Classical and Quantum Magnetic Ground States on an Icosahedral Cluster
aa r X i v : . [ c ond - m a t . o t h e r] N ov Classical and Quantum Magnetic Ground States on an Icosahedral Cluster
Shintaro Suzuki and Ryuji Tamura
Department of Material Science and Technology,Tokyo University of Science, Katsushika, Tokyo 125-8585, Japan
Takanori Sugimoto ∗ Department of Applied Physics, Tokyo University of Science, Katsushika, Tokyo 125-8585, Japan (Dated: November 30, 2020)Recent discovery of various magnetism in Tsai-type quasicrystal approximants, in whose alloysrare-earth ions located on icosahedral apices are coupled with each other via the Ruderman–Kittel–Kasuya–Yosida interaction, opens an avenue to find novel magnetism originating from the icosa-hedral symmetry. Here we investigate classical and quantum magnetic states on an icosahedralcluster within the Heisenberg interactions of all bonds. Simulated annealing and numerical diago-nalization are performed to obtain the classical and quantum ground states. We obtain qualitativecorrespondence of classical and quantum phase diagrams. Our study gives a good starting point tounderstand the various magnetism in not only quasicrystal approximants but also quasicrystals.
PACS numbers: Valid PACS appear here
I. INTRODUCTION
Long-range magnetic orders in quasi-periodic latticehave been fascinating and challenging targets since thediscovery of quasicrystals [1]. First investigation of mag-netism in quasicrystals was performed in Al-Mn basedalloy in 1986 [2–4]. Next, Bergman-type quasicrystalswere examined thanks to the discovery of Zn-Mg-RE qua-sicrystals [5, 6] (RE = rare earth). However, no magneticlong-range ordering has been observed in these qusicrys-tals so far [7, 8]. On the other hand, another quasicrys-tal with containing rare-earth elements was discoveredby A. P. Tsai in 2000 [9–12]. To attain the targets, atpresent, Tsai-type quasicrystals are investigated [13], dueto a key observation of antiferromagnetism in an approxi-mant Cd Tb [14–16], which have the same local structureas the quasicrystals but have a periodicity.The Tsai-type quasicrystal approximants in commonwith the quasicrystals, consist of rhombic triacontahedralclusters. The cluster includes a concentric tetrahedron, adodecahedron, an icosahedron, and an icosidodecahedronfrom center out, whose apices constituent ions are locatedon [17]. Among polyhedra, the rare-earth ions placed onthe icosahedron only contributes magnetism helped byso-called the Ruderman–Kittel–Kasuya–Yosida (RKKY)interaction [18–20], resulting in novel magnetic orders,e.g., multifarious magnetism discovered in Tsai-type 1/1approximants [21–23]. Interestingly, the magnetism iscontrolled by constitutional ratio of ions in ternary al-loys of the approximants via electron density, becausethe RKKY interaction depends on the Fermi wavenum-ber which is a function of the electron density in Fermigas approximation. Actually, the Curie–Weiss tempera-ture observed in the 1/1 approximants shows an oscilla- ∗ [email protected] tion as a function of estimated electron density indicatingthe RKKY interaction [21, 22].Surprisingly, the latest experimental study on a Tsai-type 2/1 approximants has reported almost the same be-haviors of magnetism as the 1/1 approximants despitedifference of crystal structure between 1/1 and 2/1 ap-proximants [24, 25]. This result suggests importance ofthe common local structure, i.e., the rhombic triaconta-hedral cluster including the magnetic icosahedron. Thenumerical calculations of the magnetic ground state andphysical properties in a single icosahedral cluster bothin the classical and quantum Heisenberg model are al-ready performed with nearest-neighbor exchange inter-action [26–28]. However, these studies do not considerthe interaction between the 2nd and 3rd neighbor spins.Therefore, we investigate magnetism of an isolated icosa-hedron within the 2nd and 3rd neighbor interactions.Especially, we focus our examination on the magneticground states to understand low-temperature physics inthe Tsai-type approximants. Since magnitude of mag-netic moment depends on rare-earth ions, we considerboth quantum and classical spins corresponding to smalland large magnetic moments, respectively. II. MODEL AND METHOD
