Unconventional states of bosons with synthetic spin-orbit coupling

# Unconventional states of bosons with synthetic spin-orbit coupling

## Abstract

Spin-orbit coupling with bosons gives rise to novel properties that are absent in usual bosonic systems. Under very general conditions, the conventional ground state wavefunctions of bosons are constrained by the “no-node” theorem to be positive-definite. In contrast, the linear-dependence of spin-orbit coupling leads to complex-valued condensate wavefunctions beyond this theorem. In this article, we review the study of this class of unconventional Bose-Einstein condensations focusing on their topological properties. Both the 2D Rashba and 3D -type Weyl spin-orbit couplings give rise to Landau-level-like quantization of single-particle levels in the harmonic trap. The interacting condensates develop the half-quantum vortex structure spontaneously breaking time-reversal symmetry and exhibit topological spin textures of the skyrmion type. In particular, the 3D Weyl coupling generates topological defects in the quaternionic phase space as an SU(2) generalization of the usual U(1) vortices. Rotating spin-orbit coupled condensates exhibit rich vortex structures due to the interplay between vorticity and spin texture. In the Mott-insulating states in optical lattices, quantum magnetism is characterized by the Dzyaloshinskii-Moriya type exchange interactions.

###### pacs:
Keywords: Bose-Einstein condensation, spin-orbit coupling, Landau level, time-reversal symmetry, spin texture, skyrmion, quaternion, Dzyaloshinskii-Moriya.

## I Introduction

Spin-orbit (SO) coupling plays an important role in interdisciplinary areas of physics. In quantum mechanics, SO coupling arises from the relativistic effect as a low energy approximation to the Dirac equation. Its semi-classic picture is the Thomas precession that electron spin moment couples to a velocity-dependent effective magnetic field generated by the Lorentz transformation of the electric field. In atomic physics, SO coupling constitutes one of the basic elements to the formation of the atomic structures. The development in condensed matter physics shows that SO coupling is indispensable for important phenomena ranging from spintronics ifmmodecheckZelsevZfiutiifmmodeacutecelsecfi2004 (), anomalous Hall effects Nagaosa2010 (); Xiao2010 (), spin Hall effects Dyakonov1971 (); Hirsch1999 (); Murakami2003 (); Sinova2004 (), to topological insulators Hasan2010 (); Qi2011 (). In particular, topological insulators have become a major research focus of current condensed matter physics.

Most current studies of SO coupling are considered for fermionic systems of electrons. On the other hand, the ultra-cold atomic systems have opened up a whole new opportunity to explore novel states of matter that are not easily accessible in usual condensed matter systems. In particular, it currently becomes experimentally possible to implement various kinds of SO coupled Hamiltonians in ultracold atomic gases for both fermions and bosons Galiski2013 (); Lin2009 (); Lin2009a (); Lin2011 (); Zhang2012a (); Wang2012 (); cheuk2012 (); Qu2013 (). The high controllability of these systems makes them an ideal platform to explore novel SO coupled physics with bosons.

An important property of bosons is the “no-node” theorem, which states that in the coordinate representation the many-body ground state wavefunctions are positive-definite Feynman1972 (). This theorem is valid under very general conditions such as the Laplacian type kinetic energy, arbitrary single-particle potentials, and coordinate-dependent two-body interactions. It applies to most of the known ground states of bosons, including Bose-Einstein condensations (BECs), Mott insulators, and supersolids. Technically, it indicates that the ground state wavefunctions of bosons can be reduced to positive-definite distributions, and thus imposes a strong constraint on bosonic states. For example, it rules out the possibility of time-reversal (TR) symmetry breaking ground states in traditional boson systems. Considerable efforts have been made in exploring unconventional BECs beyond the “no-node” theorem Wu2009 (). One way is the meta-stable state of bosons in high orbital bands in optical lattices Isacsson2005 (); Liu2006 (); Kuklov2006 (); Cai2011 (), because the “no-node” theorem does not apply to excited states. Unconventional BECs with complex-valued condensate wavefunctions have been experimentally realized Muller2007 (); Wirth2010 (); Olschlager2011 ().

The kinetic energies of SO coupled systems are no longer Laplacian but linearly depend on momentum, which invalidates the necessary conditions for the “no-node” theorem as pointed out in Ref. [Wu2009, ]. This provides another way towards unconventional BECs. For instance, the Rashba SO coupled BECs were early investigated for both isotropic Wu2011 () and anisotropic cases Stanescu2008 (). In the isotropic case, Rashba coupling leads to degenerate single-particle ground states along a ring in momentum space whose radius is proportional to the SO coupling strength. Such a momentum scale is absent in usual BECs in which bosons are condensed to the zero momentum state, and thus bears certain similarities to Fermi momentum in fermion systems. If interaction is spin-independent, the condensates are frustrated in the free space at the Hartree-Fock level, and quantum zero point energy selects a spin-spiral state based on the “order-from-disorder” mechanism. Imposing the trapping potential further quantizes the motion around the SO ring which leads to Landau level type quantization of the single-particle levels. Under interactions, condensates spontaneously break TR symmetry exhibiting topologically non-trivial spin textures of the skyrmion type Wu2011 (). All these features are beyond the framework of “no-node” theorem.

