Chiral Topological Orders in an Optical Raman Lattice

We find an optical Raman lattice without spin-orbit coupling showing chiral topological orders for cold atoms. Two incident plane-wave lasers are applied to generate simultaneously a double-well square lattice and periodic Raman couplings, the latter of which drive the nearest-neighbor hopping and create a staggered flux pattern across the lattice. Such a minimal setup is can yield the quantum anomalous Hall effect in the single particle regime, while in the interacting regime it achieves the $J_1$-$J_2$-$K$ model with all parameters controllable, which supports a chiral spin liquid phase. We further show that heating in the present optical Raman lattice is reduced by more than one order of magnitude compared with the conventional laser-assisted tunneling schemes. This suggests that the predicted topological states be well reachable with the current experimental capability.


Introduction
Generation of synthetic gauge fields for cold atoms opens a new direction in the study of exotic topological states beyond natural conditions. Two different scientific paths have been followed in the experiment to create synthetic gauge fields via optical means. One is to adopt Raman couplings between different internal hyperfine levels (atomic spins) [1][2][3][4][5][6], which has recently been used in experiments to generate synthetic spin-orbit (SO) coupling for cold atoms [7][8][9][10]. Another is to adopt laser-assisted hopping between neighboring lattice sites without spin flip, which can generate U(1) fluxes by imprinting the phases of Raman lasers into the hopping matrix elements [11][12][13][14][15]. Compared with the technique using spin-flip Raman couplings, the latter strategy can be achieved with far-detuned lasers, and therefore can avoid the spontaneous decay of excited states.
Realization of a gapped insulating topological state typically necessitates an optical lattice and synthetic gauge fields which satisfy proper conditions [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28]. In the conventional techniques, the optical lattice and gauge fields are generated through different atom-laser couplings. In such cases the topological regimes are achieved with careful manipulations of parameters, which might be challenging for the experimental observation. Recently, it was proposed that creations of the optical lattice and SO couplings can be integrated through the same standing wave lasers, and this new technique can have explicit advantages in realizing topological phases with minimal setups and without complicated manipulations [21,29]. Nevertheless, generating SO couplings requires near-resonant light which heats up the system by spontaneous emission [7][8][9][10]. A possible resolution of this difficulty is to consider lanthanide atoms which can have less heating due to large fine structure splitting and narrow natural linewidth in the excited levels [30].
In this paper, we introduce the model of optical Raman lattice without SO coupling to observe chiral topological phases for cold atoms. The setup includes a double-well square lattice and periodic Raman couplings generated simultaneously through two incident plane-wave beams. We show that this scheme can naturally realize chiral topological phases without fine tunings, and may have advantages in the experimental observation including the minimized heating and full controllability in parameters.
The manuscript is organized in the following way. In section 2 we present the model realization, and discuss the properties of the model; in section 3 we give the tight-binding model, which shows the quantum anomalous Hall (QAH) effect; in section 4 we show that the QAH states can be detected by measuring only Bloch states at two symmetric momenta of the first Brillouin zone (FBZ). The heating effect is discussed in section 5, and the chiral spin liquid phase is studied in section 6. Finally, we present the conclusions in section 7.

Model
In this section we present the details of the model realization. The system includes a double-well square optical lattice generated by an incident plane-wave light with both in-plane and out-of plane polarized components, and the two-photon Raman potentials which induce hopping between nearest-neighboring sites and create a staggered flux pattern across the double well lattice. The lattice and Raman potential profiles exhibit the relative antisymmetry in the spatial inversions.