In this paper, we examine magnetic ground states inthe following model Hamiltonian, H = X n =1 J n X h i,j i n S i · S j (1)where J n and h i, j i n ( n = 1 , ,
3) represent the ex-change energy and n -th neighbor bonds, respectively.For simplicity, instead of J n , we use other angle pa-rameters θ J and φ J to control the exchange energies J n via ( J , J , J ) = J (sin θ J cos φ J , sin θ J sin φ J , cos θ J ) FIG. 1. (a) Schematic icosahedral spin cluster and its connec-tivity of interactions. Balls (circles) denote apices assigningspins. The red, green, and blue bonds correspond to 1st, 2nd,and 3rd neighbor interactions. An ellipse represents a pair ofspins located on opposite apices of icosahedron. (b), (c) Con-nectivity of 1st or 2nd neighbor interaction. These graphs areequivalent. with the energy unit J = p J + J + J = 1. Thespin degree of freedom S i on i -th site is regarded as aunit vector in the classical model and a spin-1 / K , , , , , . Figure1(b)-(c) shows connectivity of the 1st and 2nd neighborinteractions. Since these graphs are the same, there issymmetry with respect to permutation of the site in-dexes corresponding to Fig. 1(b)-(c); i.e., backgroundphysics of two different points in the parameter space( θ J , φ J ) = ( θ J , π/ − δ ) and ( θ J , π/ δ ) are essentiallyequivalent, while the spin configurations are different ata glance. Such the correspondence between two differ-ent models is often called duality. Hence, we call thecorrespondence J - J duality.To obtain the ground state, we numerically apply sim-ulated annealing method to the classical model and exactdiagonalization method to the quantum model. The sim-ulated annealing is a Monte-Carlo method where vectorspins are updated one by one with a certain probabil-ity based on the statistical mechanics at each tempera- ture. The temperature is gradually lowered to zero likethe annealing process in heat treatment of real materi-als. On the other hand, the exact diagonalization is gen-uinely a quantum method at zero temperature to takequantum fluctuations into account. In this method, amatrix form of the Hamiltonian represented in the ba-sis of spin wavefunctions is exactly diagonalized to ob-tain the eigenstate of the minimal energy. We have con-firmed accordance of ground-state energies obtained bythe numerical methods and analytical energies at exactlysolvable points in the parameter space, e.g., ( θ J , φ J ) =(0 , , ( π, , (tan − √ , π/ III. CLASSICAL MAGNETIC STATESA. Simmulated annealing
In this section, we first show numerical results in theclassical model with | S i | = 1. In this case, the Hamilto-nian can be described as H = X n =1 J n N n h cos α ij i h i,j i n (2)where N n and α ij denote the number of n -th neighborbonds and an angle of two spin vectors S i and S j , respec-tively. The bracket h cos α ij i h i,j i n represents mean valueof inner product S i · S j connected with the n -th neighborbonds, N − n P h i,j i n cos α ij .Figure 2(a)-(c) shows the h cos α ij i h i,j i n in the param-eter space ( θ J , φ J ) obtained by the simulated annealing.The mean values h cos α ij i h i,j i and h cos α ij i h i,j i takeonly four values − . , − . , .
44 and 1 .
