Phase diagram of the interacting persistent spin-helix state
Hong Liu, Weizhe Edward Liu, Stefano Chesi, Robert Joynt, Dimitrie Culcer
PPhase diagram of the interacting persistent spin-helix state
Hong Liu , Weizhe Edward Liu , Stefano Chesi , Robert Joynt and Dimitrie Culcer School of Physics and Australian Research Council Centre of Excellence in Low-Energy Electronics Technologies,UNSW Node, The University of New South Wales, Sydney 2052, Australia School of Physics and Australian Research Council Centre of Excellence in Low-Energy Electronics Technologies,Monash Node, Monash University, Melbourne 3800, Australia Beijing Computational Science Research Center, Beijing 100193, China and Physics Department, University of WisconsinMadison, Madison, Wisconsin 53706, USA (Dated: February 25, 2020)We study the phase diagram of the interacting two-dimensional electron gas (2DEG) with equalRashba and Dresselhaus spin-orbit coupling, which for weak coupling gives rise to the well-knownpersistent spin-helix phase. We construct the full Hartree-Fock phase diagram using a classicalMonte-Carlo method analogous to that used in Phys.Rev.B , 235425 (2017). For the 2DEG withonly Rashba spin-orbit coupling it was found that at intermediate values of the Wigner-Seitz radius r s the system is characterized by a single Fermi surface with an out-of-plane spin polarization,while at slightly larger values of r s it undergoes a transition to a state with a shifted Fermi surfaceand an in-plane spin polarization. The various phase transitions are first-order, and this shows upin discontinuities in the conductivity, and the appearance of anisotropic resistance in the in-planepolarized phase. In this work we show that the out-of-plane spin polarized region shrinks as thestrength of the Dresselhaus spin-orbit interaction increases, and entirely vanishes when the Rashbaand Dresselhaus spin-orbit coupling strengths are equal. At this point the system can be mappedonto a 2DEG without spin-orbit coupling, and this transformation reveals the existence of an in-plane spin polarized phase with a single, displaced Fermi surface beyond r s > .
01. This is confirmedby classical Monte-Carlo simulations. We discuss experimental observation and useful applicationsof the novel phase, as well as caveats of using the classical Monte-Carlo method.
I. INTRODUCTION
The two-dimensional electron gas (2DEG) with spin-orbit coupling and many-body electron-electron interac-tions is a paradigmatic system in semiconductor physicsand technology, in addition to being one of the funda-mental models in condensed-matter physics. Much ofthe interest in spin-orbit coupling centers around the factthat it enables spin generation, spin manipulation andspin detection without using external magnetic fields ormagnetic materials , while at the same time being man-ifest in a great variety of spin textures in solids , manyof which are associated with topological effects , un-conventional states of matter and novel phases .Acquiring a full understanding of the spin-orbit cou-pled 2DEG is key to our ability to utilise the electronspin degree of freedom in semiconductors to control thespin states and transfer spin information, which is a fun-damental requirement for future spintronic devices andquantum computing among other applications.Keeping in mind both basic science and potential tech-nological interest, identifying many-body ground stateswith novel spin textures and polarizations is one of thegoals of present-day condensed matter research. Thisproblem is notoriously difficult analytically even in theabsence of spin-orbit coupling. The Hartree-Fock (HF)method often provides simple analytical solutions, andthough it entirely ignores the effect of correlations it gen-erally provides useful insights into the structure of thesingle-particle levels . In addition to this, the pastfew decades have seen dramatic improvements in ourability to simulate complicated physical systems using Monte Carlo (MC) approaches . At the same time,the transition of the electron gas into either a partiallyor a fully polarized fluid is not fully understood .At variance with the 3D case, there is the possibility of apartially polarized phase , while for the 2D case there isno evidence for a stable partially polarized phase . Fur-thermore, calculations at intermediate densities are diffi-cult because very small energy differences are importantand any approximation has to treat the various phases ofthe gas with equal accuracy .The situation becomes even more complicated in spin-orbit coupled 2DEGs. In general the spin-orbit cou-pling (SOC) in semiconductor 2DEGs can take severalforms. In realistic semiconductor nanostructures boththe Rashba and the Dresselhaus spin-orbit coupling areoften present. Rashba SOC is present primarily becausequantum wells frequently have a built-in asymmetry ,and has been experimentally observed in semiconductorheterostructures, where it has been proved to be tunablein strength by means of a gate voltage . The Dres-selhaus SOC reflects the inversion asymmetry inherentin zincblende lattices, which includes the crystal struc-ture of many III-V and II-VI semiconductors such asGaAs, InSb, and CdTe . In 2DEGs the SOC can be de-scribed by an effective momentum-dependent magneticfield. This effective field favors spin textures that havezero net moment, while the electron-electron interactionsfavor ferromagnetism. This gives rise to a multitudeof phases, many of which remain to be explored in de-tail. An insightful theoretical approach adopted in earlierstudies of interacting spin-orbit coupled systems involvedapplying a unitary transformation to leading order in the a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b spin-orbit strength, which yields a transformed Hamilto-nian whose eigenstates are also spin and angular momen-tum eigenstates . More recently it has been shownthat in addition to the well known out-of-plane spin po-larized phase, the interacting 2DEG with Rashbaspin-orbit coupling exhibits an in-plane spin-polarizedphase with a shifted Fermi surface , somewhat resem-bling a Pomeranchuk instability. It is caused by an ex-change enhancement of the current-induced spin polar-ization. This phase appears already at intermediate val-ues of r s , the Wigner-Seitz radius, which represents therelative strength of the electron-electron interactions tothe average kinetic energy. The same result is expectedin systems with Dresselhaus SOC, since the Rashba andDresselhaus interactions are related by a spin rotation.Several additional recent works have examined the inter-play between Rashba spin-orbit coupling and electron-electron interactions in 2DEGs .Motivated by these observations we examine the phasediagram of the interacting 2DEG with Rashba and Dres-selhaus spin-orbit interactions, focusing on systems withnearly equal Rashba and Dresselhaus SOC . The sys-tem with exactly equal Rashba and Dresselhaus interac-tions is an interesting special case with SU(2) symmetry.This symmetry is robust against spin-independent disor-der and interactions, and is generated by operators whosewave vector depends on the coupling strength. It rendersthe spin lifetime infinite at this wave vector, giving riseto a persistent spin helix, which has been realisedexperimentally . When the Rashba and Dresselhausinteractions are of equal magnitude, the effective mag-netic field describing the spin-orbit interaction singles outa well-defined direction in momentum space. There isa single spin-quantization axis, and all the spins in thesystem point either parallel or antiparallel to this axis.We determine the full Hartree-Fock phase diagram of the2DEG with equal Rashba and Dresselhaus interactions.When either the Rashba or the Dresselhaus interac-tion is dominant we expect the same phase diagram asin Ref. 48, while the case when the two interactions areequal is qualitatively different. Experimentally, the phasetransitions are observe in transport properties, most no-tably the DC conductivity. We present analytical resultsfor this case and perform classical MC simulations alongthe lines of Ref. [48], focusing on the zero-temperatureproperties of the electron gas. Our analytical calculationsreveal that the out-of-plane spin polarized phase shrinksas the magnitudes of the Rashba and Dresselhaus inter-actions approach each other, i.e. , as the persistent spinhelix state is approached. When the Rashba and Dressel-haus spin-orbit couplings are equal the out-of-plane phasedisappears altogether and only an in-plane spin-polarizedphase exists for all r s > .
