Physica E 1 (1997) 54–58

Shape and phase transitions in quantum-Hall skyrmions

Madan Rao a, Surajit Sengupta b, R. Shankar a;∗

aInstitute of Mathematical Sciences, C.I.T Campus, Madras-600113, IndiabIndira Gandhi Centre for Atomic Research, Kalpakkam-603102, India

Abstract

The shape deformability of quantum-Hall skyrmions leads to a rich phase diagram in the �–g space. We study thelong-wavelength physics of a collection of interacting skyrmions using a nonlinear sigma model. At zero temperaturethe ground state is crystalline with generalized N�eel order. As a function of the �lling factor �, the skyrmion crystal undergoesa sequence of structural transitions driven by a change of shape of the individual skyrmions. Quantum e�ects lead to meltingand orientational disordering transitions at high and low skyrmion densities, respectively. We show that moment of inertiaof skyrmions also arises from the shape deformability of the skyrmions. ? 1997 Elsevier Science B.V. All rights reserved.

Keywords: QHE; Skyrmions; Crystal; Structural transitions

In the past work [1–4], a case has been made that thecharged quasiparticles about the � = 1 quantum-Hallstate in GaAs are extended spin textures or skyrmions.The novel feature of these excitations is that their spinis signi�cantly greater than 1

2 . This leads to the pre-diction that the spin polarization would fall sharplyas � deviates from 1. Experimental evidence for thishas been discussed extensively in this workshop[5–9]. Though the e�ect has been seen by severalgroups there is no consensus as yet.If, indeed, the charged excitations are skyrmions

then the question is: What is the ground state of asystem of interacting skyrmions in two dimensions?At very low temperatures, a crystalline state seems aplausible candidate. Speci�c heat measurements [10]on GaAs heterojunctions, show a sharp peak at a tem-

∗ Corresponding author. E-mail: shankar@imsc.ernet.in.

perature T ≈ 30 mK. It has been suggested [10] thatthis anomaly may be associated with the freezing ofskyrmions into a crystal lattice. Previous theoreticalwork [11, 12] have analysed the crystalline state ofskyrmions. Using a mean-�eld analysis of electronscon�ned to the lowest Landau level, Brey et al. [11]claim that at T = 0, the skyrmions form a squarelattice with a N�eel orientation ordering. On the otherhand, in the framework of a nonlinear sigma model,Green et al. [12] conclude that the skyrmions form atriangular lattice with a generalized N�eel ordering.In this paper, we study the T = 0 phase diagram of

a system of interacting skyrmions in the �–g plane,using the e�ective classical O(3) nonlinear sigmamodel (NLSM). The Land�e g factor can be tuned bythe application of hydrostatic pressure [13]. We workat g = g0 and g = 0:1g0, where g0 is the zero-pressurevalue for GaAs.

1386-9477/97/$17.00 ? 1997 Elsevier Science B.V. All rights reservedPII S 1386-9477(97)00010 -6

M. Rao et al. / Physica E 1 (1997) 54–58 55

The NLSM is the low energy, long-wavelengththeory written in terms of the local spin polariza-tion which is represented by a unit vector �eld,n(x). The basic di�erence between an ordinary anda quantum-Hall ferromagnet is that in the latter,charge density, �(x) is proportional to the topologi-cal charge density, 4�q(x) = n · (9xn × 9yn) in thelong-wavelength limit [1]. The topological chargeQ =

∫q(x) d2x is always an integer and counts the

number of times the spin con�guration n(x) wrapsaround the unit sphere. The energy functional E[n][2, 14] is a sum of three terms, the exchange energy,

E0 =∫d2x( =2)9in9in; (1)

the Zeeman energy,

EZ = (g�BB=2�)∫d2x(1 + n3)=2 (2)

and the Coulomb energy,

Ecoul = (e2=2�)∫x; yq(x)|x− y|−1q(y): (3)

is the spin wave sti�ness and is equal to (1=16√2�)

