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Large and exact quantum degeneracy in a Skyrmion magnet B. Douçot,1 D. L. Kovrizhin,2,3 and R. Moessner4 1LPTHE, CNRS and Université Pierre and Marie Curie, Sorbonne Universités, 75252 Paris Cedex 05, France 2T.C.M. Group, Cavendish Laboratory, J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom 3RRC Kurchatov Institute, 1 Kurchatov Square, Moscow 123182, Russia 4Max-Planck-Institut fu¨r Physik komplexer Systeme, 01187 Dresden, Germany (Dated: January 19, 2016) We identify a large family of ground states of a topological Skyrmion magnet whose classical 6 degeneracy persists to all orders in a semiclassical expansion. This goes along with an exceptional 1 robustnessoftheconcomitantgroundstateconfigurations,whicharenotatalldressedbyquantum 0 fluctuations. We trace these twin observations back to a common root: this class of topological 2 ground states saturates a Bogomolny inequality. A similar phenomenology occurs in high-energy n physics for some field theories exhibiting supersymmetry. We propose quantumHall ferromagnets, a where these Skyrmions configurations arise naturally as ground states away from integer filling, as J thebest available laboratory realisations. 8 1 PACSnumbers: 73.43.Lp,71.10.-w,73.21.-b ] l e Introduction. Degeneracies in quantum mechanics sical ground state configurations. These degeneracies, - arenotoriouslyfragile. Thishasvariouswell-knownman- however, are not believed to persist when nonlinear in- r t ifestations, such as level repulsion in many-body spec- teractions between the spin-wave modes are taken into s . tra1, or the third law of thermodynamics which puts a account12,13. t a limit on the ground-state degeneracy of generic many- Thepurposeofthis lengthyexpositionis toemphasize m body systems2. Two stable ways of arranging for per- the unusuallylargeandrobustdegeneracywhichwe find - mitted subextensive degeneracies in the thermodynamic in a topological magnetic system with Skyrmion excita- d limit involve broken symmetries and topological order3. tions14. Such systems arise in a quantum Hall effect set- n Models,however,arenotso constrainedandmany ex- ting inpresenceofinternal(spin, valleyorlayer)degrees o hibit degeneracies of considerable size. Such degenera- of freedom. When a Landau level is fully occupied, the c [ cies are labelled as accidental4 in that they tend to re- latter enter a ferromagnetic ground state, spontaneously quirefine-tuningofmodelparametersfortheirexistence. breaking the symmetry between directions in the space 1 From this perspective, a non-vanishing Casimir force5 is of internal degrees of freedom, provided anisotropies do v 5 amanifestationofthe non-degeneracyofthevacuumen- notdo soexplicitly15. Varyingthe occupancyawayfrom 4 ergy with respect to some tuning parameter. the charge-flux commensurability point leads to a nu- 6 Sometimes, fine-tuning can be natural. This for in- cleation of charged spin textures known as Skyrmions 4 stance is achieved in geometrically frustrated magnets, (a.k.a. baby-Skyrmions in high-energy physics16). 0 where classical ground-state degeneracies appear in sys- Here, we show that a semiclassical degeneracy of the . 1 temsconsistingofhighlysymmetricbuildingblockssuch resulting multi-Skyrmion ground states is robust to all 0 as tetrahedra or triangles, where the choice of which orders in the semiclassical expansion. This family of 6 symmetry-equivalent bonds to frustrate can give rise ground states is further special in that it is not sub- 1 to extensively degenerate classical ground states6. Yet ject to dressing by quantum fluctuations at any order. : v again, these are famously fragile via a (class of) mecha- It appears generally in CPN 1 models17 with a Wess- − Xi nism(s) known as order by disorder7,8. Zumino-Novikov-Witten(WZNW) term20. A well-studied instance is quantum order by disor- Weidentifythesourceofthisunusualbehaviourwhich r a der8, the lifting of a classical ground state degeneracy appearstobedistinctfromthecasesofsymmetrybreak- by quantum fluctuations. Here, a semi-classical ’dressing and topological order mentioned above. Rather, the ing’ of a classical ground-state configuration generically fact that this class satisfies the Bogomolny inequality, distinguishesbetweendifferentnon-symmetry-equivalent and concomitant analytic structure of the classically de- groundstates,withanapparenttendencytoselectacon- generate ground state manifold17, underpins these twin figuration exhibiting some form of symmetry breaking. properties. We emphasize that the degeneracy discussed Extensivestudiesofthismechanismhaveunearthedin- in this paper is large, in that it comprises an extensive stancesofmodelswhichatleastpartiallyevadequantum number of independent continuous degrees of freedom. orderbydisorder. Themostprominentofthese–e.g.the Our study connects to the question of quantization of Heisenberg magnet on the kagome9, checkerboard10, or topological excitations, which been a subject of exten- the fully frustrated dice lattices – exhibit emergent dy- sive investigations in high energy physics. It was found namical gauge-like symmetries11 between the excitation early on that in certain supersymmetric