Electronic Liquid Crystalline Phases in a Spin-Orbit Coupled Two-Dimensional Electron Gas

0.89 MB·English

by  Erez Berg

ElectronicLiquidCrystallinePhasesinaSpin-OrbitCoupledTwo-DimensionalElectronGas
Erez Berg,1 Mark S. Rudner,1,2 and Steven A. Kivelson3
1Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
2IQOQIandInstituteforTheoreticalPhysics, UniversityofInnsbruck, 6020Innsbruck, Austria
3Department of Physics, Stanford University, Stanford, California 94305, USA
(Dated:January25,2012)
Wearguethatthegroundstateofatwo-dimensionalelectrongaswithRashbaspin-orbitcouplingrealizesone
ofseveralpossibleliquidcrystallineorWignercrystallinephasesinthelow-densitylimit,evenforshort-range
repulsiveelectron-electroninteractions(whichdecaywithdistancewithapowerlargerthan2). Dependingon
specificsoftheinteractions,preferredground-statesincludeananisotropicWignercrystalwithanincreasingly
anisotropic unit cellas the density decreases, a stripedor electron smectic phase, and a ferromagneticphase
whichstronglybreaksthelatticepoint-groupsymmetry,i.e.exhibitsnematicorder. Meltingoftheanisotropic
2
Wignercrystalorthesmecticphasebythermalorquantumfluctuationsgivesrisetoanon-magneticnematic
1
0 phasewhichpreservestime-reversalsymmetry.
2

n I. INTRODUCTION short-rangeinteractions,V(r) 1/rαwithα>2.(Inpartic-
a ∼
ular,notethattheCoulombinteractionscreenedbyametallic
J
gateisdescribedbyα=3.)Theinstabilityinthiscaseoccurs
4 Enhancingtheroleofelectron-electroninteractionsrelative
atanelectrondensitynforwhichtheFermienergyissmaller
2 to that of the kinetic energy often leads to interesting many-
thananenergyscalesetbytheSOC.
body effects. Electron crystallization is an extreme example

] of this phenomenon. At low densities, where electrons are Wehaveinvestigatedcandidateorderedstatesbyconstruct-
l
e far apart and the kinetic energy cost of localization is low, ing variational wave-functions, determining the patterns of
-
r Coulombinteractionsdominateandtheelectronsformanor- broken symmetry which minimize their variational energies,
st deredstate,knownasaWignercrystal1. and then comparing the energy to that of the UFL. We con-
.
t The nature of the crystallized electronic state has been in-
a
tensely investigated through Quantum Monte Carlo (QMC)
m
calculations2,3.Thesecalculationssupporttheexistenceofthe (a) (b)
-
crystallinephase,thoughthedensityatwhichtheyfindcrys-
d
n tallization is typically significantly lower than what heuris-
o tic arguments would suggest. The Wigner crystal has also k)
c been sought experimentally. Despite difficulties associated E(
[
withreachingtheultra-lowdensityregimewherecrystalliza-

3 tionisexpected, evidencethattheWignercrystalphasemay k k0 k
v havebeenrealizedhasbeenreportedforexperimentsonever- x y
2 cleaner samples of two-dimensional electron gases (2DEGs) (c) (ii)
2 insemiconductorquantumwells4.
2
In this paper, we study the low density limit of the 2DEG (i) (iii)
1
. system with Rashba spin-orbit coupling (SOC), which is
8 presentwheneverthe2DEGlacksinversionsymmetry5. This
0
is the case, for example, when the 2DEG is confined in an 2 a
1
1 asymmetricquantumwell,orifitisformedatthesurfaceofa
: three-dimensionalmaterial. AsshowninFig.1a,theresulting
v
dispersionrelationhasanextended(highlydegenerate)mini- Figure 1: (a) Dispersion of a particle with Rashba SOC. The min-
i
X mumwhichformsaringinmomentumspace.Thelow-energy imum of the dispersion occurs on a ring in k space, marked by a
r density of states exhibits a divergent van Hove singularity, redcircle.Theredarrowsshowthespinpolarizationofthedifferent
a ρ(ε) ε 1/2,akintothebehaviorofaone-dimensionalsys- Bloch states. (b) Density of states as a function of energy, corre-
−
tem(∼seeFig.1b). Thisisinstrikingcontrasttotheusualbe- spondingtothedispersionshownin(a). Nearthebandbottom,the
densityofstatesdivergesasρ(E)∼(E)−1/2. (c)Schematicphase
havior ρ(ε) const, familiar for two-dimensional systems
∼ diagraminthelow-densitylimitwithrepulsiveelectron-electronin-
without spin-orbit coupling. Thus the Rashba SOC greatly
teractionswhichdecayatlongdistancesasV ∼ r−α. Forα ≤ 2,
enhancestheroleofinteractionsrelativetothatofkineticen-
thegroundstateisanisotropicWignercrystal(i.e.itpreservesadis-
ergyinthelow-densitylimit.
creterotationalsymmetry,C withn > 2). Forα > 2,stateswith
n
ForasystemwithCoulombinteractions, V(r) 1/r, the afurtherbrokenrotationalsymmetryarefavored. (i)representsan
∼
groundstateatlowdensitiesisaWignercrystal,justasfora anisotropicWignercrystalwithaunitcellwhichbecomesparamet-
2DEGwithoutspinorbitcoupling. Remarkably,however,we rically anisotropic in the low–density limit. (ii) and (iii) represent
findthatwithRashbaSOC,brokensymmetrystatesappearto snapshotsofasmecticstateandaferromagneticnematicliquidstate,
respectively.
befavoredovertheuniformFermiliquid(UFL)stateevenfor
2
sider the following types of broken symmetry states: 1)
Table I: Scaling of the ground state energy per particle as a func-
Wignercrystalline(WC)states,i.e.insulatingstateswithonly
tionoftheelectrondensityninthelowdensitylimit,forthevarious
discrete translational symmetry corresponding to one elec-
candidategroundstatesconsideredinthispaper: thenematicferro-
tronperunitcell, allowingforvariouspossiblecrystalstruc-
magnetic(FM)state,theanisotropicWignercrystal(AWC),andthe
turescorrespondingtodifferentpatternsofrotationsymmetry smectic. For α < 2, the isotropic Wigner crystal always has the
breaking; 2) an electron smectic state which breaks transla- lowestenergy.
tional symmetry in only one direction, and can be viewed as
a partially melted version of an anisotropic WC; 3) a ferro- state 2<α≤3 3<α≤4 4<α
magneticnematicstatewhichpreservestranslationsymmetry, nematicFM n2(1−α1) n2(1−α1) n32
but breaks time reversal symmetry and rotational symmetry
- this state is invariant under time reversal followed by a ro- AWCorsmectic n2(1−α1) n43 n34
tation by π around the symmetry axis. Note that we refer
to a WC as anisotropic when only a discrete 2-fold rotation
symmetry(C )remainsunbroken. Inthelimitoflowdensity, state. Thescalingofthegroundstateenergyasafunctionof
2
each of these ordered states has parametrically lower energy the electron density for each type of state considered in this
(inpowersofthedensityn)thantheUFL.Thisstronglysug- paperislistedinTableI.