(e2=�lc).Before describing our calculation in detail, we

would like to demarcate the precise regime of va-lidity of NLSM in this context. It has been shown[15, 16] that single-skyrmion properties computedfrom the NLSM approaches that obtained from elec-tron Hartree–Fock and exact diagonalization (withinthe lowest Landau level approximation) when spin islarge. The di�erences in energy of a single-skyrmioncomputed within these three schemes is negligiblewhen the spin is greater than about 10, while it isabout 5% when the spin is ∼ 4. At g = 0:1g0, thespin of the skyrmions is¿10 for all values of � con-sidered. We emphasize that at these values of g, theNLSM provides a quantitatively accurate descrip-tion. At g = 1, the spin is signi�cantly less than 10,and so the precise position of the transitions may beinaccurate. However, we believe that the qualitativescenario described above will continue to hold.It is convenient to work in a notation where

the unit vector �eld n(x; y) is replaced by a com-plex �eld w(z ≡ x + iy; �z ≡ x − iy), obtained bythe stereographic projection of the unit sphere onto

the complex plane. Thus, w = cot(�=2)ei�, where �and � are the polar angles of the unit vector n. Inthe absence of Coulomb and Zeeman interactions,any (anti)meromorphic function w(z) is a solutionof the resulting Euler–Lagrange equations [17]. Thetopological charge is simply Q =

∑i ni, where i runs

over the poles of w and ni is the degree of the ithpole. The one skyrmion solution given by

w0( �z; z;; �) =�0ei

z − � ; (4)

clearly has a Q = 1. The spin and charge distributionsare centred at � and fall o� as power laws with ascale set by �0. The XY component of the spin atx is oriented at an angle to the position vector x.The Z component of the spin SZ, however, divergeslogarithmically.In the presence of the Zeeman and Coulomb inter-

actions, Eq. (4) is no longer the minimum energy so-lution. These terms destroy scale invariance, and gen-erate a ‘size’ for the optimal skyrmion, leading to a�nite SZ. To motivate a variational ansatz, we exam-ine the asymptotics of the solution w. As |z| → ∞,w → 0 and the Euler–Lagrange equation reduces to

− 9z9 �zw + g∗

2�w = 0; (5)

where g∗ = g�BB. It is easy to see that the coulombterm is suppressed by a factor of 1=|z|. Thus, w ∼�0e−�|z|=|z| as |z| → ∞. A similar analysis at |z| →0 shows that w ∼ 1=z. Thus, the most general varia-tional ansatz for the single skyrmion which leads tocircularly symmetric spin and charge densities can bewritten as

w( �z; z;; �) =ei

z − �e−�|z−�|[�0 + F(|z − �|)]; (6)

where F(r)→ 0 as r → 0 and ∞. Since F is asmooth function, the con�guration in Eq. (6) can besmoothly deformed to the con�guration in Eq. (4)and so the topological charge remains 1. We ex-pand F in a complete set of functions, F(r) =e−(br)

2=2∑∞n=1 �n(br)

2n=√2n!, where b is a suitably

chosen scale parameter. Minimization of the energywith respect to the set {�; �0; {�n}}, would lead tothe exact solution. In practice, however, we �ndthat including more than the �rst �ve parameters

56 M. Rao et al. / Physica E 1 (1997) 54–58

(�; �0 : : : �3), changes the minimized energy by lessthan 0.1%. With just two parameters � and �0, theerror in the energy is typically around 1%, and atworst 3%.A system of N identicalQ = 1 skyrmions centred at

{�I} with orientations {I}, can now be parametrizedby

w( �z; z) =N∑I=1w( �z; z;I ; �I ): (7)