theories, quan- spectra (and hence zero-point energies) of different clas- tum correctionsto classicalsolutions which saturate Bo- 2 gomolny inequality vanish due to a remarkable cancella- classical action. This result is in apparent contradiction tion of bosonic and fermionic contributions in the loop to our findings, and also to conclusions of Ref. 29 that expansions18,21. It was later shownby Witten and Olive the energy of a Skyrmion is independent of its size in that the Bogomolny bound is in fact exact19. While the the case of short-range-interactingquantum Hall Hamil- Bogomolnybound is saturatedquantum mechanicallyin tonian. This discrepancy possibly originates from the our case as well, the model that we study does not con- ambiguities (divergences) in the quantization procedure tain fermions, although it does of course originate from usedthere27,andsubtleissuesencounteredintheprocess the gradientexpansionofa fermionic theory. This raises of quantising systems with constraints in the continuum a question of whether the mechanism which we find is limit. linked with the one appearing in supersymmetric theo- The remainder of this paper is structured as follows. ries22. After introducing the model, we first crisply state our Quantum corrections to classical Skyrmions have also central results. This is followed by an outline of their been studied in condensed matter physics context, in derivation,withtechnicaldetailsrelegatedtotheAppen- e.g. applications to high-temperature superconductiv- dices. We close by placing these results in the broader ity24, and in the theory of topological magnets25–28. In contextofsemiclassicaltreatmentsofCasimirforcesand particulartheauthorsofRef.27studytheCasimireffect quantummagnets,anddiscussthe relevanceofquantum in an isotropic ferromagnetic SU(2) sigma-model with Hall experiments for probing these phenomena. a single Belavin-Polyakov Skyrmion. By calculating the Model. We study the non-relativistic form of the contribution to the zero-point energy of the Skyrmion quantum CPN 1 model, describing a system without − from magnon excitations they find a non-vanishing cor- Lorentz-invariance (in other words a ferromagnet rather rection, which was further dependent on the Skyrmion than an antiferromagnet)which is defined by the follow- radius, thus breaking the conformal invariance of the ing action in 2+1 dimensional space-time: i ψ ∂ ψ ∂ ψ ψ ψ ψ ψ ψ ψ ψ = dtd2r h | t i−h t | i E h∇ |∇ i h∇ | ih |∇ i . (1) S 4πl2 ψ ψ − ex ψ ψ − ψ ψ 2 Z (cid:20) h | i (cid:18) h | i h | i (cid:19)(cid:21) In this expression, l denotes the magnetic length of the canbeviewedasaBerryphase,andinthe N =2caseis underlying electronic system in the presence of a strong identical to the usual SU(2) spin Berry phase. The sec- magnetic field, and E is the spin exchange energy, ondtermis the potentialenergy,whichisentirelydue to ex which originates from the combined effect of a short- Coulomb repulsion, after the orbital electron degrees of range part of the Coulomb repulsion and the Pauli prin- freedom have been quenched in the lowest Landau level. ciple. The bras and kets are compact notations for an The quantization of (1) can be performed naturally us- N-component complex spinor field ψ (r,t), 1 a N, ing coherent-state path integrals31–35. Note that a non- a ≤ ≤ which represents an internal electronic degree of free- relativistic nature of the above classical action has far- dom (e.g. spin and valley in a graphene sheet) and the reaching consequences for the quantization, as explained ∂ ,∂ operators involve only two spatial deriva- for example in the context on some non-linear sigma x y t∇ive≡s.{This a}ction exhibits global SU(N) symmetry, in models in two-dimensional space-time30. Indeed, this addition to a local gauge symmetry, according to which leads to drastic simplifications in the diagrammatic ex- theactionisunchangedundertransformationsψ (r,t) pansion of scattering amplitudes between magnons, and a → f(r)ψ (r,t), wheref(r) is anarbitrarycomplexfunction to a suppressionof many expected quantum corrections, a of position. Because of this symmetry, it is natural to as we find to be the case in the 2+1 dimensional version view the local spinor ψ (r,t) as a representative of the studied here. a complex line thatit generatesinCN, so the targetspace is the complex projective space CPN 1 rather than CN. InthefollowingwemostlyusetheHamiltonianversion − of the model. In order to obtain it, first it is convenient Inmostexperimentalimplementationsvarioussymme- to discretize the two-dimensional physical (coordinate) try breaking terms are present, which usually eliminate space, assuming that its area divided by 2πl2 is equal thecontinuousdegeneracyofclassicalground-statesthat to an integer N , which can be interpreted as the total φ is our main interest here. When the strength of these number of magnetic flux quanta in the system, or equiv- symmetry breaking terms is small compared to E one alently, the total number of states in the lowest Landau ex should first compare their size with the magnitude of level. Note that for anelectronic system close to the fill- possibledegeneracyliftingCasimir-likeforcesinducedby ing factor ν = 1, the total number of electrons N is el quantum fluctuations in the fully symmetric model (1). equalto N N , where N is the topologicalcharge φ top top − Thefirsttermintheaction(1)istheWZNWterm,which associated with the texture ψ (r). At each of the N a φ 3 lattice sites R , we place a quantum degree of freedom mentsofCPN 1. BygeneralizingEq.