Emergent pseudospin-1 Maxwell fermions with threefold degeneracy in optical lattices

Emergent pseudospin-1 Maxwell fermions with threefold degeneracy in optical lattices

Yan-Qing Zhu National Laboratory of Solid State Microstructures and School of Physics, Nanjing University, Nanjing 210093, China    Dan-Wei Zhang zdanwei@126.com Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, SPTE, South China Normal University, Guangzhou 510006, China    Hui Yan Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, SPTE, South China Normal University, Guangzhou 510006, China    Ding-Yu Xing National Laboratory of Solid State Microstructures and School of Physics, Nanjing University, Nanjing 210093, China Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China    Shi-Liang Zhu slzhu@nju.edu.cn National Laboratory of Solid State Microstructures and School of Physics, Nanjing University, Nanjing 210093, China Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, SPTE, South China Normal University, Guangzhou 510006, China Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
July 5, 2019
Abstract

The discovery of relativistic spin-1/2 fermions such as Dirac and Weyl fermions in condensed matter or artificial systems opens a new era in modern physics. An interesting but rarely explored question is whether other relativistic spinal excitations could be realized with artificial systems. Here, we construct two- and three-dimensional tight-binding models realizable with cold fermionic atoms in optical lattices, where the low energy excitations are effectively described by the spin-1 Maxwell equations in the Hamiltonian form. These relativistic (linear dispersion) excitations with unconventional integer pesudospin, beyond the Dirac-Weyl-Majorana fermions, are a new kind of fermions named as Maxwell fermions. We demonstrate that the systems have rich topological features. For instance, the threefold degenerate points called Maxwell points may have quantized Berry phases and anomalous quantum Hall effects with spin-momentum locking may appear in topological Maxwell insulators in the two-dimensional lattices. In three dimensions, Maxwell points may have nontrivial monopole charges of with two Fermi arcs connecting them, and the merging of the Maxwell points leads to topological phase transitions. Finally, we propose realistic schemes for realizing the model Hamiltonians and detecting the topological properties of the emergent Maxwell quasiparticles in optical lattices.

I Introduction

Discovery of new particles in nature or new quasiparticles in condensed matter systems is at the heart of modern physics Wilczek (). One of the remarkable examples is the discovery of relativistic Dirac fermions emerged from graphene, which has attracted great interest in physics Castro (). Furthermore, it was demonstrated that Weyl fermions, which are massless spin-1/2 particles according to quantum field theory and never before observed as fundamental particles in nature, can emerge as quasiparticles in condensed matter Wan (); Huang (); Weng (); Xu1 (); Lv1 (); Xu2 (); Lv2 () or photonic crystals Lu (); Chan (). Most interestingly, Dirac and Weyl fermions have rich topological features Castro (); Wan (); Huang (); Weng (); Xu1 (); Lv1 (); Xu2 (); Lv2 (); Lu (); Chan (); Hasan (); Qi (). However, quasiparticles with higher spin numbers are also fundamentally important but rarely studied in condensed matter physics or artificial systems Bradlyn (); Lan (); Liang (). Recently, a pioneer work in this direction theoretically predicted that ”new fermions” (fermionic quasiparticle excitations) beyond the Dirac-Weyl-Majorana classification can emerge in some band structures with three- or more-fold degenerate points in the presence of time-reversal symmetry Bradlyn (). Very recently, the observation of three-component new fermions in the topological semimetal molybdenum phosphide has been reported Lv2017 (). Although it may be difficult to find these (and other) new fermions in condensed matter systems, they may also emerge from well-designed and tunable ultracold atomic systems, especially since Dirac and Weyl fermions have already been well studied in the field of cold atoms Zhu (); Zhang2012 (); Tarruell (); Duca (); Bermudez (); Dubcek (); Zhang2015 ().

