Vortices in spinor cold exciton condensates with spin-orbit interaction

Vortices in spinor cold exciton condensates with spin-orbit interaction

H. Sigurdsson,1, 2 T. C. H. Liew,1 O. Kyriienko,1, 2 and I. A. Shelykh1, 2

1Division of Physics and Applied Physics, Nanyang Technological University 637371, Singapore2Science Institute, University of Iceland, Dunhagi-3, IS-107, Reykjavik, Iceland

(Dated: January 8, 2014)

We study theoretically the ground states of topological defects in a spinor four-component conden-sate of cold indirect excitons. We analyze possible ground state solutions for different configurationsof vortices and half-vortices. We show that if only Rashba or Dreselhaus spin-orbit interaction(SOI) for electrons is present the stable states of topological defects can represent a cylindricallysymmetric half-vortex or half vortex-antivortex pairs, or a non-trivial pattern with warped vortices.In the presence of both of Rashba and Dresselhaus SOI the ground state of a condensate representsa stripe phase and vortex type solutions become unstable.

PACS numbers: 71.35.Lk,03.75.Mn,71.70.Ej

I. INTRODUCTION

An existence of topological phases and excitationscan be seen as a manifestation of unique and univer-sal laws of physics. Being widely studied in varioussystems, a remarkable understanding of topological de-fects was attained for Bose-Einstein condensates of ul-tracold atoms,1,2 where quantization of angular momen-tum was experimentally observed.3 The resulting quasi-particles — quantum vortices — consist of a vortex core,where the condensate density reaches its minimum andphase becomes singular, and a circulating superfluid flowaround, with phase winding being an integer number of2π.4 Other examples of topological defects include do-main walls, solitons,5 warped vortices,6 skyrmions,7 andfractional vortices which can appear in multicomponent8

or spinor condensate systems.9

The usual scheme for generation of vortices in atomicphysics is based on the effective Lorentz force appear-ing due to rotation of the condensate.10,11 Recently analternative approach with an optically-induced artificialgauge field generation was realized.12 The next importantstep forward in manipulation of atomic condensates wasperformed with implementation of an artificial spin-orbitcoupling between several spin components.13 Followedby numerous theoretical proposals,14–16 this system wasshown to be an excellent playground for studying diversespin-related topological phases and excitations,17 includ-ing single plane wave and striped phases,18 hexagonally-symmetric phase,19 square vortex lattice,20 skyrmionlattice,21 and even a quasicrystalline phase for cold dipo-lar bosons.22

A major drawback in the study of cold atom systems isthe ultralow temperature (< 1 nK) required for conden-sation of atoms in magnetic traps. However, solid-statephysics offers a large variety of systems, where bosonicquasiparticles with small effective mass can condense atcomparably high temperatures. They include QuantumHall bilayers,23 magnons,24, indirect excitons,25–27 andcavity exciton-polaritons.28–31 Moreover, the latter sys-tem possesses a spinor structure being formed by two po-

lariton spin components with ±1 spin projection.32 Com-plementary to the full quantum vortices in the polaritonfluids,33–35 this allows one to study half-integer quantumvortices36–38 and their warped analogs.6,36,39 An intrigu-ing spin dynamics there is caused by an analog of spin-orbit interaction (SOI) given by momentum-dependentTE-TM splitting.40–42

Even higher spin degeneracy can be achieved for thesystem of indirect excitons — bound pairs of electronsand holes which are spatially separated in two parallelquantum wells [Fig. 1].43–45 Due to the small overlapbetween the wavefunctions of electrons and holes thesequasiparticles possess very large radiative lifetime (upto microseconds), which allows them to thermalize andconsequently form a macroscopically coherent state withproperties similar to a Bose-Einstein condensate.27 An-other important feature of indirect exciton gases is for-mation of a so-called macroscopically ordered state man-ifesting itself as a fragmented exciton ring.46–48

Accounting for four possible ±1,±2 cold indirect exci-ton spin projections, an ambiguous choice of condensateground state is possible.49,50 This results in non-trivialcondensate topology and the possibility for generationof various topological defects.51 Moreover, complex spintextures around fragmented beads of cold exciton con-densates were observed.52 They were explained with aninfluence of SOI of various types, which affects the center-of-mass exciton motion.50,53,54 This assures that physicssimilar to atomic spin-orbit coupled condensates, includ-ing artificial magnetic field generation,55 can be studiedwith cold indirect excitons.