(18)toacollection j − whichlivesinthe fundamentalrepresentationofSU(N). of N sites, a matrix element of the evolution operator φ As explained in Appendix 1 such degrees of freedom can can be expressed in terms of a coherent state path inte- be described in terms of coherent states labelled by ele- gral Nφ Nφ ψ e iHˆt ψ = [ψ (t)]exp dt α[ψ (t)]∂ ψ (t) iE [ψ (t)] , (2) h out| − | ini D j  j t j − var j  Z jY=1 Z Xj=1    wherethesinglesiteBerryphaseformα[ψ (t)]isdefined ago14,15,36–38 that the leading term in this semi-classical j as expansion of E is precisely the CPN 1 energy func- var − tional,andthattheBerryphaseformforthiscontinuous 1 ψ dψ dψ ψ α[ψ (t)]= h | i−h | i, (3) family of Slater determinants gives the first term in the j 2 ψ ψ action defined in Eq. (1). h | i AremarkablefactabouttheCPN 1 energyfunctional and E [ψ (t)] is the expectation value of the quantum − var j is that it satisfies the Bogomolny bound17 Hamiltonian of the system taken on the tensor product of coherent states ψ (t) at sites R , for 1 j N . If Evar[ψj(t)] is cho|sejn toi be a discrejtized ver≤sion≤of thφe Evar ≥2π|Ntop|Eex. (6) CPN 1 energy functional then in the limit N : − φ →∞ For a fixed topological charge, this bound is reached for holomorphic (resp. anti-holomorphic) textures when ψ ψ ψ ψ ψ ψ Evar[ψj(t)]→EexZ d2r(cid:18)h∇hψ||∇ψi i − h∇ h|ψi|ψhi2|∇ (i4(cid:19)) Ngrtooupn≥d-s0t,a(treessopf.thNetoCpP≤N−01).enTehregryeffournectiifoNnatlopfo6=rm0a, tdhee- generate family with an extensive number of continuous and the path integral assumes the form parameters. We now consider classical ground states of Skyrmion ψ e iHˆt ψ = [ψ(r,t)]ei [ψ] (5) h out| − | ini D S textures in quantum Hall ferromagnets as representing Z quantum coherent states, see Appendix 1. Let us pick with [ψ] defined in Eq. (1). Note that there are known suchastate,denotedby Ω . Itischaracterized,asusual, subtleSties in coherent state path integral quantizationof bytheconditionsaˆ (r)Ω| =i 0forj =1...N 1bosonic j spin, whose discussion we omit here as these do not af- annihilation operators|aˆ i(r) which define a co−mplete set j fectourresults,seee.g.Ref.34. InAppendix 2,weshow of SU(N) Schwinger bosons. What plays the role of the that the above energy functional can be seen as the con- classical energy E is then the quantum mechanical ex- 0 tinuumlimitofthe variationalenergyforalatticemodel pectation value of the two body interactionHamiltonian of ferromagnetically coupled SU(N) spins. Hˆ : E = ΩHˆ Ω . The question which arises nat- int var int While this won’t be necessary for the statement, and urally is whethhe|r the| diegeneracy of the E functional var for the derivation of our main results, it is nevertheless is preservedatthe quantumlevel. To addresssuchques- usefultorecallhowsuchquantummodelemergesfroma tion,oneusuallyexpandsthequantumHamiltonianHˆ int non-relativisticsystemofelectronswithN internalstates in powersof these bosonic operators12. Let us denote by at quantum Hall filling factor ν close to 1. The assump- Hˆ the term which contains normal ordered products of n tion of a strong magnetic field allows one to project or- exactly n single bosonic operators. By assumption that bital degrees of freedom onto the lowest Landau level. Ω is a minimum of E , we observe that Hˆ =0. Lin- var 1 Forany classicaltexture, describedbythe N component |eari spin wave theory, and most microscopic theories for spinor field ψa(r) , it is possible to write a Slater de- Casimir forces, truncate this series to a quadratic term | i terminant |Sψi associated to this lowest Landau level in Hˆ2. Inmanycases,thisissufficienttodressthecoherent such a way that in the limit of a very strong magnetic state vacuum Ω with quantumfluctuations andto gen- field l 0 the internal degree of freedom wave-function erate a finite q|uaintum correction to the classical energy → at point r is given by a local spinor |ψa(r)i. In this functional Evar. limit, which corresponds to a small Skyrmiondensity on Results. Our first main result is that with the above the scale of the magnetic length l, the expectation value quantization of the CPN 1 energy functional, these op- − Evar = ψ Hînt ψ of the two-body interaction Hamil- timal textures are exact eigenstates, not only of the hS | |S i tonian Hˆ can be expressed as a power series of nl2, quadratic Hamiltonian Hˆ , but of the full quantum int 2 where n is the average topological charge density of the Hamiltonian Hˆ . The degeneracy within this contin- int classical texture. It has been shown already a long time uous family of optimal textures is therefore preservedto 4 all orders in quantum fluctuations. case persists to all orders in the semiclassical expansion. Our second result concerns the fate of these coherent The first item (i) implies that we do not have a stan- states in the semiclassical approximation. We find that dard, simple, structure such as boson number conserva- they