Recently, SO coupled systems with ultra-cold bosons have aroused a great deal of research interest both experimental and theoretical Galiski2013 (). Experimentally, pioneered by Spielman’s group Lin2009 (); Lin2009a (); Lin2011 (), BECs with SO coupling in the anisotropic 1D limit have been realized by engineering atom-laser interactions through Raman processes Lin2009 (); Lin2009a (); Lin2011 (); Zhang2012a (); Wang2012 (); Qu2013 (). Condensations at finite momenta and exotic spin dynamics have been observed Lin2009 (); Lin2009a (); Lin2011 (); Zhang2012a (); Wang2012 (); cheuk2012 (); Qu2013 (). Various experimental schemes have proposed to realize the isotropic Rashba SO coupling Jaksch2003 (); Ruseckas2005 (); Osterloh2005 (); Campbell2011 (); Juzeliunas2011 (); Dalibard2011 (). On the side of theory, the Rashba SO coupled bosons have been extensively investigated under various conditions, including the exotic spin structures in the free space, spin textures in harmonic traps, vortex structures in rotating traps, and the SO coupled quantum magnetism in the Mott-insulating states Stanescu2008 (); Wu2011 (); Wang2010 (); Ho2011 (); Anderson2011 (); Burrello2010 (); Burrello2011 (); Kawakami2011 (); Li2011 (); Sinha2011 (); Xu2011a (); Yip2011 (); Zhu2011 (); Anderson2012 (); Anderson2012a (); Barnett2012 (); Deng2012 (); Grass2012 (); He2012b (); Hu2012a (); LiYun2012 (); Ramachandhran2012 (); Ruokokoski2012 (); Sedrakyan2012 (); Xu2012 (); Xu2012b (); Zhang2012 (); Zhang2012c (); Zhang2012d (); Zheng2012 (); Cui2012 (). Furthermore, a recent progress shows that a 3D -type SO coupling can also been implemented with atom-laser interactions Li2012c (); Anderson2012a (). This is a natural symmetric extension of Rashba SO coupling to 3D dubbed Weyl SO coupling below due to its similarity to the relativistic Hamiltonian of Weyl fermions Weyl1929 (). The Weyl SO coupled BECs have also been theoretically investigated Li2012b (); Kawakami2012 (); Zhang2013 (); Anderson2012a ().

In addition to the ultra-cold atom systems, recent progress in condensed matter systems has also provided an SO coupled boson system of excitons. Excitons are composite objects of conduction electrons and valence holes, both of their effective masses are small, thus relativistic SO coupling exists in the their center of mass motion. The effects of SO coupling on exciton condensations have been theoretically investigated Yao2008 (); Wu2011 (), including the spin texture formations Wu2011 (). An important experiment progress has been achieved in Butov’s group High2011 (); High2012 (), that spin textures in a cold exciton gas have been observed in GaAs/AlGaAs coupled quantum wells from the photoluminescence measurement.

In the rest of this article, we review the current theoretical progress of studying SO coupled bosons including both the 2D Rashba and 3D Weyl SO couplings. Our emphases will be on the non-trivial topological properties which are absent in conventional BECs. The single-particle spectra will be reviewed in Section II. They exhibit a similar structure to the Landau level quantization in the sense that the dispersion with angular momentum is strongly suppressed by SO couplings Wu2011 (); Burrello2010 (); Burrello2011 (); Li2012c (); Hu2012a (); Sinha2011 (); Ghosh2011 (); Li2012b (). However, a crucial difference from the usual magnetic Landau levels is that these SO coupling induced Landau levels maintain time-reversal symmetry, and thus their topology belongs to the class Wu2011 (). The interplay between interactions and topology gives rise to a variety of topological non-trivial condensate configurations and spin textures, which will be reviewed in Section III. In particular, the 3D condensates with the Weyl coupling exhibit topological defects in the quaternionic phase space. It is exciting to find an application of the beautiful mathematical concept of quaternions. In Section IV, we review the SO coupled BECs in rotating traps, which are subject to both the Abelian vector potential due to the Coriolis force and the non-Abelian one from SO coupling. The combined effects of the vorticity and spin topology lead to rich structuresZhouXF2011 (); Radic2011 (); Xu2011 (); Liu2012 (); Zhao2013 (), including half quantum vortex lattices, multi-domain of plane-wave states, and giant vortices. In Section V, we summarize the current progress on strongly correlated spin-orbit coupled systems Ramachandhran2013 (); grass2012b (); grass2012c (); palmer2011 (); komineas2012 (). Furthermore, in the strongly correlated Mott-insulators, SO coupling effects exhibit in the quantum magnetism as the Dzyaloshinskii-Moriya type exchange interactions Cai2012 (); Cole2012 (); Radic2012 (); Gong2012 (); Mandal2012 (); grass2011 (), which will be reviewed in Section VI. Conclusions and outlooks are presented in Section VII.

Due to the rapid increasing literatures and the limit of space, we will not cover other important topics, such as SO coupled fermions Ghosh2011 (); Iskin2011 (); Jiang2011 (); Li2011 (); Zhou2011 (); Doko2012 (); He2012 (); He2012a (); Hu2012 (); Liu2012a (); Maldonado-Mundo2012 (); Martone2012 (); Orth2012 (); Vyasanakere2012 (); Zhang2012b (); cheuk2012 (), the SO coupled dipolar bosons Deng2012 (); Zhao2013 (). and the proposals for experimental implementations Jaksch2003 (); Ruseckas2005 (); Osterloh2005 (); Campbell2011 (); Juzeliunas2011 (); Dalibard2011 (); tagliacozzo2012 (); goldman2013 ().

## Ii The SO coupled single-particle spectra and the Landau level quantization

We begin with the single-particle properties. Consider the following Hamiltonian of 2D two-component atomic gases with an artificial Rashba SO coupling defined as

 H2D,R0 = →p22M+Vtp(→r)−λR(σxpy−σypx), (1)

where ; the pseudospin components and refer to two different internal atomic components; is the Rashba SO coupling strength with the unit of velocity; is the external trapping potential, and is the characteristic frequency of the trap. Another SO coupled Hamiltonian will be considered is the 3D Weyl SO coupling defined as

 H3D,W0 = →p22M+Vtp(→r)−λW→σ⋅→p, (2)

where is the SO coupling strength.