We first introduce the generation of a 2D double well square lattice depicted in figures 1(a) and (b), with an onsite energy difference Δ between A and B sites. Based on the experiments by NIST group [31], this lattice configuration can be realized via an incident plane-wave laser beam which has both nonzero in-plane and outof-plane linearly polarized components, and with the assistance of three mirrors [see figure 1(a)]. The total electric field of the incident laser beam can be described as where the polarization angle α determines the magnitudes of in-plane and out-of-plane polarization components with wave-vector k 1 , and the out-of-plane polarized field has another component with its wave vector ( k 2 1 ) being twice of that of the former ones. Note that all the components of the incident beam can be generated from a single laser source through an optical frequency doubler, with which one can control the ratio of the field strengths E E 0 0 |˜| in the experiment [32]. With the reflection by mirrors the yˆ-polarization Figure 1. (a) The laser beam incident from x direction has nonzero polarization components along y and z axes, with the zˆpolarization field having two components of wave-vectors k 1 and k 2 1 . With the reflections of the three mirrors M j (j 1, 2, 3 = ), a double-well lattice is generated and can be controlled by the 1/4-wave plate and electro-optic modulator (EOM); (b) the created double-well lattice has an energy off set between A and B sublattices; (c) an additional linearly-polarized running laser beam, with polarization components in x and y directions, is applied along z direction. This beam, together with the laser components used to create square lattice, can induce spatially periodic Raman couplings (V V , The amplitudes are taken that V 0 D < , and the above potential describes a double-well square lattice illustrated in figure 1 (b), with the staggered onsite energy offset Δ between A and B sites well controlled by the polarization angle α. When Δ is large compared with the bare hopping couplings between neighboring s-orbitals of the A and B sites, the effective tunneling between them is suppressed, while the diagonal AA/BB hoppings (denoted by t A B ¢ ) are allowed along dashed lines in figure 1(b). The second term in equation (4) reduces the difference in height of the barriers along the AB-bond and the diagonal (AA/BB) directions. Thus it can enhance t A B ¢ relative to the hopping coupling between A and B sites, providing vast tunability in parameters. The tunneling between neighboring A and B sites (denoted by t i j  ) can be restored by two-photon Raman couplings. A key ingredient of the present scheme is that the in-plane blue-detuned laser beam which generates the square lattice also takes part in the generation of Raman couplings. For this we apply an additional planewave laser beam with frequency w dw -(dw » D), propagating along the perpendicular z direction and having linear polarization components along x and y axes ( figure 1(c)). This beam is described by where x y f is the initial phase of the x/y-axis polarization component. With the assistance of both E xy and E xỹ , two independent Raman couplings are induced by the x-and yˆ-polarization components, respectively (figure 1(d)). In particular, the x y (ˆ)-components of the lights E xy and E xỹ generate the Raman potential V Rx (V Ry ) which takes the form . We shall see below that a finite magnitude of x y f f -, controllable in experiment, gives rise to a nonzero staggered flux pattern for the square lattice, as illustrated in figure 1(c).
From equations (4)- (6) we can see that the zeros of V Rx Ry , are located at the lattice-site centers, which implies that the Raman potentials are parity odd relative to each lattice-site center ( figure 1(c)). With this key property the present blue-detuned optical Raman lattice can naturally realize topological states and exhibit essential advantages in minimize heating effects for experimental studies. The symmetry properties of s-orbitals and V Rx Ry , lead to two important consequences. First, the Raman potential V Rx (V Ry ) only induces the nearestneighbor hopping along x (y) direction. The hopping along x/y axis is associated with a phase y x f ( y x f -) if the hopping is toward (away from) B sites. In experiment, one can set that 2 x y 0 Then the hopping along the directions depicted by arrows in figure 1(c) acquires a phase 0 f , resulting in a staggered flux pattern with the flux 4 0 f F = . Secondly, the hopping from one site to its leftward (upward) neighboring site has an additional minus sign relative to the hopping to its rightward (downward) neighboring site. It is important that all these interesting properties are obtained automatically by using the two incident beams without complicated fine tunings.

Quantum anomalous Hall effect with large gap-bandwidth ratio
Now we give the tight-binding Hamiltonian. Based on the previous analysis we can obtain the s-band tightbinding Hamiltonian in the following form  the creation and annihilation operator, and the Zeeman term represent the summations over the nearest-and nextnearest-neighboring sites. The hopping phase 0 f determines the real (proportional to cos 0 f ) and imaginary (proportional to sin 0 f ) parts of the nearest hopping coefficients, with the staggered factor 1 i j n =  (−1) for hopping along (opposite to) the marked direction in figure 1(c). From the periodic profiles of the Raman potentials, the nearest-neighbor hopping coefficients satisfy t t t t 1 , 1 ) the s-orbital wave function at the j  -th site. We note that the staggered sign factor 1 i x -( ) is due to the periodicity of Raman potentials and staggered position distribution of A and B sites, and can be absorbed by redefining the annihilation operator of B sites to be c c e ¢  -¢. The tight-binding model can now be obtained directly and in k space the Bloch Hamiltonian reads t ka ka t k 2 cos sin sin 2 with n being integer. The QAH phases [17,33,34] Figure 2 provides the numerical estimate with V E V E 4, 1 show a large ratio (∼4.9) between the band gap and bandwidth E width in the range from t . It is noteworthy that a large gap-bandwidth ratio can enable the study of correlated topological states like the fractional QAH effect [35] in the interacting regime.

