Disorder and the Quantum Hall Ferromagnet. A. G. Green Department of Physics, Princeton University, New Jersey NJ 08544 The distinguishing feature of the quantum Hall ferromagnet is the identity between electrical and 8 topologicalchargedensitiesofaspindistortion. Inadditiontothewealthofphysicsassociatedwith 9 SkyrmionicexcitationsofthequantumHallferromagnet,thisidentificationpermitsarathercurious 9 coupling of spinwaves to the disorder potential. A wavepacket of spinwaves has an associated, 1 oscillating dipole charge distribution, due to the non-linear form of the topological density. We n investigate the way in which this coupling modifies the conductivity and temperature dependence a of magnetization of the quantumHall ferromagnet. J 0 3 The distinguishing feature of the quantum Hall ferro- these terms is an interactionwith the disorderpotential, ] magnet (QHF) is the identity between the topological U(x), and the second, V[J0(x)], is the Coulomb energy l density of a spin distortion and the associated electrical of the charge distribution, J (x). Eq.(1) describes both l 0 a chargedensity. Thisidentificationpermitsachemicalpo- the low energy spin and charge dynamics of the quan- h tential to stabilize topologically non-trivial groundstate tum Hall system. The quantization of Hall conductivity - s spin configurations, known as Skyrmions1. The theoret- follows from the final term, the Hopf term4. e ical prediction of these states has received substantial Here, we are concerned with the effect of the dis- m experimentalsupport2 andpromptedagooddealofthe- order potential upon small fluctuations, l = (l ,l ,0), 1 2 t. oretical speculation. The link between topological and about the ferromagnetic groundstate, n¯ = (0,0,1); n = a electrical charge densities also produces a curious cou- (l ,l , 1 l2). The effective action and current, ex- m 1 2 −| | pling of spinwaves to the disorder potential. Although a panded to lowest order in these fluctuations, are - planewavespindistortioncarriesnocharge,awavepacket p d n ofspinwaveshasanoscillatingdipole chargedistribution S = d2xdt1¯l ρ¯∂ ρ 2 ρ¯gB l o associatedwithit,duetothenon-linearformofthetopo- 2 2 t− s∇ − c logical density. Spinwaves couple to the disorder poten- Z (cid:16) (cid:17) [ tial through this charge distribution. In this work, we d2xdtJ0(x)U(x), − v1 icnovnedsuticgtaivteitythaendwatyeminpewrahtiuchretdheispecnoduepnlicnegofmmodaigfineestitzhae- Jµ =i8eZπνǫµνλ∂ν¯l∂λl. (3) 4 tion of the quantum Hall state. 2 The low energy effective action for the QHF at filling We use the complex notation, l = l +il , ¯l = l il . 1 2 1 2 3 fractionsν =1 and the Laughlinfilling fractionsis given Both the Coulomband statistical interactions have−been 1 by1,3 neglected in writing down Eq.(3). Although important 0 in determining the size and shape of the Skyrmion ex- 8 ρ¯ ρ 9 S = dtd2x ∂tn.A[n] s ∂µn2+ρ¯gB.n citations, the former is less relevant than the remaining 2 − 2 | | t/ Z h i terms in its effect upon spinwaves5. We will show later a dtd2xJ (x)U(x) dtV [J (x)] that the quantization of Hall conductivity, produced by 0 0 m − − the Hopf term, is unaffected by weak disorder. The cal- Z Z - xi yi culationspresentedinthisworkconcerntheperturbative d +ν dtd2xd2yJ0(x)ǫij x −y2Jj(y), (1) effects of weak disorder. It is worth noting that the ef- n Z | − | fective action, Eq.