geststhattheUFLisunstableatlowdensities. Incontrast,the Thispaperisorganizedasfollows. Themodelisdescribed
energybalancebetweendifferentbrokensymmetryphasesis in Sec. II. In Sec. III, we explain the basic physics leading
moredelicate, andmaywelldependonlong-distancefluctu- to anisotropic Wigner crystal formation. We analyze three
ationaleffectsthatarenotwellcapturedbyvariationalwave- cases: contactinteractions, extendedshort-rangeinteractions
functions; we will return to this point in the final section of (which fall off with distance with a power which is larger
thepaper. than 2), and long-range interactions. Considerations related
The nature of the low-density instability of the UFL, and tothemagneticstructureoftheWignercrystalphasearedis-
the origin of the strong tendency of the system to a nematic cussed in Sec. IV. In Sec. V we discuss melting the Wigner
patternofrotationsymmetrybreaking(whetherornotitisac- crystal partially to obtain a smectic state. In Sec. VI we dis-
companied by other patterns of symmetry breaking) can be cusstheferromagneticnematicstate. Sec.VIIpresentsapro-
most easily seen by studying candidate WC wavefunctions. posedschematicphasephasediagramfora2DEGwithSOC
A schematic version of the resulting WC phase diagram is and screened Coulomb interactions. In Sec. VIII we discuss
showninFig. 1casafunctionoftheexponentα. Forα > 2 possible realizations in electronic and atomic systems. Ap-
(short-range interactions), the unit cell of the Wigner crystal pendixApresentsthesolutionofasingleRashbaparticleina
becomesincreasinglyanisotropic,withanaspectratiothatdi- boxproblem,whichiscrucialfortheargumentsregardingthe
verges in the low-density limit. This unusual behavior can anisotropic Wigner crystal and the smectic phases, and Ap-
be traced back to the form of the single-particle dispersion, pendixBpresentsthedetailsoftheHartree-Fockanalysisof
Fig.1a,whichisstronglyanisotropicinthedirectionsperpen- theferromagneticnematicstate.
dicularandtangentialtothering-likeminimum.
A complimentary view can be obtained by considering a
ferromagnetic state. Because of the Rashba SOC, the ori-
II. MODEL
entation of the magnetization vector (which is always in-
plane) necessarily defines a preferred nematic axis - any fer-
We consider a 2DEG with Rashba SOC and repulsive
romagneticstatemustnecessarilybenematic,althoughanon-
electron-electron interactions, described by the Hamiltonian
magneticnematicphaseispossible. Forexample,amagnetic
(inunitswith(cid:126)=1)
momentintheydirectionimpliesaspecialroleforthepoint
(cid:126)kdi0spinerFsiigo.n1wa,itwhhtihcehldaergfiensetsptohsesipbolienvtaolnuethoefrkinxg. oAftmloinwimduenm- H = (cid:88)(cid:26)21m(cid:20)−∇2j − 2ki0(∇j ×zˆ)·(cid:126)σj(cid:21)+E0(cid:27)
sity, the resulting Fermi surface forms an ellipse encircling
j
this special point. As in the usual Stoner theory of ferro- 1(cid:88)
magnetism,spinpolarizationlowerstheinteractionenergyvia + V((cid:126)rl (cid:126)rj ). (1)
2 | − |
the Pauli-exclusion principle, which helps electrons to avoid l=j
(cid:54)
each other at short distances. However, the cost in kinetic
energy is parametrically smaller than in a conventional FL, Here,mistheelectroniceffectivemass,k0isaparameterthat
owing to the divergent density of states. The variational en- characterizesthestrengthoftheSOC,E0 ≡ k02/(2m),(cid:126)σj is
ergywefindfortheferromagneticnematicstatediffersfrom thevectorofPaulimatriceswhichactonthespinofelectron
that of the anisotropic crystal only by a numerical constant j, and V((cid:126)r ) is the (repulsive) electron–electron interaction
| |
for 2 < α 3, so it is not possible, on the basis of the potential. TheHamiltonian(1)isinvariantundertranslations,
presentconsi≤derations,toconfidentlydeterminewhich(ifei- underrotationsaroundthezaxis,andundermirrorreflections
ther)isthepreferredgroundstate. Forα > 3,theferromag- aboutthexandy axes,Mx andMy,butnot underinversion
netic nematic state has parametrically lower energy than the (cid:126)r (cid:126)r.
→−
anisotropicWC,suggestingthatitisabettercandidateground Below we consider cases in which, at large inter-particle
3
separationr,theinteractionpotentialdecaysasapowerlaw: Thepotentialenergyperparticle,ontheotherhand,is
ˆ
V 1
V ((cid:126)r ) 0. (2) ε¯pot = d(cid:126)rd(cid:126)r(cid:48)V ((cid:126)r (cid:126)r(cid:48)) n((cid:126)r)n((cid:126)r(cid:48)) , (7)
| | ∼ rα 2nΩ − (cid:104) (cid:105)
We distinguish between long-range and short-range interac- whereΩisthetotalareaofthesystem,n((cid:126)r)isthelocalden-
tions, which are characterized by α 2 and α > 2, respec- sityatposition(cid:126)r,and denotesaveragingintheuniform
tively. ThebareCoulombinteraction≤isdescribedbyα = 1, (Fermi gas) state. In(cid:104)t·h·e·(cid:105)low–density limit, for short-range
while screening can lead to α > 1. In particular, screening interactions (α > 2 in Eq. 2), ε¯pot n. For long-range
∝
duetoanearbymetallicgatetopgateleadstoα=3. interactions, the right hand side of Eq. (7) diverges. Here a
In the absence of interactions, V((cid:126)r ) = 0, Eq. (1) yields neutralizing background must be taken into account, leading
thesingle-particledispersionlaw(see|F|ig.1a): to ε¯pot nα/2. We see that in all cases, in the low-density
∝
limit, ε¯ ε¯ , suggesting that the uniform state is un-
pot kin
E((cid:126)k)= 1 (cid:0)k2 2k k(cid:1)+E , (3) stable to fo(cid:29)rming some sort of order. One possibility is that
0 0
2m ± at asymptotically low densities, the ground state is a Wigner
crystal,asina2DEGwithnoSOC.Note,however,thathere,
where(cid:126)kistheelectronmomentum((cid:126)=1),andk = (cid:126)k . The inthepresenceofRashbaSOC,thisinstabilityoccursevenfor
| |
minimum kinetic energy occurs for any value of momentum short-rangeinteractions.
falling on a ring of radius k = k , with a minimal value of
0
E =0.Thecorrespondingdensityofstates,showninFig.1b,
isgivenby B. Contactinteractions
(cid:40) (cid:113)
D(E)= mπ |EE0| (E <E0), (4) For simplicity, we begin by considering the case of the
m (E >E ). shortest range interactions: repulsive “contact” interactions.