In this paper, we propose and analyze an exotic kind of pseoduspin-1 fermions in two-dimensional (2D) square and three-dimensional (3D) cubic optical lattices, dubbed “Maxwell fermions” as they are analogous to massless spin-1 photons described by the Maxwell equations. We first rewrite the Maxwell equations in an anisotropic medium in the form of the Schrödinger equation and then construct 2D and 3D tight-binding models, where the low-energy excitations are described by the Maxwell Hamiltonian. With a tunable parameter, the systems can be varied in different quantum phases: topological or normal Maxwell insulator, and topological or normal Maxwell metal. The topological Maxwell metal is characterized by threefold degenerate points, known as Maxwell points, which have nontrivial monopole charges or quantized Berry phases and the low-energy excitation near the Maxwell point behaved like a photon described by the Maxwell equations. In the 2D system, we find interesting spin-momentum locking edge states in the Maxwell insulating phase, which is in analogy with the circularly-polarized polarization of the photons. In the 3D system, the topological properties of Maxwell fermions are similar to those of the Weyl fermions in Weyl semimetals, and the Maxwell points have monopole charges of with two Fermi arcs connecting them. The experimental scheme for implementation (and detection) of our models using three-component ultracold atoms in optical lattices is presented. Although some threefold band degeneracies were revealed in solid-state systems with the body-centered lattice structure and time-reversal symmetry in Ref. Bradlyn (), our proposal is essentially different. First, our threefold Maxwell points in 3D exist when the time-reversal symmetry of the system is broken, in which case the threefold band degeneracies in Ref. Bradlyn () will split into a number of Weyl points. Thanks to the broken time-reversal symmetry, the minimal number of threefold degeneracies in our 3D system can be two (which is thus a minimal model), instead of four in Ref. Bradlyn (). In addition, we use three atomic internal states to form the pseudospin-1 basis and thus only the simple cubic lattice is used in our proposal, in contrast to the required body-centered cubic lattices for spin-1/2 electrons in real materials. This enables us to realize exotic threefold fermions in a lattice of simplest geometry.

The paper is organized as follows. In Sec. II, we rewrite the Maxwell equations in the Schrödinger’s form and then present the general idea of realizing Maxwell fermionic quasiparticles in lattice systems. Section III introduces the 2D square-lattice model for realizing the Maxwell metals and insulators, and explores the topological properties of the emergent Maxwell fermions. In Sec. IV, we proceed to study the properties of 3D Maxwell fermions in the cubic-lattice model. In Sec. V, we propose schemes for realization of the model Hamiltonian and detection of the topological Chern numbers in the optical lattices. Finally, a brief discussion and a short conclusion are given in Sec. VI.

Ii Maxwell Hamiltonian and Maxwell fermions in lattice systems

In this section, we first rewrite the Maxwell equations in an anisotropic medium in the form of the Schrödinger equation, and then describe the general scheme for realizing the Maxwell fermions in artificial lattice systems.

ii.1 Maxwell equations in the Schrödinger’s form

In a region absent of charges and currents, the well-known Maxwell equations in matter are given by

(1)

where the displacement field with being the electric field, and the magnetic field with being the magnetizing field. Here () is the permittivity (permeability) of free space, and and are the relative permittivity and permeability, respectively. In an anisotropic medium, and are tensors rather than numbers. To simplify the proceeding analysis, we assume that the tensors and are simultaneously diagonalized, then the relationships between and , and now become

(2)

Thus Eq. (1) can be rewritten as

(3)

where , , and . Then we define the photon wave function as Oppenheimer (); Good ()

we have , and

(4)

where , , . When , that is, [the condition for obtaining a hermitian Hamiltonian, see Eq. (6)], then we can further rewrite Eq. (4) as

(5)

We hence obtain the following Schrödinger’s equation

(6)

This corresponds to the Maxwell equations in the anisotropic medium in the Schrödinger’s form

(7)

where the Hamiltonian is given by

(8)

Here , and is the Levi-Civita symbol. Noted that, , , and are the necessary and sufficient condition to obtain a hermitian Hamiltonian in Eq. (6). Typical case when , it returns to the free space situation and we obtain the related Hamiltonian of single photon in vacuum as . Here are the spin matrices for a particle of spin-1, which are defined as

(9)

One can check that , , and with . These matrices are the three generators of SU(3) group which have eight generators called Gell-Mann matirces. Equation (8) is a relativistic Hamiltonian to discuss a particles with pseudospin-1, which is analogous to the Weyl equation for the massless relativistic fermions with spin-.