(3), is very similar to that of electrons o where in a random potential, aside from the unusual form of c eν : J = ǫ n.(∂ n ∂ n). (2) the current density and the bosonic nature of the fields. v µ −8π µνλ ν × λ This similarity is suggestive of the possibility of weak i X n(x) is an O(3)-vector order parameter of unit length, localization effects. These are not considered here. r describingthelocalpolarizationofthequantumHallsys- We represent the bare, momentum space propagators, a tem. ThefirstlineofEq.(1)istheusuallowenergyeffec- ¯l(q,ω˜)l( q, ω˜) and ∂µ¯l(q,ω˜)∂νl( q, ω˜) , by the h − − i h − − i tiveactionforaferromagnet. A[n]isthevectorpotential diagrams ofaunitmonopoleinspinspace,ρ¯istheelectrondensity 1 (ρ¯ = ν/2πl2, where l is the magnetic length), ρs is the ¯l(q,ω˜)l( q, ω˜) = = h − − i iρ¯ω˜/2 E(q) spinstiffnessandgistheZeemancoupling,intowhichwe − q q have absorbed the electron spin and the Bohr magneton ∂ ¯l(q,ω˜)∂ l( q, ω˜) = = µ ν for ease of notation. The second line of Eq.(1) contains h µ ν − − i iρ¯ω˜/2 E(q) − terms arising due to the identity of charge and topolog- ical charge (which is embodied in Eq.(2)). The first of where E(q) = ρs q2 +ρ¯gB is the spin energy density. | | 1 The disorder interaction is given by d2q e−2d|q| mΣ(ω,p)=Kρ2 (p q)2 I s (2π)2 × q2 Z | | S = dω˜ d2q1 d2q2 i eν ǫij i U( q1- q2) j , ×πδ(ρ¯ω/2−E(q+p)). (8) i (2π)2(2π)2 8π   Z (cid:16) (cid:17) q1,ω q2,ω Therealpartoftheself-energycanbeapproximatedfrom   Eq.(7) in the limit ρ p2, ρ¯ω 2gB /2 ρ /d2. The where the frequency integral, dω˜, is a shorthandno- leading order contribsu|ti|on i|s p−roport|iona≪l tos p2 and | | tation for the bosonic Matsubara frequency summation provides a correction to the spinwave stiffness, ∆ρs = iT1mpu∞nr=it−y∞po..t.e|ωn˜=ti2aπlni/sTe.ntNiroetliyceeltaRhstaitctih.ee.stchaetteenreinrggyolffabtehles RtoeΣp/2|pln|2.pF2ordeρps|epn|d2e>ncρe¯|.ωW−e2gfiBnd|/2,thereisacrossover P | | | | on the propagators are conserved. Kρ 4ρ¯gB ρ¯ω/2d2 In GaAs heterostructures, the disorder potential felt eΣ(ω,p) s p2ln | − | by the electrons in the 2DEG is due mainly to Coulomb R ≃ 8π | | (cid:20) ρs (cid:21) interaction with ionized donor impurities in the n-type for ρ p2 <ρ¯ω 2gB /2, s region6. This region is separated from the 2DEG by an | | | − | Kρ insulating spacer layer of width d. One may obtain an s p2ln 4p2d2 ≃ 8π | | | | expression for the correlations in the disorder potential for ρ p2 >(cid:2) ρ¯ω (cid:3)2gB /2. (9) by modeling this situation with the potential due to a s | | | − | randomplanardistributionofchargeatadistancedfrom The first of these expressions has been calculated by ex- the 2DEG. The correlations in the disorder potential in panding Eq.(7) to lowest order in p2 and by replacing this model are given by the exponential factor, e−2d|q|, wit|h|an ultra-violet cut- e√n 2 e−2|q|d off, 1/2d. The second expression is calculated exactly hhUqUq′ii=(2π)2δ(q+q′) 2ǫd q2 from Eq.(7), setting ω =2gB. (cid:18) (cid:19) | | The imaginary part of the self energy may be calcu- q =(2π)2δ(q+q′) , (4) lated exactly when d=0, with the result (cid:16) (cid:17) K where nd is the area density of donor impurities. This mΣ(ω,p)= ρ¯(ω/2 gB)θ(ω/2 gB) simplemodelofdisordersomewhatoverestimatesthepo- I − 8 − − tential felt by the 2DEG. Due to Coulomb interactions for ρs p2 >ρ¯ω 2gB /2, | | | − | between the donors, the size of the fluctuations in the K = ρ p2 disorderpotentialisusuallymuchlessthanwouldbe ex- − 8 s| | pectedforatotallyuncorrelateddistributionofchargein for ρ p2 <ρ¯ω 2gB /2. (10) the disorder plane. We follow Fogler et al.7 and assume s| | | − | thatthiseffectmaybetakenintoaccountbyinterpreting The integral for finite d is much trickier and cannot be nd in Eq.