Continuous ground-state degeneracy of classical dipoles on regular lattices
Dominik Schildknecht, Michael Schütt, Laura J. Heyderman, Peter M. Derlet
CContinuous ground-state degeneracy of classical dipoles on regular lattices
Dominik Schildknecht,
1, 2, 3, ∗ Michael Sch¨utt,
1, 4, 5
Laura J. Heyderman,
2, 3 and Peter M. Derlet † Condensed Matter Theory Group, Paul Scherrer Institute, 5232 Villigen PSI, Switzerland Laboratory for Mesoscopic Systems, Department of Materials, ETH Zurich, 8093 Zurich, Switzerland Laboratory for Multiscale Materials Experiments, Paul Scherrer Institute, 5232 Villigen PSI, Switzerland Department of Theoretical Physics, University of Geneva, 1211 Geneva, Switzerland Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland (Dated: July 24, 2019)Dipolar interactions are crucial in the modeling of many complex magnetic systems, such as thepyrochlores and artificial spin systems. Remarkably, many classical dipolar-coupled spin systemsexhibit a continuous ground-state degeneracy, which is unexpected as the Hamiltonian does notpossess a continuous symmetry. In this paper, we explain how such a finite point symmetry leadsto a continuous ground-state degeneracy of specific classical dipolar-coupled systems. This worktherefore provides new insight into the theory of classical dipolar-coupled spin systems and opensthe way to understand more complex dipolar-coupled systems.
I. INTRODUCTION
In the early 20th century, adiabatic demagnetizationassociated with the magnetocaloric effect was exploitedto reach temperatures below 1 K, in particular in theparamagnetic salts . The magnetic order limits thecoldest temperatures achievable with this method ,which called for a better understanding of the orderedstates in such systems. The difficult problem of theground state determination in dipolar-coupled spin sys-tems was, however, not successfully tackled until thepioneering work of Luttinger and Tisza (LT), who in-troduced a theory to determine the ground state oftranslationally invariant systems. While the construc-tion scheme provided by LT can be extended beyonddipolar-coupled systems , its original purpose was to findthe ground-state configuration of arrangements of clas-sical dipoles on lattices such as those in the paramag-netic salts. The ground-state configuration was foundto be strongly dependent on the geometry of the lattice,and it is even sample-shape dependent for ferromagneticalignment of the spins . While the LT construc-tion scheme does not apply to all lattices, it enables thedetermination of the ground-state configuration of com-mon systems such as dipolar-coupled spins placed on thesquare lattice .Remarkably, dipolar-coupled spin systems exhibit acontinuous ground-state degeneracy in many different ge-ometries . The origin of this degeneracy is still notfully understood, although it has become clear that thedegeneracy is not protected by symmetry so that evensmall perturbations, such as temperature or disorder,lift the degeneracy entirely through an order-by-disorder transition .In recent years, the interest in dipolar systems has in-creased due to experimental work on the pyrochlore spinices , leading to theoretical studies on systems withsimilar spin arrangements . Furthermore, the de-sire to better understand the physics governing the spin-ices provided the motivation to explore correlated mag- netic behavior in artificial spin systems with nanomag-netic moments taking on the role of the spins . Suchartificial spin systems are, in contrast to the pyrochlores,neither restricted in lattice geometry nor the single par-ticle magnetic anisotropy. Therefore, even though forinitial investigations the focus was on Ising degrees offreedom , there has since been an increased interestin nanomagnets with continuous degrees of freedom .The theory for the artificial spin systems with con-tinuous degrees of freedom, experimentally addressed inin Refs. , has been discussed in previous works .However, the field lacks a generalization that is free fromassuming a specific lattice. In this paper, we provide amore general approach via a detailed symmetry discus-sion, which gives a framework to determine the ground-state