ii.2 Maxwell fermions in lattice systems

The Maxwell Hamiltonian in Eq. (8) originally describes a massless relativistic boson (photon) with spin one. Moreover, in quantum field theory, bosons are identical particles with zero or integer spins, while fermions are particles with half integer spins. So it seems that the Maxwell Hamiltonian cannot be used to describe the fermionic particles. However, there is a fundamental difference between particles in a lattice and those at high-energy. Rather than constrained by Poincare symmetry in high-energy physics, quasiparticles in a lattice system are constrained only by certain subgroups (space groups) of the Poincare symmetry Bradlyn (). So there is the potential to find free fermionic excitations in lattice systems for which Hamiltonian is written in the form of Eq. (8). In the subsequent sections, we demonstrate that the Bloch Hamiltonian of certain well-designed lattice models can be written as

(10)

where denotes the Bloch vectors. Some threefold degenerate points exist in the bands of the model Hamiltonian where low-energy physics should be described by the Schrödinger equation with the Hamiltonian (8), and thus we call such quasiparticles Maxwell quasiparticles. Potential candidates include atoms in the optical lattices, electrons in certain crystals, and photons in photonic lattices. Here we focus on the fermionic atoms in optical lattices. For these fermionic atoms, the Maxwell quasiparticles are fermions instead of bosons (spin-1 photons) in the original Maxwell equations. In principle, Maxwell fermions can be realized with two different schemes. First, we can use non-interacting fermionic atoms in a square or cubic optical lattice and choose three atomic internal states in the ground state manifold to encode the three spin states (). Notably, the use of the atomic internal degree of freedom enables us to implement our model in a lattice of simplest geometry, i.e., a primitive square or cubic lattice. Alternatively, Maxwell fermions can be realized by using single-component fermionic atoms in optical lattices with three sublattices, where the pseudospin-1 basis is replaced by the three sublattices in a unit cell. For conceptual simplicity, we discuss the first scheme in the main text, and the realization of the second scheme is addressed in Appendix A.

Iii Maxwell fermions in 2D lattice systems

In this section, we construct a 2D tight-binding model on a square lattice and then investigate the intrinsic properties of the emergent pseudospin-1 Maxwell fermions in different topological phases.

iii.1 The 2D model

The 2D model Hamiltonian we considered is given by

(11)

where and represent the spin-flip hopping along the and axis with the tunneling amplitude , respectively. is the fermionic annihilation operator on site for the spin state , and with the tunable parameter is the strength of the on-site spin-flip.

Under the periodic boundary condition, Hamiltonian (11) can be rewritten as

where is the annihilation operator in momentum space . The Bloch Hamiltonian has the form of Eq. (10), where the Bloch vector is given by

(12)

with the lattice spacing and hereafter. The energy spectrum of this system is given by , which has a zero-energy flat band in the middle of the three bands. In the following, we illustrate that this model has rich phase diagrams: it is a normal insulator for with Chern number ; it is a topological insulator for except of when with Chern number ; it is a topological metal for with a quantized Berry phase ; it is a trivial metal for with Berry phase (see Appendix B).

iii.2 2D Maxwell fermions in Maxwell metals

The three bands touch at a single point when is or , and touch at two points when is . For , the three bands touch at in the energy spectrum shown in Fig. 1(a). We expand the Bloch Hamiltonian in the vicinity of the threefold degenerate point and obtain the following effective Hamiltonian for the low-energy excitations in the system

(13)

where is the effective speed of light and . This effective Hamiltonian takes the Maxwell Hamiltonian in Eq. (8) in 2D, and thus the dynamics of the low-energy excitations can be effectively described by the Maxwell equations. In this sense, we name these low-energy excitations Maxwell fermions and the threefold degeneracy point Maxwell point. When the Fermi level lies near the Maxwell point, the system can be named Maxwell metal, which is a metallic state due to the existence of the zero-energy flat band.

Figure 1: (Color online) (a) The energy spectrum for ; (b) The Berry curvature for and ; (c) The energy spectrum for ; (d) The Berry phase as a function of the parameter , which corresponds to the Chern number when the 2D system is in the insulating phase with .