Detection with minimal measurements
Detection of the QAH insulating phase can be carried out with several different measurement strategies in the edge [17,19] and bulk [36][37][38][39][40]. In particular, it was proposed recently that the topology of a QAH insulator can be determined by measuring Bloch eigenstates at only two or four highly symmetric points of the first Brillouin zone [41]. It was shown that this approach is valid for QAH insulators which satisfy the inversion symmetry . For 87 Rb atoms using 532 Hz. The Chern number C 1 1 = + . defined by P P R 2D = Äˆ, where R 2D is a 2D spatial inversion operator transforming the Bravais lattice vector R R and P is a parity operator acting on the (pseudo)spin space [41]. In the present lattice system the unit cell is doubled relative to the original square lattice, and from the Hamiltonian k ( ) one can check that no parity symmetry can be satisfied. Nevertheless, we show below that this minimal measurement method can be still applied to the present system with a highly nontrivial generalization.
In the physical Hamiltonian k ( ) the Pauli matrices x y z , , s operate on the sublattice space. To complete our proof, we construct an artificial Hamiltonian which is formally equivalent to k The only difference is that in the new Hamiltonian we assume that x y z , , s act on a spin space which is independent of the position space. In this way, we know that the new Hamiltonian k  ( ) is invariant under the following 2D inversion transformation on both the position and spin space ). Therefore, at the four symmetric points 0, 0 , 0, , , 0 , , Bloch states are also parity eigenstates with P u u In the third line of the above equation we have defined that Therefore the invariant for the constructed system 0 n õ . This indicates that the Chern number for the Hamiltonian k  ( ) should always be even C N 2 1 = [41]. This result is easy to understand. As pointed out previously, for the original physical system the unit cell is doubled. Accordingly, the FBZ of the original square lattice, denoted by FBZ W is only half of the FBZ FBZ W for the constructed artificial system. The two momenta k k , From the number ñ one cannot tell the difference of a topological phase from a trivial phase. In the next step, we shall show that the topology of the artificial system can also be determined by the invariant ν which is defined in equation (12) with the parity eigenvalues at 0, 0 ), half of the four parity-symmetric points in FBZ W . The magnitudes 0 n = and +1 correspond to the topologically trivial and nontrivial states, respectively. Then, together with the above relation, we can further use this invariant to characterize the topology of the original physical system.
The proof is straightforward and is valid for any two-band system satisfying the following two conditions. First, the quantum anomalous Hall phases are characterized by low Chern numbers. In particular, for the artificial system it is C 0, 2 1 =  { }and for the original physical system C 0, 1 1 =  { }. Second, the system can be adiabatically connected to the one obtained under a four-fold C 4 rotational transformation on such system. In other words, the topology is not changed under the C 4 transformation in position and (pseudo)spin space It is easy to see that the inversion symmetry in equation (10) is given by P By a direct check one can verify that the constructed system in our consideration belongs to the class of Hamiltonians satisfying the above conditions. What we need to prove is that the transition between a trivial phase and a topological phase must be associated with the change from 0 n = to 1 n = + . Let the system be initially a trivial insulator. To have topological phase transition, the bulk gap should close and reopen at some momentum points. Around such momenta the bulk can be described by massive Dirac Hamiltonians, with the Dirac masses changing signs during the transition. We denote one of the Dirac momentum as k k k , Then, from the C 4 symmetry we know that there are four-fold of such Dirac points k Dj (j 1, 2, 3, 4 = ). Moreover, from the relation between the artificial and original physical systems, we have that the momentum k , Dj p p + ( ) is also a Dirac point. With these results in mind, we get that when a topological phase transition occurs, the Dirac masses simultaneously reverse signs at following momenta (not necessarily independent) It is easy to know that there must be even number (denoted as N 2 ) of Dirac points in the above formula which are independent. On the other hand, from the symmetry we know that all these Dirac points contribute the same Chern number to the whole bulk invariant. Before and after the phase transition the Chern number changes by  (15) are independent. This implies that for the later two situations the Chern number changes by 4 and 8, respectively, while in the first case the Chern number changes by 2. Therefore, for a system with low Chern number, only the first situation can happen, namely, the bulk gap must close and reopen at two inversion symmetric momenta 1 L and 3 L (or 2 L and 4 L ). Note that the Dirac masses at these points are equivalent to the parity eigenvalues, so the topological phase transition must be associated with the sign change of corresponding parity eigenvalues, leading to the change of the invariant ν. Furthermore, it is easy to verify that the trivial phase with C 1 correspond to 0 n = , and then the topological phases with C 2 1 =  are given by 1 n = + . Together with the relation(13) we conclude that the invariant ν classifies the topology of the original physical system. This completes our proof.