degeneracy for some generic lattices. This leadsto a guide for the determination of whether a particularclassical dipolar system has a continuous ground-statedegeneracy. We provide the essence of this discussion inthe flow diagram shown in Fig. 1.The remainder of this paper is structured as follows:In Section II, the model of classical dipolar spins is in-troduced through the Hamiltonian with an emphasis onsymmetries. After a brief review of the LT method in Sec-tion III A, we extend this method in Section III B usingthe representation theory for the point symmetry groupof the lattice to determine the ground-state degeneracy.We then illustrate this method for several examples inSection IV. Finally, we summarize our results in Sec-tion V, where we give an outlook on how the methodpresented here can be generalized to include the order-by-disorder transitions commonly found in dipolar-coupledsystems. a r X i v : . [ c ond - m a t . s t a t - m ec h ] J u l Classical dipolar-coupled spin systemLuttinger-Tiszamethod [8,9]works?Commen-surateorder?Is V irreducible? Decays toonly 1Dirreps?Has lowestenergyblock d b > d b Continuousdegeneracyof dimen-sion d NocontinuousdegeneracyNo generalstatementpossible noyesyes noyes no yesnoyes noFIG. 1. (color online) This flow diagram summarizes the find-ings of this article. Applying this scheme to a generic classicaldipolar-coupled spin system, the LT method is extended withan additional classification based on reduction of the vectorrepresentation V in the point symmetry group P , which de-termines the nature of the ground-state degeneracy. II. MODEL & SYMMETRIES
Here, we introduce the classical model of dipolar-coupled spins with the Hamiltonian H = D (cid:88) i (cid:54) = j | (cid:126)r ij | (cid:104) (cid:126)S i · (cid:126)S j − (cid:16) (cid:126)S i · ˆ r ij (cid:17) (cid:16) (cid:126)S j · ˆ r ij (cid:17)(cid:105) , (1)where D is the dipolar interaction strength, defined for | (cid:126)S i | = 1. The vector (cid:126)r ij is the difference vector betweenthe positions of the sites i and j on a regular lattice, andˆ r ij is the normalization of (cid:126)r ij to unit length.A classical spin (cid:126)S i is typically described by a vector on the unit sphere, i.e., a Heisenberg spin. However, addi-tional anisotropies can lower the effective degree of free-dom of the spins. For example, in artificial spin ice, shapeanisotropy can give rise to Ising-like behavior or XY-like behavior , and the presence of magnetocrystallineanisotropy can result in clock-model-like behavior . Forthe remainder of the article, we will focus on spins withXY or Heisenberg behavior, in order to determine whencontinuous ground-state degeneracies arise.Regardless of whether the spin is Heisenberg or XY,the dipolar Hamiltonian (1) is geometrically frustrated.Namely, the first term (cid:126)S i · (cid:126)S j is minimized for antiparallelspin alignment, whereas the second term − (cid:126)S i · ˆ r ij )( (cid:126)S j · ˆ r ij ) is minimized for parallel alignment if the spins canalign along their bond. As a consequence, in a systemwith dipolar interactions given by Eq. (1), the alignmentof spins follows the “head-to-tail” rule, i.e., spins alignparallel if they can align along their bond, and antipar-allel if the spins are orthogonal to the bond.In dimension, d = 3, the r − dipolar interaction islong-range and, if the local magnetic configuration has anet magnetic moment, as in a ferromagnet, the energydensity will grow with system size. As a result, sample-shape dependent corrections are expected and Weissdomains are formed through the minimization of thestray field energy. This sensitivity to the sample shapeof “ferromagnetic” configurations of dipolar systems is aresult of Griffiths’ theorem . This theorem implies that“ferromagnetic” configurations cannot be single domainin the thermodynamic limit since sample-shape depen-dent corrections in the form of the demagnetization fac-tor arise. However, if the local magnetic configurationdoes not have a net magnetic moment, no demagnetiza-tion factor arises and the magnetic stray fields typicallyself-screen such as in an antiferromagnet or in the dipolarspin ice model . For the remainder