It is interesting to note that the Maxwell point in the 2D lattice system has topological stability characterized by a quantized Berry phase. To study the topological stability, we calculate the Berry phase for a Maxwell fermion circling around the Maxwell point

(14)

where the Berry curvature with the Berry connection defined by the wave function in the -th () band . For this three-band system described by the Bloch Hamiltonian , the lowest-band Berry curvature in the - space can be rewritten as He1 ()

(15)

The distributions for fixed and typical parameters are plotted in Fig. 1(b), and the results show that is a Dirac- function at the Maxwell point. The numerical integration of over the Brillouin zone for gives the Berry phase , which is confirmed by analytical calculation. When , the single Maxwell point moves to the Brillouin edge with a quantized Berry phase , and the low-energy effective Hamiltonian becomes . When and with the energy spectrum shown in Fig. 1(c), there are two Maxwell points at and with the effective Hamiltonian

(16)

In this case, the Berry phase for both Maxwell points is , which corresponds to a trivial metallic state. So we can conclude that the single Maxwell point with linear dispersion relationship carrying a Berry phase is a unique topological property of this 2D Maxwell metallic state, which is different from the Dirac points in monolayer or bilayer graphene Yao ().

Figure 2: (Color online) Energy spectra and edge states in 2D lattices for (a) ; (b) ; (c) ; (d) ; and (e) . The inset in (c) shows the density distributions of four typical edge modes. The edge modes in (a-e) are plotted in red. The lattice site is under the open boundary condition.

iii.3 Maxwell edge modes in Maxwell insulators

The system is an insulator when since there is a gap between any two subbands. Under this condition, we can calculate the corresponding Chern number for the three bands with the band index :

(17)

We find that nonzero Chern numbers for and for , and thus the zero Chern number for the flat band (see Appendix B). Figure 1(d) shows the Berry phase of the lowest band as a function of the parameter , which indicates topological phase transition with band closing in this system when .

To further study the topological properties, we numerically calculate the energy spectrum of the system under the periodic boundary condition along the direction and under the open boundary condition along the direction with the length . The results in Fig. 2 show the variation of the energy spectra by changing the parameter . For [Fig. 2(a)], there is no edge mode between the two band gaps in this trivial insulating state with the Chern number . When decreases to critical values [Figs. 2(b) and 2(e)], the band gaps close and the system is in the nontrivial Maxwell metallic phase with Berry phase (corresponding to the Chern number ) and a branch of edge modes connecting the lowest (third) band and the middle flat band. For [Figs. 2(c) and 2(d)], the spectra contain two pairs of asymmetric branches of edge modes connecting the lowest (third) band and the middle flat band, which is consistent with bulk-edge correspondence with the bulk Chern number . The density distributions of some edge modes are shown in the inset of Fig. 2(c) for typical .

Figure 3: (Color online) (a) Expectation value of as a function of and with lattice sites under the open boundary condition; (b) Density distribution of for and ; (c) Schematic diagram for Maxwell edges states and in the Maxwell topological insulator with opposite momenta, both corresponding to the right circularly-polarized photons with helicity .

The edge modes in the topological insulator phase have novel properties. Without loss of generality, we explore the edge modes in the first band gap for parameter . We find a correspondence between the helicity of these edge states and the polarization of photons, so we named them Maxwell edge modes in this so-called Maxwell topological insulator. In particular, we reveal that this system exhibits the analogous quantum anomalous Hall effect Nagaosa (), with the edge modes being strong spin-momentum locking as eigenstates of the spin operator . This means that the two bunches of quasiparticle streams on the two edges (in the direction) can be treated as the streams of polarized Maxwell quasiparticles moving along the axis.