π 0 Westartwithavariationalwavefunctionwhichminimizesthe
For E > E , the density of states is independent of energy, interaction energy, taking each electron to be confined to a
0
just as for a usual 2DEG without spin-orbit coupling. How- rectangularboxofdimensionsLx Ly,withdifferentboxes
×
ever, for E 0, the density of states diverges as 1/√E. non-overlapping. This is a zero energy eigenstate of the po-
Because this→divergence will play a crucial role in the anal- tentialenergyoperator.Inorderfortheboxestotiletheplane,
ysisbelow,wecommentbrieflyonitsorigin. Thedivergence LxandLy areconstrainedbythecondition
comesfromthefactthattheminimumofkineticenergyisin-
1
finitelydegenerate,occurringeverywhereonaringinmomen- L L = . (8)
x y
n
tumspace,ratherthanatasinglepointorfinitesetofpoints.
Inthepresenceofcrystallineanisotropy(whichmanifestsit- Thus, we have a single variational parameter, the aspect ra-
selfthroughcorrectionstotheeffectivemassapproximationin tioη L /L ,whichweusetominimizethekineticenergy
x y
≡
realmaterials),thedivergenceiscutoffnearthebandbottom ofthetrialstate. Thekineticenergyperparticleinthevaria-
andthedensityofstatesgoestoaconstant. However,aslong tionalstateisgivenbythegroundstateenergyofasinglepar-
ask a/π 1,whereaisthelatticeconstant(i.e.aslongas ticleinaboxwithRashbaSOC.Thisproblemisinvestigated,
0
(cid:28)
the spin-orbit coupling is weak), the crystal field anisotropy both analytically and numerically, in Appendix A. Surpris-
terms are small and Eq. (4) provides a good approximation ingly, unlike the case with no SOC, the ground state energy
downtoenergiesoforder E (k a)2abovethebandbottom. inthelow–densitylimitisminimalforη = 1. Intheη 1
0 0
| | (cid:54) (cid:29)
limit, we find the following expression for the ground state
energyasafunctionofηandn:6
III. WIGNERCRYSTAL
(cid:18) (cid:19)
n Bn
ε(n,η)= Aη 1+ η2 , (9)
A. InstabilityoftheFermiliquidstate 2m − k02
whereAandB arenumbersoforderunity,seeEq.(A14)in
Webeginbyconsideringthestabilityoftheuniform(Fermi Appendix A. Minimizing Eq. (9) with respect to η, we find
liquid)state. AccordingtoEq.(4),atlowdensities,theFermi thattheoptimalaspectratioη(cid:63)scalesas
energyε is
F
η(cid:63) (cid:0)n/k2(cid:1)−13 , (10)
π2n2 ∼ 0
ε = , (5)
F 4m2 E andthegroundstateenergyperparticlescalesas
0
| |
where n is the density of electrons per unit area. Thus, the ε(cid:63)(η(cid:63)) E (cid:0)n/k2(cid:1)43 . (11)
kineticenergyperparticleinthehomogeneousstateis ∼| 0| 0
ˆ Therefore,inthelow–densitylimit,wegetthattheenergyper
1 εF π2n2 particle of this anisotropic Wigner crystal state is parametri-
ε¯ = εD(ε)dε= . (6)
kin n 12m2 E callylowerthanthatoftheuniformstate, whichscalesasn.
0 | 0|
4
Notethat,consistentwithourassumptions,theoptimalaspect case,asinthecaseofcontactinteractions,theWignercrystal
ratiooftheunitcellintheWignercrystalbecomesparametri- stateisextremelyanisotropicinthelow–densitylimit.
callylargeatlowdensities.
Inthecaseofextendedinteractions,thepotentialenergyin
The fact that the kinetic energy is minimal for an
theWignercrystalphasecannotbeneglected. Toestimatethe
anisotropic box can be understood as follows. Suppose that
potentialenergy,weconsiderthesamevariationalwavefunc-
the ground state wavefunction is a superposition of plane
tionasbefore,inwhichtheparticlesarelocalizedinanarray
waves with wavevectors close to some wavevector (cid:126)k(cid:63) of ofnon-overlappingL L boxes. Toestimatethepotential
length k0, for which the dispersion (3) is minimal. Near(cid:126)k(cid:63), energy,wewillreplacxe×thewyavefunctionofeachparticlebya
the dispersion is quadratic in the radial direction, while it is constant,suchthatthedensityisuniform,n=1/(L L );the
x y
anomalouslyflat(quartic)inthetransversedirection. There- parametric dependence of the energy on n and η should not
fore confinement in the direction perpendicular to(cid:126)k(cid:63) is less depend on this assumption. Let us focus on the anisotropic
costlythanconfinementparallelto(cid:126)k(cid:63),andtheoptimalaspect limit, L L , assumingthatthisistheoptimalconfigura-
x y
(cid:29)
ratio is such that the box is long in the direction of(cid:126)k(cid:63), and tion. The interaction energy of a given particle with all the
shortinthetransversedirection. otherparticlesis
C. Extendedshort-rangeinteractions
ε¯ (n,η) 2(U +U ), (12)
v 1 2
≈
We now turn to the case of extended, short–range interac-
tions,whichcorrespondsto2<α< . Weshowthatinthis whereU andU aregivenby
1 2
∞
ˆ ˆ ˆ ˆ
U1 = L2x1L2y 0Lxdx 0Lydy −∞∞dx(cid:48) L∞y dy(cid:48)[(x−x(cid:48))2+V(0y−y(cid:48))2]α/2 =V0nα2ηα2−1C1 (13)
and
ˆ ˆ ˆ ˆ
U2 = L2x1L2y 0Lxdx 0Lydy L∞x dx(cid:48) 0Lydy(cid:48)[(x−x(cid:48))2+V(0y−y(cid:48))2]α/2 =V0nα2ηα2−2(cid:2)C2+O(cid:0)η2−α(cid:1)(cid:3). (14)
C andC aredimensionlessconstantswhichdependonα.In We see that, for 2 < α < 3, η still becomes parametrically
1 2
theη 1limit,wegetthatU U ,andthereforewene- largeinthen 0limit.InsertingEq.(17)backintoEq.(16),
1 2
glectt(cid:29)helatter. Now,ifweassum(cid:29)ethatη n−13,asEq.(10) weget →
∼
suggestsinthecaseofcontactinteractions,wefind 1
ε(cid:63)tot ∼ m(2mV0)2/αn2(1−α1) (2<α<3). (18)
ε¯v ∼n31(1+α). (15) Thusfor2 < α < 3theanisotropicWignercrystalhaspara-
metricallylowerenergyperparticlethantheuniformstate.
