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Theory of ground states for classical Heisenberg spin systems I Heinz-Ju¨rgen Schmidt1 ∗ 1Universität Osnabru¨ck, Fachbereich Physik, Barbarastr. 7, D - 49069 Osnabru¨ck, Germany We formulate part I of a rigorous theory of ground states for classical, finite, Heisenberg spin systems. The main result is that all ground states can be constructed from the eigenvectors of a real,symmetricmatrix withentriescomprisingthecouplingconstantsofthespin systemaswell as certain Lagrange parameters. The eigenvectors correspond to the uniquemaximum of theminimal eigenvalueconsideredasafunctionoftheLagrangeparameters. However,therearerarecaseswhere all ground states obtained in this way have unphysical dimensions M > 3 and the theory would 7 1 havetobeextended. Furtherresults concernthedegree of additionaldegeneracy,additional tothe 0 trivial degeneracy of ground states due to rotations or reflections. The theory is illustrated by a 2 coupleof elementary examples. b e I. INTRODUCTION parts of which the present paper will be the first one. F The first four sections after this Introduction con- 5 In quantum mechanics the ground states of a system tain general results illustrated by elementary examples ] are the eigenvectorsof the HamiltonianH corresponding whereas the proofs are given in a separate section VI. r e to the lowest energy eigenvalue. Thus there is a clear Thismakesit,hopefully,possibletoobtainageneralsur- h recipe how to find ground states: Just diagonalize the vey without dwelling upon mathematical details. The ot Hamiltonian. Inpracticethismayturnouttobenumer- mathematics used in the proofs is rather elementary . ically difficult, neverthelessit is a straightforwardproce- and presumably known to all physicists with a moder- t a dure. The analogous problem for a classical Heisenberg ate background in mathematics. One exception might m spin system cannot be solved in an analogous fashion. bethe useofcertainconceptsofconvexanalysisthatare - Although the definition of ground states is clear (states not much common in a physical context (except in the d where the classical Hamiltonian assumes its minimum) foundationsofquantummechanics). Herewe haveto re- n the groundstates canonly be analyticallydeterminedin ferthereadertothe pertinentliterature,e.g.,to[8]. We o special cases. Numerical procedures are available, but will summarize the central results of this paper in a the- c [ theymayconvergeslowly,andprovidenoguaranteethat orem 5, see section VII, that contains also the pertinent theobtainedstaterepresentsaglobal,notonlylocalmin- definitions and can be read independently of the main 2 imum of energy. After all, one is never sure that the text. v numerical procedures will find all ground states, which 9 Themethodweadopttotacklethegroundstateprob- 8 may be crucial for calculations of thermodynamic prop- lem I have dubbed the “Lagrange variety approach”, 4 erties at low temperatures. It may be that there are ad- see subsection IIA. It is based on the observation that 2 ditional ground states that cannot be obtained from the the ground states satisfy the“stationary state equation” 0 knownonesbyrotationsorreflections. Anotherproblem (SSE) involving certain Lagrange parameters due to the . 1 isthe dimensionalityofgroundstates. Under whichcon- constraints of constant spin lengths. The SSE can be 0 ditions there exist 1-dimensional, 2- dimensional or only castintotheformofaneigenvalueequationforsomema- 7 3-dimensional ground states? The latter problem is also trix that we call the“dressed -matrix”. Its entries are 1 connected with“frustration”: Classical spin systems are J the coupling constants between the spins in the Heisen- : v frustrated if they do not possess 1-dimensional ground berg model plus certain Lagrange parameters λ in the i states (but not vice versa, see example 5 in section IV). X diagonal. The set of eigenvalues of the dressed -matrix Existing theories mainly focus on spin lattices, see the depending on λ is called the “Lagrange varietyJ” . In r seminal work of Luttinger and Tisza [1], followed by [2], V a this way we obtain a 1 : 1 connection between the solu- [3], [4] and the more recent publications [5], [6] based on tions of the SSE and certain points of , called“elliptic this approach. An alternative approachis [7], but this is V points”. In section III we give a geometrical characteri- mainly focussedon finite systems with large point group zationoftheellipticpointsof thatessentiallysaysthat symmetries and does not cover the general case. V in an infinitesimal neighborhood of these points the La- Hencethereistheneedforageneraltheoryofclassical grangevarietyisgivenbythesurfaceofa“verticaldouble groundstatesthatsettlesthementionedproblems. Iwill cone”, see Figure 3 for an illustration. For the minimal trytooutlinesuchatheoryalthoughacoupleofquestions eigenvalue of the dressed -matrix there exists a unique J will remain open. Since this theory exceeds the format pointof