In this paper we study the ground states of varioustopological defects in an indirect exciton condensate.We show that the presence of the SOI leads to drasticchanges in the ground state of the topological defects inthe indirect exciton condensate. Using the imaginary-time Gross-Pitaevskii equations for the spinor macro-scopic wave function, we find that in the presence of onlyone type of SOI half-vortex solutions are possible, whilefor both Rashba and Dresselhaus SOI the only possiblestable solution is a striped state with zero vorticity. Westudy the numerical solutions of the equations and de-

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FIG. 1: (Color online) Sketch of the system. (a) A het-erostructure with biased coupled quantum wells, where anelectron from the right quantum well (RQW) is coupled witha heavy hole in the left quantum well (LQW), forming an indi-rect exciton (IX). (b) The energy structure of an electron-holebilayer showing spatial separation of electron (e) and hole (h)wave functions.

rive analytical estimates for the boundaries, which definetopological charge stabilities. The results are consistentwith recent experimental observations of spin textures ina diluted coherent gas of cold indirect excitons.

II. THE MODEL

An indirect exciton is a composite boson consisting ofa spatially separated electron and hole [Fig. 1]. Its spinis defined by electron and heavy hole spin projectionson the structure growth axis, being ±1/2 and ±3/2, re-spectively. The resulting four combinations correspondto possible exciton spin projections, Sz = ±1,±2. Thestates with Sz = ±1 are called the bright excitons, sincethey can be optically excited by an external pump. Incontrast, the states with Sz = ±2 spin are optically in-active due to angular momentum conservation selectionrules and are typically referred to as dark excitons. How-ever, they can appear due to exchange interaction be-tween bright states or as a result of spin-orbit interac-tion. In the case of direct excitons the bright and darkstates are typically split by short range electron-hole ex-change, with dark states lying at lower energies.56 Thiscan possibly lead to the dark or gray condensation in thecorresponding systems, which prevents direct observa-tion of macroscopic coherence in the photoluminescencemeasurements.57,58 Moreover, the effects of spin-orbit in-teractions where shown to interplay with a bright-darksplitting, leading to unconventional pairing effects in thedense BCS-like direct exciton condensates.59 In the caseof indirect excitons the small overlap between electronand hole wave functions leads to approximately equal en-ergies of all four indirect exciton states. The dark statesstill play an important role and cannot be excluded fromthe consideration.60,61

To describe a coherent state of indirect excitons,we can use the mean-field treatment similar to Refs.[50,51], where the Gross-Pitaevskii equation for the four-component wave function Ψ = (Ψ+2,Ψ+1,Ψ−1,Ψ−2) wasintroduced. In the general form it can be derived varying

the Hamiltonian density over the macroscopic wave func-tion, i~dΨ/dt = ∂H/∂Ψ∗. The Hamilton density can bewritten as a sum of a linear single particle and nonlinearinteraction parts, H = H0 +Hint.

The single particle part of the Hamiltonian densityis composed of the kinetic energy and SOI. The lat-ter appears as a consequence of spin-orbit interactionacting on the electron or hole spin. In the followingwe will account only for the part of SOI affecting thespin of electron. It consists of two terms. The Dres-selhaus term arises from bulk inversion asymmetry andfor a [001] quantum well is described by the Hamilto-nian HD = β(σxkx − σyky), where kx,y are Cartesiancomponents of the electron wave vector, σx,y are Paulimatrices, and β denotes the strength of the Dresselhausinteraction. The Rashba term appears due to structureinversion asymmetry and is described by the Hamilto-nian HR = α(σxky −σykx), with α being the strength ofthe Rashba interaction.

The single particle term in the Hamiltonian densitythus reads50:

H0 = Ψ†TΨ, (1)

where

T =

(T0 ∅∅ T0

), (2)

with ∅ being a null matrix and

T0 =

(~2K2/2mX SK

S∗K ~2K2/2mX

). (3)

Here

SK = χ[β(Kx + iKy) + α(Ky + iKx)], (4)

where χ = me/mX is the electron-to-exciton mass ratioand K = −i∇ denotes the center of mass wave vector ofthe indirect exciton. Note that in the described Hamilto-nian we neglect the bright-dark splitting of the indirectexciton states. This however can be straightforwardlyintroduced for the systems, where such a splitting wasobserved.54,58