We see that, for α > 3, ε¯ becomes negligible compared to
v
the kinetic energy ε(cid:63)(η(cid:63)) n4/3, Eq. (11), in the n 0
∼ →
limit. Therefore, Eqs. (10) and (11) are not modified in this
D. Long-rangeinteractions
case. For α < 3, ε¯ dominates in the low-density limit, and
v
we should consider both the kinetic and potential energies,
Forα < 2(longrangeinteractions), theWignercrystalin
Eqs.(11),(12):
the low density limit has the same hexagonal (C ) symmet-
6
rictriangularstructureastheclassicalcrystallinephasewhich
ε¯ = ε¯ (n,η)+ε(n,η)
tot v minimizesthepotentialenergy.Wewillrefertothehexagonal
(cid:18) (cid:19)
≈ 2nm Aη−1+ Bk2nη2 +C1V0nα2ηα−22. (16) cdreyssctrailbeads“pirseovtirooupsilcy”,,washiocphphoasesdatloowtheer“saynmismotertorpy.icT”ocrsyhsotwal
0
thattheWignercrystalisisotropicforα<2,wenotethatthe
Minimizingwithrespecttoη andkeepingonlythemostsin- potentialenergyinaclassicalcrystalscalesas
gulartermasn 0gives
→ εWC nα2. (19)
∼
η(cid:63) 1 nα2−1 (2<α<3). (17) If,intheWignercrystalphase,eachelectronisconfinedtoa
∼ (2mV )2/α region whose dimension is some fraction of the mean inter-
0
5
electrondistance,thekineticenergycostofformingthecrys- 0.5
talscalesasthedensityn,asinthecasewithoutSOC7.(Here,
unlikebefore,weassumethattheregiontowhichtheelectron 0.4
S
is confined has an aspect ratio of order unity.) Therefore, in y
thelow-densitylimit,crystallizationyieldsapotentialenergy
0.3
gainwhichoverwhelmsthekineticenergycost. Thus,tofirst y,z
approximation,wemayignorethekineticenergy.Theground Sx,
0.2
stateisthereforeahexagonalWignercrystal,andisnotqual-
S
itativelyaffectedbytheRashbaSOC. x
0.1
S
z
0
1 1.5 2 2.5 3
IV. MAGNETICPROPERTIESOFTHEWIGNER η
CRYSTAL
Figure2:ThequantitiesS ,definedinEq.20vs.theaspectratio
x,y,z
So far, we have ignored the magnetic degrees of freedom. η=L /L ofaboxwithfixedareaΩ=75/k2.WhenL /L =1,
x y 0 x y
ThegroundstateofaRashbaparticleinaboxistwo-foldde- S =S ;asL /L increases,S becomescloseto1andS →0.
x y x y y x
generate, according to Kramers’ theorem. Correspondingly,
the variational wavefunction we considered (in which elec-
trons occupy non-overlapping boxes) is 2N-fold degenerate, V. SMECTICSTATE
whereN isthenumberofelectrons. Thisdegeneracyislifted
byexchangeinteractions. The anisotropic Wigner crystal variational wavefunction
First,weelucidatethenatureoftheKramerspairofground described above assumes that translational symmetry is bro-
statesofasingleelectroninabox. Becauseofthespin-orbit ken. However, long-range quantum fluctuations can restore
coupling, these states are not eigenstates of the spin opera- translational symmetry, either partially or fully, resulting in
tor(cid:126)σ. However, in the anisotropic (η = L /L 1) limit, eitherasmecticstate(whichbreakstranslationalsymmetryin
x y
there are particular linear combinations of the t(cid:29)wo ground only one direction), a nematic state which breaks rotational
stateswhichareapproximatelypolarizedinthe yˆdirections, symmetry but is translationally invariant, or an isotropic liq-
whereyˆisthenarrowdirectionoftheunitcell.±Theexpecta- uid.Sincethegroundstateenergyismostlysensitivetoshort-
tionvaluesofallotherspincomponentsaresmall;thisistrue rangecorrelations,thecrystalandthevariousliquidstatescan
foranychoiceofbasisinthegroundstateHilbertspace. havecloseenergies.Below,wedemonstratethatonecanwrite
an explicit wavefunction describing a smectic state with the
Todemonstratethis,wecalculatethequantities
same parametric dependence of the ground state energy on
(cid:115) density as that of the anisotropic WC. In the next section, a
1 (cid:88)
S αs β 2, (20) ferromagnetic nematic variational wavefunction will be de-
i i
≡ 2 |(cid:104) | | (cid:105)| scribed.
α,β=1,2
Letusconsider,forsimplicity,thecaseofshort-range(con-
tact) interactions. To construct a wavefunction for the smec-
where α = 1,2 are the two ground states obtained from
| (cid:105) tic,weconsideratrialHamiltonianinwhichtheelectronsare
the numerical solution of the particle in a box problem (see
confined to move along an array of strips of width L with
AppendixA),ands = σ /2whereσ arePauli(spin) y
i i i=x,y,z
infinitely hard walls separating differentstrips. The problem
matrices. Asdefined,S isthemaximumexpectationvalueof
i
isthenreducedtofindingtheelectronicgroundstateofastrip
thespincomponentiinthegroundstatemanifoldspannedby
with a linear density of nL . The single-particle dispersion
theKramerspair α = 1,2 . ThevaluesofS asfunc- y
i=x,y,z
| (cid:105) in the strip is derived in Appendix A (Eq. A12). The dis-
tionsoftheaspectratioηareshowninFig.2. Asηincreases,
persion of the each transverse sub-band has two degenerate
Sy becomes close to 1/2, while Sx,z → 0. This can be un- “valleys”at (k +δk(cid:63)),whereδk(cid:63) 1/(cid:0)k L2(cid:1). Wewill
derstood as a consequence of the fact that when Lx Ly, ± 0 x x ∼ 0 y
(cid:29) assumethatonlythelowestsub-bandofthestripisoccupied.
the ground states contain mostly components with momenta
close to(cid:126)k = k xˆ, with spin polarizations close to the yˆ (ForafixedLy,thisisvalidforasufficientlysmalldensity12.)
± 0 ± Moreover,wemayignorethe“valley”degeneracysinceevery
directions.
electroncanbe assumedtobeinone ofthetwovalleys, and
ThemagneticdegreesoffreedomintheanisotropicWigner
exchange processes between the valleys are suppressed. We
crystalcanthereforebethoughtofasIsing-likespins, which
thusobtainaneffectiveone-dimensionalHamiltonianforthe
arepolarizedinthe yˆdirections. Thesespinsarecoupledby
motionalongastrip:
exchangeprocesses±, whichgenerateN-bodyinteractions8–10.