of this paper, we ne-glect the influence of sample-shape dependent correctionsand boundary terms as in previous studies . Thus, for d = 3, the generality of our work is restricted to “non-ferromagnetic” systems. However, it is noted that evenlow-dimensional d < , are known to be sensitive to the sampleshape or the truncation of the Hamiltonian .In addition to the long-range nature of the dipolarHamiltonian in Eq. (1), the Hamiltonian also possessesthe symmetry group Z × T × P , which is rather unusualfor spin Hamiltonians. In this group, the time-reversalsymmetry is reflected by Z , the translational invarianceby T , and P corresponds to the point-group symmetryof the lattice. The time-reversal symmetry follows di-rectly from the invariance of Eq. (1) under (cid:126)S i (cid:55)→ − (cid:126)S i .The translational invariance T is explicitly given by themapping ( (cid:126)r i , (cid:126)S i ) T (cid:55)→ ( (cid:126)r i (cid:48) , (cid:126)S i (cid:48) ) = ( (cid:126)r i − (cid:126)t, (cid:126)S i (cid:48) ) . (2)Since only relative coordinates appear in the dipolarHamiltonian in Eq. (1), a shift of the system by a vector (cid:126)t is irrelevant whenever (cid:126)t is a lattice vector. Finally, thepoint symmetry group P is inherited by the underlyinglattice. If we denote the vector representation of P inthe d -dimensional vector space with V , where d is thedimension of a spin, then a vector (cid:126)v ∈ R d transformsunder the action of g ∈ P according to V ( g ) (cid:126)v , such thatthe Hamiltonian (1) stays invariant under( (cid:126)r i , (cid:126)S i ) P (cid:55)→ ( V ( g ) (cid:126)r i , V ( g ) (cid:126)S i ) . (3)Here, V acts on both the lattice and the spin simulta-neously. This simultaneous action of P on both vectors (cid:126)r i and (cid:126)S i is required by the second term in the Hamilto-nian given in Eq. (1), which tightly connects real-spaceand spin-space. We discuss specific examples of completesymmetry groups in Section IV.Formally, the model incorporates two different dimen-sions, the real-space lattice dimension d lattice and thespin-space dimension d spin . The two spaces are coupledas a result of the dipolar interaction described by theHamiltonian given by Eq. (1). Hence, it is useful to in-troduce a working dimension d , which is the dimensionof the space in which both the spins and the lattice canbe embedded. For some simple situations, it can be suf-ficient to work in the smaller of the two spaces. This canbe seen, for example, for in-plane XY spins on the cubiclattice. Here, the XY anisotropy reduces the point sym-metry group of the system to the point symmetry groupof the square lattice. Therefore, the problem of XY spinson a cubic lattice reduces to the problem of XY spins onsquare-lattice layers . III. GROUND STATES
In this section, we use the symmetries of the dipo-lar Hamiltonian to explain the origin of the ground-statedegeneracy. For this purpose, we first summarize theLT method and subsequently extend the LT method byusing the representation theory for the point symmetrygroup to determine the nature of the ground-state degen-eracy. A. Luttinger-Tisza construction
The LT method is based on an ansatz for the magneticunit cell that stays invariant under lattice symmetry op-erations ( T and P ) and subsequently minimizes the dipo-lar energy associated with the magnetic unit cell. Thegeneralized LT method builds on the original method andis based on a Fourier transformation of the interaction,and finding the ordering vector of the ground state inthe Fourier space rather than in real-space . In general,commensurate and incommensurate order can be foundusing the generalized method, although the original LTmethod is only applicable to systems with commensurate (cid:126)S (cid:126)S (cid:126)S (cid:126)S (cid:126) S FIG. 2. (color online) Schematic illustrating the LT method,which is based on an ansatz for the magnetic unit cell (shadedin blue). The magnetic unit cell contains the spins (cid:126)S , . . . , (cid:126)S N (here N = 4), which form the effective spin configuration (cid:126) S .The aim of the LT method is to provide the so-called basicarrays, namely a symmetry-guided basis for (cid:126) S . order, irrespective of the unit cell size. Indeed, the gen-eralized LT method finds commensurate order for manydipolar systems such as those listed in Table I. For thesesystems, the magnetic unit cell is at most double thestructural unit cell.If the LT method can successfully be applied to adipolar-coupled spin system, then minimization of thedipolar energy for a suitable magnetic unit cell leads tothe exact ground-state of the system. When unphysicalsolutions appear, then the LT method fails. We will dis-cuss the issue of unphysical solutions towards the end ofthis section after introducing the use of the LT methodfor finding the ground-state of dipolar-coupled spin sys-tems.For a general dipolar-coupled spin system, one startsby making an ansatz for the magnetic unit cell that re-spects the point symmetry group of the lattice. Subse-quently, the N spins in the magnetic unit cell are col-lected into one vector (cid:126) S = ( (cid:126)S , (cid:126)S , . . . , (cid:126)S N ) as illustratedin Fig. 2. Since the Hamiltonian in Eq. (1) is quadratic,the effective Hamiltonian for the magnetic unit cell canbe written in terms of (cid:126) S as H = − (cid:126) S † H (cid:126) S , where H (cid:126) S is theinduced dipolar field of the configuration (cid:126) S . This finite-dimensional diagonalization problem is further simplifiedby taking into account the translational invariance T ofthe Hamiltonian: Using the representation theory for thetranslational invariance in the magnetic unit cell, one canobtain a symmetry-guided basis for (cid:126) S , the so-called basicarrays. This basis is constructed using the irreduciblerepresentations of T in the magnetic unit cell, which cor-respond to the discrete Fourier states. Therefore, typi-cal basic arrays are, for example, the ferromagnetic con-figuration ( (cid:126) S ferro ,x = (ˆ e x , ˆ e x , . . . )) or the antiferromag-netic configuration ( (cid:126) S afm ,x = (ˆ e x , − ˆ e x , . . . )) . Thesesymmetry-guided configurations are mutually orthogo- TABLE I. Overview of previous theoretical treatments ofdipolar-coupled coupled spins placed at the sites of variouslattices and whether the ground state can be determined bythe LT method. Lattice LT?chain lattice Yesrectangular lattice Yessquare lattice
Yeshoneycomb lattice Yeskagome lattice
Nocubic lattice
Yes“fcc-kagome” lattice No nal by construction and, because of the translational in-variance, the different sectors such as the ferromagneticsector (ferro) or the antiferromagnetic sector (afm) arenot mixed. This leads to a further simplification of theground state, since H = ⊕ i H i , i.e., H is block-diagonal.Each of the blocks H i describes the coupling betweenthe basic arrays with one type of ordering. For exam-ple, one block describes the coupling between the ferro-magnetic configurations (cid:126) S ferro ,x , (cid:126) S ferro ,y , . . . and anotherdescribes the coupling between antiferromagnetic config-urations (cid:126) S afm ,x , (cid:126) S afm ,y , . . . . Hence, each of the blocks is d -dimensional. Therefore, with the LT method, we find H = H ferro . . . H afm . . . ... ... . . . , (4)which significantly simplifies the problem since only asmall number of explicit lattice summations have to becarried out. Here, it should be noted that, if the magneticunit cell size is increased, then H becomes larger, but thismethod can still be applied.Finally, the LT method only guarantees the “weak con-dition” (cid:126) S = (cid:80) Ni (cid:126)S i = N , and can therefore give unphys-ical solutions where the “strong condition” | (cid:126)S i | = 1 is vi-olated. If an unphysical lowest-energy configuration (cid:126) S isidentified by this method, then the method fails to pro-vide the ground state. For such systems, one can eitherintroduce Lagrange-multipliers or resort to numericalmethods . While Lagrange-multipliers render theproblem non-linear, using numerical methods one typi-cally finds non-orthogonal states as ground-state config-urations. When the method fails, it is not clear if the sys-tem possesses a continuous degeneracy or a discrete de-generacy . Nevertheless, as seen from Table I, the LTmethod works for many important systems, and we showin the next section that, for these systems, P uniquelydefines the type and dimension of the degeneracy. B. Continuous ground-state degeneracy