Even though we will mostly consider bosons for the Hamiltonians of Eq. (1) and Eq. (2), they possess a Kramer-type TR symmetry satisfying , where is the complex conjugate operation. At the single particle level, there is no difference between bosons and fermions. Both Hamiltonians are rotationally invariant but break the inversion symmetry. The 2D Hamiltonian Eq. (1) still possesses the reflection symmetry with respect to any vertical plane passing the center of the trap. For the 3D Hamiltonian Eq. (2), no reflection symmetry exists.

These two typical types of SO interactions have received a lot of attention recently in the community of ultra-cold atoms due to their close connection to condense matter physics. There have already been great experimental efforts on realizing spin-orbit coupling through atom-light interactions Lin2009 (); Lin2009a (); Lin2011 (); Wang2012 (); Zhang2012 (); Qu2013 (). In fact, several proposals for experimental implementations of Eq. (1) and Eq. (2) have appeared in literatures Ruseckas2005 (); Juzeliunas2011 (); Anderson2012a (); Dalibard2011 (); LiYun2012 ().

In this section, we review the single-particle properties of Eqs. (1) and (2) focusing on their topological properties. In Sec. II.1, their Berry phase structures in momentum space are presented. When the quadratic harmonic trap potential is imposed, the Landau-level type quantization on the energy spectra appears with TR symmetry as shown in Sec. II.2. This Landau level quantization provides a clear way to understand novel phases of bosons after turning on interactions. In Sec. II.3, wavefunctions of the lowest Landau levels of Eq. (1) and Eq. (2) are explicitly provided. The topology of these Landau level states are reviewed through edge and surface spectra in Sec. II.4.

### ii.1 Berry connections in momentum space

Both Eq. (1) and Eq. (2) possess non-trivial topology in momentum space. Let us begin with the 2D Rashba Hamiltonian Eq. (1) in the free space, i.e., . Its lowest single-particle states in free space is not located at the origin of momentum space but around a ring with the radius . The spectra read

 ϵ±(→k)=ℏ22M(k∓kRso)2, (3)

where refer to the helicity eigenvalues of the operator . The corresponding two-component spin wavefunctions of plane-wave states are solved as

 |ψ→k±⟩=1√2⎛⎜ ⎜⎝ce−iϕ→k2∓ieiϕ→k2⎞⎟ ⎟⎠, (4)

where is the azimuthal angle of in the -plane.

For bosons, the lower energy branch states with a fixed helicity are important. The Berry connection of positive helicity states is defined as

 →A(→k)=⟨ψ→k+|i→∇k|ψ→k+⟩=12k^eϕ→k, (5)

where is the unit vector along the azimuthal direction. The Berry curvature is defined as . For a loop winding around the origin , the Berry phase is

 ∮d→k⋅→A(→k)=π. (6)

This is because a two-component spinor after rotating 360 does not return to itself but acquires a minus sign. Consequently, is zero everywhere except contributing a -flux at the origin of momentum space.

Next we consider the 3D generalization of the Rashba SO coupling of the type, i.e., the Weyl coupling. Now in the free space without the trap, the lowest energy single-particle states are located around a sphere in momentum space with the radius also denoted as with the value of , and the spectra are

 ϵ±(→k)=ℏ22M(k∓kWso)2, (7)

where the subscripts refer to the helicity eigenvalues of the operator . The corresponding eigenstates are solved as

 |ψ→k−⟩=⎛⎜⎝−sinθ→k2cosθ→k2eiϕ→k⎞⎟⎠,  |ψ→k+⟩=⎛⎜⎝cosθ→k2sinθ→k2eiϕ→k⎞⎟⎠, (8)

where and are the azimuthal and polar angles of in the spherical coordinates. The Berry connection of the positive helicity states is

 →A(→k)=12tanθ→k2^eϕ→k, (9)

which is the vector potential for a unit magnetic monopole located at the origin of momentum space, and is the azimuthal direction of . Defining , the corresponding Berry curvature is , where is the radial direction of .

### ii.2 Landau-level quantization in the harmonic trap from SO couplings

The SO couplings in Eqs. (1) and (2) introduce a SO length scale even in the free space defined as . Here and the following, we omit the superscripts of and without loss of generality. The physical meaning of is as follows: the low energy states of Eqs. (1) and (2) are not of long-wave length as usual but featured with large magnitude of momentum depending on the SO coupling strength. is the length scale of wavepackets that can be formed by using the low energy states on the SO ring of Eq. (1) or the SO sphere of Eq. (2). On the other hand, in the typical experimental setup with ultra-cold quantum gases, a harmonic trap is used to confine atoms. The trapping potential introduces another length scale as as the typical system size. The trap energy scale is .

It is useful to define a dimensionless parameter to describe the relative strength of SO coupling with respect to the trapping potential. Physically, is the number of wavepackets which can be packed in the trap length. In the limit of large values of , the trapping potential gives rise to Landau level type quantizations in both 2D and 3D spin-orbit coupling systems Wu2011 (); Hu2012 (); Ghosh2011 (); LiYun2012 ().

The terminology of Landau levels in this section is generalized from the usual 2D magnetic case as: topological single-particle level structures labeled by angular momentum quantum numbers with flat or nearly flat spectra. On open boundaries, Landau levels systems develop gapless surface or edge modes which are robust against disorders. We will see that the low energy states of both Eq. (1) and Eq. (2) satisfy this criterion in the case .