withaverticaldoubleconeandhenceaground V of a single article I have decided to split it into different state living on the corresponding eigenspace, see section IV. In the special case where the minimal eigenvalue has a smooth maximum we obtain a 1-dimensional ground state. However,itmayhappenthatallgroundstatesob- ∗[email protected] tained in this way will be M-dimensional, M > 3, and 2 hence unphysical. In this case one has to look for other operators”. The operatoranalogousto the statisticalop- ellipticpointsof inordertofindphysicalgroundstates. erator is the “Gram matrix”defined in subsection IIB. V We provide an example in section V. Nevertheless, these The Gram set approach is not a substitute for the La- examples are rare in practice and the approach of the grange variety approach but a supplement that deepens present paper seems to be useful. the understandingofthe groundstate problem. Further, This approach also gives interesting results for the it will be useful to illustrate the complete solutionof the problem of degeneracy, see subsection IIB. All ground groundstate problemfor the generalclassicalspintrian- states of Heisenberg systems are trivially degenerate in gle. the sense that arbitrary rotations/reflections are always In the presentpaper we have mainly providedelemen- possible. But sometimes“additional degeneracy”occurs, taryexampleswherethesetofgroundstateswasalready for example, if independent rotations of a subset of spin knowninordertoillustrateourtheory,example5insec- vectors are possible. The theory tells us how the degree tion IV being an exception. What is still missing are ofadditionaldegeneracycanbereadofffromanyground more applications to systems where the complete set of state of maximal dimension. One simple example is the ground states is either completely unknown or only par- anti-ferromagnetic bow tie that can be viewed as result- tially known. A possible candidate for the latter case is ing fromthe“fusion”oftwotrianglesandshowsanaddi- the anti-ferromagnetic cuboctahedron, where additional tional degeneracyof degree one,see subsectionIIB. The degeneracydue to independent rotationshas been found general process of fusion is sketched in subsection IIC. [9]. If we also admit unphysicalgroundstates with M >3 it can be shown that no further degeneracy occurs, i. e. , Another question is to what extent the present re- allgroundstateshavethesameLagrangeparameters,see sults could be generalized to spin systems where the subsection IV. Hamiltonian is no longer of Heisenberg type, but, say, This has important practical consequences. Assume still bilinear in the spin components. This would in- that we consider a certain Heisenberg spin system and clude dipole-dipole interactions as well as corrections of look for ground states. As mentioned above there ex- Dzyaloshinsky-Moriya type. Whereas the first steps fol- istsimple codestonumericallydeterminecertainground lowingthe SSEcanbe accordinglygeneralized,see,e.g., states. For example, we can start with a random 3- [10], I am pessimistic about the possibility to generalize dimensional spin configuration and fix a certain spin central parts of the theory to non-Heisenberg systems. number µ = 1,...,N. Then we choose the spin vec- But there is a specialcase of non-HeisenbergHamilto- tor s such that the energy of the interactionof the spin µ niansthatisparticularlyinterestingforphysicalapplica- µ with all other spins is minimized. We consider the tions, namely a Heisenberg Hamiltonian plus a Zeeman next spin µ+1 and repeat the process until the change termdescribingtheinteractionofthespinswithanouter of the total energy is smaller than a given ǫ > 0. If the magnetic field B. This case in some sense can be traced repetitionofthewholeprocedurewithdifferentinialcon- back to the pure Heisenberg case. First, one observes ditions gives reproducible results we can be rather sure thatinthepresenceofamagneticfieldthegroundstates that we havefound some groundstates. But how to find will be among the“relative ground states”, i. e. , ground allgroundstates? Applicationofthe presenttheorysug- statesforagiventotalspinS. Thelattersatisfyananal- geststocalculatetheLagrangeparametersofthenumer- ogousSSEwithanadditionalLagrangeparameter,say,α icallydeterminedgroundstateandtoexaminetheeigen- due tothe additionalconstraintS2 = const.. The terms value and the corresponding eigenspace of the dressed involvingαcanbedistributedtothedressed -matrixin -matrix. If the eigenvalue is minimal (and this will be J J such a way that one obtains an SSE of the pure Heisen- the typical case), we have no problems with unphysical berg form and the present theory can be applied. The groundstates: Wecaneasilysolvethe“additionaldegen- onlydifferenceisthattheentriesofthedressed -matrix eracyequation”(ADE),seesubsectionIIB,andthusfind J proportional to α have a different physical meaning and all additional degeneracies, provided the degree of addi- α is not a given constant but may vary over some do- tional degeneracyis not