are not dressed at all by quantum fluctuations, in tion, which could have arisen from an underlying, pos- sharp contrast to, for example, a simple Néel state of a sibly hidden, U(1) symmetry. Such a symmetry alone Heisenberg antiferromagneton a square lattice. would already have explained the absence of anomalous These two salient features originate from a single fea- terms, but it would impose much stronger restrictions tureofthesemiclassicalexpansion. Namely,wefindthat: on the type of terms appearing in the semiclassical ex- (i) There exist terms in this expansion violating boson pansion. The second feature (ii) interdicts any non-zero number conservation, such as aˆ aˆ aˆ. (ii) Nonetheless, † † differencebetweentheclassicalandsemiclassicalenergies there are no anomalous terms consisting exclusively of atquadraticandallhigherordersinthesemiclassicalex- creation (annihilation) operators, such as aˆm for any in- pansion, but also demonstrates that the state Ω itself teger m>0, or hermitian conjugate. | i is never dressed by excitations. In other words the state Indeed, with all aˆ aˆ terms absent from the quadratic † † Ω is the vacuum of normal ordered Hˆ for all n. bosonic Hamiltonian Hˆ2 for any optimal texture, this | i n obviates the need for a Bogoliubov transformation, and Sketch of proof. To be specific, let us assume that henceimpliestheabsenceofsqueezing,whichamountsto N ispositive,andpickaclassicalminimalenergycon- top an admixture of the bosonic ’excitations’ to the ground figuration ψ of the CPN 1 energy functional. Then cl − state. (Such anomalous terms appear in standard ex- ψ (z) is a holomorphic spinor, z = x+ iy. Now we cl | i pansionsnearnon-collinearmagneticgroundstates,that expand the spinor field ψ(r) around ψ (z) , writing cl originate from e.g. SˆxSˆz terms in the rotating basis, see ψ(r) = ψ (z) + χ(r) .|Usinigthefactt|hat∂iψ =0, cl z¯ cl | i | i | i | i e.g. Ref. 13). Note that the absence of squeezing in our we rewrite the r.h.s. of Eq. (4) as ∂ χ∂ χ ( ∂ χψ + ∂ χχ )( ψ ∂ χ + χ∂ χ ) E =2π N E +4E d2r h z | z¯ i h z | cli h z | i h cl| z¯ i h | z¯ i . var | top| ex ex ψ ψ + ψ χ + χψ + χχ − ( ψ ψ + ψ χ + χψ + χχ )2 Z (cid:26)h cl| cli h cl| i h | cli h | i h cl| cli h cl| i h | cli h | i (cid:27) (7) The power of this expression lies in the fact that the term containing solely creation (annihilation) Schwinger presence of the small perturbation χ does not modify bosonsoperatorsappearintheexpansionofthequantum the total topological charge N , and the holomorphic Hamiltonian, see Appendix 4 for more details. top nature ofψcl producesa remarkablesimplification ofthe Relation to other flavours of semiclassics. The second term, which can be stated as follows: the Taylor analysis presented above amounts to adopting a slightly expansion of the CPN−1 energy functional in powers of different viewpoint comparedto conventionalCasimir or χa(r) and χ¯a(r) does not contain terms involving only spin-wave discussions. In the latter, once the classi- χa(r)’s, or terms involving only χ¯a(r)’s. This is the key cal ground-state with energy E0 is chosen, the quantum property which underlies the above stated results. Hamiltonian is expanded to a quadratic order in terms of normal coordinates pˆ,qˆ, (1 i N ), which obey i i s The rest of the proof takes the following steps. First, canonical commutation rules [pˆ≤,qˆ]≤= i~δ . It is con- i j ij we apply a unitary transformation which maps a coher- venient to introduce creation and annihilation operators ent state associated with ψcl into the state |Ωi. The lat- aˆ+j ,aˆj, defined as ter corresponds to a trivial ferromagnetic configuration ψ (r)=δ . Thisisanalogoustochoosing,inaspinwave a a0 aˆ =(ipˆ +qˆ)/√2~, aˆ+ =( ipˆ +qˆ)/√2~. analysis,arotatingspinquantizationaxisparalleltoalo- j j j j − j j cal spin in a given classical ground-state. We emphasize LetusnowassumethataquadraticHamiltonianinterms that, because such transformation maps a topologically of these operators takes the form non-trivial state ψ into a trivial one, it cannot be a cl | i continuous function of spatial coordinates. However the 1 transformationis holomorphicwith respectto the target Hˆ =E + A aˆ+aˆ +A¯ aˆ aˆ++B aˆ+aˆ++B¯ aˆ aˆ . CPN−1 manifold, and therefore, the expansion of the 2 0 2Xij ij i j ij i j ij i j ij j i transformed energy functional around a ferromagnetic (8) configuration also satisfies the key property mentioned Here, the complex numbers A and B are the entries ij ij above. Furtherdetailsonthisunitarytransformationare of two N N matrices, A and B, which are respec- s s × presented in Appendix 3. Having reduced the discussion tively hermitian, and symmetric. Such a form occurs to the ferromagnetic configuration,it is then rather easy when classical quadratic monomials like p q are quan- i i to translate the key property into a statement that no tized according to a symmetric ordering prescription, 5 i.e. as (pˆqˆ + qˆpˆ)/2. After diagonalization via a Bo- Hamiltonian i i i i goliubov transformation39, the Hamiltonian Hˆ may be 2 written, in terms of the normal modes γˆα+,γˆα, as Hˆ2 =LS2f0+ S (fq+gq) 2 q 1 Ns Ns X Hˆ2 =E0+ 2 ~ωα+ ~ωαγˆα+γˆα. (9) +S {(fq+gq−2f0)aˆ+q aˆq α=1 α=1 q X X X 1 The ground-state energy E2 of