axis. Solving the first equality leadsto ϕ + k ( − , · ( r,
0) = k (cid:32) , − √ (cid:33) · ( r, , (74)which gives r = 2 ϕ/ k . To achieve acceleration a alongthe x -direction, one needs ϕ ( t ) = 3 k (cid:18) at (cid:19) = 3 kat . (75)Assuming that the frequency variation is applied at t = 0,the condition can be translated to (cid:90) tt =0 ∆ ω ( t )d t = ϕ ( t ) , (76)which on differentiating yields the frequency differenceas a function of the time required to achieve the phasedifference ϕ ( t ) at the origin,∆ ω ( t ) = (cid:18) ka (cid:19) t. (77)Therefore, to achieve acceleration α along + x direction,the frequency of the third beam needs to be changed ata linear rate, with d∆ ω/ d t = 3 ka/ M point can be calculated as follows. The rate ofchange of crystal momentum is given by (cid:126) d k d t = − ma, (78) so that to reach ( k,
0) from (0,0), the time required is T = (cid:126) kma , (79)where m is the mass of the atom. The acceleration a can be chosen to be arbitrary, but the adiabaticity con-dition [69] must be satisfied, T (cid:29) (cid:126) (cid:107) d H/ d t (cid:107) ∆ E . (80)Typically, we expect the spin-orbit coupling strength t so (cid:28) t hop , so that the energy gap is determined by t so and the bandwidth by t hop . The condition on a thenbecomes (cid:126) kma (cid:29) (cid:126) t hop t = ⇒ a (cid:28) kt mt hop . (81)If the Hamiltonian in Eq. (18) is realized usingfermionic atoms, and assuming that the density of atomsis adjusted so that the band is half-filled, then the Chern-spin polarization relation can still be leveraged to mea-sure the Chern number experimentally. In contrast tobosons, the fermions at half-filling occupy the entire lowerenergy band due to Fermi-Dirac statistics. The spin po-larization at both the Γ and the M points can then bedirectly obtained by standard TOF Stern-Gerlach imag-ing, which comprises first turning off the lattice and Ra-man lasers to let the atoms evolve freely for a time t inan external magnetic field, and then imaging separatelythe population of each hyperfine level. At the end of thetime interval, the atoms with crystal momentum k reachapproximately the point k t/m (cid:126) in real space, so that thespin amplitude at k can be obtained from the populationdifference at the point k t/m (cid:126) in real space. In this case,the spatial profile of the population difference betweenthe two levels will resemble the pattern in Fig. 2. IV. CONCLUSIONS
In this work, we have shown that the origin of some |C| = 2 , |C| = 2 phase on a triangular lattice usingRaman-induced spin-orbit coupling in ultracold atomicgases. The trivial and topological phases can then bedistinguished using TOF Zeeman imaging for fermions1and a combination of Bloch oscillations and TOF Zee-man imaging for bosons. Our scheme for the detectionof the Chern number suggests that Bloch oscillations andZeeman spectroscopy could be adapted to a large class ofSO coupled systems for the detection of topological orderin ultracold atoms.The Chern-spin polarization relation and the tight-binding models that we constructed illustrate that sym-metries with larger values of m induce favorable condi-tions for the realization of higher Chern number states,which should galvanize the search for such states in sys-tems with unconventional SO coupling. In the presentanalysis, the values of |C| are restricted to 2 , ACKNOWLEDGMENTS
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