The point symmetry group P determines the type ofdegeneracy in the following way: Since P is a symme- try of the Hamiltonian, as described by Eq. (3), it istherefore also a symmetry of the effective interactionmatrix H . Hence, symmetry-group operations have tocommute with H , formally expressed as [ R ( g ) , H ] = R ( g ) H − H R ( g ) = 0 for all point symmetry group el-ements g ∈ P , where R is a representation of P . Therepresentation R can be found considering that eachblock matrix H i ∈ {H ferro , H afm , . . . } has dimension d . Indeed, given one spin, for example (cid:126)S , all otherspins in the magnetic unit cell are defined by the index i ∈ { ferro , afm , . . . } . Therefore the representation of P acting on the subspace for H i is V , the vector representa-tion of P . Hence, the representation for the entire matrix H is given by R = ⊕ N V .The symmetry condition implies that, for one blockmatrix, the reduced symmetry condition is [ V ( g ) , H i ] = 0for all g ∈ P . For the case where V is irreducible,the first lemma of Schur implies that H i = h i sothat there are d mutually orthogonal configurations (cid:126) S , . . . , (cid:126) S d , all having the same energy. Hence, any su-perposition (cid:126) S (cid:126)α = (cid:80) di =1 α i (cid:126) S i , with the normalization con-straint (cid:80) di =1 | α i | = 1, yields the same energy as the ba-sis states since the Hamiltonian from Eq. (1) is quadraticin (cid:126)S i . The normalization constraint itself is the equationof a ( d − d − V is reducible, the block matrices H i decompose into smaller block matrices. The explicitsummation over the lattice identifies the smaller blockmatrix with dimension d b that is lowest in energy. Thenthe ground-state manifold is described by the reduced( d b − d b = 1, the degeneracy isdescribed by the 0-sphere, which is equivalent to Z andtherefore only a discrete degeneracy is recovered and nota continuous degeneracy that is found for systems where d b > IV. EXAMPLES
We now illustrate the concepts presented in Sec-tion III with some examples. Specifically, we considerthe dipolar-coupled XY spins on the square lattice inSection IV A, Heisenberg spins on the (tetragonally dis-torted) cubic lattice in Section IV B, and XY spins onthe triangular lattice in Section IV C.
A. XY spins on the square lattice
Here we determine the ground-state of dipolar-coupledXY spins on the square lattice, as this example hasalready been well studied . Here, it isexpected that the ground state exhibits a continuousdegeneracy equivalent to the 1-sphere, independent ofwhether a truncation is applied to the Hamiltonian ornot . The symmetry group of this system is given by Z × T sq × C v , where Z is the time-reversal symmetryand C v is the point symmetry group of the square lat-tice. The translational invariance T sq can be parameter-ized via vectors (cid:126)t = x ˆ e x + y ˆ e y , with x, y ∈ Z and ˆ e x , ˆ e y being the unit vectors along the x -axis and the y -axis,respectively. Therefore, T sq is isomorphic to Z × Z .In the next step, the LT method is applied to a two-by-two magnetic unit cell, so that (cid:126) S = ( (cid:126)S , (cid:126)S , (cid:126)S , (cid:126)S ). Since C is a symmetry of the system, it is sufficient to onlyconsider basic arrays with spins parallel or antiparallel toˆ e y . The LT method then suggests a suitable basis basedon the translational invariance T sq , which is, however,broken by the two-by-two magnetic unit cell. Hence, thebasic arrays correspond to (discrete) Fourier componentsthat arise due to the reduced translational invariance.Since the translational invariance is reduced by a factorof two in every direction, the basic arrays are formed bythe square root of unity in every direction. The resultingbasic arrays are depicted in Fig. 3, with the Fourier vectorthat characterizes the elements given below each figure.From explicit calculation, it can be observed that theconfiguration depicted in Fig. 3c is the basic array withthe lowest energy .Finally, we need to validate if V is irreducible (for de-tails of how this is done, see for example Ref. ). Thereduction is shown in Table II, and we indeed observethat V ≡ E is irreducible in C v . Hence, we know thatthe basic arrays corresponding to Fig. 3c, with spins ei-ther aligned along ˆ e x or aligned along ˆ e y , have the sameenergy. Hence, we have found a continuous ground-statedegeneracy described by the 1-sphere, which is depictedin Fig. 4, in agreement with previous studies . B. Heisenberg spins on the (distorted) cubic lattice