(4) as a density of ‘uncorrelated’ donors,which carried out analytically. For large d it is exponentially is much less than the actual density of donors. suppressed by a factor e−2d|p|. The lowest order contribution of disorder to the self- Taken at face value, Eq.(9) implies a threshold disor- energy is der strength at which the renormalized spin-stiffness is zero at zero frequency. We interpret this as indicative of a depolarization transition to a paramagnetic state. A similar suggestion has been made by Fogler et al.7 in Σ(iω˜,p)= order to explain the breakdown of spin splitting in high =Kρ2 d2q (p×q)2 e−2d|q|, (5) Landau levels. Strictly, the calculations presented here s (2π)2iρ¯ω˜/2 E(q+p) q2 apply only for weak disorder and small ∆ρ . That the Z − | | s threshold behaviour suggested here does indeed occur, where may be seen in a number of ways. The most elegant 1 eν 2 e√n 2 of these is through a Bogomolnybound type argument8. d K = (6) The presenttreatmentenablesone to investigatethe ap- ρ2 8π 2ǫ s (cid:16) (cid:17) (cid:18) (cid:19) proach to this threshold. is a dimensionless measure of the disorderstrength. The Optical conductivity. The longitudinal and trans- retardedself-energy is obtained by analytic continuation verse conductivities are given by the Kubo formula9: torealfrequencieswiththesubstitutioniω˜ ω+iδ. The → i real and imaginary parts of the self-energy so obtained σ (ω)= J (0,ω˜)J (0, ω˜) (11) ij i j are ωh − i(cid:12)iω˜→ω+iδ (cid:12) eΣ(ω,p)=Kρ2 d2q (p×q)2 e−2d|q|, (7) In order to determine the longitudin(cid:12)(cid:12)al conductivity, we R s (2π)2ρ¯ω/2 E(q+p) q2 must evaluate the following diagram: Z − | | 2 J(0,ω˜).J(0, ω˜) or diagrammatically, h − i q ,Ω (cid:0)(cid:1)α(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) γ (cid:0)(cid:1)µ(cid:0)(cid:1)q ,Ω µ q ,Ω =−(cid:16)8eπν(cid:17)2ǫiαβǫiγδ(cid:0)(cid:1)β(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) q ,ω +Ω δ (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)βα(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)q ,ω +Ω = νµ + (cid:0)(cid:0)(cid:1)(cid:1)αβ(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) q ,ω +Ω eν 2 d2q ν . = ǫiαβǫiγδ dΩ˜ q q˜ q q˜ ν − 8π (2π)2 µ ν γ δ (cid:16) (cid:17) Z This Dyson’s equation may be recast in terms of the Γαβ,µν(q,iΩ˜,iΩ˜ +iω˜) (q,iΩ˜ +iω˜) (q,iΩ˜), (12) scalar vertex function, γ(q,iΩ˜,iω˜+iΩ˜): × G G where q = (iΩ˜,q), q˜ = (iΩ˜ + iω˜,q) and (q,iΩ˜) is µ µ G d2k (q.k)2 the full thermodynamic Green’s function. The vertex γ(q,iΩ˜,iω˜+iΩ˜)=1+ W function, Γ , is given by the summation (2π)2 q4 qk αβ,µν Z | | Γαβ,µν(q,iΩ˜,iΩ˜ +iω˜) α µ ×G(k,iΩ˜ +iω˜)G(k,iΩ˜)γ(k,iΩ˜,iω˜+iΩ˜), (15) =δ δ + eν 2ǫα′µǫβ′να where αµ βν 8π β β (cid:16) (cid:17) ν eν 2 e2n 2 e−2d|q−k| W = d (q k)2 . qk − 8π 2ǫ × q k2 (cid:16) (cid:17) (cid:18) (cid:19) | − | +... +... +... (13) In order to obtain Eq.