Ordering and Partial Ordering in Holmium Titanate and Related Systems

arXiv:cond-mat/0009265v1 [cond-mat.stat-mech] 18 Sep 2000 n tteLbrtied hsqedsSlds Université Solides, des France. Fran Orsay, Physique 91405 05, de 510, Cedex Bat. Laboratoire Paris Sud, the 75231 at Lhomond, and rue Supérieure, 24 male atal ree rudsae hc sdgnrt but degenerate is which has state, it ground that ordered suggest partially titanate a holmium pyrochlore argu- Ising those the take we here 2]; [3, for papers further. reasons ments earlier the our explored in low- had this different We similar very behaviour. has the temperature titanate long- Moreover, holmium a pyrochlore really Ising is interaction. interaction dipole-dipole dominant the ranged nearest- quite here: not had a is simple for Anderson story so the happen However, should ice. pyrochlore. this Ising of neighbour dys- that that predicted ground that like ago a [3] much long exhibits very discovered pyrochlore, entropy recently Ising especially state We an is titanate, it by prosium model. approximated and Ising well and [1], pyrochlores the years disorder consider recent to of interesting in physics attracted attention the have great magnets in pyrochlore interest magnetism, frustrated wide the With Introduction 1 lattice. fcc in- the states, on ground model Ising their known in well magnetic ordering the show applied which full cluding an models or other partial of several similar systems discuss we these Finally, on field. dis- effect sim- We why the behaviour. to also ice-like cuss as show is true experiments explanations titanate the and some dysprosium ulations that offer of show and model ordered, also our fully We of state. state ground these ground separating true barriers the energy states from infinite or- ordered are fully partially there be reach must because only system simulations this but of state dered, ground analytically, true show, the We that to interactions. compared neighbour strong nearest are the spins ice- ranged between display long interactions the dipolar not because have temperatures does low we at namely titanate behaviour As like holmium ice”, titanate. observed, “spin dysprosium earlier of and titanate realizations been have holmium possible which compounds as two at discussed look another take We Abstract reigAdPrilOdrn nHlimTtnt n Relat And Titanate Holmium In Ordering Partial And Ordering eosre ale 2 htsmltoso oe of model a of simulations that [2] earlier observed We ∗ rsnl tteLbrtied hsqeThéorique, Ecole Physique de Laboratoire the at Presently 2 elLbrtre,Lcn ehoois 0 onanAven Mountain 600 Technologies, Lucent Laboratories, Bell 1 eateto hsc,Ida nttt fSine Bangal Science, of Institute Indian Physics, of Department Siddharthan R 1 ∗ Shastry S B , coe 5 2018 25, October Paris- Nor- ce, 1 ueetdn napwee ape[] ecompare We simulations [3]. with mea- samples sample heat powder on powdered specific data experimental a the the of on basis done was the surement ordering on dependent suggested direction direction initially very strongly is A field magnetic on dependent. a done in zero- behaviour titanate were the the dysprosium experiments that of as suggest same the simulations the While samples, not powdered is state. esti- field ground our strong in field a state and ground of field) the presence the Moreover, the is with superexchange. (which the couples of moment that mate dipole thing the only of the experi- calculation quantitatively, reproduce our and qualitatively field confirms well, magnetic quite that results a fact mental with The in- simulations superexchange. magnetic our the isotropic dipole-dipole and an field, are and magnetic system teractions a the of presence in the interactions in a decreased is titanate, there is dysprosium that it in namely observed entropy [3], work temperature earlier low our from model ing ti- state. holmium ground for ice-like model the an have our to should that suggestions tanate that recent namely to [4], contrast contrary to in transition a ordering, of support partial suggestions simulations, original our our ground also substantiate true and and the results, into exact can- this Our and from K), state. dislodged 1 easily (over has be temperature then system higher not this a at an because state into is ice-like settles and suggest, interactions finite we nearest-neighbour a stronger This, nearest-neighbour suggest of ice”. [3] characteristic “spin experiments entropy the state and but ground titanate; model the dysprosium the also of model is both our this of Moreover, state ground ground thenceforth. true these true time the of pre- reach finite one to in (and, in impossible state simulation and stuck the cooling, get on to for states compound) easy real is infinite the by It sumably, state ground there barriers. true but the energy from complete, of separated fact states are ordering metastable in which ordered is the partially low-lying here that numerous recourse state are show without ground we it true Actually, nature the explain the and clarify simulations. ordering, we to article partial par- this this above In of the the phase. to from ordered transition phase tially phase paramagnetic order temperature first a high be to appears there a oetoyprpril tedgnrc en fteor- the of being 2 degeneracy der (the particle per entropy no has eas usataetemjrrslsadteunderly- the and results major the substantiate also We 1 , 2 L n Ramirez P A and where L e uryHl,N 77,USA 07974, NJ Hill, Murray ue, stesse egh ahrta 2 than rather length, system the is r 602 India 560012, ore 2 dSystems ed L 3 ,and ), averaged over large numbers of directions, compare sharp 2 Ordering in our model of features in both, and make important predictions for pos- Ho Ti O sible future experiments on single crystals. 2 2 7 Next, we use the insight from the ground state analysis We briefly recapitulate our model of holmium titanate. to progressively reduce the pyrochlore Ising model to a se- The underlying lattice is the pyrochlore lattice, a lattice quence of simpler models which display similar behaviour: of corner-sharing tetrahedra of two possible orientations, namely, a six-vertex model on the “diamond lattice” with which is well visualised as an fcc lattice of tetrahedra (Fig- non-local interactions, which reproduces all the essential ure 2). It may be generated by taking a single tetrahe- behaviour of holmium titanate, and has a fully ordered dron of one orientation and translating it by the primi- ground state and several metastable partially ordered low- tive basis vectors of the fcc lattice (Figure 3); the tetrahe- lying states (section 4; a six-state magnetic model on the dra of the other orientation emerge automatically by this fcc lattice, which actually has the sort of partially or- procedure—see Figure 3. Thus we use the lattice vectors dered ground state that the simulations had suggested for holmium titanate, and also reproduces the important a1 = (r, √3r, 0), behaviour of holmium titanate (section 6); and the well- a2 = r, √3r, 0 , known Ising model on the fcc lattice, which has been stud- − ied before [5] and is known to have exactly the same sort a3 = (0, 2r/√3, 2r 2/3). (1) of partial ground state ordering that concerns us here (sec- − tion 7). Along the way, we also introduce a square-lattice with a basis of atoms located at p vertex model, by analogy with the above diamond-lattice model, which may be worthwhile to study in its own right. x0 = (0, 0, 0) Actually, the simplest example of partial ordering in an x1 = (r, 0, 0) , Ising system is perhaps the triangular lattice antiferromag- 1 √3 net, with interaction J1 along bonds in one direction (say x2 = r, r, 0 , 2 2 parallel to the x axis), J2