Inaddition,becauseofthespin-orbitcoupling,VanderWaals-
likeinteractionsgeneratespin-spinterms11,whicharenotex- H =(cid:88)(cid:18)∆ ∂x2 (cid:19)+ 1(cid:88)V (x x ), (21)
ponentiallysuppressedintheWignercrystalphase.Adetailed − 2m(cid:63) 2 i− j
i i,j
estimate of these interactions is complicated, and we defer
theiranalysisforlaterwork. where,fromEq.(A12), ∆ = A2/8mk L4 andm(cid:63) m(A
1 0 y ∝ 1
6
isadimensionlessconstant). Inthen 0limit,theinterac-
→
tionbecomesstrongcomparedwiththeFermienergy,andthe (cid:40)
edleencttlryonosftbheehiravvealalesyeifnfedcetxiv).eTlyhehasrydstceomrecapnarbtieclmesap(ipneddeopnento- εFM ∼ nn232(,1−α1), αα≤>44. (24)
anon-interactingspinlessfermionproblem. Thegroundstate
Comparing this to Eqs.(11),(18), and (23), we see that the
energyperparticleisthus
ground state energy of the ferromagnetic state is parametri-
A (πnL )2 cally smaller than that of either the anisotropic WC or the
ε(L )= 1 + y . (22) smectic states for α > 3, making it the best candidate for
y 8mk2L4 6m
0 y ∗ the ground state. For 2 < α 3, the scaling of the ground
≤
MinimizingthisexpressionwithrespecttoL ,weobtainthe stateenergywithdensityofallthreestateshasthesameexpo-
y
groundstateenergyperparticleofthesmecticstate: nent. Theenergiesdifferonlybytheprefactor,whichcannot
be estimated reliably within the simple variational approach
usedhere.
k2 (cid:18) n (cid:19)43 Theferromagneticnematicstatespontaneouslybreakstime
ε 0 . (23)
SM ∼ m k0 reversal (T) and rotational symmetry (Rθ) about the z axis,
as well as the mirror symmetry, , for the plane parallel
y
The scaling of the energy of the smectic state with n is thus M
totheferromagneticmoment. However,itpreservestheprod-
thesameasthatoftheanisotropicWignercrystal,Eq.(11)12.
uct, ,oftime-reversalandrotationbyπ(hencethename,
π
Thenumericalprefactor,whichcannotbedeterminedreliably TR
“nematic”) and reflection through the plane perpendicular to
fromsuchsimpleconsiderations,isthereforeimportantinde-
themoment, .Thelattersymmetryinsuresthatthereisno
x
termining which of these two states is favored in the n 0 M
out-of-planecomponentofthemagnetization,andnoanoma-
→
limit.
lous Hall effect. Note that, even though this state carries a
In the case of extended interactions which decay with an
finitecrystalmomentum,itdoesnotcarryafinitecurrentden-
exponent α, one can use the same variational wavefunction sity,asrequiredbyatheorembyF.Bloch13–15. Thereis,how-
forthesmectic,inwhichtheexpectationvalueofthepotential
ever,alargeanisotropyinthein-planeDrudeweight(effective
energy is finite, and then minimize the total energy over L .
y mass).FromourHartree-Fockstate(seeAppendixB),wefind
TSehce.cIaIIlcCu,laatniodnwperowciellednsotinrepesesaetntthiealdlyettahilesshaemree. Twhaeyraessuilnt thattheanisotropyscalesasn2−α4 forα < 4,andasn−1 for
α 4,inthelow-densitylimit.
is that the parametric dependence of the smectic variational ≥
Thephysicsbehindtheformationofthein-planeferromag-
energyonnisthesameasthatoftheWignercrystal,Eq.(18),
neticstateissimilartotheusualStonerpictureforferromag-
foranyα.
netism: the system gains exchange energy by polarizing, at
theexpenseofkineticenergy.InasystemwithRashbaSOCat
lowdensity,theexchangeenergygainexceedsthekineticen-
VI. FERROMAGNETICNEMATICSTATE
ergycostduetothehighdensityofstates. Inthelow-density
limit, the Fermi surface becomes parametrically anisotropic.
Finally, weconsideracompletemeltingoftheanisotropic
Qualitatively,theshort-rangecorrelationsinthisstatearesim-
Wigner crystal phase, preserving its preferred orientation.
ilar to those of the anisotropic Wigner crystal. This explains
This results in a nematic state. Similarly to the situation in
whythesestatesarecloseinenergy, atleastforα 3. The
the Wigner crystal and the smectic states, we expect that in ≤
long-range correlations, however, are very different: the fer-
the low-density limit, only states in the vicinity of two op-
romagnetic state is a fluid, whereas the Wigner crystal is an
posite points (cid:126)k0 on the ring of minimal dispersion will be insulator.
±
occupied. For simplicity, we will assume the occupation is
limitedtothevicinityofonlyonepointonthering,(cid:126)k ,which
0
makesthenematicstatealsoferromagnetic(withanin-plane VII. PHASEDIAGRAM
magnetization perpendicular to(cid:126)k ). Such a state is particu-
0
larly easy to describe within a Hartree-Fock approximation. So far, we have argued that for sufficiently low density
Weemphasize,however,thataparamagneticnematicstateis andforshort-rangedinteractions,thesystembreaksrotational
also possible, although it is not easily captured by a simple invariance, going into either an anisotropic Wigner crystal,
variationalwavefunction. smectic, or a nematic state. We now discuss the global fea-
The Hartree-Fock analysis of the ferromagnetic nematic turesofthephasediagramasafunctionofdensityandthein-
state is straightforward, and is described in Appendix B. At teractionrange. Forconcreteness,letusdiscussa2DEGwith
asufficientlylowdensity,aspontaneousin-planemagnetiza- Rashba SOC with screened Coulomb interactions, where the
tiondevelops,andtheFermisurfacebecomesasymmetric. At screeningisfromanearbymetallicgateatadistanceξ away.
asymptotically low densities, the Fermi surface becomes an The effective electron-electron interaction is V (r) e2/κr
≈
ellipsecenteredaroundoneofthepointsontheminimaldis- forr ξ,whereκisthedielectricconstantofthesurround-
(cid:28)
persionringinmomentumspace.Thetotalvariationalground ingmaterial,andV (r) e2ξ2/κr3forr ξ.
∼ (cid:29)
stateenergyperparticleinthislimitscaleswithdensityas(see Thebrokensymmetrystateformsatadensityn(cid:63) atwhich
Eq. B10) variousscalesbecomecomparabletoeachother. Wedefinea
7
hexagonal symmetry forms when r > r 35.2 Imagine
s s,c
≈
thatwestartdeepintheWignercrystalphase,witharbitrarily
1/r 1/r ~ a /x
s s 0 large r and ξ. Upon decreasing ξ, while keeping r fixed,
s s
1/r ~ (x /a )-1/2 eventuallywereachξ (cid:46) r a , wheretheinteractionsareef-
s 0 s 0
fectivelyshort-rangedandthekineticenergybecomesimpor-
tant. At some point along this path, we expect a phase tran-
UFL Isotropic sitionfromthehexagonalWignercrystaltooneofthephases
WC of lower rotational symmetry: either an anisotropic Wigner
1/r ~ k a crystal,asmectic,oranematic,whichcanalsobeferromag-
s 0 0
netic. Whichofthesephasesisrealizedcannotbedetermined
Highly reliablyonthebasisofthepresentanalysis.
anisotropic phases At higher densities, such that r < r < 1/(k a ), the
s,c s 0 0
x/a SOC can essentially be ignored. Then, upon decreasing ξ
0 fromtheWignercrystal,weexpectatransitiontoaFermiliq-
uid. The reentrant tip of the Wigner crystal phase originates
fromthesamephysicalreasoningasthatdescribedinRef16.