00 in Fig. 2(a)-(b). In consideration of these mean values, the spin con-figuration can be classified principally into four ground-state phases. Besides, h cos α ij i h i,j i = ± J - J duality in Fig. 1(b) and (c).(c) Ferromagnetic (F) state: all spins are parallel.(d) Parallel pairs’ antiferromagnetic (PPA) state: spinslocated on opposite apices of icosahedron are par-allel, whereas total moment is zero. FIG. 2. Numerical results of the classical model: (a) h cos α ij i h i,j i , (b) h cos α ij i h i,j i and (c) h cos α ij i h i,j i . In theparameter space, we take 41 ×
201 sample points for calcu-lation. See Fig. 3 for the acronyms, HA, DHA, PPA, andF. (e) Antiparallel pairs’ antiferromagnetic (APA) state:spins located on opposite apices of icosahedron areantiparallel.In the following, we explain these states in detail. InFig. 3(a), since two spins with a 3rd neighbor bond areantiparallel, total spins are cancelled, indicating antifer-romagnetism. In addition, mean values of the 1st and2nd neighbor inner products imply a distinct order inFig. 2(a)-(b). More certainly, we find that the spin vec-tors in this phase correspond to the normal vectors ofcircumscribed sphere of icosahedron with applying an ap-propriate global O(3) rotation, which conserves all anglesof spins and total energy. We thus call this hedgehogantiferromagnetic (HA) phase. Based on the hedgehogstructure, we can estimate the inner product as follows h cos α ij i h i,j i = − h cos α ij i h i,j i = ττ + 2 = 0 . .., (3)where τ is the golden ratio. The spin configuration inthe HA order is also discussed in Sec. 3.2. As men-tioned in Sec. 2, there is a duality of the 1st and 2ndneighbor interactions with respect to permutation of thesite indexes corresponding to Fig. 1(b)-(c). Reflecting this duality, the HA phase corresponds to the dual HAphase, and thus, we call this ordering as dual hedgehogantiferromagnetic (DHA) phase. Note that the HA phaseis quite similar to the cuboc order reported in numericalstudy on the RKKY magnetism in Tsai-type 1/1 approx-imant [23], whereas symmetry group of an icosahedron isbasically different from that of cubic. Furthermore, wemention that although the DHA phase is also similar tothe magnetic structure determined by neutron scatter-ing on Au-Al-Tb 1/1 approximant [30], those should bein general discussed individually because of difference ofthe models.In the ferromagnetic (F) phase, h cos α ij i h i,j i n = 1 forall n implies all spins are parallel as shown in Fig. 3(c).This numerical calculation does not reproduce the spinconfiguration of ferromagnetism determined by neutrondiffraction in Au-Si-Tb 1/1 approximants [31]. Thiscan be an evidence that the anisotropy caused by to-tal angular momentum of Tb ion dominates the low-temperature magnetism [32], though we do not con-sider the anisotropy in this study. In the parallel pairs’antiferromagnetic (PPA) phase, the spin configurationchanges one by one in many trials of calculation, whiletwo spins on opposite apices of icosahedron, which areconnected with a 3rd neighbor bond, are always paral-lel [see Fig. 3(d)]. Therefore, we call this PPA phase.Undoubtedly, Fig. 2(a)-(b) displays that the same meanvalues h cos α ij i h i,j i = h cos α ij i h i,j i = − .
20, implyingno distinct orders. In fact, if a certain spin’s angles to1st and 2nd neighboring spins are completely randomvalues and the mean values are the same, the mean valuereads,Similarly, at the boundary between the HA andDHA phases ( φ J = π/ , π/ h cos α ij i h i,j i and h cos α ij i h i,j i , h cos α ij i h i,j i = − E nc as a function of ( θ J , φ J )in Fig. 4(a), which is obtained numerically. To evaluatethe E nc , this result is compared with the energy function E est , which is estimated by the spin configurations infour phases: 30[ τ / ( τ + 2)]( J − J ) − J for the HAphase, 30[ τ / ( τ + 2)]( J − J ) − J for the DHA phase,30( J + J ) + 6 J for the F phase, and − J + J ) + 6 J for the PPA phase. Note that the APA state appearsonly at J = J , so that this is merged into the HA andDHA phases. Energy difference E nc − E est displayed inFig. 4(b) gives a good coincidence between E nc and E est except for phase boundaries, where numerical accuracyis not enough. Furthermore, the phase boundaries areapparently obtained by the derivative of total energy withrespect to φ J in Fig. 4(c). FIG. 3. (a)-(d) Examples of ground-state spin configuration in hedgehog antiferromagnetic (HA), dual hedgehog antiferro-magnetic (DHA), ferromagnetic (F), parallel pairs’ antiferromagnetic (PPA) phase. (e) Spin configuration of antiparallel pairs’antiferromagnetic (APA) state appearing at the boundary of the HA and DHA phases.