Since the parity operator is z s , the parity eigenvalues are the pseudospin eigenstates. To measure the parity  figure 3(a) and (b)). On the other hand, in the trivial regime m t t 2 ), the sign of the polarization keeps unchanged during the Bloch oscillation ( figure 3(c) and (d)). Only qualitative measurements at the two symmetric momenta are needed for the experimental detection.

Heating
The experimental feasibility of observing topological states, especially the correlated topological states, crucially depends on heating effects in the realization. Note that in the present scheme all applied lights are far-detuned. Thus the heating due to spontaneous decay from excited atomic levels is negligible. On the other hand, due to the energy difference between the Raman photons, the two-photon Raman processes have two main consequences. The first one is that it induces hopping between neighboring sites by compensating the energy difference between A and B sites, leading to the time-independent effective tight-binding Hamiltonian(7) and the topological band as discussed in the previous sections. This process includes both the energy transfer from Raman photons to atoms (hopping from B to A sites) and energy transfer from atoms to Raman photons (hopping from A to B sites). Thus in average it does not have net energy transfer between Raman photons and the atoms. Another main consequence is that if the two-photon Raman process drives an onsite transition rather than the hopping, it can transfer the energy from Raman photons to atoms. This process leads to the increase of the total mean mechanical energy, i.e. the expectation value of the kinetic and potential energies [42], of an atom trapped in the lattice potential. Note that the laser-assisted tunneling scheme without spin flip does not suffer from large spontaneous decay from excited states. Thus the main heating is induced by onsite two-photon Raman transitions which do not drive neighboring-site hopping but convert the energy difference between two Raman photons to mechanical energy of the lattice system [13,14]. Note that in the present optical Raman lattice, due to the antisymmetry of the Raman potentials the onsite intraband scattering (s s « bands) is forbidden, and only the interband scattering (s p « bands) can heat up the system. This distinguishes essentially from the conventional schemes which apply plane-wave and red-detuned Raman beams and have both inter-and intraband onsite transitions [11][12][13][14][15]. The life time of the trapped cold atoms can be estimated by calculating the change rate of the mean-mechanical energy of an atom. For comparison, we consider both the present optical Raman lattice system (with the heating rate denoted as E t d d

OR
) and the conventional laserassisted schemes (denoted as E t d d CO ). The rates of change of the mean-mechanical energy are given by ss sp k k k k CO , , where N is the number of lattice sites, and w ss (w sp ) represents the s-s (s-p) band scattering rate, obtained by the product of the two-photon Rabi-frequencies ss W ( sp W ) and the transition probabilities. In the above formulae we have denoted by OR G and CO G the heating rates of the optical Raman lattice system and the conventional lattice systems, respectively. Note that the s-s band onsite transition, e.g. from an initial state with momentum k to the final state k ¢ has the two-photon detuning , with E s k, the s-band spectrum. Similarly, for s-p band onsite transition the corresponding two-photon detuning reads ¢ the energy spectrum for the p-band states. Typically the bandwidths of s and p bands are much less than the s-p band gap which is E V E 2  Namely, for the conventional laser-assisted-hopping schemes, the minimum heating (denoted as CO min G ) requires that the frequency difference between Raman lasers be close to the half of the s-p band gap [13,14]. With these results in mind we obtain directly from the above two equations the following relation The life time can be estimated by V V , . In the present optical Raman lattice we can set that E sp D  , and then we have OR CO min G G  , which shows that the present optical Raman lattice has much less heating than that in the conventional schemes. In particular, with the parameter regime used in figure 2 s, which is extremely long enough for realistic experiments. We note that other more complicated possible heating mechanisms, such as the multiband effects and atom-atom interaction, are not considered in the present study, but deserves future efforts in more systematic investigations.