In Fig. 3(a), we numerically calculate the expectation value of the wave function with . The results show that the distribution of has two peaks localized at both the left and right edges with an opposite sign. To be more precise, we plot for and in Fig. 3(b), respectively. The result indicates that only the two edge states for each edge are the eigenstates of . The edge states on the left with positive eigenvalue are , and the ones on the right edge with negative eigenvalue are , where () are the unit vectors of Cartesian coordinates. So the effective Hamiltonian of edge states is given by

(18)

This effective Hamiltonian is none other than the 1D Hamiltonian of circularly-polarized photons. The helicity operator defined as

(19)

is the projection of the spin along the direction of the linear momentum Lan (). Thus, the edge quasiparticle-streams in this Maxwell topological insulator can be treated as Maxwell fermion-streams with the same helicities for opposite momenta, which satisfies the helicity conservation of massless photons in quantum field theory, as shown in Fig. 3(c). In addition, the momentum can also be considered the wave vector of the plane electromagnetic wave propagated along the axis. Both edge states and with the same helicities can be regarded as right circularly-polarized waves which constitute the two independent transverse polarization vector and with opposite momenta. We can see from Fig. 3(c) that the Maxwell edge modes moving along the () direction correspond to the right circularly-polarized waves rotating anticlockwise (clockwise) in the plane (along the axis) propagated along the () direction. Likewise, when with , the Maxwell edge modes on the left (right) edge with correspond to the left circularly-polarized waves propagated along the () direction. Because the electromagnetic waves are transverse waves, there is no longitudinal component and no edge mode with helicity , which corresponds to the unit wave vector of plane waves. So, our Maxwell edge modes with strong spin-momentum locking correspond perfectly to the circularly-polarized photons.

Iv Maxwell fermions in 3D lattice systems

In this section, we generalize the proposed model and results of Maxwell fermions to the 3D lattice system. We first construct a 3D lattice model by adding spin-flip hopping term along axis into the previous 2D model Hamiltonian in Eq. (11), and then study the topological properties of the Maxwell fermions near the 3D Maxwell points.

Figure 4: (Color online) (a) The phase diagrams with regimes I, II, III denote two pairs of Maxwell points and , one pair of Maxwell points , and normal band insulators, respectively. (b) The energy spectrum as a function of and for ; (c) The vector distribution of the Berry curvature at the plane. The figures in the red circles show around the Maxwell points as a sink or source in momentum space; (d) and (e) show Fermi arcs appear at and under the open boundary condition along the direction, respectively. The two black solid lines on the left represent two Fermi arcs connecting the two red circles which are the bulk states near the Maxwell points with opposite monopole charges at the top surface. The two black dot lines on the right represent Fermi arcs in the bottom surface. The two red lines in (e) also denote bulk states. The parameter in (b-e) is .

iv.1 The 3D model

The generalized 3D tight-binding model on a simple cubic lattice Hamiltonian is given by

(20)

where is the additional hopping term along the axis. The Bloch Hamiltonian of the 3D system takes the same form as Eq. (10), and the Bloch vectors with are given by

(21)

One can check that the spin-1 matrices satisfy the following relationship under the inversion operation and the time-reversal operation :

(22)

where and with and being the complex conjugate operator. Thus the Bloch Hamiltonian has an inversion symmetry represented by

(23)

but it does not have the time-reversal symmetry since

(24)

In this system, the Maxwell points can be manipulated through the tunable parameter . The phase diagram with respect to is shown in Fig. 4(a): the system is a Maxwell metal for , while it is a normal insulator for . Moreover, there are two pairs of Maxwell points denoted by and for ; there are a single pair of Maxwell points at () for (). At the critical points of , the two Maxwell points merge and then disappear by opening a gap when , corresponding to the normal insulating phase.

The three bands can touch at certain points to form threefold degeneracy points under the condition of . Considering the typical case of , we find that the band spectrum hosts two threefold degeneracy points in the first Brillouin zone at , as shown in Fig. 4(b). The low-energy effective Hamiltonian now becomes

(25)

where is the effective speed of light. This Hamiltonian takes the form of the isotropic 3D Maxwell Hamiltonian in Eq. (8), and thus these low-energy excitations are named 3D Maxwell fermions.

iv.2 Topological properties

To further study the topological properties of the 3D Maxwell fermions, we plot the vector distribution of the Berry curvature at the plane in Fig. 4(c). One can find that the Maxwell points behave as a sink and source of the Berry flux (see the 3D distribution of near the two points). Thus the Maxwell points behave like ¡°magnetic¡± monopoles in the momentum space with topological charges defined by the Chern numbers (see Appendix B)