too large. Some of these ground main. At any case, the extension of the present theory states may be unphysical, but all physical ones are in- to the case of B =0 seems to be highly desirable. cluded. Thenwearedone: Thetheorytellsusthatthere 6 are no further ground states. Another realm of possible future work would be the After havingoutlined the contentof the presentpaper specialization of the present theory to cases with a large withtheoptimisticnumberIinitstitleitwillbeinorder symmetrygroupandthecomparisontotheknownresults tosayafewwordsaboutpossibleextensionsthatmaybe of[1]–[6]or[7]. Afewremarksaboutthesymmetriccase coveredbyforthcomingpapers. BesidestheLagrangeva- already can be found in section IV as well as a Theorem riety approach there exists another approach that I will 4 about the existence of symmetric ground states. Since call“Gram set approach”. Its main idea is to linearize we have assumed finite spin systems from the outset an the energy function that is bilinear in the spin vectors, application to infinite spin lattices could probably only analogous to the linearization of the expectation value be made in the sense of approximating the lattice by a in quantum mechanics by the introductionof“statistical finite system with periodic boundary conditions. 3 II. GENERAL DEFINITIONS AND RESULTS The Hamiltonian (4) does not uniquely determine the symmetric matrix J: Let λ , µ = 1,...,N be arbitrary µ A. The Lagrange variety approach real numbers subject to the constraint N Theclassicalphasespace forthesystemsofN spins λ =0, (5) P µ under consideration consists of all configurations of spin µ=1 X vectors (or“states”) and define a new matrix (λ) with entries J s , µ=1,...,N , (1) µ J(λ) J +δ λ , (6) µν µν µν µ ≡ subject to the constraints then N s s =1, µ=1,...,N . (2) µ· µ H˜(s) J(λ)µνsµ sν (7) ≡ · From a physical point of view one is only interested in µX,ν=1 thosecaseswherethevectorsoccurringin(1)and(2)are N N at most 3-dimensional. However, this restriction turns = Jµνsµ sν + λµsµ sµ (8) · · out to be mathematically unnatural and hence will be µ,ν=1 µ=1 X X cancelled. Thus the vectors occurring in (1) and (2) are = H(s), (9) assumed to be elements of RM where M is some natural due to (2) and (5). The transformation J J(λ) numberthatmayassumedifferentvaluesthroughoutthe µν → µν accordingto(6)hasbeencalleda“gaugetransformation” paper. The corresponding phase space is the N-fold M P in [7] accordingto the close analogywith other branches product of unit spheres of physics where this notion is common. In most prob- SM 1 ... SM 1 (3) lems the simplestgaugewouldbe the“zerogauge”,i.e., M − − P ≡ × × setting λ =0 for µ=1,...,N. However,in the present µ andhence compact. We will use the naturalembeddings context it is crucial not to remove the gauge freedom by vPeMcto⊂rsPfMo′rfmoraMthe<maMtic′.alErxetaesnodnisndgotehsendoimt menesaionnthoaftspwine eaxcpelrictaitilnycshtroeiscsetohfetdheepλeµndbeuntcetoofretthaeincoiut.plWinegwmialltrhiexnocne ignore the fact that in physical applications this dimen- the undetermined λµ by using the notation (λ). (λ) J J sion must not exceed 3. We have still the possibility to will be called the“dressed -matrix”and its entries will J retrievethephysicalspinconfigurationsfromalargerset be, as above, denoted by J(λ)µν. The rationale is that of mathematical configurations by looking at their di- we want to trace back the properties of ground states to mensions. The exactdefinitionof“dimension”is givenin the eigenvalues and eigenvectors of (λ) and these in a J the following paragraph. non-trivial way depend on λ. The“undressed”matrix J Let s denote the N M-matrix with entries s , µ= withoutλwillalwaysdenoteasymmetricN N-matrix 1,...,N, i=1,...,M×. According to the differenµt,iuse of in the zero gauge. × Greek and Latin indices it will be always clear that s Let Λ denote the N 1-dimensional subspace of RN de- µ − denotes the µ-th row of s and s its i-th column. The fined by i “dimension”dim(s) of s is simply defined as its matrix N rank. Hence it is equal to the maximal number of lin- Λ λ RN λ =0 (10) µ emaarxlyiminadlenpuemndbeenrtorfolwinsesaµrlyofinsd,eopre,nedqeunitvacloelnutmlyn,stosithoef ≡( ∈ (cid:12)(cid:12)(cid:12)µX=1 ) sth.eIpthfyoslilcoawlsptahralatnacelwwayeswdililmsp(se)ak≤ofN“c.olAlincecaorrdsitnagtetso” Anesnctsooλrid,iina=te1s,.in..