Hˆ2 is given by + 2(fq−gq)[aˆ+qaˆ+−q+aˆ−qaˆq]}. (12) 1 Ns The classical energy E = LS2f is the leading term of E =E + ~ω . (10) 0 0 2 0 2 α Hˆ2 in the large S limit. After a Bogoliubov transforma- α=1 X tion, we have The sum over normal mode frequencies in this expres- 1 sion is generic for Casimir energies due to quantum fluc- Hˆ =LS(S+1)f + ω + ω γˆ+γˆ , (13) tuations. If we have a continuous family of degenerate 2 0 2 q q q q q q X X ground-states(i.e. withthe samecommonvalue forE ), 0 this degeneracy is preserved, at the level of quadratic where γˆ+, γˆ are the quasiparticle creation and annihi- q q fluctuations, only if the frequency sum αωα is inde- lation operators, and ωq =2S[(fq f0)(gq f0)]21 is the pendent of the classical ground-state within the family. − − eigenfrequency of mode q. So Eq. (10) is modified into P Such a condition is very difficult to satisfy generically, ubynleasnseaxnaycttwsyomdmeegternyeroafttehgerqouuanndt-ustmatHesamarieltocnoniannecHˆte2d. E2 =E0+LSf0+ 1 Ns ~ωα. (14) This viewpoint is the one which has been taken in the 2 α=1 X earlier references on the subject of quantum corrections totheenergyofspintexturesintopologicalmagnets25–28. TheextratermLSf0 isusuallyabsorbedintoaquantum Note that in the case of spin systems Eq. (10) has to renormalizationofthespinlength,S2becomingS(S+1), be slightly revisited. For concreteness, let us consider in the expression of E0. The variational energy Evar is a chain of L spins S with spin length S, and let us theexpectationvalueofthe quantummechanicalHamil- n assume that a classical Hamiltonian admits spiral con- tonian Hˆ2 in the vacuum state of Holstein-Primakoff figurations with uniform twists as stable local minima. bosons, which is nothing but the classical spiral configu- We then choose the spin quantization axis so that the ration. Then, Evar−E0 = S2 q(fq+gq),andisapurely x direction in spin space is everywhere aligned with the localterm,i.e.itinvolvesonlytheselfinteractionofeach P local classical spin configuration. We also assume that spin in the xy plane. This may be at the origin of the the initial spin Hamiltonian is rotationallyinvariant and difference between our result, which states the absence that z is the rotation axis of the spiral configuration. In of any quantum correction to the variational energy of the frame used for quantization, there remains a man- holomorphic textures, and the result of a direct evalu- ifest U(1) symmetry corresponding to z-rotations, and ation of the sum of magnon frequencies27, according to the Hamitonian takes the form which degeneracy lifting occurs. Subtle renormalization issues may occur when one replaces a lattice system by Hˆ = fq[SˆqxSˆxq+SˆqySˆyq]+gqSˆqzSˆzq+ihqSˆqxSˆyq. (11) a continuous field theory as the lattice size goes to zero. q − − − − Adetailedstudy ofthe problemoftakingthe continuum X limit is an interesting direction for future work. Because H is Hermitian, the three functions f , g and q q QuantumHallexperiments. Wewouldliketomen- h are real. It is convenient also to assume that f , g q q q tion that such a striking resilience of coherent states as- are even in q and that h is odd. Let us now use the q sociated to holomorphic textures in the non-relativistic Holstein-Primakoff bosonic representation31. To leading quantumCPN 1 modeldefinedbyEq.(1)hasbeensug- − order in 1/S expansion, we get: gested to us by a remarkable observation made twenty years ago by MacDonald et al.29, which was further ex- 1 Sˆx = √L(Sδ aˆ+aˆ ), ploited by Pasquier40,41. They noticed that in a model q q,0− L k k+q k with short-range, delta-function, repulsive interactions X projectedontothelowestLandaulevel,holomorphictex- S Sˆqy = r2(aˆq+aˆ+−q), tduerteesrm|ψinaa(zn)tis can be put in correspondence with Slater S Sˆz = i (aˆ aˆ+ ). q − r2 q− −q Nel |zi|2 Sψ(z1a1,··· ,zNelaNel)= (zi−zj) ψai(zi)e− 4l2 . To a quadratic order in Holstein-Primakoff bosons, this i<j i=1 Y Y gives the following normal-ordered expression for the (15) 6 Due to presence of a Jastrow factor in this expression, a Skyrme crystalscales with the square rootof the devi- which prevents two particles with opposite spins to oc- ation from commensurate filling in units of spin stiffness cupy the same position, the wave-function(15) is clearly of the ferromagnet. This scale can thus in principle be an exact zero energy eigenstate of the quantum Hamil- tuned to be parametrically small by studying systems tonian with delta-force repulsive interaction, see also closeto integerfilling. While the electrondensity canbe Ref. 36,37 for a discussion of corrections to the Hartree- uniformly tuned by changing a gate potential or varying Fock energy functional compared to Ref. 15. On the the duration of photodoping, it is nonetheless not possi- other hand, we have already pointed out that the non- ble to approach integer filling arbitrarily closely as non- relativistic quantum CPN 1 model appears as the lead- uniformities, for example arising from stray fields due to − ingterminthesemi-classicalexpansion(inpowersofnl2) ionised donor impurities, lead to an effectively variable of most electronic models with repulsive interactions af- electrochemical potential which will lift the degeneracy ter