The nonlinear part of the Hamiltonian density Hint de-scribes interactions between indirect excitons. Since ex-citons are composite bosons, there are four possible typesof interactions corresponding to the exchange of electrons(Ve), exchange of holes (Vh), simultaneous exchange ofelectron and hole (or exciton exchange, VX), and directCoulomb repulsion (Vdir). Introducing the interactionconstants V0 = Ve + Vh + Vdir + VX and W = Ve + Vh,the interaction part of the Hamiltonian density becomes

Hint =V0

2

(|Ψ+2|2 + |Ψ+1|2 + |Ψ−1|2 + |Ψ−2|2

)2+W

(Ψ∗+1Ψ∗−1Ψ+2Ψ−2 + Ψ∗+2Ψ∗−2Ψ+1Ψ−1

)(5)

−W(|Ψ+2|2|Ψ−2|2 + |Ψ+1|2|Ψ−1|2

).

3

We mainly focus on the weakly depleted Bose-Einsteincondensates of indirect excitons where the biggest in-teraction contribution comes from vanishing transferredmomentum q, thus working in the long wavelength limit(q → 0), where Vdir = VX and Ve = Vh (s-wave ap-proximation). The interaction parameters can be furtherestimated using a narrow QW approximation.50

As a specific system corresponding to our model weconsider the indirect exciton system studied in Ref. [27],

where macroscopic coherence of indirect exciton gas wasreported. The studied sample is high quality doublequantum well structure with 8 nm GaAs QWs and a 4nm Al0.33Ga0.67As barrier. The observation of nontrivialspin structures in the same sample presumes an impor-tance of spin-orbit interaction in the described system.52

The dynamics of the system is described by a set of fourcoupled nonlinear equations of Gross-Pitaevskii type:

i~∂Ψ+2

∂t= EΨ+2 + SRΨ+1 + V0Ψ+2|Ψ+2|2 + (V0 −W )Ψ+2|Ψ−2|2 + V0Ψ+2(|Ψ−1|2 + |Ψ+1|2) +WΨ∗−2Ψ+1Ψ−1, (6)

i~∂Ψ+1

∂t= EΨ+1 − S∗RΨ+2 + V0Ψ+1|Ψ+1|2 + (V0 −W )Ψ+1|Ψ−1|2 + V0Ψ+1(|Ψ−2|2 + |Ψ+2|2) +WΨ∗−1Ψ+2Ψ−2, (7)

i~∂Ψ−1

∂t= EΨ−1 + SRΨ−2 + V0Ψ−1|Ψ−1|2 + (V0 −W )Ψ−1|Ψ+1|2 + V0Ψ−1(|Ψ+2|2 + |Ψ−2|2) +WΨ∗+1Ψ+2Ψ−2, (8)

i~∂Ψ−2

∂t= EΨ−2 − S∗RΨ−1 + V0Ψ−2|Ψ−2|2 + (V0 −W )Ψ−2|Ψ+2|2 + V0Ψ−2(|Ψ+1|2 + |Ψ−1|2) +WΨ∗+2Ψ+1Ψ−1. (9)

Here E = −~2∇2/2mX is the exciton kinetic energy op-erator and

SR = χ[β(∂y − i∂x) + α(∂x − i∂y)

](10)

is the SOI operator accounting for both Rashba (α) andDresselhaus (β) contributions.

III. NUMERICAL METHOD

We use the imaginary time method to find the statecorresponding to the local minima of the Hamiltonian ofthe interacting exciton system described by Eqs. (6)-(9).Fourier spectral methods are used in space and a vari-able order Adams-Bashforth-Moulton method in time toachieve accurate discrete gradient flow towards a possiblelow energy state. Note, that the energy profile can havemultiple minima, and the one that is reached in numeri-cal procedure strongly depends on the initial conditions.In particular, one can suppose that if a stable vortex ispresent in the system the corresponding solution will befound if the initial distribution contains non-zero vortic-ity, while the ground state with zero vorticity (homoge-neous or striped) will be found if one does not have aninitial angular momentum. If the system does not possessany stable solutions in the form of vortices, state with noangular momentum will be recovered independently ofthe initial condition.