. It is clear that the third order K-term emerges due to the time-reversal-symmetry breaking. The summation in the third term means that each set of i j k , , ( ) consists of a minimum triangular. It can be verified that 2 In this case all spins experience a uniform magnetic field and the spin system respects the emergent translational symmetry which, however, is not respected by the original free fermion system. We note that the next-order coupling are four-spin interacting terms, with the four spins located in the four sites of a plaquette. Since the flux across each plaquette is π, the four-spin interacting terms do not break time-reversal symmetry, and are expected to have much weaker effect on the chiral spin liquid phase. The magnitudes of J 1,2 are fully controllable by tuning t 0,1 through V 0 and Raman potentials.
We solve the spinon mean-field phase diagram, as shown in figure 4(b) (details can be found in the appendix). It can be read that three different phases are clearly dominated by different interacting terms in H eff . The antiferromagnetic (Neel) or stripe order is obtained when the J 1 -or J 2 -term dominates. In the stripe phase the staggered spin order exists only in the x or y direction. On the other hand, when the three-spin interactions (K-terms) dominate, the CSL phase results [48][49][50][51]. In this regime, no symmetry-breaking order exists and the spin degree of freedom is captured by the bosonic 1 2 n = Laughlin state which has bulk semion excitations and chiral gapless spinons in the edge [52]. When K=0, the transition between Neel and stripe orders occurs at J J 2 , which is in the CSL phase region. We note that the spinon mean-field calculation only shows qualitative results of the predicted phases, while recent numerical simulation using density matrix renormalization group method also confirms the CSL phase in a similar spin model [53]. More advanced investigations of the current J 1 -J 2 -K model are necessary and will be presented in the next publication.

Conclusions
In conclusion, we have introduced the model of optical Raman lattice without SO coupling to observe chiral topological phases for cold atoms. We predict the QAH effect with a large gap-bandwidth ratio in the singleparticle regime, and in the interacting regime we realize the J 1 -J 2 -K model which supports the chiral spin liquid phase. The QAH state can be detected with a minimal measurement strategy, say, by only measuring the Bloch states in the two symmetric momentum points of the FBZ. The minimized heating in our scheme and vast tunability in parameters imply high feasibility for the observation of both the single-particle and strongly correlated topological states. Generalization of the present optical Raman lattice scheme to other situations, e.g. the high-orbital bands, 3D systems, and more exotic lattice configurations, shall realize different classes of topological states which might even have no prior analogue in solids. Especially, the correlation effects on such topological phases should be particularly interesting. This work opens a broad avenue in both theory and experiment for the studies of exotic topological states with cold atoms. 1 1 The initial relative phase between the light components of frequencies ω and 2w is irrelevant for the present study and is neglected. The in-plane polarized components (x and yˆpolarization components) generate the Here θ is the phase acquired through the path from mirror M 3 to lattice center, f represents the phase acquired by the laser beam propagating along the path from lattice center to the mirror M 1 , then to M 2 , and finally to the lattice center again (refer to figure 1 of where the additional 2 p -phase shift in the later term is due to the 1/4-wave plate for the zˆ-component light with frequency ω placed in the path from mirror M 3 to lattice center ( figure 1 (a) of the main text). Note that there is no interference between the two components with frequencies ω and 2w. It can be verified that the magnitudes of the phases ( , q f df -) only lead to global shift of the lattice, and therefore are irrelevant to our present study (they also do not affect the relative phase x y f f relating to the Raman potentials V Rx and V Ry ).
We then set these phase factors as zero to facilitate the description.
Together with the periodic Raman potentials induced through both the in-plane laser and the one propagating in z direction and having frequency w dw -, the total effective Hamiltonian for the optical Raman lattice is given by The third term in V x y , sq ( ) leads to an onsite energy offset V 1 D = between A and B sites, with V 1 being small compared with V 0 . The second term with amplitude V 0 reduces the difference in height of the barriers along the AB-bond and the diagonal (AA/BB) directions. Thus it can enhance the diagonal tunneling relative to the hopping coupling between A and B sites, providing vast tunability in parameters. The neighboring hoppings between A and B sites are restored when dw » D. Notice that both the Neel order M n and the stripe order M s are collinear, so they don't appear in decoupling the spin chirality interaction term S S S i j ḱ · . In the chiral spin liquid phase M M 0 n s = = , and from the above expressions we find that the spinons experience a uniform magnetic field which leads to the quantum Hall effect for spin degree of freedom. Then the ground state of the chiral spin liquid phase is captured by the bosonic 1 2 n = Laughlin state [52] which has chiral gapless anyonic spinon excitations in the edge [43,44]. The phase diagram can solved self-consistently based on the above decoupled formulas.