projection onto the lowest Landau level in quantum in favour of a Skyrmion glass47. Needless to say, in the Hall systems at filling factor ν 1. The striking re- caseofneutralquantumHallsystems,proposedlongago semblancebetweenthe ground-sta∼tesectorsof the quan- in the field of coldatoms48, such electrostatic effects can tum Hamiltonian with delta repulsion, and of the non- be avoided. relativistic quantum CPN−1 model raises an intriguing As is common in cases with high degeneracies, one questionabouttheirpossibleequivalence,alsoforexcited needs to probe their presence at the temperature, which states. Unfortunately, the methods that we use here do scaleisabovetheleadinginstabilityscale;indeed,astudy not seem to provide any straightforwarddirection to ad- of such instabilities is a worthy research subject in itself dress this question. inthecontextofunderstandingvariousorder-by-disorder Another motivation for the present study was to pro- mechanisms. The question about which experimental vide a theoretical basis for our recent work on periodic probes to use of course again depends on the precise de- textures in the case of SU(N) symmetric repulsive in- tails of model systems. teractions42,43. The absence of quantum degeneracy lift- Forsemiconductorheterostructuresaninnocuousther- ing mechanisms among holomorphic textures, which we modynamicprobesuchasspecificheatisproblematicon demonstrate here, justifies the approach developed in account of the low thermal mass of the quantum Hall Ref. 42,43. In the latter study the degeneracy lifting layer in a 3D bulk system. However in semiconduc- mechanism was due to next-to-leading term in the ex- tors other probes are readily available, such as electrical pansionoftheclassicalenergyfunctionalEvar,thatarises transport,whichcanbe measuredexquisitelysensitively. from the long-range tail of the Coulomb potential14,15. For example, it allows a study of the low-energy spec- Experiment. This naturally leads us to the ques- trumofthe quantumHallferromagnetviaresistivelyde- tion about the context in which such Skyrmion crystals tected NMR measurements49. The large degeneracy of may be observed. The experimental challenge is to min- Skyrmion systems discussed here should therefore show imise the effect of possible symmetry-breaking interac- upinaverylargelow-energydensityofstates,andthere- tions, which are present in real systems in addition to fore, a relatively fast dynamics compared to a magnet theonesalreadycapturedinourmodel. Thesetermswill confinedto remainneararobustunique groundstate. A generically not leave the degeneracy intact – in experi- systematic study of this physics is clearly a worthy goal ment the third law of thermodynamics tends to reassert for future experimental efforts. itself eventually. Summary. We have identified a particularly robust In semiconductor-based quantum Hall systems such large degeneracy in topological isotropic SU(N) ferro- terms include most simply a ‘one-body’ anisotropy, such magnets. We have discussed origin and ramifications of as a Zeemanfield, breaking the SU(2) symmetry in spin this degeneracy. We believe that these observations are space;orinthecaseofvalleyisospin,anisotropiceffective of conceptual importance for the broad and fundamen- masses can break a continuous SU(3) symmetry down tal question of how degeneracies arise, and how they are to a discrete Z . Such anisotropies remove the SU(N) lifted. Our work seems to point to a novel mechanism, 3 symmetry, which underpins the degeneracy constitutive on a superficial level perhaps most closely related to su- to ourmodel. One thus needs to consider systems where persymmetric ideas from high-energy physics. It has the anisotropies are small, e.g. in semiconductors where it is addedbonusofbeing approximatelyrealizableinexperi- possible to tune effective electronic g-factor to zero by ment,andthus expandsthe zoooffrustratedanddegen- applying hydrostatic pressure44–46. The effect of resid- erate systems not only by a new mechanism but also by ualsymmetry-breakingterms canthen, atleastapproxi- a new approximate materials realisation. Our findings mately, be taken into account within the CPN 1 model. pose intriguing questions about relationof our results to − The physics of Coulomb interactions also enters nat- supersymmetricfieldtheories,andprovideinterestingin- urally, as the topological charge of the spatially non- sights into a long-standing problem of the quantization uniform textures goes along with spatial modulations of of systems with constraints. the electronic charge density. The minimization of this Acknowledgements: We would like to thank David functional within the family of holomorphic textures is Tong and Costas Bachas for interesting discussions an interesting problem in itself. The cohesive energy of regarding possible connections between the absence 7 of quantum corrections in the non-relativistic CPN 1 Withthis definition,the overlapbetweencoherentstates − model and supersymmetry. D.K. acknowledges EPSRC reads Grant No. EP/M007928/1. Appendix 1: CPN−1 coherent states. Let us con- (1+ v v )m ′ sider the following natural SU(N) generalization of the ev¯′ ev¯ = h | i h | i m! Schwinger boson construction of SU(2) spin