01. It is characterized by a sin-gle Fermi surface, which is displaced from the Brillouinzone center, and has a non-trivial spin texture, which be-comes more pronounced at higher values of r s and of thespin-orbit interaction strength. Our expectations basedon this analytical treatment are confirmed by classical Figure 1. Two-Fermi surface state at ˜ α = ˜ β = 0 . r s =1 . MC simulations, which rely on the same method as thatused in Ref. 48.We find, however, two caveats related to the applica-tion of the classical MC method to this system. Firstly,in a narrow parameter regime the bare
MC phase di-agram appears to display a phase with a small out-of-plane spin polarization, which we have referred to as OP ∗ throughout this manuscript. To determine whether thisphase is physical or not we varied the number of pointsin the MC simulations and studied the evolution of theground state in this region as a function of the numberof k-points N . We have found that in the thermody-namic limit N → ∞ the ground state has an in-planespin polarization in exact agreement with the analyticalcalculations. The energy difference, while larger than thenumerical error, remains relatively small. Secondly, thepersistent spin helix system exhibits a spin-density wavephase that is degenerate with the in-plane spin-polarizedphase and that is not captured accurately by our MCmethod. The existence of this phase can be determinedby analytical arguments, and we believe it is related tothe appearance of the OP ∗ phase in the MC phase dia-gram. The application of the classical MC depends onusing a class of HF wavefunctions for which the groundstate energy can be reduced to a classical minimizationproblem. The method will not find ground states outsidethis class such as the spin-density wave.The organization of this paper is as follows: In Sec. II,we describe the Hamiltonian of the system and the spintexture as well as the phase diagram obtained from an-alytical arguments using a gauge transformation. InSec. III, the phase diagram of the interacting 2DEGwith equal Rashba and Dresselhaus spin-orbit couplingis given by our classical MC simulation. In Sec. IV, wecompare the results obtained from the two preceding sec-tions and discuss the possibilities for experimental obser-vation. In Sec. V, we summarize our results. Figure 2. OP ∗ state at ˜ α = ˜ β = 0 . r s = 2 . II. HAMILTONIAN AND GAUGETRANSFORMATION
The general many-body Hamiltonian including bothRashba and Dresselhaus SOC reads: H α,β = (cid:88) k ss (cid:48) (cid:104) k s | H (0) α,β | k s (cid:48) (cid:105) c † k ,s c k ,s (cid:48) + V ee , (1)where c k ,s is the annihilation operator for a single elec-tron with wave vector k and spin index s = ± , and c † k ,s isthe corresponding creation operator. Here we consider atranslationally invariant system, which is permissible aslong as localization effects are negligible. Hence, only di-agonal matrix elements in k appear in the single-particleHamiltonian (first term). Including both Rashba ( α ) andDresselhaus ( β ) spin-orbit couplings, H (0) α,β reads: H (0) α,β = p m + α ( p y σ x − p x σ y ) + β ( p x σ x − p y σ y ) , (2)where σ i are Pauli matrices. The electron-electron inter-action V ee takes the standard form V ee = 12 A (cid:88) kk (cid:48) ss (cid:48) (cid:88) q (cid:54) =0 V q c † k + q ,s c † k (cid:48) − q ,s (cid:48) c k (cid:48) s (cid:48) c k s . (3)where A is the area of the system. The specific formof V q is not important for the the arguments developedin this section, as long as V ee has the usual coordinaterepresentation V ee = (cid:80) i 2, indepen-dent of k . The single-particle spectrum is immediatelyobtained as ε , k ± = (cid:126) ( k ∓ q α ) m − mα , (6)where q α = √ αm (cid:126) ( e x + e y ). Note that the ± q α shift inmomentum appearing in Eq. (6) gives rise to two distinctFermi surfaces, displaced in opposite directions.Besides the existence of the conserved quantity ˜ σ y , itwas recognized early on that a spatially-dependent spinrotation: U α = exp (cid:104) i mα (cid:126) ( σ y − σ x )( x + y ) (cid:105) , (7)relates the non-interacting Hamiltonain H (0) α to the fa-miliar case without SOC, and this is still true after in-cluding spin-independent potentials. The many-bodyform of U α acts as U α c k ,s U † α = c k + s q α ,s and commuteswith the electron-electron interaction. Therefore, thewhole family of many-body Hamiltonians H α = H α,α may be related