B. Analytical explanations
To explain the ground-state phase transitions in theclassical model, we consider two specific conditions, (I) J = J and (II) J > J >
0) with J = J = 0( J = J = 0). Condition (I) corresponds to φ J = π/ π/ θ J , φ J ) = ( π/ ,
0) or( π/ , π/ Condition (I)— At the symmetric line of J = J , theclassical Hamiltonian corresponds to the following form, H = J X i =0 S i ! − + ( J − J ) X h i,j i S i · S j . (4)We can obtain the ground state in four cases, (i) J > J > J , (ii) J > J < J , (iii) J < J < J , and(iv) J < J > J . In case (i), which corresponds to θ J < tan − √ φ J = π/
4, the minimum energy isobtained with S i = − S ¯ i , where ¯ i denotes the oppositesite from i -th apex of icosahedron. Therefore, the groundstate in case (i) is understood by antiferromagnetic basedon six pairs of antiparallel spins connected by 3rd neigh-bor bond [see Fig. 1(a)], obtained as the APA state [alsosee Fig. 3(e) for the spin configuration]. This case how-ever seems unstable and corresponds to a boundary be-tween the HA and DHA phases obtained by numericalcalculation [see Fig. 3(a), (b) and Fig. 4(a)]. The groundstate in case (ii) is similar to case (i), whereas this caseis stable in numerically-obtained phase diagram, corre- sponding to θ J > tan − √ φ J = π/ S i = S ¯ i and P i S i = 0, where spins on oppositeapices of icosahedron are parallel under zero net momentof icosahedron, leading antiferromagnetism. Case (iii)is more trivial. Spin configuration of the minimum en-ergy is ferromagnetism, i.e., S i = S j , corresponding to θ J > tan − √ φ J = 5 π/ α and that every composite vector of the spinson opposite apices is the same, the energy is given by, E = J (cid:2) (12 cos α ) − (cid:3) + 6( J − J ) cos α (5)With decreasing J , the ground state changes from an-tiferromagnetic ( α = π ) to ferromagnetic ( α = 0) at J = − J corresponding to θ J > tan − ( √ /
5) with φ J = 5 π/
4. The antiferromagnetic state is also the APA,so that this antiferromagnetic state is also unstable andthe boundary between the HA and DHA phases. On theother hand, the ferromagnetic state is merged into thephase of case (iii).