(26)

which is twice as that of a Weyl point in Weyl semimetals. Once we fix as a parameter, can be viewed as a 2D Maxwell insulator, with the -dependent Chern number when and when otherwise. So there are always two Fermi arcs of surface states connecting a pair of Maxwell points with opposite topological charges, as shown in Figs. 4(d) and 4(e), in contrast to one Fermi arc Wan (); Huang (); Xu1 (); Lv1 () in Weyl semimetals. Here the surface and bulk states are plotted in black and red, and the surface states lie at the top and bottom surfaces along the direction are respectively denoted by solid and dot lines. It is interesting to note that the Fermi-arc surface states in this Maxwell metal diffuse to the bulk states due to the zero-energy flat band, which leads to non-zero energy Fermi arcs. In addition, the 3D Maxwell points in Maxwell metals has linear momenta along all the three directions, in contrast to the quadratic form of double-Weyl points Fang ().

Figure 5: (Color online) The merging of 3D Maxwell points. (a) When , two pairs of Maxwell points locate at and with a monopole charge of . (b) When , four Maxwell points appear at , and with a zero monopole charge. (c) Typical case of , where there are two Maxwell points located at with a monopole charge of . (d) Critical point when . The two Maxwell points with opposite monopole charges emerge at and then disappear by opening band gaps for .

We can change the positions of the Maxwell points inside the Brillouin zone by changing the parameter . Without loss of generality, we discuss the merging process of the Maxwell points for . For , there are two pairs of Maxwell points at and with monopole charges , as shown in Fig. 5(a). The two pairs of Maxwell points move together along the axis as we increase the parameter . They then emerge at and , respectively, with a monopole charge of for . Meanwhile, there are another pair of Maxwell points both with a monopole charge of created at , as shown in Fig. 5(b). When we continuously increase , two Maxwell points located at and disappear by opening gaps, and the single pair of Maxwell points denoted by move to the point with a monopole charge of , with the typical case for shown in Fig. 5(c). At the critical point in Fig. 5(d), the two Maxwell points with opposite monopole charges emerge at and then disappear by opening gaps when .

Finally in this section, we brief discuss the topological stability of the 3D Maxwell points. Unlike Weyl points that are strongly robust Wehling (), Maxwell points can only be stabilized by certain symmetry. In our proposed lattice systems, the band gaps will be opened and thus Maxwell points are gapped when the inversion symmetry is broken by introducing a perturbation term with one of the other five Gell-Mann matrices. This is due to the fact that the Maxwell Hamiltonian only takes three of the eight Gell-Mann matrices. So the topological stability of Maxwell points are weaker than Weyl points. However, Maxwell points would still be stable as long as the system is protected by certain symmetry, such as the inversion symmetry in our 3D system. In this case, the perturbations under the same symmetry would not open the band gaps and only change the positions of Maxwell points in the Brillouin Zone (see Fig. 5) until the topological phase transition occurs.

V Proposal for experimental realization and detection

Now we proceed to propose a realistic scheme for realizing the 2D and 3D model Hamiltonians with ultracold atoms in the square and cubic optical lattices by using the Raman-assisted tunneling method LAT1 (); LAT2 (); LAT3 (); LAT4 (); LAT5 (); LAT6 (); LAT7 (), respectively. We also suggest that the Chern numbers can be revealed from the shift of the hybrid Wannier center of an atomic cloud, based on a generalization of topological pumping in the optical lattices Thouless (); Marzari (); Smith (); Wang (); Pumping1 (); Pumping2 (); Pumping3 ().