Figure3:Sketchofthephasediagramofa2DEGwithRashbaSOC,
As drawn, a sliver of UFL is shown between the isotropic
as a function of 1/r ≡ a (πn)1/2 and the screening length ξ,
s 0 WC and broken rotational symmetry phases. Such a region
where a = κ/me2 is the Bohr radius. The three regions cor-
0 may or may not exist, depending on details of the numerical
respond to the uniform Fermi Liquid (UFL), the isotropic Wigner
factorswhicharebeyondthescopeofthecalculationhere.
Crystal (WC), and a phase featuring a high degree of anisotropy,
which can be either an anisotropic Wigner crystal, a smectic, or a
nematic. The red dashed lines correspond to r ∼ ξ/a , where
s 0
the crossover between effectively short-range (screened) and long- VIII. POSSIBLEREALIZATIONS
rangeCoulombinteractionsoccurs; 1/r ∼ k a , wheretheSOC
s 0 0
lengthscalebecomescomparabletotheinter-particledistance; and
Thephysicsdescribedherecouldberelevanttoelectronsin
1/r ∼(ξ/a )−1/2,wheretheenergiesoftheuniformFermiliquid
s 0
two-dimensionalheterostructureswhichlacksinversionsym-
andtheanisotropicphasesbecomecomparable,seeEq.(25).
metry, such as GaAlAs quantum wells. The magnitude of
the Rashba spin-orbit coupling in these systems, however,
densityn(cid:63) atwhichtheenergyofthebrokensymmetrystate is rather small. As discussed above, a necessary condition
1 for realizing the phases with broken rotational symmetry is
perparticleiscomparabletothatoftheuniformFermiliquid
r (cid:38) 1/k a , or equivalently, k (cid:46) k ; in typical GaAlAs
state(whichisdominatedbyCoulombenergy): s 0 0 F 0
quantum wells, k /k 0.5 or less, and the characteris-
0 F
∼
e2ξn(cid:63)1 1 (cid:18)me2ξ2(cid:19)23 (n(cid:63))43 , (25) taicfeewnedreggyresecsalKeeolfvitnh1e7,1S8O. CM,oEre0pirsomatisminogstsyosftethmesoarrdeersuorf-
κ ∼ m κ 1 face states of heavy metal surface alloys. For instance, the
boundarybetweenBi Pb andAg(111)supportsasurface
where e is the electron charge, and κ is the dielectric con- state with very strongxRa1s−hxba SOC, with k 2nm 1 and
stant of the host material. On the left hand side we have E 0.1eV.19,20Moreover,itwasdemonstra0te≈dthatb−yvary-
used that, for short range interactions, the potential energy 0 ≈
ing x, the Fermi level of the surface state can be tuned to
of the uniform state scales linearly with density (assuming
belowerthanE .21 TheCoulombinteractionsonthesurface
that (n(cid:63)1)−12 (cid:29) ξ), and on the right hand side we have used are naturally scr0eened by the metallic bulk22. Detecting the
Eq.(18) for the energy of the broken symmetry state with
brokenrotationalsymmetryonthesurfaceposesachallenge,
V e2ξ2/κ. Notethattheenergiesofallthedifferentcandi-
0 because transport measurements would be dominated by the
∼
datestatesarethesameuptoanumericalprefactorinthecase
bulk. Onepossibilityistolookforsignaturesofanisotropyin
α = 3;compareEqs.(18)and(B10). Thisgivesn(cid:63) (a ξ),
1 ∼ 0 thefinite-frequencyresponse,e.g. intheopticalconductivity,
wherea κ/me2istheeffectiveBohrradius.
0 assuming that the anisotropic domains can be aligned (e.g.,
≡
In addition, we define a density n(cid:63)2 at which the inter- by application of an in-plane magnetic field). Scanning tun-
electrondistanceiscomparabletothescreeninglengthξ,and neling microscopy can be done on metallic surface alloys23,
adensityn(cid:63)3 atwhichtheFermiwavevectoriscomparableto and used to detect anisotropy in the electronic structure. Fi-
k0. These characteristic densities are given by n(cid:63) < n(cid:63)2 ≡ nally, magnetic spectroscopy can be used to detect the fer-
1/ξ2 andn(cid:63) < n(cid:63)3 ≡ k02. Thestronglyanisotropicstatesare romagnetic state, which has a large in-plane moment. Such
favoredatdensitieswhichsatisfyn<n(cid:63) =min[n(cid:63)1,n(cid:63)2,n(cid:63)3]. measurements have recently been done24 on the conducting
We can now speculate about the structure of the zero interface between LaAlO and SrTiO , and indeed, large in-
3 3
temperature phase diagram of a 2DEG with Rashba SOC, planemomentswerefound. Whetherthesearerelatedtothe
sketched in Fig. 3, as a function of the dimensionless inter- mechanismdescribedinthispaperremainstobeseen.
electron spacing rs ≡ (cid:0)πna20(cid:1)−1/2 and the screening length Ithasbeenproposed25thatsimilarphysicscanariselightly
ξ. Let us consider large ξ, for which the Coulomb interac- dopedbilayergraphenewithaperpendicularelectricfield,in
tionsareeffectivelyunscreened. Then,aWignercrystalwith whichthesingleparticledispersionhasaminimumonaring
8
ink-space,evenwithoutSOC.Inbilayergraphenethesingle theanisotropicWignercrystalphase,sincetheycombineex-
particledispersionisvalleyandspindegenerate. Theground tremely strong Rashba SOC, a tunable Fermi energy, and
state is likely to have additional broken symmetries, lifting screeningduetothenearbymetal. Inarealsystem,however,
thesedegeneracies. disorder will inevitably play a major role. As a result, both
Itisinterestingtonotethattheconsiderationswhichleadto thepositionalandorientationalorderareexpectedtobeshort-
broken rotational symmetry at low densities are independent ranged. To detect the broken symmetry state on the surface,
oftheparticlestatistics,andarethusvalidfortwo-component one can either resort to local probes (such as scanning tun-
bosonswitheffective(isotropic)Rashba-likespinorbitinter- neling spectroscopy), or find a way to align the orientational
actions. Recently, various techniques were proposed to real- domains,e.g. byanin-planemagneticfield.
ize effective SOC in systems of trapped ultracold atoms26,27.