Our crystalline ansatz corresponds to placing the {�I}on a triangular or a square lattice. This leads to spinand charge densities commensurate with the crystalsymmetry. We have studied both the ferro-oriented(I = 0; ∀I) and generalized N�eel oriented con�gu-rations. Since the square lattice is bipartite, the N�eelcon�guration is characterized byI = 0 for the A sub-lattice and I = � for the B sublattice. Likewise, forthe tripartite triangular lattice, the generalized N�eel or-dering is obtained by assigning I = 0; 2�=3; 4�=3 tothe A, B and C sublattices, respectively.To �nd the classical ground states, we numeri-

cally compute the energy (accurate to 1 part in 106)using our crystalline ansatz and minimize with respectto the variational parameters, for a given lattice andorientational ordering. As for the one-skyrmion case,retaining only two parameters, � and �0, leads to en-ergies accurate to better than 3%. We have calculatedthe phase diagram using this two-parameter ansatz.The existence of the structural transition discussed inthe introduction has also been veri�ed using the moreaccurate �ve-parameter ansatz.We have chosen typical values of the carrier con-

centration (n = 1:5× 1011 cm−2) and the magnetic�eld (B = 10T) and have varied � by tilting the mag-netic �eld with respect to the normal keeping of itsmagnitude �xed. The lattice spacing a for a given lat-tice type is �xed by the value of � (e.g., for a squarelattice, a ≡ √

2�=|1− �|).Fig. 1 shows the percentage di�erence in the mini-

mized energies of the N�eel square and the N�eel trian-gular crystals as a function of � for g = g0 and 0:1g0(the ferro-oriented crystals have much higher energiesin this region). When � is away from 1, a triangularlattice (�1 phase) has the lowest energy. The smallerZeeman energy of the competing square lattice is o�-set by E0 and Ecoul. As � approaches 1, Ecoul decreases,

Fig. 1. Percentage di�erence of the energy densities of the trian-gular (E� 1 ) and square (E ) crystals at B = 10T as a function of�. E± ≡ E ± E� 1 . The symbols correspond to � : g = 0:1g0,+ : g = g0.

resulting in a weak �rst-order transition (slope discon-tinuity) to a square lattice. The structural transitionbetween the �1 and the square phase is accompaniedby a jump in the size of the individual skyrmions.As |�− 1| gets very close to zero, the energy of

the ferro-oriented triangular lattice becomes compa-rable to the Ne�el-oriented lattices. For g = g0, theferro triangular lattice becomes the ground state when|�− 1|¡ 0:025. For g = 0:1g0, the square Ne�el lat-tice remains in the ground state until at least |�− 1| =0:005. In the limit of extreme dilution, �→ 1, the en-ergy of the N -skyrmion con�guration can be evalu-ated as an expansion in (�a)−1,

EN = NE1 + e∗∑i¿j

1|xi − xj| + O

(1�a

); (8)

where E1 is the single skyrmion energy and the cou-lomb interaction is between point charges. To leadingorder, EN is clearly minimized by placing the chargeson a triangular lattice (�2 phase), however, it couldhave any orientational order. Higher-order contri-butions favour the ferro-orientational ordering [18],consistent with our numerical calculations. Thus,a second structural transformation occurs betweenthe square N�eel lattice and a triangular ferro lattice(�2 phase) via a weak �rst-order phase transition as� approaches 1. Note that the �1-square structural

M. Rao et al. / Physica E 1 (1997) 54–58 57

Fig. 2. Spin polarization P as a function of � at B = 10Tand g = 0:1g0. The structural transition predicted by our the-ory is accompanied by a discontinuity in the spin polarization at|�− 1| ≈ 0:04.

transition moves towards � = 1 as g decreases. In-deed, at g = 0 the �1 phase always has the lowestenergy.In the vicinity of the �1-square transition, we �nd

that the size of the skyrmions ∼ a. A simple scalingargument shows that the shape sti�ness of a singleskyrmion is of the same order as the elastic sti�nessof the crystal in the neighbourhood of the transition.The single skyrmion energy is of the form