We introduce a weak harmonic 2D-trapping potentialVtrap in the Hamiltonian to keep the condensate localizedwithin the system. The trap profile is given by Vtrap =u0r

2⊥, where u0 = mXω

2/2 represents the trap strength.A choice of initial conditions is not always trivial when

dealing with a nonlinear set of equations controlled bymany parameters. In our case the typical initial conditioncorresponds to the vortex solution:

Ψσ(r) = R(0)(r)r/χ√

r2/χ2 + 1ei(mσθ+kσπ). (11)

Here R(0)(r) is a Gaussian function corresponding to thetrapped exciton gas, σ is the spin index and χ is the heal-ing length of the vortex in a one component BEC givenby χ = ~/

√2mXV0n,62 where V0 is the nonlinear inter-

action parameter defined before and n is the 2D densityof the exciton gas. The effective mass of the exciton istaken to be mX = 0.21me, where me is the free electronmass.46 We assume that the healing length of a vortex ina four component BEC is comparable with one compo-nent BEC case.

We stress that the initial condition is used here onlyto set different topologies in the system. The final resultof imaginary time propagation obtains the minimum en-ergy state for a given topology (if such a state exists),that is, the ground state of a given topological defectcharacterized by winding numbers mσ. We checked thatsuch solutions are unchanged for different topologicallyinvariant spatial profiles of the initial conditions; chang-ing the specific shape of the radial wave function does not

4

change the final result. One could start the calculationwith just uniform density subject to some circulation andthe density dip of the vortex appears in the ground stateresults.

Note that the relative phases between the componentsin the initial condition (set by kσ) can affect the solu-tion. Where this is so, we minimize over different valuesof kσ to find the minimum energy state. Finally, we con-firm that our results are stationary states by propagatingthem in real time numerically.

IV. VORTICES, HALF VORTICES, AND HALFVORTEX-ANTIVORTEX PAIRS

Let us first consider the cylindrically symmetric sta-tionary wave function of the Gross-Pitaevskii equationas a possible minimal energy state for a rotating BECaround the z-axis,62

Ψσ(r, θ, t) = Rσ(r)ei(mσθ+kσπ)e−iµt, (12)

where µ is the chemical potential of the condensate. Thecirculation of the tangential velocity over a closed contourfor quantum vortices is quantized in units of 2π~/mX

controlled by the winding number mσ, also known asvorticity or topological charge. Recent works on spinorexciton condensates have concluded that one of the sim-plest vortex solutions is of opposite vorticity in the Ψ±1

components (half vortex-antivortex pair) and zero vor-ticity in the dark components (or vice versa).39,51 Thiswill later be shown to be indeed a possible low energysolution amongst other interesting vortex solutions fordifferent mσ and kσ.

The radial part is taken to be purely real and is relatedto the total density of the condensate as

|R+1|2 + |R−1|2 + |R+2|2 + |R−2|2 = n, (13)

where ∫ ∑σ

|Rσ|2 d2r = N (14)

is the total number of excitons in the system. In thispaper we use the exciton density in the harmonic trapbeing n ∝ 108 cm−2. The lateral size of the system of20 µm was chosen corresponding to localized bright spotsobserved in past experiments on exciton condensates.52

The total number of particles was estimated as N ≈ 100.The phase difference kσπ becomes essential in whether

the vortex solution is present in the condensate or not.Adding π phase difference switches the sign of the wavefunction and thus switches the sign of the second lineterm in the nonlinear part of the Hamiltonian density[Eq. (5)] corresponding to bright to dark exciton conver-sion. Moreover, Eq. (12) reveals that for the solution tobe cylindrically symmetric in the spinor exciton conden-sate the winding numbers need to satisfy the following

bound:

m+1 +m−1 = m+2 +m−2. (15)

Let us rewrite Eqs. (6)-(9), in the limit that the SOIstrength is zero:

i~∂Ψ+2

∂t= EΨ+2 + V0nΨ+2 +WΨ∗−2Ψ2

∆, (16)

i~∂Ψ+1

∂t= EΨ+1 + V0nΨ+1 −WΨ∗−1Ψ2

∆, (17)

i~∂Ψ−1

∂t= EΨ−1 + V0nΨ−1 −WΨ∗+1Ψ2

∆, (18)

i~∂Ψ−2

∂t= EΨ−2 + V0nΨ−2 +WΨ∗+2Ψ2

∆, (19)

where we used definition Ψ2∆ ≡ Ψ+1Ψ−1 − Ψ+2Ψ−2.