represen- tations35. We start from the infinite-dimensional Fock spaceassociatedto N bosonicdegreesoffreedom,whose Here, we use a compact notation v v′ to denote h | i creation and annihilation operators are denoted by aˆ+i , Ni=−11v¯ivi′. The Cauchy-Schwartz inequality implies aˆ , here 0 i N 1. For any positive integer m, we thattheoverlapbetweennormalizedcoherentstatesvan- i select a ph≤ysica≤l sub−space defined by the constraint iPshes exponentially fast in the classical limit m , → ∞ provided v =v . ′ N 1 6 − a+a =m. (16) Itistheninstructiveto computethe Berryphaseform i i α associated to an infinitesimal variation from v to v = i=0 ′ X v+dv. We get: The dimension of this subspace is given by the binomial m+N 1 coefficient N 1− . This subspace is isomorphic ev¯′ ev¯ 1 v dv dv v (cid:18) − (cid:19) α dv′ h | i = h | i−h | i to the fully symmetrized m-fold tensor product of the ≡ hev¯′|ev¯′ihev¯|ev¯i!v′=v 2 1+hv|vi fundamental representation of SU(N). In the special It is interesting to try to extend α to the whole of N =2 case with m=2S we recoverthe standardspin-S CP(N 1)p. For this, we use homogeneous coordi- − representation of SU(2). An orthonormal basis is given nates ψ , ,ψ . U correspondsto the subsetwhere 0 N 1 0 by the states ψ =0 an·d··on U−, v = ψj. Then: 0 6 0 j ψ0 (aˆ+)n0 (aˆ+ )nN−1 ~n = 0 ··· N−1 0 , (17) | i n ! n ! | i m ψ dψ dψ ψ 0 ··· N−1 α= 2 h | iψ−ψh | i −imdargψ0 with {ni} non-negativpe integers, and Ni=−01ni =m. h | i Anovercompletecoherentstatebasisisconstructedas Thepresenceofthesecondtermisrequiredtoensurethat follows. Let us consider an open subPset U0 in CPN−1 a pure gauge transformation ψ eiθψ , which doesn’t j j composed of complex lines, which are generated by vec- → generate any new state, doesn’t produce a Berry phase tors of the form (1,v , ,v ), with v C. In U , tchoeorNdin−a1tecosymsptelemx.nLue1mtb·u·es·rsdevNfi1,n−·e1·· ,vN−1 pir∈ovideagoo0d teeitnhdeerd. tBoutthtehiwsheoxlperCesPsiNon−1shmowansiftohldat: αitcisansninogtublaereoxn- the complement of U , which is nothing but the hyper- 0 v¯n1 v¯nN−1 plane at infinity characterizedby ψ0 =0. ev¯ = 1 ··· N−1 ~n , A quantum Hamiltonian Hˆ is completely specified | i n ! n !| i X 0 ··· N−1 by its expectation value on coherent states Evar(ψ) = where the sum is over Np-vectors ~n, whose components hψ|Hˆ|ψi50. It is possible to write the corresponding evo- ψψ are non-negative integers which satisfy Ni=−01ni = m. luhti|oni operator in the form of a coherent path integral: P ψ exp( iHˆt)ψ = ψ(t) exp dt α ∂ ψ iE [ψ(t)] (18) out in ψ(t) t var h | − | i D { − } Z (cid:20)Z (cid:21) As usual, the justification for this expression is the exis- haves as (m/π)(N 1), which is consistent with the basic − tence of a representation of the identity operator as an quantum mechanical rule that a quantum state of a sys- averageoverprojectorsoncoherentstates withthe stan- temwithN 1degreesoffreedomoccupiesaphase-space dard SU(N)-invariant measure on CP(N 1): volume of o−rder ~(N 1), provided we take the effective − − Planck’s constant to be proportional to 1/m. It is also I= (m+N −1)! Nj=−11dvjdv¯j |ev¯ihev¯| interesting to mention the fact that changing the choice π(N−1)m! Z Q(1+hv|vi)N hev¯|ev¯i opfhathseeboypeannsinutbesgeetrUm0umltiopdleifioefs2tπhme .inTtehgirsaalmofbitghueitByehrrays Notethatintheclassical limitm ,theprefactorbe- →∞ 8 no influence as long as m is an integer. This is reminis- to SU(N) matrices M. In homogeneous coordinates, cent of Dirac’s quantization of the magnetic monopole the homography h sends the complex line through M charge placed inside a sphere. (ψ , ,ψ ) into the line through (ψ , ,ψ ), asAapSpUen(Ndi)xfe2r:roTmheagnnoent-.rLeleattuivsicsotnicsiCdePrNa−s1ysmteomdeofl wh0er·e··ψi′ =N−1 Nj=−01Mijψj. In the open su0′b·s·e·t u0N′,−t1he complex coordinates v = ψj are transformed according two SU(2) spins Sa,Sb. We introduce Schwinger bosons P j ψ0 to: a+, a (with σ = , ) associated to S and b+, b asso- ciσatedσto Sb31. Th↑en↓it is easy to checak that:σ σ Mi0+ j=0Mijvj Hˆferro =−Sa.Sb =−21 a+σb+σ′aσ′bσ+S2 (19) vi′ = M00+Pj6=6 0M0jvj (23) Xσ,σ′ If M SU(N), then: P ∈ where S is the size of these two spins. This is easily generalized to SU(N) spins defined above in Appendix |M00+ M0jvj|2(1+hv′|v′i)=(1+hv|vi) (24) 1. In this generalization we keep the first term in the Xj6=0 aboveHamiltonian,wherethe indicesσ,σ nowrunfrom ′ In the Schwinger boson Fock space, we define the quan- nn0ootwromeNavlai−zleuda1t,etaetnnhdseotrehxpeprceoocdtnuasctttiroanoinfvtac(olu1h6ee)rEeinsvatrensotffoartHˆceesfedr|r.eov¯Liinetatnhudes TttûoMmc|h0miec=ekcht|0ahinaitacnaTˆldoTˆpiMseraua+jntoiTˆtrMa−rT1ˆyM.