To provide a higher dimensional example, we con-sider dipolar-coupled Heisenberg spins on the cubic lat- (a) (0 ,
0) (b) (0 , π )(c) ( π,
0) (d) ( π, π ) FIG. 3. The four basic arrays for the two-by-two magneticunit cell on the square lattice with moments aligned alongthe y -axis. The Fourier vector that generates the basic ar-ray is indicated below each figure. The lattice summationsassociated with the dipolar Hamiltonian reveal that the con-figuration shown in (c) has the lowest energy.TABLE II. Character table for the point symmetry group ofthe square lattice C v and the reduction of V in this group. C v C C σ h σ d A A − − B − − B − − E − V − ≡ E tice with lattice constants a = b = c . Here, a continuousground-state degeneracy described by the 2-sphere is ex-pected independent of whether a truncation is applied tothe Hamiltonian or not . The symmetry of thissystem is given by Z ×T cu × O h where Z is time reversal, O h is the point symmetry group of the cubic lattice and FIG. 4. (color online) The magnetic unit cell for the groundstate of dipolar-coupled XY spins on the square lattice for ageneral θ . FIG. 5. (color online) Ground-state configuration for dipolar-coupled Heisenberg spins on the cubic lattice according toRef. . The shaded area indicates the magnetic unit cell.TABLE III. Character table for point group O h and the re-duction of V in this group. O h C C C C i S S σ h σ d A g A g − − − − E g − − T g − − − − T g − − − − A u − − − − − A u − − − − − E u − − − T u − − − − T u − − − − V − − − − ≡ T u T cu ∼ = Z × Z × Z parametrizes the translational invariancewith vectors (cid:126)t = x ˆ e x + y ˆ e y + z ˆ e z . Subsequently, the LTmethod requires the evaluation of 2 = 8 lattice sum-mations analogous to the ones for the square lattice .From this calculation, Luttinger and Tisza found thatthe striped configuration depicted in Fig. 5 has the low-est energy of all of the basic arrays. Finally, we reducethe vector representation V in the group O h in Table III,which results in V ≡ T u , so that the vector representa-tion is once more irreducible. This yields a continuousground-state degeneracy corresponding to the 2-sphere inaccordance with previous studies . In Ref. a graphicalrepresentation of this ground-state manifold is providedin their Fig. 1.If the cubic lattice has a tetragonal distortion, whereone lattice constant (e.g. c ) is different from the othertwo ( a = b ), then the three-dimensional block matrixdescribing the ground state of the undistorted cubic lat-tice H striped reduces to two block matrices of dimensions1 and 2, respectively. To determine which block ma-trix is lower in energy, one can consider the two cases c < a = b and c > a = b . If c > a , the system be- FIG. 6. (color online) Ground-state configuration for dipolar-coupled XY spins on the triangular lattice according to Ref. .The shaded area indicates the magnetic unit cell, which is alsothe structural unit cell due to the ferromagnetic ground-stateconfiguration. haves as weakly interacting layers, that follow the sym-metry constraints given by a square lattice. As a con-sequence, the two-dimensional representation is lower inenergy, which yields a continuously degenerate groundstate whose manifold resembles the unit circle. This isanalogous to the manifold found for XY spins on a squarelattice. If c < a , the system consists of chains of spinswith weak interaction between the chains, whose low-energy sector is described by the one-dimensional block.Therefore, no continuous degeneracy emerges. Both ofthese cases are in agreement with previous literature . C. XY spins on the triangular lattice