Λ,Nwseinwciellt(cid:12)(cid:12)uhseeNth-tehficrostmNpon−en1tccoamnpboe- or ”Ising states” in case of dim(s) = 1, and “co-planar expressed by the others via λN =− Ni=−11λi. states”in case of dim(s) = 2. The case of dim(s) = 3 P has not yet received a particular denomination and will A“ground state”of the spin system is defined as any be referred to as s being a“3-dimensionalstate”. configurations N whereH(s)assumesitsglobalmin- ∈P The Heisenberg Hamiltonian H is a smooth function imum Emin. We will also say that s is the ground state defined on PM of the form of the Hamiltonian H or of J. The restriction to PN does not exclude any ground state of whatever dimen- N sion since always dim (s) N. The existence of ground H(s)= Jµνsµ sν , (4) states is guaranteedsince≤the continuous function H de- · µ,ν=1 fined on the compact set assumes its minimum at X N P some points s of . Let us define the set of ground N where the coupling coefficients J are consideredas the P µν states by entriesofareal,symmetricN N matrixJwithvanish- ing diagonal. × ˘ s H(s)=E . (11) N min P ≡{ ∈P | } 4 In general there exist a lot of ground states. For ex- and the deviations from the mean value ample, a global rotation or reflection of a ground state is again a ground state due to the invariance of the λ κ κ¯, µ=1,...,N . (16) µ µ Hamiltonian (4) under rotations/reflections. The group ≡ − ofrotations/reflectionsR of M definedby the property We denote by Λ Λ the set of vectors λ with com- R⊤ = R−1 is usually denoteRd by O(M); hence we will ponents (16) resu0lt⊂ing from (13) in the case of a ground also speak of O(M)-equivalence of ground states. Later state s ˘. Later we will prove that Λ consists of a 0 we will present examples that show additionaldegenera- singlepo∈inPtΛ = λˆ butatthemomentwewillnotuse cies of the ground states apart from the “trivial”rota- 0 { } this fact. λ Λ will be called a“ground state gauge”. tional/reflectional degeneracy. If there is no additional ∈ 0 It can be used for a gaugetransformationJ J(λ) degeneracy, i. e. , if any two ground states are O(M)- µν → µν whichrenders(14)intheformofaneigenvalueequation: equivalent we will also say that the ground state is“essentially unique”. Let M˘ be the maximal dimension of N ground states, i. e. , J(λ) s = κ¯s , (17) µν ν µ − M˘ Max dim (s) s ˘ . (12) νX=1 ≡ ∈P n (cid:12) o or, in matrix form, It can be shown that for any (cid:12)(cid:12)ground state s ˘ ∈ P there exists an R O(N) such that Rs ∈ ∈ PM˘ ⊂ PN (λ)s= κ¯s. (18) w. r. t. the above-mentioned natural embedding of J − phase spaces. Hence we can always assume that ground states s are N M˘-matrices. Nevertheless, it will be This means that each column si, i = 1,...,M of the often more conv×enient not to fix M = M˘ but to use an matrix s will be an eigenvector of the matrix J(λ), λ ∈ Λ corresponding to the eigenvalue κ¯. undetermined integer M in the pertinent definitions. 0 − Since this situation will occur very often through- out the paper we will use the abbreviating phrase “ϕ It is well-known that a smooth function of M N × is an eigenvector of (A,a)” iff the eigenvalue equation variables has a vanishing gradient at those points where Aϕ = aϕ holds for ϕ = 0. We note that a global rota- itassumesits(localorglobal)minimum. Ifthedefinition domain of the function is constrained, as in our case, tion/reflection s 7→ s′6 where s′µi = Mj=1Rijsµj, R ∈ its gradient no longer vanishes at the minima but will O(M), does not affect the eigenvalue κ¯ and the P − only be perpendicular to the“constraint manifold”. For ground state gauge λ Λ0. In this sense, the ro- ∈ arigorousaccountsee,e.g.,[11]. The resultingequation tational/reflectional degeneracy is factored out by the reads, in our case, present approach. TheconnectionbetweentheminimalenergyE and min N the eigenvalue κ¯ is given by Jµνsν = κµsµ, µ=1,...,N . (13) − − ν=1 X N N (9) Heretheκ aretheLagrangeparametersduetothecon- E = J s s = J(λ) s s(19) µ min µν ν µ µν ν µ · · straints (2). This equationis only necessarybut not suf- µ,ν=1 µ,ν=1 X X ficient for s being a groundstate. If it is satisfiedwe call N the corresponding state a ”stationary state”and will re- (1=7) κ¯ s s (=2) Nκ¯ . (20) µ µ ferto (13)asthe“stationarystate equation”(SSE).This − · − µ=1 X wording of course reflects the fact that exactly the stationary states will not move according to the equation It will be instructive to consider the reverse process. of motion for classical spin systems, see, e. g., [7], but Let s , i = 1,...n, be the eigenvectors of (λ) for some i we will not dwell upon this here. All ground states are J λ Λcorrespondingtoann folddegenerateeigenvalue. stationarystates but there are stationarystates that are ∈ − Then the eigenvectorsneed not lead to a spin configura- not ground states. Let us rewrite (13) in the following tion since n s2 may depend on µ. If n s2 is way: i=1 µ,i i=1 µ,i independent of µ and hence can be taken as 1, the spin P P vector will generally be n dimensional. Even if n 3, N − ≤ J s =(κ¯ κ )s κ¯s = λ s κ¯s , (14) we have only obtained a stationary state that need not µν ν µ µ µ µ µ µ − − − − beagroundstate. Thisillustratesthe problemsinherent ν=1 X to a general theory of ground states. wherewehaveintroducedthemeanvalueoftheLagrange We introduce some more notation. Let λ Λ be arbi- parameters trary and s ˘ be any ground state of the s∈pin system. ∈P 1 N Further, let jα(λ) denote the α−th eigenvalue of J(λ) κ¯ κµ , (15) and jmin(λ) its lowest eigenvalue. Application of the ≡ N Rayleigh-Ritz variational principle to the present situa- µ=1 X 5 tion yields From this one concludes the following, see [8], Cor. 10.1.1: N M Emin = Jµνsν,isµ,i (21) Corollary 1 jmin is a continuous function. µ,ν=1i=1 X X Since the set j (λ)λ Λ is bounded from min (=9) N M J(λ)µνsν,isµ,i (22) saubpovej by(λN1)Eλmin{Λ, seeexis(t2s|1.)