The properties of such systems have been the subject of in-
tensestudy28–31. Highlyanisotropicphasesmaybeaccessible Acknowledgments
in such systems at sufficiently low densities. Indeed, it was
found31 that the ground state breaks rotational symmetry at We thank E. I. Rashba, E. Demler, A. Amir, I. Neder,
low densities, and that the ground state energy per particle and B. I. Halperin for useful discussions. We are particu-
scales as n4/3 in the limit n 0, consistently with our re- larly indebted to Srinivas Raghu for his part in stimulating
→
sultsfortheanisotropicWCandsmecticphaseswithcontact this project. This work was supported by the NSF under
interactions. grantsDMR-0757145,DMR-0705472(E.B.),DMR-090647
and PHY-0646094 (M. S. R.), and by DOE grant # AC02-
76SF00515atStanford(S.A.K.).M.R.thankstheIQOQIfor
IX. CONCLUSIONS their hospitality. Work at the IQOQI was supported by SFB
FOQUSthroughtheAustrianScienceFund,andtheInstitute
forQuantumInformation.
In the presence of strong Rashba SOC, even short-range
electron-electroninteractionsbecomeimportant. Asaresult,
the system is expected to form a broken symmetry state at
low enough densities. In this work, we have shown that for
AppendixA:Rashbaparticleinabox
sufficiently short-range interactions, states which break rota-
tionalsymmetryarefavoredinthelow-densitylimit. Thisisa
The arguments presented in this paper rely on the solu-
consequenceofthefactthatthesingle-particledispersionhas
tion of the quantum mechanical problem of a single parti-
a minimum on a ring of finite radius in k-space, rather than
cle with Rashba SOC in a rectangular box of size L L
at a single point. This physics is not limited to the case of x y
×
with infinite potential walls. While the corresponding prob-
Rashba SOC; for instance, a similar situation arises in spin-
lemwithoutspin-orbitcouplingistrivial,withspin-orbitcou-
imbalancedfermionicsuperfluidswithnoSOC,inwhichthe
pling the problem of boundary-condition matching with the
majority-spinquasiparticleshaveadispersionwhichismini-
multi-componentwavefunctionishighlynon-trivial. Therea-
malneark ,32orinbilayergraphenewithatransverseelectric
F
sontheproblemwithSOCismoredifficultisthatinthiscase,
field25.
theHamiltonianisnolongerseparable(i.e.,itcannotbewrit-
We believe that the variational wavefunctions and phys-
tenasasumoftwocommutingterms,oneofwhichdepends
ical arguments proposed above capture the correct scaling
onlyonthexcoordinateandtheotherony). Acircularwell
of the ground state energy, which is found to be paramet-
canbesolvedexactly33,thankstoitsrotationalinvariance.
rically lower than other states (e.g. a uniform Fermi liquid
In this Appendix, we combine several approaches to de-
or an isotropic Wigner crystal). However, this approach is
duce the asymptotic form of the ground state energy in the
toocrudetoanswersomeimportant,moredetailedquestions,
anisotropic, low–density limit, Eq. (9) of the main text. We
such as discriminating between the different broken symme-
firstsolvetheproblemexactlyintheL limit(keeping
try states considered here. For sufficiently short-range inter- x →∞
L fixed),andthenprovideanargumentwhichyieldstheform
actions (which fall off with an exponent larger than 3) a ne- y
of the leading corrections for finite L . Finally, we present
matic ferromagnetic state has a parametrically lower energy x
numericalresultssupportingtheanalyticalarguments.
than all the other states considered here, and is therefore the
bestcandidateforthegroundstate.Itisnotclear,atthispoint,
whetheranon-magneticnematicstateisacompetitorornot;
1. SolutionintheL →∞limit
such a state is harder to capture within a simple variational x
approach. More detailed calculations will be needed to de-
InthelimitL ,theproblembecomestranslationally
terminethephasediagramforinteractionswhichfalloffwith x
→∞
invariantinthexdirection. Theeigenfunctionsthentakethe
distancewithapowerof3orless. Forinstance,itmaybein-
form
terestingtotreatthisprobleminanunrestrictedHartree-Fock
approximation, whichcanbeusedtosystematicallyimprove ψ(x,y)=eikxxϕ(y), (A1)
thevariationalwavefunctionsusedinthiswork.
Edge states of surface alloys, such as the one discovered where ψ, ϕ are two–component spinors. We choose coor-
by Ast et al.19, seem to be promising candidates to realize dinates such that the walls are at y = L /2. Then ϕ(y)
y
±
9
satisfiestheboundaryconditions where
(cid:32) (cid:33)
0
ϕ( L /2)= . (A2)
± y 0 eiθη1,2 = kx+iky,η1,2 (A6)
k
Seeking a solution with energy E = ε, we get that the
(cid:113)
wavevectormodulusk = k2+k2satisfies
x y
anda (η ,η = )arecoefficientswhicharedetermined
η1,η2 1 2 ±
by the boundary conditions. We may reduce the number of
(k k )2
ε= − 0 . (A3) coefficients by using symmetry. Under reflection, y y
2m the spinor ϕ(y) transforms as ϕ(y) σ ϕ( y). Req→uir−ing
y
→ −
thatthewavefunctionsareeitherevenoroddunderreflection
Fixingk ,wefindfourallowedvaluesofk ,whichwedenote
x y
gives
by k :
y,
± ±
(cid:114)
(cid:16) (cid:17)2
k = k √2mε k2. (A4)
y,± 0± − x aη1,η2 =±a−η1,η2 (A7)
Notethatk canbeimaginary.
y,
Thewavef±unctionforthetransversemotiontakestheform
Imposingtheboundarycondition,Eq.(A2),onthewavefunc-
(cid:32) (cid:33) tioninEq.(A5), fortheevensectorgivesthefollowing(im-
ϕ(y)= (cid:88) aη1η2eiη1ky,η2y ei−eiiηη11θθηη22//22 , (A5) plicit)equationforε:
η1,η2=±
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
k L θ k L θ k L θ k L θ
cos y,+ y + cos y,− y + − cos y,+ y + + cos y,− y − =0, (A8)
2 − 2 2 2 − 2 2 2 − 2
wherek ,θ aregivenbyEqs.(A4)and(A6). Fortheodd Eq.(A8)asfollows. Tolowestorderinεandδk k k ,
y, x x 0
sector,we±get±anidenticalequationwithcosreplacedbysin. this equation depends on ε, δk , and L through≡the fa−ctors
x y
Equation (A8) determines the dispersion ε(k ). The disper- k L :
x y, y
sionofthelowestsubband,nearkx = k0,isshowninFig.4, ± (cid:114)
forwidthsk0Ly/2rangingfrom10to20. k L = (cid:16)k √2mε(cid:17)2 (k +δk )2L
y, y 0 0 x y
Theformofthelowenergydispersioncanbededucedfrom ± ± −
(cid:113)
2k √2mε 2k δk L . (A9)
0 0 x y
≈ ± −
ThereforeEq.(A8)hasthefunctionalform
-3
0.04 -4 DD ~ Ly-4 F (cid:16)2k0√2mεL2y,2k0δkxL2y(cid:17)=0, (A10)
00.0.3035 DD)log()log(--65 k0Ly= whereF issomefunctionoftwovariables. Closetotheband
5 minimum,thedispersionhasthefollowingform:
-7
k)x00.0.2025 2.l5og(Ly) 3 5.5 2k0√2mεL2y =A1+A2(cid:2)2k0δkxL2y−A3(cid:3)2, (A11)
E( 6 whereA aredimensionlessconstants. Therefore
1,2,3
0.015
0.01 7.5 ε(δk ) 1 (cid:34)A1 +A (cid:18)2k δk A3(cid:19)2L2(cid:35)2
DD 10 x ≈ 8mk2 L2 2 0 x− L2 y
0.005 0 y y
1 A2 A A
0 1 + 1 2 (δk δk(cid:63))2, (A12)
-0.05 0 0.05 0.1 ≈ 8mk2 L4 m x− x
k - k 0 y
x 0
whereδk(cid:63) = A /2k L2. Weseethedispersionminimum∆
x 3 0 y
Figure4:Dispersionofthelowestsubbandforinfinitestripsofvary- scalesas1/L4, whiletheeffectivemassinthexdirectionis
y
ingwidthLy.Theinsetshowsthedispersionminimum∆vs.Lyon Lindependent.Weconfirmtherelation∆ 1/L4bysolving
alog-logplot,showinggoodagreementwith∆∼1/L4. ∼ y
y Eq.(A8)numerically,seeinsetofFig.4.