#### TR invariant Landau levels from the 2D Rashba SO coupling

Let us briefly recall the usual 2D Landau level arising from the magnetic field. In the symmetric gauge with , its Hamiltonian is simply equivalent to a 2D harmonic oscillator in a rotating frame as

 H2D,LL=(→p−ec→A)22M=p22M+12Mω2r2−ωLz, (10)

where and . Inside each Landau level, the spectra are degenerate with respect to the magnetic quantum number . Non-trivial topology of Landau levels comes from the fact that does not take all the integer values. For example, in the lowest Landau level, starts from and runs all the positive integer number. This chiral feature is a TR symmetry breaking effect due to the magnetic field.

Next let us consider the 2D Rashba SO coupling of Eq. (1) in the limit of . The physics is most clearly illustrated in momentum representation. After projected into the low energy sector of the positive helicity states, the harmonic potential in momentum space becomes a Laplacian subjected to the Berry connection as

 Vtp(→∇→k)=M2ω2(i→∇→k−→A→k)2, (11)

where is given in Eq. (5) with a -flux at the origin of the 2D - plane. In momentum space, the trapping potential quantizes the motion on the low energy spin-orbit ring with radius , and is mapped to a planar rotor problem. The moment of inertial in momentum space is where is the mass in momentum space defined as , and the angular dispersion of energy is . Due to the -flux phase at , is quantized to half-integer values. On the other hand, the radial component of the trapping potential in momentum space is just the kinetic energy for the positive helicity states

 HK=12Mkω2(k−kso)2. (12)

For states near the low energy spin-orbit ring, the radial motion can be approximated as 1D harmonic oscillations, and the energy gap remains as . Combining the radial and angular dispersions together, we arrive at

 Enr,jz≈{nr+j2z2α2+12(1−α2)}Etp, (13)

where is the radial quantum number, and is the constant of zero point energy.

The degeneracy over angular momentum quantum numbers is a main feature of Landau level quantization. For the Hamiltonian Eq. (1), although its spectra Eq. (13) are not exactly flat with respect to , they are strongly suppressed at , thus these low energy levels are viewed as Landau levels. The radial quantum number serves as the Landau level index and the gaps between Landau levels are roughly . For states in the -th Landau level with , their energies remain lower than the bottom of the next Landau level, thus they can be viewed as gapped bulk states. Actually, the similarities of these SO coupled states to Landau levels are more than just spectra flatness but their non-trivial topology will be explained in Sec. II.4.

#### 3D Landau levels from the Weyl SO coupling

The 3D Landau level systems are not as well-known as the 2D case of Eq. (10). Recently, a large progress has been made in generalizing Eq. (10) to 3D with exactly flat energy dispersions Li2011 (); Li2012d (). In particular, they can be constructed with the full 3D rotation symmetry by coupling spin- fermions with the SU(2) gauge potential. The Hamiltonian is equivalent to a 3D harmonic oscillator plus SO coupling as

 H3D,LL=p22M+12Mω2r2−ω→L⋅→σ. (14)

Excitingly, the lowest Landau level wavefunctions of Eq. (14) possess elegant analytic properties, satisfying the Cauchy-Riemann-Fueter condition of quaternionic analyticity. Just like that the complex analyticity is essential for the construction of fractional quantum Hall Laughlin states, the quaternionic analyticity is expected to play an important role in high dimensional fractional topological states. These 3D Landau level states preserve both TR and parity symmetry. The 3D Landau levels have also been generalized to the relativistic Dirac particles Li2012 ().

Now let us come back to the Hamiltonian of Eq. (2) with the 3D Weyl SO coupling and a trap potential. The parallel analysis to the 2D Rashba case applies. Again, in the limit of , after the projection into the sector of the positive helicity states, the trap potential becomes and takes the form of a magnetic monopole one in Eq. (9). The problem is reduced to a spherical rotor problem in momentum space on the low energy SO sphere with the radius . The monopole structure of the Berry connection quantizes the total angular momentum to half-integer values. Similarly to the 2D Rashba case, the low energy spectra are approximated as

 Enr,j,jz≈{nr+j(j+1)2α2+12(1−α2)}Etp. (15)

Again the angular dispersion is strongly suppressed by SO coupling at . These spectra exhibit quasi-degeneracy over the 3D angular momentum good quantum numbers of and , and thus can be viewed as a 3D Landau level quantization with TR symmetry. The length scale of these Landau level states is also the SO length . Topological properties of these Landau level states will be studied in Sec. II.4.

### ii.3 Lowest Landau level wavefunctions and parent Hamiltonians

The Landau level energy spectra of Eq. (13) in 2D and Eq. (15) in 3D are not exactly flat but with weak dispersions over angular momentum quantum numbers. Nevertheless, parent Hamiltonians based on slight modification on Eqs. (1) and (2) can be constructed. Their lowest Landau level spectra are exactly flat and their wavefunctions can be solved analytically as shown in Eqs. (17) and (19) below. These wavefunctions maintain TR symmetry but break parity. In the limit of and for Landau level states with angular momenta in 2D or in 3D, the lowest Landau level wavefunctions of Eqs. (1) and (2) are well approximated by these expressions.

For the 2D case, the parent Hamiltonian is just

 H2D,P0 = H2D,R0−ωLzσz, (16)

where and the coefficient is the same as the trap frequency. As shown in Ref. [Li2012c, ], its lowest Landau level wavefunctions are solved as

 ψLLL2D,jz(r,ϕ)=e−r22l2T(eimϕJm(ksor)−ei(m+1)ϕJm+1(ksor)), (17)

where is the azimuthal angle; ; is the -th order Bessel function. The lowest Landau level energy is exactly flat as .

In the case of and for small values of , the decay of the wavefunctions Eq. (17) is controlled by the Bessel functions rather than the Gaussian factor. Their classic orbit radiuses scale as . Since linearly depends on , the effect of the term compared to that of the Rashba one is at the order of . Thus Eq. (16) is simply reduced to Eq. (1) whose lowest Landau level wavefunctions are well approximated by Eq. (17). In this case, the length scale of Landau level states is determined by the SO length instead of the trap length . The reason is that these Landau levels are composed from plane-wave states with a fixed helicity on the low energy Rashba ring. The confining trap further opens the gap at the order of between SO coupled Landau levels.