Eqs. (16)-(19) show that the only difference betweenthe equations describing bright and dark excitons is thesign of the W term describing bright to dark exciton con-version. This symmetry between bright and dark com-ponents means that if topological defects exist for thebright excitons then the same defects can exist for thedark excitons. Of main interest are configurations suchas (m+2,m+1,m−1,m−2) = (0,1,-1,0), (1,1,1,1), (1,0,1,0)satisfying Eq. (15). We observe that if Eq. (15) is notsatisfied, then there is no energy minimum for a trappedstate of the considered topological defect, cylindricallysymmetric or not. The real time propagation revealedthat if for example a stable solution of mσ = (0, 1,−1, 0)was suddenly switched to (0, 1, 1, 0) by conjugating theΨ−1 component then the solution became immediatelynon-stationary and the localized topological defect wasdestroyed.

The vortices with high topological charges, |mσ| > 1,were shown to be unstable in single component BECsdepending on interaction strength.63 This holds as wellin our case: single topological defects are no longer ob-served for |mσ| > 1 in the case when SOI is absent. Thissituation changes, however, if SOI is taken into accountas it will be discussed in the next section.

One should note that in the four component BEC theterm vortex commonly applies when all components arerotating. The half vortex pair corresponds to circularmotion of two components in the same direction, and halfvortex-antivortex pair to two components with oppositedirection of rotation.

In Fig. 2 we show four cases of low energy solutions forvortex topological defects in the four-component excitoncondensate. The top plots correspond to a half vortex-antivortex pair in Ψ±1. The second from the top corre-sponds to a basic vortex composed of two half vortex-antivortex pairs in both bright and dark components.The second from the bottom corresponds to a basic vor-tex composed of two half vortex pairs in both bright and

5

FIG. 2: (Color online) Density and phase profiles of the exci-ton condensate with different topological defects. Top: mσ =(0, 1,−1, 0). Second from top: mσ = (1,−1, 1,−1). Sec-ond from bottom: mσ = (1, 1, 1, 1) and kσ = (1,−1, 0, 0).Bottom: mσ = (1, 0, 1, 0) and kσ = (1, 0, 0, 0). In the topthree: V0 = 1 and W = −0.1 µeVµm−2. Bottom: V0 = 1 andW = 0.1 µeVµm−2. In all pictures: u0 = 1 µeVµm−2.

dark components — both with a π phase difference. Bot-tom plots correspond to a half vortex pair in Ψ−1 andΨ+2 components. One can see in the top and bottomlines that the vortex core stabilizes at a greater heal-ing length due to the other components trying to fill in

the density dips. The densities of bright and dark exci-tons try to complement each other, staying close to theThomas-Fermi profile.

The existence of a low energy solution with vortices isdetermined by the last term in Eqs. (16)-(19) and thekinetic energy term. This can be seen from analysis ofthe Hamiltonian density (5). In the case of W = 0 thereis a competition between the kinetic energy term in thetotal Hamiltonian and the interaction energy term

Hint =V0

2

(|Ψ+2|2 + |Ψ+1|2 + |Ψ−1|2 + |Ψ−2|2

)2.

If interactions are weak then it will be energetically fa-vorable to transfer intensity from a component with avortex to a component without one, since a componentwith a vortex has higher kinetic energy. For this reason,there may be no minimal energy states with vortices intwo components only — numerical calculations give in-stead a depletion of components containing vortices infavor of those without vortices. We can then expect thatthe only possible stable states with non-zero topologicalcharges (in components with non-zero intensity) containvortices in all components.

One should keep in mind that while the kinetic energycontribution can be reduced by transferring intensity toa component without a vortex, this may increase the po-tential energy due to interactions. The term proportionalto V0 in the Hamiltonian can be reduced if the spatialoverlap of the intensity distribution of components is re-duced. Thus the V0 term favors formation of vortices, butit must be strong enough to overcome the correspondingincrease of kinetic energy for the states with vortices intwo components only.

In the case W 6= 0 and (m+2,m+1,m−1,m−2) =(0, 1,−1, 0) the wave functions can be written:

Ψ+2 = U(r)eiφ+2 , Ψ+1 = V (r)eiθeiφ+1

Ψ−1 = V (r)e−iθeiφ−1 , Ψ−2 = U(r)eiφ−2

where U(r) and V (r) are real functions. The W depen-dent part of the Hamiltonian can then be written as

HW = W[2U2V 2 cos (∆φ)− U4 − V 4

], (20)

where ∆φ = φ+2 + φ−2 − φ+1 − φ−1.In the case that W > 0, the phases can be chosen to

minimize the Hamiltonian to HW = −W(U2 + V 2

)2=

−W(|Ψ+2|2 + |Ψ+1|2 + |Ψ−1|2 + |Ψ−2|2

)2. That is, the

W term has the same form as the V0 term. Consequently,the same arguments as considered in the W = 0 caseapply: if the strength of the interaction V0−W is unableto overcome the kinetic energy term, then a state withvortices in two components only (with non-zero intensity)is not stable.