(15), we have used the relation Γ (k,iΩ˜,iω˜+iΩ˜)q q k k =γ(k,iΩ˜,iω˜+iΩ˜)(k.q)2, ab,cd a b c d Infact,allcontributionsto the vertexfunction containa which follows from the definition of γ and the symmetry factor of q q˜ and there is considerable simplification in α β defining a new, scalar vertex function, γ(q,iΩ˜,iΩ˜ +iω˜); of the disorder interaction; since the disorder potential couples only to the charge density and not to any other q q˜ γ(q,iΩ˜,iΩ˜ +iω˜)=Γ (q,iΩ˜,iΩ˜ +iω˜)q q˜ . components of the current density, Γ = δ δ if α β αβ,µν µ ν αβ,µν αµ βν either or both of µ or ν are time-like. This definition of the vertex function is then substituted In order to calculate the real part of the optical con- into Eqs.(11,12)to find the conductivity. After perform- ductivity we require γ(q,ǫ iδ,ǫ+ω+iδ). Evaluating ing the summation over bosonic Matsubara frequencies − this is a very difficult task. However, several simplifying andafewotherstandardmanipulations9,therealpartof assumptions may be made. Firstly, we assume that the the longitudinal conductivity is given by the expression frequencydependenceoftheopticalconductivityisdom- eν 2 d2q ∞ dǫ inatedbytermsinEq.(14)otherthatthevertexfunction. σ(ω)=ω q2 [n (ǫ+ω) n (ǫ)] 8π (2π)2| | 4π B − B Secondly,thetermsGAGRinEq.(14)arestronglypeaked (cid:16) (cid:17) Z Z−∞ withinω ofǫ=2E(q)/ρ¯. Therefore,wehaveonlytocal- e GA(q,ǫ)GR(q,ǫ+ω)γ(q,ǫ iδ,ǫ+ω+iδ) ×ℜ − culateγ(q,2E(q)/ρ¯)=γ(q,2E(q)/ρ¯ iδ,2E(q)/ρ¯ iδ). (cid:2) GR(q,ǫ)GR(q,ǫ+ω)γ(q,ǫ+iδ,ǫ+ω+iδ) , Using Eq.(15), we find − − − (14) (cid:3) γ(q,2E(q)/ρ¯) where n (x) is the Bose occupation number. The con- B d2k (q.k)2 tribution to the Hall conductivity is zero, on symmetry =1+ W GR(k,ǫ)GA(k,ǫ)γ(k,ǫ) grounds,sincethecurrent-currentcorrelator J J gives (2π)2 q4 qk (cid:12) h × i Z | | (cid:12)ǫ=2E(q)/ρ¯ risetoafactorofq qinthe integrand. Comparedwith (cid:12) theanalogousresult×forelectronicconductivity9,Eq.(14) =1+ d2k (q.k)2W A(k,ǫ) ,γ(k,ǫ) (cid:12)(cid:12) , (16) contains an additional factor of ω2, which ensures that (2π)2 q4 qk2∆(k,ǫ) (cid:12) the d.c. conductivity is zero. This is due to the factthat Z | | (cid:12)ǫ=2E(q)/ρ¯ (cid:12) the charge fluctuations in the QHF are dipolar. (cid:12) where ∆(k,ǫ) = mΣ(k,ǫ) and(cid:12) A(k,ǫ) = Vertex corrections. In the ladder approximation, 2 mGR(k,ǫ) is the sp−ecItral function. In the limit of the vertex function is given by the following Dyson’s − I very weak disorder, A(k,ǫ) 2πδ(ρ¯ǫ/2 E(k)). The equation: ≈ − deltafunctionimposestheconstraint k = q and,since | | | | Γ (q,iΩ˜,iω˜+iΩ˜)=δ δ γ(q,2E(q)/ρ¯) γ(q), Eq.(16) reduces to an algebraic αβ,µν αµ βν ≡ | | equation. The solution is eν 2 e2n 2 d2k e−2d|q−k| d ǫbνǫdµk k k k − 8π 2ǫ (2π)2 a b c d q k2 ∆(k,2E(q)/ρ¯) (cid:16) (cid:17) (cid:18) (cid:19) Z | − | γ(q,2E(q)/ρ¯)= (17) (k,iΩ˜ +iω˜) (k,iΩ˜)Γαβ,ac(k,iΩ˜,iω˜+iΩ˜), ∆T(k,2E(q)/ρ¯) ×G G 3 d2k TheenergyandmomentumintegralsinEq.(20)maythen ∆(k,2E(q)/ρ¯)= W δ(E(q) E(k)) (18) (2π)2 qk − be carried out with the result Z d2k 1 eν 2 ∆ (k,2E(q)/ρ¯)= W δ(E(q) E(k)) σ(ω) ω2e−2gB/T for g T. (22) T (2π)2 qk − ≈ πρ2K 8π ≫ Z s (cid:16) (cid:17) (q.k)2 Fortypicalexperimentalsystemsatν =1,anupperesti- 1 (19) × − |q|4 ! mnat=efρ¯o)ratnhdedtihseorsdpeirnssttrieffnngetshsiρsK ∼4K0.1. (Tahpeprcooxnidmuacttiinvg- d s ∼ Eq.(18)issimplyare-writingofEq.(8)fortheimaginary ities predicted by Eqs.(21,22) are vanishingly small and part of the spinwave self-energy. The final term in the probably unmeasurable. integrand of Eq.