On the contrary, in the opposite limit, i.e., , the term dominates and the Rashba term can be neglected. In this case, Eq. (16) is reduced into with conserved. In each spin eigen-sector, it is just the usual Landau level Hamiltonian in the symmetric gauge with opposite chiralities for spin up and down, respectively. Nevertheless, at , the approximation of the projection into the Rashba ring for Eq. (1) is not valid, and the eigenstates are no longer Landau levels. For the intermediate values of , the physics is a crossover between the above two limits.

Following the same logic, the 3D parent Hamiltonian with exactly flat SO coupled Landau levels is

 H3D,P0=H3D,W0−ω→L⋅→σ, (18)

where is the 3D orbital angular momentum, and the coefficient of the term is the same as the trap frequency. Again, as shown in Ref. [Li2012c, ], the lowest Landau level wavefunctions of Eq. (18) are solved analytically as

 ψLLL3D,jjz(→r) = e−r22l2T{jl(ksor)Y+,j,l,jz(Ωr)+ijl+1(ksor) (19) × Y−,j,l+1,jz(Ωr)},

where is the -th order spherical Bessel function; ’s are the SO coupled spherical harmonics with total angular momentum quantum numbers and , which are composed of the spherical harmonics and spin- spinors. These lowest Landau level states are degenerate over all the values of with .

Following the same reasoning as in the 2D case, in the limit of , we can divide the lowest Landau level states of Eq. (19) into three regimes as , , and , respectively. At , the classic orbit radius scales as , and thus comparing with is a perturbation at the order of . In this regime, the lowest Landau level wavefunctions of Eq. 2 are well approximated by Eq. (19). On the contrary, in the regime of , dominates over , thus the eigenstates of Eqs. (18) and (2) are qualitatively different. In this case, Eq. (18) is reduced to the 3D Landau level Hamiltonian Eq. (14).

### ii.4 The Z2-stability of helical edge and surface states

Non-trivial topology of the 2D Landau level manifests from the appearance of robust gapless edge states. The classic radius of each Landau level state expands as increases. For example, in the lowest Landau level, where is the cyclotron radius. With an open boundary, as becomes large enough, states are pushed to the boundary halperin1982 (). Unlike the flat bulk spectra, the edge spectra are dispersive, always increasing with , and thus are chiral and robust against external perturbations. Each Landau level contributes one branch of chiral edge modes. If the system is filled with fermions, when chemical potential lies in the gap between Landau levels, the chiral edge states give rise to the quantized charge transport.

For Landau levels of the Rashba SO coupling in Eq. (1), a marked difference is that these states are TR invariant. The angular momentum in Eq. (13) takes all the half-integer values as , and thus these states are helical instead of chiral. Since the system described by Eq. (1) does not possess translation symmetry, the usual method of calculating topological index based on lattice Bloch wave structures in Brillouin zones does not work kane2005 (); fu2007 (); moore2007 (); roy2010 ().

Nevertheless, the non-trivial topology should exhibit on the robustness of edge states. The trap length can be used as the sample size by imposing an open boundary condition at . States with are bulk states localized within the region of . States with are pushed to the boundary, whose spectra disperse to high energy rapidly. For a given energy lying between Landau level gaps, each Landau level with bulk energy blow contributes to a pair of degenerate edge modes due to TR symmetry. Nevertheless, these two edge modes are Kramer doublets under the TR transformation satisfying . The celebrated Kane-Mele argument for translational invariant systems kane2005 () can be generalized to these rotation invariant systems by replacing linear momentum with the angular momentum. If a given energy cuts odd numbers of helical edge modes, then any TR invariant perturbation cannot mix these modes to open a gap. Consequently, the topological nature of such a system is characterized by the index.

When loading fermions in the system, and if the Fermi energy cuts the edge states, these helical edge states become active. The effective helical edge Hamiltonian can be constructed by imposing an open boundary at . The effective helical edge Hamiltonian in the basis of can be written as

 Hedge=∑jz(ℏvflT|jz|−μ)ψ†nr,jzψnr,jz, (20)

where is the chemical potential. If the edge is considered locally flat, Eq. 20 can be rewritten in the plane-wave basis. Due to the reflection symmetry with respect to the plane perpendicular to the edge, the spin polarization for momentum along the edge direction must lie in such a plane. Also combining with TR symmetry, we have

 Hedge = v[(→p×^n)⋅^z](σzsinη+(→σ⋅^n)cosη),

where is the local normal direction on the circular edge in the 2D plane; is the linearized velocity of the edge modes around Fermi energy; is a parameter angle depending on the details of the systems. There are the only terms allowed by rotation symmetry, TR symmetry, and the vertical mirror symmetry in this system. Each edge channel is a branch of helical one-dimensional Dirac fermion modes.

Parallel analysis can be applied to the helical surface states of the 3D Hamiltonian Eq. (2). Again, due to the TR symmetry, surface states are helical instead of chiral. The topological class also belongs to . If the surface is sufficiently large, and thus can be locally taken as flat, we can construct the surface Dirac Hamiltonian around the Fermi energy by using plane-wave basis basing on symmetry analysis. First, due to the local rotational symmetry around , the in-plane momentum and cannot couple to , thus spin polarization for each in-plane momentum has to lie in the -plane. Generally speaking, the spin polarization vector form an angle with respect to and such an angle is determined by the details of the surface. Combine all the above information, we arrive at

 Hsfc = v{sinη(→p×→σ)⋅^n+cosη[→p⋅→σ−(→p⋅^n)(→σ⋅^n)]},

where is the local normal direction to the 2D surface.