In the case that W < 0, the phase can be chosen to

minimize the Hamiltonian to HW = −W(U2 − V 2

)2(a

6

positive quantity since W < 0). This term favors stateswith vortices in two components (see Fig. 2, top case)since it is minimized if all components stay populated.

Note that it is not possible to say definitively whethervortices will or will not be stable for the cases where Wis positive or negative without use of numerical calcu-lation because of the tricky interplay between potentialand kinetic energy terms.

In the case of W 6= 0 and (m+2,m+1,m−1,m−2) =(1, 0, 1, 0) the wave functions can be written:

Ψ+2 = V (r)eiθeiφ+2 , Ψ+1 = U(r)eiφ−1

Ψ−1 = V (r)eiθeiφ+1 , Ψ−2 = U(r)eiφ−2 .

The W dependent part of the Hamiltonian density is

HW = 2WU2V 2 [cos (∆φ)− 1] . (21)

In the case W > 0, the phases can be chosen to minimizethe Hamiltonian to HW = −4WU2V 2. This term maystabilize the state, since it provides a reduction of theenergy when all components are populated (see Fig. 2,bottom case); if one component is depleted then this termcan no longer contribute to minimization of the energy.

In the case W < 0, the phases can be chosen to mini-mize the Hamiltonian to HW = 0. In this case we recoverthe result for the W = 0 case. While it is not possibleto say definitively whether vortices will be stable for theW > 0, we can say that for W < 0 they are unstable ifthey are also unstable for the W = 0 case.

V. CYLINDRICALLY SYMMETRIC GROUNDSTATE SOLUTIONS UNDER SPIN-ORBIT

INTERACTION

When SOI of Rashba or/and Dresselhaus type is in-cluded in the Hamiltonian, the analysis of low energystate solutions becomes more tricky. Prior studies in thefield of atomic condensates revealed a plethora of phe-nomena emerging due to spin-orbit coupling.13,17 Indirectexciton condensates can be expected to show also a greatvariety in possible low energy solutions with phase sepa-ration between components and density modulations.

Let us first analyze the possibility of the cylindricallysymmetric solutions. Using the ansatz Ψσ = Rσ(r)eimσθ

in Eqs. (6)-(9) we find that if only Dresselhaus SOI ispresent, the winding numbers should satisfy the followingbound [in addition to those given by Eq. (15)]:

m+2 = 1 + n, m+1 = n, (22)

m−1 = 1 +m, m−2 = m.

On the other hand, if only Rashba SOI is present thebound is:

m+2 = n, m+1 = 1 + n, (23)

m−1 = m, m−2 = 1 +m,

FIG. 3: (Color online) The difference between Dresselhausand Rashba SOI displayed. Density and phase profiles of thecondensate components with different vortex defects as initialcondition. In the top two cases: β = 1 µeVµm and α = 0.In the lower two cases: β = 0 and α = 1 µeVµm. Top:β = 1 µeVµm and α = 0, initial configuration correspondsto mσ = (0,−1, 1, 0). The bound (22) is satisfied and cylin-drically symmetric vortex type solution is obtained. Secondfrom top: β = 1 µeVµm and α = 0, mσ = (0, 1,−1, 0).The bound (22) is not satisfied, and as a result warped vor-tex corresponding to mσ = (+2,+3,−3,−2) is formed ina stationary regime. Second from bottom: β = 0 andα = 1 µeVµm, mσ = (0, 1,−1, 0). The bound (23) is satis-fied and cylindrically symmetric vortex type solution is ob-tained. Bottom: β = 0 and α = 1 µeVµm, initial config-uration corresponds to mσ = (0,−1, 1, 0). The bound (23)is not satisfied, and as a result warped vortex correspondingto mσ = (−2,−3,+3,+2) is formed in stationary regime. Inall pictures: kσ = (0, 0, 0, 0), V0 = 22 µeVµm−2, W = 2µeVµm−2 and u0 = 1 µeVµm−2.