E1 = c1 +g∗c2R2

2+e∗c3R; (9)

where R sets the scale for the size of the skyrmionand c1; c2 and c3 are ∼1 and weakly dependent on R.At �R ∼ a, the shape sti�ness, 92E1=9R2|R= �R, (where �Rminimizes E1) is comparable to the elastic sti�ness ofthe crystal ∼ e2=Ka3. The novel feature of this struc-tural transition is that it is caused by the shape de-formability of the “atoms”, revealing a richer physicsthan that of Wigner crystallization.The plot of the spin polarization of the skyrmion

crystals at T = 0; g = 0:1g0 is shown in Fig. 2. Aninteresting feature is that the �1 to square structuraltransition is accompanied by a discontinuity of about10% in the spin polarization, and so may be probedby accurate spin-polarization measurements. Sincethermal uctuations and the presence of quencheddisorder smear out this discontinuity, it may be neces-sary to go to very low temperatures to see this e�ect.

How do quantum uctuations a�ect the classicalT = 0 phase diagram presented above? Moving awayfrom � = 1 reduces the lattice spacing. At a criticallattice spacing, zero-point uctuations would meltthe crystal. Moving skyrmions experience a Mag-nus force equal to the Lorentz force produced bythe external magnetic �eld [19]. This implies thatif the orientation and the shape deformation of theskyrmions are ignored, then the system is analogousto interacting charged particles in a strong magnetic�eld. Wigner crystallization of charged particles ina strong magnetic �eld B occurs when the magneticlength lc ∼ 0:5a which happens at �=B ∼ 1

5 where� is the charge density. We, therefore, expect thatthe skyrmion-crystal melts at �sky=B ∼ 1

5 , which cor-responds to � ∼ 0:8. We stress that this is a crudeestimate and the value could be a�ected by the orien-tation and shape deformation of the skyrmions.In addition, we encounter a new quantum orienta-

tion disorder transition (QOD) in the limit of extremedilution. The oriented crystal is characterized bypower-law correlations in the sublattice orientation.Quantum uctuations would destroy this order whenthe uctuations in the orientation become of the orderof 2�. If the skyrmion has a rotor type of desrcriptionto describe its spinning motion then this would occurwhen ˜2=2I ∼ J (a), where I is the moment of iner-tia of the skyrmion [20] and J (a) is the energy costin changing the orientation (and is a decreasing func-tion of a). This leads to a quantum disoriented crystalin the dilute limit [21] via a Kosterlitz–Thoulesstransition.We now show that the spinning skyrmion can,

indeed, be looked upon as a rotor and that its shapedeformability gives rise to its moment of inertia. Theclassical equations of motion are

ddtn(x; t) = n(x; t)× BMF(x; t); (10)

where BaMF(x; t) = �=�na(x; t)E[n]. We look for a so-

lution that corresponds to a uniformly spinning staticcon�guration, ns(x) and �nd that it satis�es the equa-tion

�(!∫x

1 + n3s (x)2

+ E[ns])= 0; (11)

58 M. Rao et al. / Physica E 1 (1997) 54–58

where ! is the angular velocity. If the energy is ex-panded about ! = 0, we have

E = E0 + 12 I!

2 + o(!3) (12)

with the moment of inertia I , being given by I =(1=(�BB)292E=9g2)−1. Thus, in the body �xed frameof the skyrmion it feels a pseudoforce correspondingto an extra Zeeman term. This makes it expand (orcontract) costing elastic energy. The shape deforma-bilty of the skyrmion, therefore, gives rise to its �nitemoment of inertia.Expanding about the classical static solution we can

derive the action for the collective spinning mode tobe

S =∫dtddt(t)

(scl +

12+ l(t)

)− 12Il(t)2: (13)

Here scl is the classical value of the third componentof the spin and the factor of 12 comes from the Hopfterm. This leads to the Hamiltonian

H =12I(� − scl)2 (14)

with [; �] = i˜ and � has eigenvalues ± 12˜;

± 32˜; : : : .To conclude, we have shown that the shape de-

formability of the skyrmions and their orientationaldegree of freedom leads to a rich-phase structure.

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