to the extensively studied 2D electronliquid without SOC: H α = U α ( H α =0 ) U † α − N e mα , (8)where N e is the total number of electrons.Many properties of the spin-orbit coupled system canbe obtained directly from the exact mapping (8), andhere we will be interested in the occurrence of a spon-taneous spin polarization. It is then useful to considerthe spin density operators ˜ S ( r ) = (cid:80) i ˜ σ i δ ( r − r i ), where i = 1 , . . . N e labels the electrons and ˜ σ are the rotatedPauli matrices introduced after Eq. (5). U α transformsthe spin polarization as follows: U α ˜ S x ( r ) U † α = ˜ S x ( r ) cos 2 q α r + S z ( r ) sin 2 q α r , (9) U α S z ( r ) U † α = S z ( r ) cos 2 q α r − ˜ S x ( r ) sin 2 q α r , (10)while U α ˜ S y ( r ) U † α = ˜ S y ( r ). Therefore, a uniformly polar-ized state without spin-orbit interaction leads to a spinwave dependence when α (cid:54) = 0. The only exception is apolarization along e ˜ y , which is kept unchanged.A family of degenerate spin waves is obtained from theSU(2) symmetry. In the system without spin-orbit inter-action, the symmetry operations are simply spin rota-tions. However, combining an arbitrary spin rotation ofthe state uniformly polarized along e ˜ y with the U α trans-formation will induces a spatial precession of the ˜ S x,z components. These collective spin-waves can become therelevant ground states through the combined action ofelectron-electron interactions and spin-orbit coupling. Inthe same way that a Stoner transition leads to ferromag-netism (with α = 0), a sufficiently strong electron in-teraction will lead to the spontaneous formation of thespin-wave states (when α (cid:54) = 0). In this sense, the Stonertransition without spin-orbit interaction corresponds tothe spontaneous formation of collective persistent spin-helix states.The paramagnetic phase is apparently less interest-ing, as there is no spin polarization. However, the ex-istence of persistent spin-helix states is reflected on thequasiparticle excitations. We first note that, withoutspin-orbit coupling, the spin-degenerate Fermi surfacesurvives the effect of the interactions (the Luttinger’stheorem ). Applying U α to the interacting state with-out spin-orbit coupling leads to two distinct Fermi sur-faces, centered around ± q α . Like for the non-interactingcase [see Eq. (6)], the two Fermi surfaces correspond toorthogonal spin directions ± along ˜ σ y .If now we consider quasiparticle excitations, at α = 0they can be generated by single-particle operators witharbitrary spin direction, cos ( θ/ c † k , + + e iφ sin ( θ/ c † k , − where k (cid:39) k F is close to the Fermi wavevector ( k F = √ πn , where n is electron density). For α (cid:54) = 0, theseoperators transform to the following form: c † k , ˆ n ≡ cos ( θ/ c † k + q α , + + e iφ sin ( θ/ c † k − q α , − , (11)which allows us to define stable quasiparticles c † k , ˆ n | F (cid:105) ontop of the (interacting) ground state | F (cid:105) .From the previous discussions, especially Eqs. (9) and(10), it is clear that the quasiparticles states c † k , ˆ n | F (cid:105) carrya spatially precessing spin polarization. In fact, the formof Eq. (11) involves a coherent superpositions of statesat both Fermi surfaces (except for θ = 0), and the ± q α displacements induce the formation of a spin-wave withwavevector 2 q α . For a Fermi liquid, the lifetime of the c † k , ˆ n | F (cid:105) states becomes infinite in the limit k → k F .Then, we see that in the paramagnetc phase the natu-ral excitations of the system are interacting persistentspin helix states.With the help of the c † k , ˆ n operators, we can give amean-field description of the Stoner transition with spin-orbit interaction. The paramagnetic phase correspondsto: | F (cid:105) (cid:39) (cid:89) k ≤ k F c † k , ˆ n c † k , − ˆ n | (cid:105) , (12)where the choice ˆ n = e ˜ y is perhaps more natural,but any other direction of ˆ n is equivalent: they allgive the same spin-unpolarized ground state of the non-interacting Hamiltonian. On the other hand, the mean-field spin-polarized states can be written as: | F (cid:105) (cid:39) (cid:89) k ≤√ k F c † k , ˆ n | (cid:105) , (13) which actually depend on the direction ˆ n , reflecting thebroken SU(2) symmetry. As discussed, only when ˆ n = e ˜ y the spin density is uniform. For other orientations, all theelectrons occupy spin-helix states and the spin density isprecessing