As a third example, dipolar-coupled XY spins on thetriangular lattice are considered. If no truncation is ap-plied to the Hamiltonian, the system orders ferromagnet-ically . Therefore, the configuration depicted in Fig. 6is one of the ground states of the system. As the localground state is ferromagnetic, a finite sample will breakup into domains in the ground state. As in previous lit-erature , however, the effect of the sample shape isneglected here, and we only consider the local configura-tion consisting of a single domain.Furthermore, in contrast to the previous examplesgiven in this section, only the non-truncated Hamilto-nian can be considered, since otherwise the ground-stateconfiguration is altered . Indeed, when a truncationis applied to the Hamiltonian, the ground state containsa magnetic structural length scale that depends on thetruncation, and it can no longer be derived by the LTmethod . Therefore, the discussion in this section is re-stricted to the case where no truncation is applied to theHamiltonian.In this case, the ground-state configuration of dipolar-coupled XY spins on the triangular lattice is ferromag-netic . This means that the structural and the magneticunit cell are the same so that the strong and the weakcondition of the LT method are equivalent. Since the LTmethod works in this case and a commensurate ordering TABLE IV. Character table for point group C v and the re-duction of V in this group. C v C C C σ v σ d A A − − B − − − B − − − E − − E − − V − − ≡ E is obtained, the next step according to Fig. 1 is to deter-mine if V is irreducible. Using the character table for thepoint symmetry group of the triangular lattice, C v (seeTable IV), one finds that V ≡ E is irreducible. Hence,a continuous ground-state degeneracy described by a 1-sphere is found, in agreement with previous studies . V. CONCLUDING REMARKS
In this work, the origin of the continuous ground-state degeneracy in classical dipolar-coupled systems wastraced back to general properties of the underlying lat-tice. Using the representation theory for the pointsymmetry group, a generic rule for the degeneracy ofLuttinger-Tisza ground states was determined. In doingso, previously known results could be recovered.In particular, we showed that the ground-state degener-acy of dipolar-coupled LT systems crucially depends onthe vector representation V of the point symmetry group.If the representation V is irreducible, as for the examplesgiven in Section IV, then a continuous ground-state man-ifold is found. In contrast, if V is reducible, a reduceddimension of the degenerate manifold or the absence ofa continuous degeneracy altogether is expected.As the degeneracy only arises in the ground state and is not protected by a symmetry in the Hamiltonian, it isnot expected to persist after introducing excitations. Weinstead expect, in analogy to Ref. , that the inclusionof positional disorder or thermal fluctuations restores thefinite symmetry of the Hamiltonian through an order-by-disorder transition . Similarly, higher-ordermultipoles, especially relevant for artificial spin ice sys-tems, have been found to affect the ground-state degen-eracy . However, to answer the question of how exci-tations and disorder affect the ground-state degeneracy,fluctuations on top of a generic system would need to beconsidered. While this is beyond the scope of the presentwork, a symmetry-guided discussion of the fluctuationsseems feasible.Finally, we have only considered systems where theLT method is applicable. While this method is valid formany systems , there exist a number of interestingsystems where the LT method does not apply. One ex-ample is the system of dipolar-coupled Heisenberg spinson the “fcc-kagome” lattice, where a continuous ground-state degeneracy is found . The ground-state manifoldfound in Ref. is not equivalent to a sphere and thebasis states are not orthogonal, which is why the LTmethod does not apply. While such phenomena lie out-side the work presented here, it seems feasible to performa symmetry-guided discussion of non-LT systems, and wehope that this work serves as an inspiration to extend thesymmetry discussion to all such systems. ACKNOWLEDGMENTS
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