–I(t∈2c3a),n}ibtes sshuopwrenmtuhmat jˆ ≡ min min µ,ν=1i=1 { | ∈ } X X assumes this supremum at some set J: N M j (λ) s2 (=2)Nj (λ). (23) ≥ min ν,i min Proposition 3 The set J λ Λbjmin(λ) = ˆ is a ≡ { ∈ | } Xν=1Xi=1 non-empty compact, convex subset of Λ. b Westressthat(21)-(23)holdsforeverygaugeλ Λ,not We close this subsection with an elementary example. ∈ onlyforagroundstategauge. Itseemsplausiblethatfor the ground state gauge, i, e. , for λ Λ the inequality 0 ∈ Example 1: The dimer (N =2) (23) can be replaced by an equality. This is indeed the case, see Theorem2, and means that a groundstate can bebuiltfromtheeigenvectorsof( (λ),jmin(λ)), λ Λ0. Inthe antiferromagnetic(AF)casethe matrices and ButitmayhappenthatallgroundJstatesobtainedin∈this (λ) assume the form J wayhaveadimensiongreaterthan3. Wewillpresentan J example in section V. If this is not the case, that is, if 0 1 λ 1 M˘ 3 we define the spin systemto be a“standard”one. J= 1 0 , J(λ)= 1 λ , (25) F≤rom (21)-(23) it follows that 1E is an upper (cid:18) (cid:19) (cid:18) − (cid:19) N min bound of the function j : Λ R. We will show and the characteristic equation of the latter is min belowthatthefunctionj assum−→esitsupperboundat det( (λ) x ) = x2 (1+λ2) = 0. It has the two so- min J − 1 − some λ Λ. lutions x = √1+λ2 and hence j (λ)= √1+λ2, Let p(∈λ,x) = det( (λ) x ) denote the characteris- see Figur±e 1. ± min − J − 1 tic polynomial of (λ). The set J = ( ) (λ,x) Λ R p(λ,x)=0 (24) V V J ≡{ ∈ × | } is a “real algebraic variety”, see, e. g., [12] and will be calledthe“Lagrangevariety”of the classicalspin system underconsiderationsincetheparameters(λ,x)arein1: 1 relation to the Lagrange parameters κ , µ = 1,...,N µ oftheSSE(13). Thegraphofthefunctionj :Λ R min −→ is a subset of the Lagrange variety. The points (λ,x) of ( ) can be divided into two disjoint subsets: (λ,x) V J willbecalled“singular”ifthegradient p(λ,x)vanishes: ∂p(λ,x) = ∂p(λ,x) = 0 for i = 1,...,N∇ 1. Otherwise, ∂x ∂λi − (λ,x) willbe called“regular”. In the neighbourhoodofa regular point ( ) will be a smooth N 1 dimensional manifold embVeddJed into RN and its ta−ngent space at (λ,x)willbeorthogonalto p(λ,x). Notethatthevan- FIG. 1: The Lagrange variety V of the AF dimer consists of ishingof ∂p(λ,x) meansthat∇theeigenvaluexof (λ)isat twodisjointcurves. Theloweroneisthegraphofthefunction ∂x J jmin(λ). It has a smooth maximum at λ = 0 corresponding least doubly degenerate. In this case we are necessarily toa collinear ground state ↑↓. at a singular point of ( ): V J The function j (λ) has a unique maximum at λ = Proposition 1 If p(λ,x) = ∂p(∂λx,x) = 0, then ∂p∂(λλi,x) = 0 of height ˆ = jmminin(0) = 1. At this maximum the 0 for i=1,...,N 1 and hence(λ,x) is a singular point dressed -matrix assumes th−e form of ( ). − J V J 0 1 (0)= , (26) The proofs of this and following propositions and the- J 1 0 (cid:18) (cid:19) orems will be given in a separate section VI. 1 andhastheeigenvectorϕ= correspondingtothe Proposition 2 j : Λ R is a concave function, 1 min −→ (cid:18)− (cid:19) i. e. , j (αλ+(1 α)µ) αj (λ)+(1 α)j (µ) eigenvalue ˆ= j (0) = 1. This yields the collinear min min min min − ≥ − − for all λ,µ Λ and α [0,1]. ground state s =1, s = 1, symbolically s= . 1 2 ∈ ∈ − ↑↓ 6 Intheferromagneticcasej (λ)isunchanged,butat Hence the representation of spin configurations by min its maximum the dressed -matrix assumes the form Gram matrices exactly removes the “trivial” rota- J tional/reflectional degeneracy of possible ground states; 0 1 (0)= − , (27) the set of Gram matrices is in 1:1 correspondence with J 1 0 (cid:18)− (cid:19) the set of O(M)-equivalence classes of states. We note in passingthat the energyH(s) of a spin configurations 1 and has the eigenvector ϕ= corresponding to the may be written in a linearized form by using the Gram 1 (cid:18) (cid:19) matrix as H(s)=Tr (G ). eigenvalue ˆ= jmin(0) = −1. This yields the collinear Next we want to give Ja more precise definition of the ground state s =1, s =1, symbolically s= . 1 2 ↑↑ phrase that a spin configuration s can be built from the vectors of some eigenspace S of (λ) or, equivalently, J that s is“livingonS”. To this end we considera general B. Degeneracy linear subspace S RN and define: ⊂ We will recapitulate and generalize some notions al- Definition 1 1. S is called“M-elliptic” iff there ex- ready introduced in [7]. As in the previous subsection ists an s such that its columns s , i = M i let s denote the N M-matrix with entries s , µ = ∈ P µ,i 1,...,M are elements of S. × 1,...,N, i = 1,...,M. Let s denote the transposed ⊤ matrix. For each s M we define the“Gram matrix” 2. If S is M-elliptic we define ∈ P uGsu≡alsisn⊤newripthroednutcrtiesofGRµMν =. Hsµen·cseν,Gwwheilrleb·edaensoytmesmtehte- PM,S ≡{s∈PM|si ∈S for all i=1,...,M}. ric N N-matrix that is positively semi-definite, G 0, 3. S is called “elliptic” iff it is