## Iii Topological spin textures and the quaternionic phase defects in a harmonic trap

In this section, we review the unconventional BECs with interactions and SO couplings, including both Rashba and the 3D Weyl types, in the harmonic trap. The 2D Rashba case is presented in Sec. III.1. The linear dependence on momentum of SO coupling invalidates the proof of “no-node” theorem. Consequently, a general feature of SO coupled BECs is the complex-valued condensate wavefunctions and the spontaneous TR symmetry breaking. For the Rashba case, the skyrmion type spin textures and half-quantum vortex were predicted in the harmonic trap Wu2011 (). Furthermore, due to the Landau level structures of single-particle states, rich patterns of spin textures have been extensively investigated in literatures Wu2011 (); Hu2012a (); Sinha2011 (). A nice introduction of topological defects in the ultra-cold atom context can be found in Ref. [Zhou2003, ].

Even more interesting physics shows in 3D Weyl SO coupling, which will be reviewed in Sec. III.3. The non-trivial topology of the condensate wavefunction is most clearly expressed in the quaternionic representation Li2012 (); Kawakami2012 (). Quaternions are a natural extension of complex numbers as the first discovered non-commutative division algebra, which has been widely applied in quantum physics adler1995 (); finkelstein1962 (); balatsky1992 (). The condensation wavefunctions exhibit defects in the quaternionic phase space as the 3D skyrmions, and the corresponding spin density distributions are characterized by non-zero Hopf invariants.

### iii.1 Half-quantum vortices and spin texture skyrmions with Rashba SO coupling

Let us consider a 3D two-component boson system with contact spin-independent interactions and with Rashba SO coupling in the -plane. Since the Rashba SO coupling is 2D, interesting spin textures only distribute in the -plane. For simplicity, the condensate is set uniform along the -direction, then the problem is reduced to a 2D Gross-Pitaevskii (GP) equation as

 { − ℏ2∇22M+iℏλR(∇xσy,αβ−∇yσx,αβ)+gn(r,ϕ) (23) + 12Mω2r2}ψβ(r,ϕ)=Eψα(r,ϕ),

where ’s with are two-component condensate wavefunctions; is the particle density; describes the -wave scattering interaction. The interaction energy scale is defined as , where is the system size along the -axis, and the dimensionless interaction parameter is defined as .

We start with weak SO coupling, , and with weak interactions. In this case, the energy of single-particle ground state with is well separated from other states. If interactions are not strong enough to mix the ground level with other levels, the condensate wavefunction remains the same symmetry structure carrying , or, , thus bosons condense into one of the TR doublets,

where and are real radial functions. In the non-interacting limit, and as shown in Eq. (17). Repulsive interactions expand the spatial distributions of and , but the qualitative picture remains. Therefore, one spin component stays in the -state and the other in the -state. This is a half-quantum vortex configuration which spontaneously breaks TR symmetry Wu2011 ().

One possibility is that the condensate wavefunction may take linear superpositions of the Kramer doublet in Eq. (LABEL:eq:TR_doublet). The superposition principle usually does not apply due to the non-linearity of the GP equation. Nevertheless, if the interaction of the GP equation is spin-independent, all the linear superpositions of the Kramer doublet in Eq. (LABEL:eq:TR_doublet) are indeed degenerate. This is an accidental degeneracy at the mean-field level which is not protected. Quantum fluctuations remove this degeneracy as shown in the exact diagonalization calculation in Ref. Hu2012a, and select either one of . In other words, quantum fluctuations can induce a spin-dependent interaction beyond the mean-field level Wu2011 (). Certainly, we can also prepare the initial state with the average per particle , say, by cooling down from the fully polarized spin up or down state, then will be reached. On the other hand, if an additional spin-dependent interaction is introduced,

 H′int=g′∫d3→r(n↑(r)−n↓(r))2, (25)

then even the mean-field level degeneracy is removed. In this case, as shown in Ref. [Ramachandhran2012, ], the condensate wavefunctions of will also be selected.

The spin distribution of a condensate wavefunction is expressed as

 →S(r,ϕ)=ψ∗α(r,ϕ)→σαβψβ(r,ϕ), (26)

which is known as the 1st Hopf map. Without loss of generality, the condensate of is considered, and its is expressed as

 Sx(r,ϕ) = ρsin2γ(r)cosϕ,  Sy(r,ϕ)=ρsin2γ(r)sinϕ, Sz(r,ϕ) = ρcos2γ(r), (27)

where , and the parameter angle is defined through

 cosγ(r)=f(r)ρ(r),  sinγ(r)=g(r)ρ(r). (28)

Since the Fourier components of and are located around the Rashba ring in momentum space, they oscillate along the radial direction with an approximated pitch value of as shown in Fig. 1 (A). Because and are of the and -partial waves, respectively, they are with a relative phase shift of . At , is at maximum and is 0. As increases, roughly speaking, the zero points of corresponds to the extrema of and vice versa, thus spirals as increases. At the -th zero of denoted , ( and we define ).

Consequently, spirals in the -plane along the -axis as shown in Fig. 1 B. The entire distribution of can be obtained through a rotation around the -axis. This is a skyrmion configuration which is a non-singular topological defect mapping from the real space to the spin orientation space of the sphere. If the coordinate space is a closed manifold , this mapping is characterized by the integer valued Pontryagin index , or, the winding number. However, the coordinate space is the open , and decays exponentially at large distance , thus the rigorously speaking the covering number is not well-defined. Nevertheless, in each concentric circle , varies from to , which contributes to the winding number by 1. If we use the trap length scale as the system size, the winding number is roughly at the order of .