=FbuyrNitt=hh−0ee1rmMreoqirjueai,r+iei.tmcIetonntissseetrhavsaeyts ew¯ . A simple calculation, using for example Eq. (27) M P | i the total number of Schwinger bosons, so it acts within below, shows that: the the quantum Hilbert space defined by imposing the (1+ v w )(1+ wv ) constraint eq. (16). In this Hilbert space, it is also easy E (v,v¯,w,w¯)= m2 h | i h | i (20) var − (1+ v v )(1+ ww ) to check thatTˆM sends a coherentstate into anotherco- h | i h | i herent state, in a manner which is consistent with the This expression holds in the open subset U of CPN 1. underlying classical homography h . Specifically: 0 − M It is possible to write it in a gauge invariant manner by ψintr=odψucin=g1tw, oψN=-covmpanondeψnts=pinworsfoψra1ndψj′sucNhtha1t. TˆM|ev¯i=(M00+ M0jv¯j)m|ehM(v)i (25) T0hen: 0′ j j j′ j ≤ ≤ − Xj6=0 From eqs. (24) and (25), we see that the classical en- ψ ψ ψ ψ Evar(ψ,ψ¯,ψ′,ψ¯′)=−m2hψ|ψ′ihψ ′|ψ i (21) ergy functional for the hamiltonian Hˆ evaluated at v, h | ih ′| ′i is equal to the classical energy functional for the trans- It is clear that this is indeed invariantunder localtrans- formed hamiltonian TˆMHˆTˆM−1 evaluated at hM(v). In equations: formations ψ λψ , ψ λ ψ . Suppose now that ′ ′ ′ | i→ | i | i→ | i ψ and ψ are very close, so that we may replace ψ by ψ−21χ a′ndψ′ by ψ+12χ. Expandingto secondorderin hehM(v)|TˆMHˆTˆM−1|ehM(v)i = hev¯|Hˆ|ev¯i (26) χ in the χ 0 limit gives: e e e e → h hM(v)| hM(v)i h v¯| v¯i E (ψ,ψ¯,χ,χ¯)= m2+m2 hχ|χi hχ|ψihψ|χi +... Letus nowassumethatthe classicalenergyfunctional var − ψ ψ − ψ ψ 2 for the hamiltonian Hˆ has a local minimum at w U , (cid:18)h | i h | i (cid:19) ∈ 0 (22) andthatitsTaylorseriesexpansionaroundv =wdoesn’t If we now take a two-dimensional lattice of such SU(N) containany termwhichis purelyholomorphic(i.e. poly- spins, and couple nearest neighbor spins by Hˆ , nomialinv w )norpurelyanti-holomorphic(i.e. poly- ferro i i − the expectation value of the lattice Hamiltonian over nomialinv¯ w¯ ). Therealwaysexistsanunitaryhomog- i i − a normalized tensor product of local coherent states raphyh suchthath w =0. Becausethishomography M M will be written as a sum of contributions of the form isaholomorphictransformationinthev coordinates,the E (ψ(r ),ψ¯(r ),ψ(r ),ψ¯(r )), wherer andr arenear- Taylor series expansion of the transformed Hamiltonian var i i j j i j est neighbor sites. Taking the continuous limit, we as- TˆMHˆTˆM−1 around v = 0 doesn’t contain any term which sume that ψ(r ) and ψ(r ) are very close, and the total is purely holomorphic nor purely anti-holomorphic. i j variationalenergyisasumoftermsasinEq.(22). Sothe The generalizationof this statement to a system com- variational energy for this ferromagnetic model appears posed of N CPN 1 degrees of freedom is straightfor- φ − asalatticediscretizationoftheCPN 1energyfunctional ward, so it will not be detailed further here. − given in Eq (4). Appendix 4: coherent states as exact eigen- Appendix 3: unitary transformation to a fer- states of the CPN 1 Hamiltonian. To keep the dis- − romagnetic configuration. Let us first consider a cussion simple, let us consider a single CPN 1 degree of − single CPN 1 degree of freedom. A natural family of freedom, quantized in the way explained in Appendix 1. − isometries on CPN 1 are homographies h associated Letus considera quantum HamiltonianHˆ suchthat the − M 9 correspondingenergyfunctionalE (ψ)isminimizedfor Now,theonlynormal-orderedSchwingerbosonmonomi- var ψ proportionaltoδ ,whichbelongstothe opensubset als which act on Ω and produce a state orthogonal to a a0 U of CPN 1. Note that this optimal coherent state is it, have the form|n i=nδ , and m =0 for at least one 0 − j j0 j (a+)m j suchthat j 1. Such monomialwo6uldthen produce a Ω = 0 0 . It is characterized by the property that | i √m! | i contribution t≥o E proportional to: it is annihilated by a , ,a . If we discretize the var 1 N 1 CPN 1Hamiltonian,eac·h··siteis−intheextremequantum − regime m=1. It is nevertheless useful to keep m explic- Nj=−11vjmj itly, because, as explained in Appendix 1, the m (1+ v v )n limit can be regardedas a classicallimit for this sy→stem∞. Q h | i We choose holomorphic coordinates v = ψj in U , and j ψ0 0 and in particular, it would generate the monomial pwoewaesrssuomfevf,uv¯rthdeoretshna’ttctohnetaTinayalonryemxopnaonmsioianlocofmEpvaorseind Nj=−11vjmj which is holomorphic in all vj’s. Such mono- j j mial is ruled out by our assumption, and we note that it onlyvj’snoronlyofv¯j’s. Thisisthesinglesiteversionof cQan only be generated by the normal-ordered Schwinger tthheatktehyepcroohpeerretnytssttaatteed aΩfteirseaqn. (e7x)a.ctTehiegnenwsteactaenofintfheer boson monomial ( Nj=−01(a+j )mj)an0. This rules out such | i operators and proves our statement. quantum Hamiltonian Hˆ. Q To show this, we write Hˆ as a power series in single It would be easy to generalize to a system composed ofN coupledquantumCPN 1 degreesoffreedom,such bosonicoperators,writteninnormalorder,i.e. withcre- φ − that E (ψ) is minimized for the configuration ψ (R ) ationoperatorsonthe left andannihilationoperatorson var a j proportional to δ . Let us denote by Ω the corre- the right. Because of the constraint eq. (16), each of a0 | i sponding quantum coherent ferromagnetic state. Then these monomials is of the form Nj=−01(a+j )mjanjj, with Ω is annihilated by the (N 1)N operators a (R ), φ j k Nj=−01mj = Nj=−01nj ≡ n. It turQns out that the classi- |fori1 ≤ j ≤ N −1 and 1 ≤ k−≤ Nφ. If we further as- calenergyfunctionalforthesenormal-orderedmonomials sumethattheTaylorseriesexpansionofE aroundthis P P var is easy to compute. We get: local minimum doesn’t contain any purely holomorphic nor any purely anti-holomorphic monomial, then Ω is hev¯| Nj=−01(a+j )mjanjj|ev¯i = m! 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