Condition (II)— We first consider the ground statewith only 1st neighbor interaction. In this condition,we assume following spin configuration with vector spins S i = (sin α i cos β i , sin α i sin β i , cos α i ), α = 0, β = 0,and α i = α ≡ α, β i +1 − β i = β − β ≡ ∆ (mod . π ) , (6) FIG. 4. (a) Numerical result of the total energy E nc in theclassical model. (b) Energy difference between E nc and E est which is analytically estimated by the ground-state spin con-figurations in Fig. 3. (c) The derivative of E nc with respectto φ J . for i = 1 , , · · · ,
4. With the assumption, the exchangeenergy E i,j = S i · S j ( i, j ≤
5) is given by E ,i = cos α, E i,i +1 = E , = cos α + sin α cos ∆ (7)If these energies are the same, cos α = 1 , cos ∆ / (1 − cos ∆). In addition, since ∆ = 2 πn/ n = 0 , ± , ± E i,j = cos α = / √ ± π/ − / √ ± π/ . (8)Therefore, because of the antiferromagnetic interaction,the third value is chosen as the minimum energy. In-terestingly, if we consider antiparallel spins on oppositeapices of icosahedron, i.e., S i = − S ¯ i , all exchange ener-gies of 1st neighbor interaction are the same. This spinconfiguration gives a good accordance with the numericalresult in the DHA phase [Fig. 3(b)]. Note that the spinconfiguration has global O(3) rotation degree of freedom.On the other hand, if there is only 2nd neighbor antifer-romagnetic interaction, we can obtain the ground stateby using the J - J duality in Fig. 1(b)-(c). Starting from FIG. 5. (a) Ground-state energy and (b) its derivative withrespect to φ J obtained by exact diagonalization of the quan-tum model. In the parameter space, we take 100 ×
100 sam-ple points for calculation. The shaded area in (a) represents S tot = 6, i.e., ferromagnetism. (c) Fidelity of the groundstate. The dotted lines drawn by hand represents the phaseboundaries among hedgehog singlet (HS), dual hedgehog sin-glet (DHS), bonding pairs’ singlet (BPS), antibonding pairs’singlet (APS), and ferromagnetic (F) phases. the spin configuration discussed above, the permutationof spin sites also gives a good coincidence with the nu-merical result in the HA phase [see Fig. 3(a)]. In fact,with an appropriate global O(3) rotation, the spin vectorhas the same direction as normal vector of circumscribedsphere of icosahedron, like a hedgehog. IV. QUANTUM MAGNETIC STATES
In this section, we show numerical results in the quan-tum model with spin-1 / S z tot = P i S zi . By checking degen-eracy of ground states between different subspaces, wecan determine magnitude of total spin S tot , e.g., if theground-state energies of only S z tot = 0 and 1 are the same,the ground state is triplet ( S tot = 1). Figure 5(a)-(b)shows the ground-state energy and its derivative with re-spect to φ J in the parameter space. In the shaded areaof Fig. 5(a), ground states are 13-fold degeneracy, i.e., S tot = 6 ground state corresponding to the ferromag-netic (F) state. Except for the shaded area, we haveconfirmed no degeneracy between different subspaces, sothat singlet ground state appears. In the singlet area, wecan see anomalous lines in Fig. 5(b), which imply phaseboundaries. To confirm the phase boundaries, we alsocalculate an overlap of ground states with neighboringsample points in the parameter space, so-called fidelity,defined byFd( θ J , φ J ; δ ) = |h gs : θ J , φ J | gs : θ J , φ J + δ i| . (9)If the ground state is continuously deformed, that indi-cates no degeneracy at ground state and no phase tran-sition between ( θ J , φ J ) and ( θ J , φ J + δ ), the fidelity con-verges to the unity lim δ → Fd( θ J , φ J ; δ ) →
1. Otherwise,the fidelity is much smaller than 1. Therefore, we candetermine the phase boundary by using the fidelity evenin the singlet area. Figure 5(c) shows the fidelity, andwe can see several lines with a dip of the fidelity, whichcorresponds to anomalous lines in Fig. 5(b). Thus, weconclude that there are four singlet phases except for theferromagnetic phase.The four singlet phases are understood as follows. Theupper region of singlet phase includes the north pole θ J = 0 and its ground state is intuitively described bythat at the north pole. Since only the 3rd neighbor an-tiferromagnetic interaction is non-zero at the north pole,two spins on opposite apices of icosahedron compose asinglet (antibonding) pair and the ground state is thedirect product of six singlet pairs. Hence, we call theupper region antibonding pairs’ singlet (APS) phase. Onthe other hand, the south pole θ J = π requires close at-tention because the south pole is a singular point betweenthe ferromagnetic phase and the lower region of singletphase. In fact, at the south pole, where J = J = 0 and J >