The radial oscillation of the spin density is in analogy to the Friedel oscillations in Fermi systems. Around an impurity in electronic systems, the screening charge distribution exhibits the radial oscillation on top of the enveloping exponential decay. The oscillation pitch is reflecting the discontinuity of the Fermi distribution on the spherical Fermi surface. Different from the usual boson systems, the SO coupled ones have a low energy ring structure in momentum space in analogous to the Fermi surface, thus in real space spin density also oscillates in the presence of spatial inhomogeneity.

In the regime of intermediate SO coupling strength, the level spacing between single particle states within the same Landau level is suppressed as shown in Eq. (13). In the case that interactions are strong enough to mix energy levels with different angular momenta in the same lowest Landau level but not among different Landau levels, condensates do not keep rotation symmetry any more. The calculated phase boundary of interaction strength v.s. SO coupling strength is plotted in Fig. 2. In this regime, the distributions are no long concentric but split into multi-centers and finally form a triangular skyrmion lattice structure as calculated in Refs. [Hu2012a, ], [Sinha2011, ]. This 2D skyrmion lattice structure is a characteristic feature brought by SO coupling.

### iii.2 Plane-wave type condensations with Rashba SO coupling

If interactions are strong enough to mix states in different Landau levels, then the influence of the confining trap is negligible. The condensate configurations in the free space was calculated beyond the mean-field GP equation level in Ref. [Wu2011, ]. Bosons select the superposition of a pair of states with opposite momenta and on the low energy Rashba ring to condense. These spin eigenstates of these two states are orthogonal, thus the condensate can avoid the positive exchange interactions. As is well known, avoiding exchange energy is the main driving force towards BEC. For spin-independent interactions, the condensate wavefunctions exhibit degeneracy at the Hartree-Fock level regardless of the relative weight between these two plane-wave components. This is a phenomenon of “frustration”. Quantum zero-point energy from the Bogoliubov quasi-particle spectra selects an equal weight supposition through the “order-from-disorder” mechanism. Such a condensate exhibits spin-spiral configuration.

Various literatures have also studied the case of spin-dependent interactions in which the Hartree-Fock theory is already enough to select either the spin-spiral state, or, a ferromagnetic condensate with a single plane-wave component Wang2010 (); Ho2011 ().

### iii.3 Quaternionic phase defects of the 3D Weyl SO coupling

Next we review the condensates with the 3D Weyl SO coupling in the harmonic trap. The corresponding GP equation is very similar to Eq. (23) of the Rashba case. Only slight modifications are needed by replacing the spatial dimension 2 with 3, and by replacing the Rashba term with . Amazingly, in this case condensate wavefunctions exhibit topological structures in the quaternionic representation Li2012b ().

#### The quaternionic representation

Just like a pair of real numbers form a complex number, the two-component spinor is mapped to a single quaternion following the rule

 ξ=ξ0+ξ1i+ξ2j+ξ3k, (29)

where . are the imaginary units satisfying , and the anti-commutation relation . Quaternion can also be expressed in the exponential form as

 ξ=|ξ|eωγ=|ξ|(cosγ+ωsinγ), (30)

where and ; is the unit imaginary unit defined as which satisfies ; the argument angle is defined as and .

Similarly to the complex phase which spans a unit circle, the quaternionic phases span a unit three dimensional sphere . The spin orientations lie in the Bloch sphere. For a quaternionic wavefunction, its corresponding spin distribution is defined through the 1st Hopf map defined in Eq. (26) as a mapping . Due to the homotopy groupswilczek1983 (); nakahara2003 () and , both quaternionic condensate wavefunctions and spin distributions can be non-trivial. The winding number of is the 3D skyrmion number, and that of the is the Hopf invariant, both are integer-valued.

Let us apply the above analysis to the lowest single-particle level with

 ψj=jz=12(r,^Ω)=f(r)Y+,12,0,12(^Ω)+ig(r)Y−,12,1,12(^Ω),

where and . As shown in Eq. (19), in the non-interacting limit, and . The corresponding quaternionic expression is

 ξj=jz=12(r,^Ω)=ρ(r)eω(^Ω)γ(r), (32)

where and are defined in the same way as the 2D case in Eq. (27); the imaginary unit,

 ω(^Ω)=sinθcosϕ i+sinθsinϕ j+cosθ k, (33)

is along the direction of .

#### The skyrmion type 3D quaternionic phase defects

The analysis on the topology of the Weyl condensates can be performed in parallel to the above 2D Rashba case. Again in the case of weak SO coupling, interactions only expand the spatial distribution of and in Eq. (LABEL:eq:3D_cond) from their non-interacting forms. The radial wavefunction and follow the same oscillating pattern as those in the Rashba case, thus so does the parameter angle which starts from 0 at the origin and reaches at the -th zero of . For the quaternionic phase , its imaginary unit is of one-to-one correspondence to every direction in 3D, thus it exhibits a non-trivial mapping from the 3D coordinate space to , which is known as a 3D skyrmion configuration.

For a closed 3-manifold, the Pontryagin index of winding number is , i.e., integer. Again here the real space is open. In each concentric spherical shell with whose thickness is at the scale of , spirals from to , thus this shell contributes 1 to the winding number from real space to the quaternionic phase manifold. If the system size is truncated at the trap length , again the winding number is approximately . In comparison, in the 2D Rashba case reviewed in Sec. III.1, exhibits the 2D skyrmion configuration Wu2011 (); Hu2012a (); Sinha2011 (), but condensation wavefunctions have no well-defined topology due to the fact that .