in space as described by Eqs. (9) and (10).It is oriented along ˆ n only at periodic positions alongthe e ˜ x direction (e.g., ˆ n can be taken as the polarizationdirection at r = 0).For the unscreened Coulomb interaction, the phase di-agram of the electron liquid at a fixed ratio β/α dependson two dimensionless parameters. The Wigner-Seitz ra-dius r s is r s = me πε r ε (cid:126) √ πn , (14)and is a measure of the interaction strength. In-stead, the Rashba spin-orbit coupling can be rescaled asfollows: ˜ α = mα (cid:126) √ πn . (15)Here, ˜ α is a measure of the Rashba spin-orbit couplingrelative to the kinetic energy. While for β = 0 the de-pendence of the phase boundaries on ˜ α is non-trivial, when α = β we find from Eq. (8) that the phase bound-ares should be independent of ˜ α . They are simply givenby the values in the absence of spin-orbit interaction.For example, it is well known that the Hartree-Fockapproximation gives a Bloch transition at r s (cid:39) . 01 andwe show in Fig. 3 the simple mean-field phase diagram,where the transition to the states of type of Eq. (13) isindependent of ˜ α . In the paramagnetic region, r s < . α > / √ s = ± Fermi discs become non-overlapping (obtained from thecondition | q α | > k F ).Finally, we comment on the average value of the mo-mentum operator P = (cid:80) i p i in the collective persistentspin helix states, giving:1 N e (cid:104) F | P | F (cid:105) = (cid:126) q α cos θ. (16)Except for θ = π/ 2, there is a finite expectation valueof P /N e along e ˜ x = ( e x + e y ) / √ 2, reflecting a displacedmomentum-space occupation of the spin-polarized states.The displacement is largest when the polarization is uni-form and oriented along ˜ σ y ( θ = 0). As usual, Eq. (16)does not imply a finite current in equilibrium, since thesingle particle velocity along e ˜ x is given by ˜ p x /m − α ˜ σ y .The finite expectation value of P /N e is exactly cancelledby the contribution from the spin polarization:2 mαn (cid:104) F | ˜ S y ( r ) | F (cid:105) = 2 mα cos θ. (17)Note from Eqs. (9) and (10) that (cid:104) F | ˜ S y ( r ) | F (cid:105) = n cos θ is the only component of the spin polarization densitywhich is independent on r . IP phase2FSs2FSs - Separate0.5 1 1.5 2 2.5 3 3.5 40.20.40.60.81.1.2 0.5 1 1.5 2 2.5 3 3.5 40.20.40.60.81.1.2 r s α Figure 3. Phase diagram of 2D electron liquid with Rashbaequal to Dresselhaus spin-orbit coupling in the Hartree-Fockapproximation, obtained by gauge transformation. Here, ˜ α and r s are dimensionless measures for the strength of Rashba(Dresselhaus) spin-orbit coupling and electron-electron in-teractions, respectively. For the paramagnetic Fermi-liquidphase 2FSs and 2FSs-separate, there is no net spin polariza-tion. The IP phase exhibits an in-plane magnetization asso-ciated with a shifted Fermi sea. III. MONTE CARLO SIMULATION OFINTERACTING 2DEG WITH RASHBA ANDDRESSELHAUS SPIN-ORBIT COUPLING In the statically screened Hartree-Fock approximation,the exchange energy of the system can be written as E ex = − L (cid:88) k (cid:54) = k (cid:48) e [ s k · s k (cid:48) + n k n k (cid:48) ]4 (cid:15) ( k TF + | k − k (cid:48) | ) , (18)where f k is the density matrix in equilibrium, while n k = Tr f k and s k = Tr( σ f k ) are the electron’s occupationnumber and net spin polarization at k , respectively. Thetotal energy can be expressed as E tot = Tr[ f k H k ] + E ex , (19)We assume that the screening effect is negligible due tothe low electron density, so k TF = 0. In the followingsubsections we use the same method as in Ref. [48] to findthe minimum-energy configuration. When r s is close toa phase boundary, the total energy is linearly dependenton r s , which allows us to use a linear fitting to determinethe transition points. A. Simulation of 2DEG with Rashba spin-orbitcoupling Using a MC simulation, we reproduce the result for2DEG with Rashba spin-orbit coupling in Ref. [48]. As r s increases, the interaction becomes more effective, pro-ducing a tendency towards ferromagnetism. When ˜ α = 0there is the classic Bloch transition that occurs at r s =2 . 