7

FIG. 4: (Color online) Top: Density and phase profiles ofthe exciton condensate components for only Dresselhaus SOI.mσ = (0,−1, 1, 0), kσ = (0, 1, 0, 0) and W = 2 µeVµm−2,V0 = 28 µeVµm−2, β = 1 µeVµm and u0 = 1 µeVµm−2.Bottom: Linear polarization of cylindrically symmetric casesfor Dresselhaus SOI only with mσ = (0,−1, 1, 0) initial con-dition. The right panel was calculated for kσ = (0, 0, 0, 0),and corresponds to the top panel in Fig. 3. For the left panelwe set kσ = (0, 1, 0, 0), corresponding to the top panel in thisfigure.

where n and m are integer numbers.We limit our consideration in this section of the pa-

per to three types of cylindrical vortex configurationsfor SOI of either Dresselhaus or Rashba type: mσ =(0, 1,−1, 0), (0,−1, 1, 0), and (1, 0, 1, 0). We show thatin the case where the bounds (22) or (23) are not sat-isfied the initial topological charge is not preserved butinstead the configuration of warped vortices with wind-ing numbers greater than one, mσ = (±2,±3,∓3,∓2),6

or density modulated stripe phase with no vorticity isestablished, depending on the initial conditions.

For Dresselhaus SOI only the configurations mσ =(0,−1, 1, 0) and (1, 0, 1, 0) satisfy Eq. (22), and we ob-serve formation of the cylindrically symmetric vortices(Fig. 3 top and Fig. 6), whereas mσ = (0, 1,−1, 0)does not satisfy the bound, the cylindrical symmetry isno longer present, and configuration with higher wind-ing numbers mσ = (+2,+3,−3,−2) is formed (Fig. 3,second from top). The similar behavior can be observedfor the case of the Rashba SOI but this time the cylin-drical symmetry is manifested for mσ = (0, 1,−1, 0) and(0, 1, 0, 1) configurations.

FIG. 5: (Color online) Density and phase profiles of theexciton condensate components for only Dresselhaus SOI.mσ = (0,−1, 1, 0), kσ = (0, 0, 0, 0), W = −2 µeVµm−2,V0 = 28 µeVµm−2, β = 1 µeVµm and u0 = 1 µeVµm−2.

FIG. 6: (Color online) Density and phase profiles of the ex-citon condensate half vortex pair for only Dresselhaus SOI.mσ = (1, 0, 1, 0), kσ = (0, 0, 0, 0), V0 = 28 µeVµm−2,W = ±2 µeVµm−2, β = 1 µeVµm and u0 = 1 µeVµm−2.

Fig. 3 shows stable solutions for the half vortex-antivortex configurations mσ = (0, 1,−1, 0), and(0,−1, 1, 0). The top two panels correspond to the casewhen only Dresselhaus SOI is present and the bottomtwo for the case when only Rashba SOI is present. In-specting the phase profiles (top and second from bottompanel) reveals that phases of the components are differentfor the cases of Dresselhaus and Rashba SOI: there is a3π/4 phase difference in Ψ+1, −π/4 difference in Ψ−1 andπ/4 difference in Ψ±2 amplitudes if one changes Dressel-haus SOI to Rashba. These phase differences result inthe π/2 rotation of the pattern of the linear polarizationdegree calculated as

PL =Ψ∗+1Ψ−1 + Ψ∗−1Ψ+1

|Ψ+1|2 + |Ψ−1|2∝ cos (2θ) (24)

if one switches from Rashba to Dresselhaus SOI. The re-sult is expectable, as the operator describing Rashba SOIcan be obtained from the operator describing DresselhausSOI by switching Kx to Ky and vice versa [see Eq. (4)].

We observe that in the cylindrically symmetric case

8

FIG. 7: (Color online) Density and phase profiles of the ex-citon condensate components for only Dresselhaus SOI andsmall nonlinear parameters. Top: mσ = (1, 0, 1, 0). Middle:mσ = (0,−1, 1, 0). Bottom: mσ = (0, 1,−1, 0). In all pic-tures the parameters were β = 1 µeVµm, V0 = 2.8 neVµm−2,W = ±0.2 neVµm−2 and u0 = 1 neVµm−2.

there are two alternative configurations of the vortexcorresponding to the same combination of the windingnumbers. One of them is demonstrated in Fig. 3 andcorresponds to the case when the phase of macroscopicwavefunction depends only on the angle φ. This solu-tion is obtained if all kσ are put to zero. However, ifone introduces phase difference between the condensatecomponents chosen as an initial condition (i.e. kσ 6= 0),another type of the vortex solution corresponding to thespiral phase pattern is obtained [see Fig. 4, top]. Thetopological charges of both solutions are the same, andto distinguish between them one needs to analyze theirlinear polarization patterns shown in Fig. 4 (bottom).As one can see, they are radically different, being fourleaf in one case and gammadion in the other.