Figure 6: Scheme for realizing the 2D model Hamiltonian. (a) Schematic diagram of the square optical lattice and the atomic hopping along the two axis denoted by and . (b) The atomic hopping configuration in the 2D model Hamiltonian in Eq. (11). The three atomic internal states form the (pseudo)spin-1 basis, such that represent spin-flip hopping and the term represent on-site spin-flip. A large linear tilt per lattice site along -direction is used to suppress the natural hopping and the hopping can then be restored by Raman lasers. (c) The Raman coupling scheme for engineering the required spin-flip hopping along each axis. The detuning in each direction matches the frequency offset of the corresponding Raman beams. (d) The Raman lasers with the corresponding polarization and propagation direction.

v.1 Proposed realization in optical lattices

To realize the 2D model Hamiltonian in Eq. (11), we consider noninteracting atoms in a titled square optical lattice with the lattice spacing and choose three atomic internal states in the ground state manifold to carry the spin states, as shown Fig. 6(a). The other levels in the ground state manifold are irrelevant as they can be depopulated by the optical pumping. The on-site spin-flip term can be easily achieved by applying a radio-frequency field or Raman beams for coupling the atomic internal states and . The major difficulty for implementing our model is to realize the spin-flip hopping terms and along each direction. By defining the superposition states

(27)

the two hopping terms can be diagrammatically visualized as

(28)

where indicates that the hopping is forbidden along this direction. The hopping terms and are shown in Fig. 6(b), which can be realized by using the Raman-assisted tunneling with proper laser-frequency and polarization selections LAT1 (); LAT2 (); LAT3 (); LAT4 (); LAT5 (); LAT6 (); LAT7 (). First, the required broken parity (left-right) symmetry is achieved by titling the lattice with a homogeneous energy gradient along the two directions, which can be created by the gravitational field or the gradient of a dc- or ac-Stark shift. Here we require different linear energy shifts per site along different directions, such as . Then the natural hopping is suppressed by the large tilt potential, and the hopping terms are restored and engineered by applying two-photon Raman coupling with the laser beams of proper configurations [see Fig. 6(c)].

We consider the realization of the hopping term to explain the Raman scheme, and first focus on the single term (here ) for details. This term corresponds to an atom in the spin state at site hopping to site while changing the spin state to with hopping strength , which can be visualized as . This hopping term can be achieved by two Raman beams polarized along -direction and with -polarization along -direction. Here the population of the excited state which is estimated by is negligible due to the large single-photon detuning . The two-photon detuning matching the linear energy shift of the lattice per site ensures that it only allows hopping to the right, and the other direction is forbidden by a large energy mismatch . We can address the spin states through the polarization selection rule since the original spin basis differ in the magnetic quantum number by one successively. Thus, a -polarized beam excites the state and a linear -polarized beam excites the superposition state as the polarization . These two beams together induce a Raman-assisted hopping between and . The hopping amplitude and phase are controlled by the corresponding Raman beam amplitude and phase LAT1 (); LAT2 (); LAT3 (); LAT4 (); LAT5 (); LAT6 (); LAT7 (), which can be written as

(29)

Here and we have used factorization of the Wannier function in the square lattice. If we adjust the interfering angle of the lattice beams to satisfy the condition , the site dependent phase term can always be reduced to . In this case, we can obtain the required hopping strength with . We note that these two Raman lasers simultaneously induce the hermitian conjugate process of the hopping , which is the desired spin-flip tunneling from to along the direction. Similarly, the second hopping term can be realized by the two Raman beams and polarized along -direction, which couple the state and since . Thus the hopping along the axis can thus be realized by three Raman beams with the configuration shown in Fig. 6(c).

The hopping terms along the axis can be realized in the similar way. Along this direction, the hopping term can be realized by three Raman beams which excites the state , and and which together effectively excite the state . The hopping term can be realized by two additional Raman beams and which effectively excite the state . Here a wave-vector difference leads to the same hopping strength along this axis. The laser configurations for realizing the desired hopping terms along the axis are also shown in Fig. 6(c). The total laser beams with the polarization and propagation directions for realizing all the hopping terms in the 2D model Hamiltonian in Eq. (11) are shown in Fig. 6(d). Note that the detuning in each direction matches the frequency offset of the corresponding Raman beams. It is also important to forbid the undesired tunneling terms that requires different linear energy shifts per site along the two axis, which can be achieved by adjusting the direction of the gradient field to be in different angles with respect to the axes of the square optical lattice.

Figure 7: (a) Schematic diagram of the simple cubic optical lattice and the atomic hopping. (b) The Raman coupling for realizing the hopping term along the axis.