10
oneperiodisamultipleof2π:
0.04
n/k2 = 2(δk δk(cid:63))L +φ=2πj, (A13)
0.035 0 2⋅10−2 x− x x
where j is an integer, and φ = φ + φ is the sum of the
y 1 2
g 0.03 two(unknown)phaseshiftsφ ,φ associatedwithreflections
r 1 2
e
ate en0.025 1.32⋅10−2 0Lfr,oymwaentdhaeskst0uw.moHeeontwhdaset.vewTrhe,eicntaontthaerleplpihlmaacisteeδsφkhx(ifδLtkyφL→is,a10f/,uk1n/cLtki0oL)nybδyk→xa,
st 0.02 x y 0 y
d constant φ(δkxLy 0,1/k0Ly 0) φ0. We then find
oun0.015 8.8⋅10−3 tφha)t/t2hLe g,rwouhnedrestja→te isisgaivnenintbeyg→eδrkcxh−o≡seδnkx(cid:63)to=mi(n2imπjizmeinth−e
r 0 x min
G 0.01 6.8⋅10−3 energybelow. InsertingthisintoEq.(A12),wefinallyget
0.005 4.4⋅10−3 ε(L ,L ) 1 A21 + A1A2(πjmin−φ0/2)2. (A14)
1 2 3 4 5 6 7 8 x y ≈ 8mk02 L4y mL2x
Aspect Ratio L /L
x y Substituting n = 1/(L L ) and η = L /L , we obtain
x y x y
Eq.(11).
Figure5: Numerical resultsforthe groundstate energyvs. aspect
ratioL /L ofaparticlewithRashbaspinorbitcouplinginarect-
x y
angularinfinitepotentialwellofsizeLx×Ly,forseveralvaluesof 3. Numericalsolution
thedensityn(measuredinunitsof1/k2).Wehaveset2m=1.
0
InordertoverifytheassumptionsthatleadtoEq.(A14),we
havenumericallycalculatedthegroundstatewavefunctionof
a Rashba particle in a box. This is done using a generaliza-
(a) (b)1.5
-2 tion of the “plane wave decomposition” technique described
in Refs. 34,35. The solution is written as a superposition of
e *~ n4/3 h *~ n-1/3
)* -4 )* 1 eigenstates of the free Hamiltonian, which are plane waves
e(g h(g withwavevectorssatisfyingEq.(A3)forsomevalueofε.The
o o
l l0.5 coefficientsofthedifferentplanewavesaredeterminedbyre-
-6 quiring that both components of the wavefunction vanish on
a set of points evenly distributed along the boundary (in this
0
-6 -5 -4 -3 -6 -5 -4 -3 case, an L L rectangle), and that the first component of
x y
log(n/k2) log(n/k2) the wavefunc×tion spinor is equal to 1 at an arbitrary point in
0 0
theinterior. Thesumofthesquaresofthewavefunction(the
Figure6:(a)Minimalgroundstateenergy,ε(cid:63),and(b)optimalaspect “tension”)atadifferentsetofpointsontheboundaryisthen
ratio, η(cid:63), asafunctionofthedensity, onalog-logscale. Thedata calculated. Theeigenvaluesεareidentifiedastheminimaof
isconsistentwiththeanalyticalpredictions: ε(cid:63) ∼ n4/3 andη(cid:63) ∼ thetension. Theeigenvaluescanbedeterminedwithanaccu-
n−1/3atlowdensities. racyof1%orbetter. Wehavetestedthetechniquebycalcu-
latingtheeigenenergiesofaRashbaparticleinacircularwell,
andfoundexcellentagreementwiththeexactresults33.
Fig. 5 shows the ground state energy vs. the aspect ratio
2. ExtensiontoFiniteL
x
η = L /L for a few values of the density 1/n = L L .
x y x y
Thegroundstateenergyisminimalforη = 1. Wefoundthis
Next, we consider that L is finite, still assuming that behavior for n (cid:46) 0.08k2; for higher den(cid:54)sities (smaller box
x 0
L L . In this case, the solution is complicated by mul- area), the minimum energy occurs at η = 1. In Fig. 6 we
x y
(cid:29)
tiple reflections from the boundaries. Nevertheless, we can showtheminimalgroundstateenergyε(cid:63)andthecorrespond-
deduce the form of the ground state energy as a function of ingaspectratioη(cid:63)fordensitiesrangingfromn/k2 =8 10 2
0 × −
L andL asfollows. ForlargeL ,weassumethatthesolu- to4 10 3. Forlowdensities, theoptimalgroundstateen-
x y x −
×
tionislargelycomposedoftravelingwavestatesnearthebot- ergy and aspect ratio follow ε(cid:63) n4/3 and η(cid:63) n 1/3, in
−
∼ ∼
tomofthedispersiongivenbyEq.(A12),whicharereflected agreementwithEqs.(10),(11),and(A14).
backandforthfromthetwoedges. Consideraright-moving
wave with momentum k = k +δk . This state can only
x 0 x
be reflected to the state k = k δk +2δk(cid:63), where δk(cid:63) AppendixB:Hartree-Fockdescriptionoftheferromagnetic
isdefinedbelowEq.(A12x)(cid:48). Note0th−attimxereverxsalsymmetrxy nematic
prohibitsscatteringtotheotherleft-movingsolutionwithmo-
mentum k δk ,sincethisstateistheKramer’spartnerof WithintheHartree-Fockapproximation,wereplacethefull
0 x
− −
theoriginalincomingwave. Theeigenstatesofthesystemare HamiltonianH,Eq.(1),bythefollowingmean-fieldHamilto-
determined by the requirement that the phase acquired over nian :
H