M-elliptic for some × ≥ and satisfies Gµµ = 1 for all µ = 1,...,N. Moreover, integer M 1. rank(G)=rank(s) M. ≥ ≤ Conversely, if G is a positively semi-definite N N- 4. S is called “completely elliptic” iff there exists an × matrix with rank M N, satisfying G = 1 for all s such that its columns s , i= 1,...,M are µµ M i ≤ ∈ P µ = 1,...,N. Then the spectral representation of G elements of S, and moreover, dim s = dim S = yields M. M G= γ , (28) iPϕi Example 2 i=1 X where the γ > 0 are the non-zero eigenvalues and In order to illustrate the wording of Definition 1 we i Pϕi denote the projectorsonto the correspondingunit eigen- consider a system of N =6 spins with -matrix J vectors ϕ of G, i = 1,...,M. Their matrix entries are i given by 0 1 2 1 1 1 − − 1 0 1 2 1 1 (Pϕi)µν =ϕiµϕiν for µ,ν =1...,N . (29) = 2 −1 −0 1 −1 −1 (31) Then we define N spin vectors sµ RM with compo- J −11 12 11 10 01 −21 nents sµi =√γiϕiµ and conclude ∈ −1 −1 1 1 2 0   − −    M M (29)(28) Itslowesteigenvalueisj = 4withatwo-dimensional sµ sν = sµisνi = γiϕiµϕiν = Gµν . (30) min − · eigenspaceSspannedbythecolumnvectorsofthematrix i=1 i=1 X X Moreover,the s areunitvectorssinces s =G =1 1 0 µ µ µ µµ · for µ=1,...,N. 1 1 The correspondence between spin configurations s  −1 0  ∈ W = − . (32) and Gram matrices G is many-to-one: Let R 1 1 POM(M), then the two configurations s and Rs , µ =∈  0 −1  µ µ   1,...,N willobviouslyyieldthesameGrammatrix. Ac-  0 1  −  tually, this is the only possibility where two configura-   tions have the same G according to the following ThesixrowvectorsofW lieontheellipsex2+y2+xy = 1, see Figure 2. It can be shown that S is also elliptic in Proposition 4 Let s(i) , i=1,2, be two spin con- ∈PM the sense of the Definition 1: Defining figurations satisfying s(µ1) · s(ν1) = s(µ2) · s(ν2) for all µ,ν = 1,...N, then √2+√3 √2 √3 there exists a rotation/reflection R O(M) such that Γ 2 −2 , (33) s(µ2) =Rs(µ1) for all µ=1,...N. ∈ ≡ √2−2√3 √2+2√3    7 we can show that another basis of S is given by the col- column vectors of an N M-matrix W. S being (com- × umn vectors of s=WΓ: pletely)ellipticthenentailsthe conditionthatsomespin configuration (s ) can be obtained by a linear µ µ=1,...,N √2+√3 √2 √3 combination of the W : 2 −2 µi 1 1   −√2 √2 M s= −21 21+√3 14 √2−1 √6 , (34) sµj = WµiΓij, µ=1,...,N, j =1,...,M , (35)  p√2 (cid:0) −√2 (cid:1) Xi=1  √2 √3 √2+√3   −2 2  or, in matrix notation,    1 √2 √6 1 2+√3   4 − −2    s=W Γ. (36) such that the six r(cid:0)ows of s a(cid:1)re unipt vectors. The corresponding Gram matrix is G = ss = ⊤ For general M-dimensional elliptic subspaces spanned WΓΓ⊤W⊤. Then the condition that the sµ are unit by the columns of some matrix W the corresponding vectors can be written as rowvectorswilllie ona centralM-dimensionalellipsoid, in general not unique, that can be transformed into 1=Gµµ = WΓΓ⊤W⊤ ,µ=1,...,N . (37) aunitspherebysomelinearsymmetrictransformationΓ. µµ (cid:16) (cid:17) Withthedefinition∆ ΓΓ⊤ 0thisconditionassumes ≡ ≥ the form M 1= W∆W = W W ∆ , µ=1,...,N , ⊤ µµ µi µj ij i,j=1 (cid:0) (cid:1) X (38) and can be considered as a system of N inhomogeneous linearequationsforthe 1M(M+1)unknownentries∆ 2 ij ofasymmetricM M matrix. Itssolutionsetwillbean affinesubspaceof ×21M(M+1),wherethelatterspacewill R be identified with (M), the space of all real, sym- SM metric M M matrices. The condition ∆ 0 restricts × ≥ the solution set of (38) to a compact convex subset of 21M(M+1) that is, by definition, non-empty for elliptic R subspaces S. We will refer to the system of equations (38) together with the condition that ∆ 0 as the“ad- ≥ ditionaldegeneracyequation”(ADE). Itssetofsolutions ∆ 0 will be denoted by . It can be shown that ADE ≥ S G = W ∆W describes a 1 : 1 correspondence between ⊤ the points of and the Gram matrices of spin con- ADE S figurations living on S. Consideranarbitrarysolution∆ . Then there ADE exists the square root γ such that∈∆S= γ2, γ 0 and FIG.2: Thesixrowvectorsofthematrix(32)thetwocolumn ≥ s Wγ will be a spin configuration living on S. Any vectors of which span an elliptic subspace. All six vectors lie ≡ on the (red) ellipse definedby x2+y2+xy=1. other spin configuration ¯s with the same Gram matrix G = W∆W must be of the form ¯s = sR, with R ⊤ ∈ O(M), see Proposition 4, and hence Returning to the general case we will show that any elliptic subspace S′ RN contains a completely elliptic ¯s=W γR=W √∆R. (39) ⊂ subspaceS S withthesamesetofstateslivingonthe ′ ⊂ two subspaces: The latter equationnicely capturesthe separationofthe degeneracy of ground states into rotational/reflectional Proposition 5 Let S RN be M -elliptic. Then there ′ ⊂ ′ degeneracy represented by R and the additional degen- exists a completely elliptic subspace S S with dim S = ⊂ ′ eracy represented by ∆. This separation anticipates M and PM′,S′ =PM,S. the result that the Lagrange parameters of the ground Accordingtothispropositionwemayconfineourselves state are unique, Λ0 = λˆ . Otherwise we would have { } to the case of a completely elliptic