0, six triplet (bonding) pairs consisting of spins onopposite apices of icosahedron are completely decoupled.With slight positive J = J >
0, which is included inthe lower region of singlet phase, six triplets antiferro-magnetically interact with each other, resulting in a sin-glet ground state. Thus, we call the lower region bondingpairs’ singlet (BPS) phase. In middle region, there aretwo singlet phases more, which include parameter pointswith only the 1st and 2nd neighbor antiferromagnetic in-teractions, i.e., ( θ J , φ J ) = ( π,
0) and ( π/ , π/ FIG. 6. Order parameters for (a) O APS (b) O BPS , and (c) O HS . The color boundaries correspond to the phase bound-aries in Fig. 5. fore, we call these two phases hedgehog singlet (HS) anddual hedgehog singlet (DHS) phases, respectively.To distinguish the quantum phases, we introduce pro-jection operators of singlet and triplet pairs of i -th and j -th sites given by, P si,j = 14 − S i · S j , P ti,j = S i · S j + 14 . (10)We first define order parameters of the APS and BPSphases as products of these projection operators of pairedspins on opposite apices of icosahedron, i.e., O APS = *Y i P si, ¯ i + , O BPS = *Y i P ti, ¯ i + , (11)where ¯ i denotes the opposite site from i -th apex of icosa-hedron. Figure 6 shows the order parameters in the pa-rameter space. We can see that the APS and BPS phasesare well distinguished, while O BPS gives a non-zero value,that is, the unity in the ferromagnetic phase. Note thatthe quantum state in the APS phase also includes tripletconfigurations, resulting in a non-zero value of the BPSorder. On the other hand, these order parameters showzero in the HS and DHS phases. We thus consider acombination of singlet and triplet defined by O HS = * Y ( i,j )=(0 , , (1 , , (2 , (cid:16) P si, ¯ i P tj, ¯ j − P ti, ¯ i P sj, ¯ j (cid:17)+ . (12)The site combinations ( i, j ) = (0 , , (1 , ,
3) cor-respond to the 1st neighbor bonds in three perpendicularplanes of icosahedron (see Fig. 1). Therefore, the HS or-der parameter represents products of singlet-triplet con-figurations on three perpendicular planes. In Fig. 6(c),we can see two finite-value regions of the HS order, i.e.,positive values for the HS phase and negative values forthe DHS phase. The difference of sign corresponds toan exchange of singlet and triplet pairs. Site exchangeof the singlet and triplet pairs on each plane induces anexchange of the first and second neighbor bonds betweenplanes. Therefore, the HS order parameter can probe anasymmetry of the first and second neighbor bonds.
V. CONCLUSION
In this paper, we have investigated magnetic groundstates in both classical and quantum Heisenberg spinmodels on an icosahedral cluster, where all bonds are con-sidered as ferromagnetic or antiferromagnetic exchangeinteractions. The ground-state phase diagrams have beennumerically determined by using simulated annealingand exact diagonalization methods. Moreover, we haveshown analytical explanations of spin configurations at specific points in the parameter space. Based on thenumerical and analytical examinations, we have char-acterized four ground-state phases, i.e., the HA, DHA,PPA, and F phases with the APA state in the classicalmodel. On the other hand, we have also classified theground-state phases in the quantum model with numeri-cal results on the analogy of the classical phases. In fact,we have successfully demonstrated the qualitative coin-cidence between the classical and quantum phases. Fur-thermore, we have found a distinctive quantum phase,the APS phase, in addition to four quantum analogs ofclassical phases, namely the BPS, HS, DHS, and F phasestogether with those order parameters. The icosahedralspin clusters are in general found in the Tsai-type qua-sicrystals and approximants. In these alloys, spins arecoupled with each other via so-called the RKKY inter-actions, and therefore, the icosahedral spin clusters arenot isolated but interact with each other. However, mag-netic properties can strongly reflect characteristics of anisolated icosahedral spin cluster if intra cluster interac-tions are relatively large enough as compared with intercluster interactions. Thus, our study can give a goodstarting point to understand the magnetic properties ex-perimentally observed in the Tsai-type quasicrystals andapproximants.
ACKNOWLEDGMENTS
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