01. There are two conventional Fermi liquid (FL) states with one and two occupied spin sub-bands, respectively.The only effect of the exchange interaction is to renor-malize upwards the strength of the Rashba term, andthere is no net spin polarization for the two Fermi liquidstates. The phase boundary of the two Fermi liquid statesis well described by the (noninteracting) critical densityequation, n c = m α π (cid:126) . As r s increases with finite ˜ α , theferromagnetic phase is modified to the OP phase, wherespins have a z component and a component along theeffective field due to the Rashba spin-orbit coupling. Atsmall k , they point nearly along the z direction, but as k increases, they follow the spin orbit-induced field. When r s is even larger, the right half of the phase diagram isthe IP phase. The key feature of the IP phase is that thespin polarization is completely in-plane and the IP phasedoes not have any symmetry on the Fermi surface, eventhough both of them only have a single band. B. Simulation of 2DEG with Rashba andDresselhaus spin-orbit coupling All realistic systems have both Rashba and Dressel-haus spin-orbit coupling occurring together. In this sub-section, we take both coupling with equal strengths intoaccount, where ˜ β represents the strength of Dresselhausspin-orbit coupling. Here ˜ β is a dimensionless variable.We find four phases and plot the phase diagram as func-tions of ˜ α and r s in Fig. 4. The number of k pointsis N = 997. The 2FSs and 2FSs-separate phases arethe conventional Fermi liquid (FL) states. The onlyeffect of the Coulomb exchange interaction is to renor-malize upwards the strength of the spin-orbit coupling.The FL state minimizes the single-particle energy by us-ing the non-interacting states and occupation numbers.When r s > . 01, the MC simulation shows that thereis a narrow region which has partially out-of-plane spin-polarization, and we call it the OP ∗ phase to distinguishit from the OP phase found in Ref. . The spin texture ofthe OP ∗ phase is shown in Fig. 5. For large r s , the righthalf of the phase diagram is the IP phase shown in Fig. 6.A key feature of the IP phase is that the spin polarizationis completely in-plane, as shown in Fig. 6. The IP stategains exchange energy through the finite polarization andthe Fermi surface is shifted. Hartree-Fock simulations areroughly consistent with the analytical results in Sec. III,but the extremely small energy difference between the IPand the OP ∗ phases is not so easy to interpret.To interpret the results from our Monte-Carlo simu-lation, we choose a small enough spin-orbit parameter˜ α = 0 . 16 and different ratios ˜ β/ ˜ α to determine the tran-sition points, which are shown in Table I. From Table I,we see that the OP ∗ region is shrinking with increasingratio ˜ β/ ˜ α and one possible interpretation is that there is anarrow region where the OP ∗ phase is the ground state.The classical MC simulation is not sensitive enough todo more than indicate the trend of the phase boundarybetween the OP ∗ and IP states, since the discretization IPOP * - SeparateBloch Trans.0.5 1 1.5 2 2.5 3 3.5 40.20.40.60.81.1.2 0.5 1 1.5 2 2.5 3 3.5 40.20.40.60.81.1.2 r s α Figure 4. Phase diagram of a 2D electron liquid withequal Rashba and Dresselhaus spin-orbit couplings, obtainedby solving the Hartree-Fock equations using a Monte Carlomethod. Here, ˜ α and r s are dimensionless measures forthe strength of Rashba spin-orbit coupling and the electron-electron interactions, respectively: ˜ α corresponds to the ratioof the Fermi wavelength and spin-precession length, and r s isthe Wigner-Seitz radius of the 2D electron system. The dis-tinguishing features for each individual phase are indicatedschematically. For the Fermi-liquid phases 2FSs and 2FSs-separate, there is no net spin polarization. In contrast, theOP ∗ phase is characterized by an out-of-plane magnetization.The IP phase exhibits an in-plane magnetization associatedwith a shifted Fermi sea. error of the k space is of the same order as the energydifference between the OP ∗ phase and IP phase. Takingthe thermodynamic limit ( N → ∞ ), we find that the IPphase has lower energy than the OP ∗ phase, see Fig. 7.So when r s > . 01, the ground state is the IP phase. OurMC code does not capture the spin density wave states,but those states certainly exist . Based on the analy-sis from our Monte-Carlo simulation, we believe that theOP ∗ phase is a remnant of the spin density wave phase. Table I. The boundary with increasing ratio ˜ β/ ˜ α , where wechoose ˜ α = 0 . β/ ˜ α ∗ OP ∗ -IP0 2.010 2 . . . 996 2 . . 992 2 . IV. EXPERIMENTAL VERIFICATIONA. Materials The phase transitions presented in this paper occur asa function of electron density r s and the two parameters α and β that characterize the strengths of the Rashbaand Dresselhaus interactions. In 2D systems r s and α Figure 5. OP ∗ state with ˜ α = 0 . 