The proposed Raman scheme can be directly extended to realize the 3D model Hamiltonian in Eq. (20). In this case, one can prepare the noninteracting atoms in a titled cubic optical lattice, as shown in Fig. 7(a). Here we require different linear energy shifts per site along different directions, such as . The additional hopping along the axis can be implemented with the similar Raman coupling. Along the axis, the hopping term can be diagrammatically visualized as

(30)

Combing with the laser that couples the state , two additional Raman beams and that couple the state can be used to induce the hopping and , respectively. The laser configurations for realizing the desired hopping term along the axis are shown in Fig. 7(b), and thus all the hopping terms along each direction in the 3D model Hamiltonian can be realized.

Although the implementation of the 2D and 3D model Hamiltonians involves a number of Raman beams, all of the lasers can be drawn from a same one with the small relative frequency shift induced by an acoustic optical modulator. In addition, one can lock the relative frequency differences of these beams by the driving fields of the modulator such that the absolute frequencies and their fluctuations are not important. In typical experiments, for instance, by using K atoms of mass in an optical lattice with the lattice constant nm, the gravity induces a potential gradient per lattice site kHz. Gravity can provide the required gradients along three directions with an appropriate choice of the relative axes of the frame to satisfy and , such that kHz. For a lattice with potential depth , where is the recoil energy, the overlap ratio and the natural tunneling rate Hz. For Raman beams with MHz and the single-photon detuning THz, one has kHz and the Raman-assisted hopping rate kHz. Thus the population of the excited state which is estimated by is negligible due to the large single-photon detuning . During the typical experimental time of the order of , the undesired off-resonant hopping probabilities with upper bounded by and the effective spontaneous emission rate estimated by with the decay rate of the excited state MHz would be negligible LAT6 (). We note that using several Raman lasers will lead to considerable heating effects in realistic experiments LAT7 (), which is the main disadvantage in our proposal. The cold atom system can be effectively described by the proposed model Hamiltonian as long as the duration of an experiment is short compared to the heating time (the typical lifetime of an atomic gas), which is about 100 ms in experiments containing the heating induced by the Raman couplings LAT4 (); LAT7 ().

v.2 Proposed detections

The 2D and 3D Maxwell points in the band structures that have related topological phase transition can be detected by the Bragg spectroscopy or Bloch-Zener oscillations, similar to the methods used for detecting Dirac and Weyl points in optical lattices Zhu (); Tarruell (); Zhang2012 (); Zhang2015 (). In addition, the Berry curvature and thus the quantized Berry phases and the Chern numbers can be measured by the newly-developed technique of tomography of Bloch bands in optical lattices BandTomography (); Li (), and the Chern numbers can also be revealed from the shift of an atomic cloud’s center-of-mass Bloch2015 (). Below we propose that the Chern numbers can also be revealed from the shift of the hybrid Wannier center of an atomic cloud, based on a generalization of topological pumping in optical lattices Thouless (); Marzari (); Smith (); Wang (); Pumping1 (); Pumping2 (); Pumping3 ().

The 2D Maxwell insulator can be viewed as a fictitious one-dimensional insulator subjected to an external parameter by using the dimension reduction method Wang (). Thus, its Chern number can be defined by the polarization

(31)

for the geometry of the underlying band structure. According to the modern theory of polarization, the Chern number defined in - space can be obtained from the change in polarization induced by adiabatically changing the parameter by :

(32)

For measuring , one can use another fact that the polarization can alternatively written as the center of mass of the Wannier function constructed for the single occupied band. In this system, the polarization can be expressed by means of the centers of the hybrid Wannier functions, which are localized in the axis retaining Bloch character in the dimension. The variation of the polarization and thus the Chern number are directly related to the shift of the hybrid Wannier center along the axis in the lattice. The shift of hybrid Wannier center by adiabatically changing is proportional to the Chern number, which is a manifestation of topological pumping with being the adiabatic pumping parameter

In the 2D lattice system, to construct and calculate the hybrid Wannier center, we can consider the Bloch Hamiltonian with parameter and recover it to the tight-binding Hamiltonian along the axis. The hybrid Wannier center can be written as Wang ()

(33)

where is the density distribution of hybrid Wannier center and denotes the atomic densities resolved along the direction as a function of . The density distribution can be written as