(19) is an angular weighting, sin2θ, for Magnetization The variation of magnetization with scatteringevents, where θ is the anglebetween incoming temperature, in the absence of disorder, has been calcu- and outgoing spinwave states. This should be compared lated by Read and Sachdev5, using a lowest order 1/N with the electronic case, where the angular weighting is expansion. We extend this calculation to include the ef- 1 cosθ. fect of disorder. Firstly, a Hopf map (n = z¯ασαβzβ, −Ignoring vertex corrections (substituting γ = 1), 2 z 2 = 1) is used to recast the effective action, α=1| α| Eq.(14) reduces to Eq.(1), into CP1 form; P σ(ω)=ω eν 2 d2q q2 ∞ dǫ S = d2xdt iρ2¯z¯∂tz+ρs|Diz|2+ρ¯gBz¯σzz (cid:16)8π(cid:17) Z (2π)2| | Z−∞ 4π Z h i [n (ǫ+ω) n (ǫ)]A(q,ǫ)A(q,ǫ+ω). (20) d2xdt U(x)J (x)+λ z 2 1 , × B − B − 0 | | − Z A similar calculation of the finite wavevector conductiv- iνe (cid:2) (cid:0) (cid:1)(cid:3) J = ǫµνλ∂ z¯ ∂ z (23) ity, neglecting vertex corrections, gives ν −2π ν α λ α 1 eν 2 d2q ∞ dǫ where Di = ∂i+iθi. θi is an auxiliary field, introduced σ(ω,k)= ωk ǫq2 inordertodecouplequartictermsintheeffectiveaction. ω 8π (2π)2 4π| − | (cid:16) (cid:17) Z Z−∞ λ is a Lagrange multiplier that imposes the constraint. ×[nB(ǫ+ω)−nB(ǫ)]A(q,ǫ)A(q+k,ǫ+ω). The indices on zα have been suppressed for clarity. To zeroth order in the 1/N expansion, the constraint In contrast to the zero wavevector conductivity, σ(ω,k) is imposed at the mean field level in order to self- maybe non-zerointheabsenceofdisorder. Eq.(20)may consistentlydeterminethe averagevalueofthe Lagrange nowbeused,inconjuctionwiththespinwaveself-energy, multiplier, λ¯10. The resulting gap equation is Eqs.(9,10), in order to calculate the contribution of dis- order scattered spinwavesto the optical conductivity. In z¯z = d2p dΩ˜ ¯(iΩ˜,p2,σ,λ¯) theabsenceofdisorder,thespectralfunctionhasasingle hh ii (2π)2 G σ=±Z delta-functionpeak,A(q,ǫ)=2πδ(ρ¯ǫ/2 E(q)). Theef- X − d2p ∞ dǫ fectofdisorderistobroadenandshiftthispeak. ForT = n (ǫ)A(ǫ,p2,σ,λ¯), (24) g,ω and weak disorder, the product A(q,ǫ)A(q,ǫ+ω≪), σ=±Z (2π)2 Z−∞ 2π B derived from Eqs.(9,10), is strongly peaked at ρ¯ǫ/2 = X E(q) and ρ¯(ǫ+ω)/2 = E(q) and may be approximated where ¯(iΩ˜,p2,σ,λ¯) indicates the disorder average by of theGz¯z-Green’s function and A(ǫ,p2,σ,λ¯) = 2 mG¯ (ǫ,p2,σ,λ¯) is the spectral function. We have ret A(q,ǫ)A(q,ǫ+ω) 2πδ(ρ¯ǫ/2 E(q))A(q,ǫ+ω) c−arℑried out the frequency summation in order to obtain ≈ − +2πδ(ρ¯(ǫ+ω)/2 E(q))A(q,ǫ). the finalexpression. The magnetizationmay alsobe cal- − culated to this order and is given by The real part of the longitudinal optical conductivity, d2p calculated within this approximation, is z¯σzz = dΩ˜σ¯(iΩ˜,p2,σ,λ¯) hh ii (2π)2 G σ=±Z K eν 2 X σ(ω) T2(1 e−ω/T)e−2gB/T. (21) d2p ∞ dǫ ≈ 32πρ2 8π − = σ n (ǫ)A(ǫ,p2,σ,λ¯). (25) s (cid:16) (cid:17) (2π)2 2π B σ=± Z Z−∞ At very small frequency, ω KT, the product X A(q,ǫ)A(q,ǫ+ω) is no longer re≪solved into two peaks. To O(1/N), Eq.(23), is identical to the sum of two The dominant frequency dependence in Eq.(20) then copies of the spinwave action, Eq.(3), with the Zee- comes from the n (ǫ) n (ǫ+ω) term. Then man term, ρ¯gB, replaced with σρ¯gB +λ¯. The expres- B B − sions for the self-energy derived above may be used di- 2πδ(ρ¯ǫ/2 E(q)) rectly with this replacement. In the absence of dis- A(q,ǫ)A(q,ǫ+ω) A2(q,ǫ)= − . ≈ mΣ(q,ǫ) order, the spectral function has a single delta-function ℑ 4 peak; A(ǫ,p2,σ,λ¯) = 2πδ(ρ¯ǫ/2 E(p2,σ,λ¯)), where Phys. Rev. Lett. 76, 680 (1996); A. Schmeller et al, Phys. E(p2,σ,λ¯)=ρ p2+σρ¯gB+λ¯. Su−bstitution of this into Rev.Lett. 75, 4290 (1995). s Eqs.