23, ˜ β = 0 . r s = 2 . α = 0 . 23, ˜ β = 0 . r s = 2 . are tunable independently by means of the application ofgate voltages and modulation doping. β is more usuallythought of as an intrinsic parameter, but even it dependssurprisingly strongly on the details of the interface and itmay therefore ultimately be variable as well. Thus the 2Dcase offers many advantages over the 3D case in the areaof tunability. Indeed, in spite of many years of searchingit is still somewhat unclear whether the ferromagnetismlong predicted at low density in 3D has been observed,though there are some interesting experimental resultsalong these lines Ref. [59].In true 2D systems it may be difficult to obtain spin-orbit coupling strengths strong enough for the interestingeffects postulated here to occur. The criterion is that thespin-orbit lengths (cid:126) /mα or (cid:126) /mβ should be comparableto the inter-electron spacing. In Si and SiGe devices thespin-orbit coupling is simply too small. In GaAs, theRashba spin-orbit lengths are around 10 − m while typ-ical devices have interparticle spacings perhaps a factorof 5 less than this. Working in this material probablythe best way to observe the new phases in the near term.Indeed, there are indications of a spin-polarized state in2D Si δ -doped GaAs/AlGaAs heterostructures .Hole systems are also promising, since the Rashba - - - - - √ N E O P * - E I P N Figure 7. Linear fit to find the ground state at ˜ α = ˜ β = 0 . r s = 2 . 10, where N is the lattice number we take in theMonte-Carlo simulation. spin-orbit energy can be as large as 40% of the Fermienergy and second order effects in charge transport canbe sizable . The drawback in these systems is disorder.The mean free path (cid:96) tends to be short and one certainlyneeds k F (cid:96) >> . On the reverse side, disorder seems tobe strong also in this case. B. DC Transport In 2D electron systems transport measurements are al-ways the the easiest to carry out. Since the various tran-sitions that are envisioned here are first order, we expectdiscontinuous changes in both the longitudinal and theHall resistances. The coupled transport equations forcharge and spin have been written down for arbitrary α and β , but have not yet found a detailed solution. How-ever, we may understand the qualitative behavior by con-sidering the linearized Boltzmann equation for a simplemodel of short-range spin-preserving impurity scattering.We calculate only the longitudinal resistance σ ij at zeromagnetic field. To begin with, we focus on the case of β < α , since in this case all 4 phases are clearly present.The conductivity is given by σ ij = e π (cid:88) ns (cid:90) d k τ n k s v i,n k s v j,n k s δ ( E n k s − E F ) . (20)Here n labels the pieces of the FS, k is the wavevector, v i,n k s and τ n k s are respectively the i-th component of thevelocity and the transport relaxation time of an electronwith the indicated quantum numbers. The delta functionpins the integrand to the FS.The relaxation time is given by1 τ n k s = e π (cid:88) n (cid:48) s (cid:48) (cid:90) d k (cid:48) W n k s,n (cid:48) k (cid:48) s (cid:48) (1 − cos θ k , k (cid:48) ) . (21)For our model the transition rate is W n k s,n (cid:48) k (cid:48) s (cid:48) = 2 π (cid:126) δ ( E ( n k s ) − E F ) |(cid:104) n k s | U | n (cid:48) k (cid:48) s (cid:48) (cid:105)| (22)= 2 πn imp u (cid:126) δ ( E ( n k s ) − E F ) |(cid:104) s ( k | s ( k (cid:48) (cid:105)| (23)Since the impurity potential U is point-like, the matrixelement u is independent of momentum transfer. Thelast, and very important, factor is the overlap of thespinors at k and k (cid:48) . Note that u has dimensions of energytimes length squared in our normalization.The amplitude for scattering from k to k (cid:48) is propor-tional to the square of the overlap of the spinors at k and k (cid:48) . In a completely polarized ferromagnetic statethis amplitude is unity and the relaxation time τ f is in-dependent of k : 1 τ f = n imp u k F (cid:126) v F . (24)where k F and v F are the Fermi wavevector and the Fermivelocity. We use τ f as a benchmark for the relaxationtimes of the various phases.In the FL1 state there is also only one FS, and τ isagain isotropic. However, the spin texture puts the spinat k at a fixed angle from the direction