A comparison can be made with the vortex in the single-component BEC. In 2D, it is a topological defect with a singular core. Moving around the circle enclosing the core, the phase winds from to , and thus the winding number is 1. The above 3D skyrmion phase defect is a natural generalization to the two-component case whose phase space is in the quaternionic representation and is isomorphic to the SU(2) group manifold. These 3D skyrmions are non-singular defects similar to a 1D ring of rotating BEC which carries a non-zero phase winding number but the vortex core lies outside the ring.

#### Spin textures with non-zero Hopf invariants

The non-trivial topology of the condensate wavefunctions leads to a topologically non-trivial distribution of spin density . The 1st Hopf map defined in Eq. (26) becomes very elegant in the quaternionic representation as

 Sxi+Syj+Szk=12¯ξkξ, (34)

where is the quaternionic conjugate of . For the condensate wavefunction Eq. (32), is calculated as

 [Sx(→r)Sy(→r)] = g(r)sinθ[cosϕ−sinϕsinϕcosϕ][g(r)cosθf(r)], Sz(→r) = f2(r)+g2(r)cos2θ, (35)

which exhibits a perfect axial symmetry around the -axis. is plotted in Fig. 3 at different cross sections. In the -plane, it exhibits a 2D skyrmion pattern, whose in-plane components are along the tangential direction. As the horizontal cross-section shifted along the -axis, remains the 2D skyrmion-like, but its in-plane components are twisted around the -axis. According to the sign of the interception , the twist is clockwise or anti-clockwise, respectively. This 3D distribution pattern of is characterized by an integer valued Hopf invariant characterized by .

As SO coupling strength increases, condensates break rotational symmetry by mixing different states with different values of in the lowest Landau level. Even at intermediate level of SO coupling, rich patterns appear. The quaternionic phase defects and the corresponding spin textures split into multi-centered pattern as plotted in Fig. 4 for different horizontal cross-sections. In the -plane, exhibits a multiple skyrmion configuration as shown in the combined pattern of the in-plane and z-components. Again this pattern is twisted by rotating around the -axis as the interception value varies. Thus the whole 3D configuration also possesses non-trivial Hopf invariant.

In particular, in the case of , it is expected that a 3D lattice structure of topological defects maybe formed, which is a generalization of the 2D skyrmion lattice configuration in Refs. [Hu2012a, ] into 3D. Again, if interaction is very strong to mix states in different Landau levels, the condensate will become plane-wave-like or superpositions of SO coupled plane-waves Li2012b ().

## Iv Vortex configurations of SO coupled BECs in a rotating trap

Next we review the vortex configurations of SO coupled unconventional BECs in rotating traps. From a more general framework, the above-considered SO coupling can be viewed as particles subject to non-Abelian gauge fields. On the other hand, the Coriolis force from rotation behaves as an effective Abelian vector potential. Therefore, in a rotating trap, atom-laser coupling provides an elegant way to study the effects of these two different effective gauge fields. We only consider the rotating systems with the Rashba SO coupling.

### iv.1 Hamiltonians of SO coupled bosons in a rotating trap

Ultracold atoms in a rotating trap share similar physics of electrons subject to magnetic fields due to the similarity between Lorentz and Coriolis forces. Depending on the experimental implementations of rotation, Hamiltonians can be of different types ZhouXF2011 (); Radic2011 (); Xu2011 (). As pointed out in Ref. [Radic2011, ], because the current experiment setup breaks rotation symmetry, rotating SO coupled BECs is time-dependent in the rotating frame, which is a considerably more complicated problem than the usual rotating BECs. Nevertheless, below we only consider the situation of the isotropic Rashba SO coupling, such that it in principle can be implemented as a time-independent problem in the rotating frame.

The effect of rotation should be described by the standard minimal substitution method as presented in Ref. [ZhouXF2011, ]. The non-interacting part of the Hamiltonian is

 H0 = ∫d3→rψ†μ(→r)[12M(−iℏ→∇+Mλ^z×→σ−→A)2−μ (36) + Vtr(→r)−12MΩ2z(x2+y2)]μνψν(→r),

where , and the last term is the centrifugal potential due to rotation.

Note that due to the presence of SO coupling, we should carefully distinguish the difference between mechanical and canonical angular momenta. The mechanical one should be defined according to the minimal substitution as

 Lmech=Lz+Mλ(xσx+yσy), (37)

where is the usual canonical angular momentum. Expanding Eq. (36), it is equivalent to Eq. (1) plus the term of angular velocity coupling to as

 Hrot=−Ωz∫d3→rψ†μ(→r)[Lmech]μνψν(→r). (38)

Thus in the rotating frame, the effect of is not only just as usual, but also an extra effective radial Zeeman term as

 →BR(→r)=ΩzMλ→r. (39)

Such a term is often missed in literatures. As will be shown below, it affects the ground state vortex configurations significantly and thus should not be overlooked.

To make the model more adjustable, an external spatially dependent Zeeman field is intentionally introduced as

 HB = −B∫d3r ψ†μ(→r) (xσx+yσy)μν ψν(→r), (40)

which shares the same form as Eq. (39). Experimentally, such a Zeeman field can be generated through coupling two spin components using two standing waves in the and -directions with a phase difference of . The corresponding Rabi coupling is

 −Ω′{sin(kLx)+isin(kLy)}ψ†↓(→r)ψ↑(→r)+h.c. (41)

In the region of , it reduces to the desired form of Eq. (40) with . Such a term compensates the non-canonical part of the mechanical momentum in , which renders the model adjustability in a wider range.

### iv.2 SO coupled bosons in rotating traps

Now we turn on interactions and obtain the ground state condensate numerically by solving the SO coupled GP equations which have been reduced into the dimensionless form as

 μ~ψ↑ = ^T↑ν~ψν+β(|~ψ↑|2+|~ψ↓|2)~ψ↑, μ~ψ↓ = ^T↓ν~ψν+β(|~ψ↓|2+|