Also we note that the sign of exchange interaction Waffects the possible states of stable topological defects.

FIG. 8: (Color online) Density and phase profile of condensatecomponents for mσ = (0, 0, 0, 0) and kσ = (0, 1, 0, 0), the pa-rameters were β = 1 µeVµm, α/β = 1/2, V0 = 28 µeVµm−2

and W = −2 µeVµm−2.

To illustrate its role, we focus on a configuration mσ =(0,−1, 1, 0) (same as in top panel in Fig. 3) and setthe parameters to: β = 1 µeVµm, α = 0 and W =−2 µeVµm−2. We observe a half vortex in a condensatehalf depleted with a spiral phase pattern resulting fromnegative W [see Fig. 5]. The results are clearly differentfrom those shown in Fig. 3 corresponding to oppositesign of the exchange interaction, W = +2 µeVµm−2.

In Fig. 6 we show a half vortex pair solution withmσ = (1, 0, 1, 0) for only Dresselhaus SOI [in case of onlyRashba it would be mσ = (0, 1, 0, 1)]. The solution re-mained the same for both signs of the mixing parameterW and was lost when kσ 6= 0.

We also investigated the ground state vortex solu-tions for the case when the nonlinearities are very weakand the impact of the SOI terms becomes dominant(V0,W � β, α). Fig. 7 illustrates the case when onlyDresselhaus SOI is present. If the bound (22) is sat-isfied the solutions have cylindrical symmetry (top twopanels). For the case mσ = (0, 1,−1, 0) the solution isnon-symmetric and resembles a hybrid of a warped vor-tex solution and a striped phase. In this weakly nonlin-ear limit the sign of W becomes irrelevant, and the samepatterns were observed for W = ±0.2 neVµm−2.

VI. PRESENCE OF BOTH DRESSELHAUSAND RASHBA SOI

When both α 6= 0 and β 6= 0 the single particle spec-trum becomes anisotropic. Different from the cases α = 0or β = 0 the minima of the energy of non-interactingparticles correspond not to a circle of constant radius inthe reciprocal space, but to the two fixed points situatedalong Kx-Ky diagonal,64

K0 = ±χmX(α+ β)

~2

(ex + ey)√2

. (25)

One can thus expect formation of a striped ground statecorresponding to the spatial modulation of the density

9

(eiK0·r + e−iK0·r) = 2 cos (K0 · r). This is indeed thecase as can be seen in Fig. 8. As the ground state ofthe condensate reveals spatial anisotropy, no cylindricallysymmetric vortex solutions can be expected to appear inthis case.

The stability of the vortex-type versus striped phasesolutions depends on the ratio α/β. Fixing the parame-ters describing nonlinearities as V0 = 28 µeVµm−2 andW = ±2 µeVµm−2, our numerical analysis shows thatfor α/β ∼ 10−3 the vortex type solutions shown in Figs.3-6 still persist. However, already at α/β ∼ 10−2 all vor-tex solutions disappear and only stripe phase solutionsare stable. This is illustrated in Fig. 8, where we haveset β = 1 µeVµm and α/β = 1/2 for a spatially uniformcondensate as a initial condition of the imaginary timemethod.

VII. CONCLUSIONS

We studied the stationary solutions describing varioustopological defects in the system of spinor indirect ex-

citons applying the imaginary time method to the setof Gross-Pitaevskii equations. We analyzed the role ofthe SOI of Rashba and Dresselhaus types in formationof single vortices, half vortices and half vortex-antivortexpairs, and described the transition between warped vor-tex and stripe phase solutions in the presence of bothRashba and Dresselhaus SOI.

Acknowledgements. We thank Yuri G. Rubo andAlexey V. Kavokin for the valuable discussions. Thiswork was supported by FP7 IRSES projects SPINMETand POLAPHEN and Tier 1 project “Polaritons for noveldevice applixcations”. O. K. acknowledges the supportfrom Eimskip Fund. H. S. thanks Universidad Autonomade Mexico for hospitality.

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