of k . We have1 τ ( F L 1) = 14 π (cid:90) d k W n k s,n (cid:48) k (cid:48) s (cid:48) (1 − cos θ k , k (cid:48) )(1+cos θ k , k (cid:48) ) / . (25)The (1+cos θ k , k (cid:48) ) / τ f . Performing the integral we find1 τ f = 4 /τ ( F L τ ( F L 2) is more complicated but thesuppression of backscattering is still present. Hence theFL2 state has a conductivity that is also significantlyenhanced over the ferromagnetic state. In general theconductivities of the FL1 and FL2 states are expected tobe rather similar.The OP state is rather close to the completely polar-ized ferromagnetic state in its spin texture, differing inthat on the FS the spin angle from the z-axis is α . Thisyields some amount of backscattering suppression andleads to1 /τ ( OP )1 /τ ( F L 1) = cos ( α/ ( α/ − cos ( α/ 2) sin ( α/ . (26)Thus the OP state has a conductivity close to, butslightly greater than, the completely polarized state.The IP state has a complex texture that is not easilyexpressed analytically. However, it is the most interest-ing in that it breaks rotational symmetry by virtue ofthe displacement of the FS. In fact, the conductivity isanisotropic: σ xx (cid:54) = σ yy . For definiteness, say that theFS moves off center along the y direction. At the sametime a ferromagnetic moment in the -x direction devel-ops. If α is small, the state is nearly completely spin po-larized and the conductance is low and nearly isotropic.As α increases, the conductivity increases and becomesanisotropic. In the y-direction the conductivity mainlycomes from the states that have k (cid:107) ± ˆ y . These stateshave a ferromagnetic character, with the spins pointingmainly along − y and backscattering is therefore allowed.For the states with k (cid:107) ± ˆ x we have an antiferromagneticconfiguration in the following sense: for k (cid:107) + ˆ x the spinsare in the -y direction, while for k (cid:107) − ˆ x the spins are inthe +y direction. Hence backscattering is suppressed andwe find an enhanced conductivity. Hence overall for theIP phase (1) the conductivity depends strongly on the socoupling, with anisotropy developing as α increases; (2) σ xx > σ yy ; (3) the jump in conductivity on passing fromthe OP to the IP phase is small at low α and increases as α increases; (4) overall, the conductivity is intermediatebetween the FL and OP states.These considerations are summarized in Table 2. Table II. Conductivity of different phases at small β Phase FL1 FL2 OP IPConductivity High High Low MediumAnisotropic? No No No Yes By means of transport measurements it should there-fore be possible not only to detect phase transitions, butalso to identify precisely which phases are involved.In the case of electrons on the surface of topological insulators, similar considerations apply, though it is of-ten difficult to disentangle surface from bulk transport.However, one may be able to perform spin-resolved pho-toemission and observe textures directly, an option thatis not usually available in true 2D systems. V. CONCLUSION The competition among the kinetic, interaction, andspin-orbit contributions to the electronic energy producesa rich variety of phases in the parameter space of therelative strengths of these energies. When we add thedimension of the relative strength of Rashba and Dres-selhaus couplings α and β , the presence of an additionalsymmetry when α = β adds to the fascination of thisphysical system. We treat the symmetric point perform-ing a canonical transformation and add the informationso obtained to our MC simulation within the HF approx-imation. When α (cid:54) = β , we identified 4 distinct groundstates: 2FSs, 2FSs-Separate, OP* and IP phases, butwhen the symmetric point is approached, then the OP*phase gets squeezed out. The various phases have dif-ferent DC transport properties, which aids experimentalidentification.The Coulomb correlation energy increases the effectivemass and the absolute value of the correlation energyof the unpolarized 2DEG ground state is greater thanits polarized counterpart . So even with the corre-lation energy taken into account, the unpolarized 2FSsand 2FSs-Separate phases still have lower energies thanthat of the IP phase. With regard to the Pomeranchukinstability, it is an instability in the shape of the Fermisurface of a material with interacting fermions, causingLandau’s Fermi liquid theory to break down . 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