(24,25), reproduces the result of [ 5]. The effect of 3K.Moon et al, Phys.Rev. B51, 5138 (1995). disorder is to broaden and shift this peak. The real part 4V.M.Yakovenko,Fizika(Zagreb)21, suppl.3,231(1989). of the self-energy produces a renormalizationof the spin 5N.ReadandS.Sachdev,Phys.Rev.Lett.75,3509(1995). stiffness, ρ ρ˜ . Upon direct substitution of Eq.(10), 6A.L. Efros Solid State Com. 70, 253 (1989). one finds thsa→t, toslowest order in K, the new position of 7M. M. Fogler and B. I. Sklovskii, Phys. Rev. B52, 17 366 the peak is at ρ¯ǫ/2=E˜ 4K2ρ˜ p2 and so the shift due (1995). − s 8N.Cooper, privatecommunication. to the imaginary partof the self-energy may be incorpo- 9For example, See Ch.7 of G. D. Mahan, Many Particle rated as a further renormalization of the spin-stiffness. Physics, Plenum press, London (1990). This is the dominant effect of weak disorder. The gap equation and magnetization are given by the disorder 10θi entersonly at thenext order of the1/N expansion. 11Asimilaranalysismaybecarriedoutforthe1/Nexpansion free expressions5 with appropriately renormalized spin- intheO(3)representation5.Theappropriategapequation stiffness11. and magnetization are modified in the same way as in the The calculationof Ref.[ 5]showsgoodagreementwith CP1 representation. experiment12 asidefromathightemperatures,wherethe 12S.E.Barrettet al,Phys.Rev.Lett.74,5112(1995); M.J. experimentally measured magnetization appears to fall Manfra et al, Phys.Rev.54, R17 327 (1996). below even the theoretical ρs = 0 prediction. Recent 13C. Timm et al,cond-mat 9710220 (1997). work13 has shown that this discrepancy cannot be ex- 14H.D. M. Davies et al, Phys.Rev. Lett.78, 4095 (1997). plained by the inclusion of higher orders in the 1/N ex- pansion. Here, we have shown that neither can it be explained by the effects of weak disorder. In fact, to ex- plain this observation would require spectral weight to be transferedbelowthe Zeemangap. This appearsto be impossible so long as the groundstate remains ferromag- netic. Two possible alternative explanations lie in the effect of Skyrmions or the inclusion of the correct spin- wave dispersion at high momenta. The latter approach has provideda goodexplanationfor the dramatic reduc- tioninmagnetizationwithincreasingtemperaturefound at ν = 1/314. It is readily incorporated into the lowest order1/Nexpansionintheabsenceofdisorder,byinsert- ing a spectral function with a delta-function peak at the correct spinwave dispersion into Eq.(24,25) and solving the resulting equations numerically. In conclusion, we have considered the effect of weak disorder upon the quantum Hall ferromagnet. The iden- tification of charge and topological charge of spinwave distortions allows a coupling of spins to the disorder po- tential. Thesignatureofthiscouplinginthetemperature dependence of magnetization is a reduction of the effec- tivespin-stiffness. Theeffectuponconductivityisrather more interesting, although unfortunately it is probably unmeasurablysmall. Wepredictaspinwavecontribution to the longitudinal optical conductivity at finite temper- ature. We acknowledge A. M. Tsvelik, J. T. Chalker and S. Sondhi for helpful comments and suggestions. 1S.L.Sondhi,A.Karlhede,S.A.KivelsonandE.H.Rezayi, Phys.Rev. B47, 16419 (1993). 2S. E. Barrett et al, Phys. Rev. Lett. 74, 5112 (1995); R. Tycko et al, Science 268, 1460 (1995); E. H. Aifer et al. 5