subspace S. We want a third kind of“anomalous”degeneracy. But note that to analyze the set . Let us assume that a basis of the result Λ = λˆ will only be proven in the sense of M,S 0 P { } S is given and the M basis vectors are written as the admitting M-dimensionalgroundstates. Insisting of the 8 condition that M 3 for physical ground states would the Hamiltonian ≤ open the possibility for anomalous degeneracy. H =2(s s +s s +s s ), (41) 1 2 2 3 3 1 · · · Wewillfurtherinvestigatethedegreeofadditionalde- generacy. Accordingtotheassumptionofcompleteellip- andisthesimplestexampleofa“frustrated”spinsystem. ticity there exists some s with dim s=M. Such Thismeansthatitsgroundstatedoesnotminimizeeach M,S ∈P anslivingonacompletelyellipticsubspacewillbecalled term of (41). This ground state is realized by any co- a state of “maximal dimension”. It follows that in the planar spin configuration with a mutual angle of 2π/3 above representation s = W Γ the matrix Γ must have between any two spin vectors. Hence it is essentially the rank M. Let ∆0 = ΓΓ⊤, then also rank ∆0 = M unique. We will use this well-known system to illustrate which implies ∆ > 0. The latter is equivalent to ∆ the considerations of this subsection. 0 0 lying in the interior of the convex set . First we note that (λ) assumes the form ADE S J Now consider the homogeneous linear system of equations corresponding to (38): λ 1 1 1 (λ)= 1 λ 1 , (42) 2 M J  1 1 λ λ  1 2 0= W ∆W = W W ∆ =Tr (P ∆), − − ⊤ µµ µi µj ij µ   i,j=1 which leads to the characteristic equation (cid:0) (cid:1) X (40) forallµ=1,...,N ,wheretherank1matricesPµarede- 0 = det( (λ) x ) (43) fined by (P ) W W , i,j = 1,...,M. The P are J − 1 theprojectoµrsijon≡totµhie 1µ-djimensionalsubspacesspaµnned = 2−λ1λ2(λ1+λ2)−x3+x λ21+λ1λ2+λ22+3 . by the µ-th row Wµ of W multiplied by Wµ 2. (cid:0) (4(cid:1)4) k k Recall that (M) denotes the M(M + 1)/2- dimensionalspacSeMofallreal,symmetricM M-matrices. It follows that jmin(λ) has its maximum ât a singular It will be equipped with the inner produ×ct A B = point of the Lagrange variety corresponding to λ = 0 TcornAveBx c.oneSMof+p(oMsit)iv⊂elySsMem(iM-de)findietneomteastrtihchees.|cloFisuedr-, saenedFtihgeurdeou3b.ly degenerate eigeVnvalue ˆ=jmin(0)=−1, ther, let P be the subspace of (M) spanned by the SM P ,µ=1,...,N, with dimension dim P =p. Then (40) µ saysthat∆islyingintheorthogonalcomplementP of ⊥ P in (M). Since the general solution of (38) can be SM writtenasthesumof∆ andthegeneralsolutionof(40) 0 we have the following result: Proposition 6 With the preceding definitions, the set of solutions ∆ 0 of the ADE is the convex set ≥ = ∆ +P (M) and has the dimension ADE 0 ⊥ + S ∩SM d M(M +1)/2 p=dim P . ≡ (cid:0) −(cid:1) ⊥ AccordingtothisPropositiondwillbecalledthe“degree of additional degeneracy”or simply the “degree”of the matrixW thecolumnsofwhichspananellipticsubspace S. It vanishes, i. e. , ∆ is unique iff the P ,µ=1,...,N µ span the total space (M). p will be called the“co- SM degree”ofW. Wewillalsospeakofthe“degreedofs”and the“co-degreep of s”in the case of a state s of maximal dimension M living on a completely elliptic subspace. It can be shown that the co-degree is never smaller than the dimension: Proposition 7 M p N. ≤ ≤ FIG. 3: The two lowest eigenvalues jmin(λ) and j2(λ) of the We close this subsection with two elementary exam- dressed -matrixfortheAFequilateraltriangle. jmin(λ)has ples. its maxiJmum at the singular point λ = 0 where the La- grange variety V can locally be approximated by a double cone (shown in red color). The coplanar ground state (50) is Example 3: The AF equilateral triangle (N =3) living on thecorresponding eigenspace of ( (0),jmin(0)). J The AF equilateral spin triangle can be described by A basis of the eigenspace of ( (0), 1) is given by the J − 9 two column vectors of 1 1 − − W = 0 1 . (45)   1 0   The solution of the corresponding ADE (38) is unique and given by 1 1 ∆= −2 . (46) 1 1 (cid:18)−2 (cid:19) Its square root √2+√3 1 √2 √6 √∆= 2 4 − (47)  1 √2 √6 (cid:0) √2+√3(cid:1) FIG. 4: The AF bow tie and a co-planar ground state indi- 4 − 2   cated by arrows with a mutual angle of 2π/3 between neigh- (cid:0) (cid:1) leads to boring spins. 1 1 −√2 −√2 s=W √∆= 1 √2 √6 √2+√3  . (48) 5. This yields a one-parameter family of ground states 4 − 2 that are not O(3)-equivalent and hence an example of  (cid:0) √2+√3(cid:1) 1 √2 √6  additional degeneracy of degree 1.  2 4 −    Itremainstoshowhowthesefacts aboutthebowtie’s This is indeed a ground state of ((cid:0)41) albeit(cid:1) in an un- groundstatesarereproducedbythepresenttheory. First usual form. To obtain a more familiar representationwe consider the dressed -matrix of the form multiply (48) with the rotation matrix J λ 1 1 0 0 1 1 1 1 λ 1 0 0 R= −√12 √12 (49) (λ)= 1 12 λ 1 1  . (52) −√2 −√2 ! J 3 0 0 1 λ 1  4  and obtain  0 0 1 1 −λ1−λ2−λ3−λ4    1 0 We have to find a λ Λ such that j (λ) assumes its min ¯s=W √∆R= 1 √3 . (50) maximum. The prese∈nt theory does not provide a silver  −2 − 2  1 √3 bullet to fulfill this task in general and we do not want  −2 2  toanticipatetheresultsofsubsectionIICconcerningthe The preceding example illustrates the construction of groundstate gauge for fused spin systems. One possibil- groundstatesfromanellipticeigenspace,butitdoesnot ity to tackle the problem would be to find any ground show any additional degeneracy since d = 0. Hence we state by whatever means (numerical or analytical) and will provide another example where additional degener- to calculate its Lagrange parameters according to (13). acy occurs. Sometimesitwillbepossibletoestimatetheexactvalues from its numerical approximations. In our case we sim- plytaketheco-planargroundstateindicatedinFigure4 Example 4: The AF bow tie (N =5) and obtain the corresponding λ as The AF“bow tie”consists oftwo corner-sharingtrian- 4 1 λ = , λ =λ =λ =λ = . (53) gles, see Figure 4, and can be described by the Hamilto- 3 5 1 2 4 5 −5 nian This leads to the maximal eigenvalueˆ=j (λ)= 6. min −5 H =2(s s +s s +s s +s s +s s +s s ), (51) It turns out that for these values the eigenspace of 1 2 1 3 2 3 3 4 3 5 4 5 · · · · · · ( (λ),ˆ) has the dimensionM =3. A basis of it is given that can be viewed as the sum of two triangle Hamilto- bJy the column vectors of the following matrix nians H , H of the kind (41) considered in Example 3. 1 2 It is possible to minimize H and H simultaneously, for 1 1 1 1 2 − example by the co-planar ground state indicated in Fig- 0 0 1 ure 4. Moreover,one canrotatethe spins withnumber 1 W = 1 1 0 . (54) − − and 2 about the axis of the central spin with number 3 0 1 0   independently ofthe remainingspinswith number4 and  1 0 0     10 The rank 1 matrices P , µ = 1,...,5 generated by C. Fusion µ the rowsof W spana 5-dimensionalsubspace of (3). SM Hence Proposition 6 yields an additional degeneracy of This subsection contains some results on a generaliza- degree tion of Example 4 in connection with the Lagrangevari- M(M +1) 3 4 etyapproach. Itillustratessomeaspectsofthisapproach d= p= × 5=1. (55) but will not be presupposed in the following sections. 2 − 2 − The bow tie example is aninstanceofthe generalpro- In accordance with this the ADE (38) has a one- cess of“fusing”two spin systems. By this we mean the parameter family ∆(δ) of solutions union of two spin systems that are disjoint except for a 1 1 δ single spin. In the Example 4 we may consider two tri- −2 ∆(δ)= 1 1 1 δ . (56) angles with spin numbers (1,2,3) and (3,4,5) with the −2 2 −  δ 1 δ 1 common spin number 3. The bow tie then results from 2 −   the union (1,2,3,4,5),see Figure 4. Returning to the general case we denote by Σ = 1 (1,...,N ) and Σ = (N ,...,N + N 1) two sets 1 2 1 1 2 − of spin numbers that are disjoint except for the common spin with number N and by Σ = (1,...,N ,N + 1 1 1 1,...,N) their fusion, where N N + N 1. The 1 2 ≡ − corresponding Hamiltonians are N1 H = J(1)s s , (58) 1 µν µ· ν µ,ν=1 X N H = J(2)s s , (59) 2 µν µ· ν µ,Xν=N1 N H = J s s , (60) µν µ ν · FIG. 5: For the AFbow tie the eigenvalues of ∆(δ) are non- µX,ν=1 negative for −1 ≤δ ≤1. where 2 J(1) : 1 µ,ν N , µν 1 ≤ ≤ lowTshbeyeiingsepnevcatliuoens,oafn∆d(cδa)naeraesislhyobwendeinrivFeidguarnea5ly.tIictaflolyl-, Jµν =  Jµ(2ν) : N1 ≤µ,ν ≤N, (61)  0 : otherwise. that ∆(δ) 0 for 1/2 δ 1. For 1/2 < δ < 1, ≥ − ≤ ≤ − ∆(δ) representsaone-parameterfamily of3-dimensional Wewillalsospeakofthe“largespinsystem”,correspond- ground states, whereas at the endpoints of the interval ing to Σ and of the two“subsystems”, corresponding to [ 1/2,1]therankof∆(δ)andhencethedimensionofthe Σ1 and Σ2, without danger of misunderstanding. Let c−orresponding ground states is reduced to 2. This com- s(1), µ = 1,...,N , and s(2), µ = N ,...,N, be states µ 1 µ 1 plieswiththegeometricpictureofadditionaldegeneracy of the two subsystems. A usual, we consider the s(i) as of the bow tie’s ground states sketched above. N M -matrices. LetS(i) betheN (M +M )-matrices i i 1 2 Tofurtherconfirmtheaccordancebetweenthegeomet- obt×ained by copying the s(i) into t×he larger matrix and ric picture and the theory’s results we give the result for paddingtheremainingentriesbyzeroessuchthatallrows the Gram matrix G(δ) = W ∆(δ)W⊤ of the considered of S(1) are orthogonalto all rows of S(2): one-parameter family: 1 1 1 δ 1 δ S(1) s(µ1,)i :1≤µ≤N1 and 1≤i≤M1, −2 −2 2 − µ,i ≡ 0 :otherwise, 1 1 1 1 δ δ (cid:26) G(δ)= −12 1 −21 2 −1 1  . (57) (62) −2 −2 −2 −2  12 −δδ 12 −δδ −−2211 −211 −211 S(µ2,)i ≡ (cid:26)0s(µ2,)i−M1 ::oNt1he≤rwµis≤e.N and M1 <i≤M1+M2,   Recall the G = s s , µ,ν = 1,...,5. One observes (63) µν µ ν · that the mutual scalar products are constant within Then there exists an R O(M +M ) such that the triangles (1,2,3) and (3,4,5) and assume the value 1 2 ∈ cos2π/3 = 1/2 corresponding to the triangle’s ground − RS(2) =S(1) . (64) state consideredin Example 2. Only the scalarproducts N1 N1 between the two groups (1,2) and (4,5) vary with δ as We set it must be if the corresponding spins are independently rotated. S¯(2) RS(2), ν =N ,...,N , (65) ν ≡ ν 1