Figure 2: The conventional unit cell of the pyrochlore, a lattice of corner-sharing tetrahedra with fcc symmetry.

We have Ising spins (the f electron states of the holmium Figure 1: Anisotropic triangular lattice, with antiferro- atoms) located at these points. The local Ising axis is the magnetic Ising interaction J along solid lines and J ′ < J line joining the centres of the two adjoining tetrahedra: along dashed lines. In ground state, each chain along solid thus, each spin points directly out of one tetrahedron, and lines is ordered, but there is no order along dashed lines. into the next one. The spins carry magnetic moments corresponding to J = 8, gs (the Land´efactor) = 1.25. Based on this, the expected nearest-neighbour dipole-dipole interaction has an energy of 2.35 K, the sign depending on ±

2 the alignment of the spins: one pointing out, one in is pre- ground states, each of which has two spins pointing out and ferred. However, the experimental compound (unlike its two in. We label these states A, A′, B, B′, C, C′ where A′ cousin, dysprosium titanate) has properties which we can is A with the directions of all spins reversed, and similarly only explain by postulating a signiﬁcant nearest-neighbour B′ and C′ (Figure 4). Consider a cluster of sites consist- superexchange, which we estimate from high temperature ing of a single “up” tetrahedron and its translation by the expansions for the susceptibility to be around 1.9 K with three primitive lattice vectors of the fcc lattice (as in Fig- opposite sign to the dipolar interaction. So, with the con- ure 3), and long-ranged dipole-dipole interactions spanning vention that an Ising spin S = 1 if it points out of an “up” the entire cluster plus a nearest-neighbour superexchange, tetrahedron and S = 1 otherwise, we write the Hamilto- as described above. We ﬁnd the GS of this “tetrahedron nian as follows: − of tetrahedra” by the unobjectionable method of enumer- ating each of the 216 allowed states on a computer, and H = Jij SiSj (3) picking out the lowest-energy ones. i,j X Jij = 0.45 Kelvin, for nearest neighbour spins (4)

µ0 2 2 2 ni nj (ni rij )(nj rij ) Jij = gs µb J ·3 3 · 5 · , 4π " rij − rij # further neighbours (5) where ni is a unit vector pointing along the Ising axis A BC at site i in the outward direction from an “up” tetrahedron. Thus, this system has a drastically reduced nearest- neighbour interaction energy, which means the importance Figure 4: Three possible ground state configurations A,B, of the further-neighbour interactions increases. We saw C of a tetrahedron. We will denote by A′, B′, C′ these that with only a long ranged dipole-dipole interaction be- respective configurations with all spins reversed. ‘+’ in- tween these Ising spins, the behaviour seems to change dicates a spin pointing out of the tetrahedron, ‘ ’ a spin − little from the nearest-neighbour Ising model which has a pointing into the tetrahedron. finite ground state entropy; we speculate that the substan- tially different behaviour of holmium titanate is due to the + significantly greater importance of further-neighbour in- + − − teractions. As noted earlier, simulations of such a model + − − + do predict a freezing of the system at around 0.7 K, in − − + + + + + + − − agreement with experiment. + +

− − − − − − − − + + + +

Figure 5: Two of the twelve allowed ground states for a a cluster of four tetrahedra. Each of the twelve states is a 2 related to the others by rotations or reﬂections. This con- 1 sideration by itself leads to partial ordering in the ground a3 state of the full system, as described in the text.

We find that the GS of this cluster is twelve-fold degenerate. Two are shown in Figure 5. All others are rotations/reflections of these (which are reflections of each other). In each GS, as one might expect, only states sat- Figure 3: The pyrochlore lattice can be generated by isfying the ice rule occur. Furthermore, only two kinds of ′ ′ translating a tetrahedron along vectors a1, a2, a3. Tetra- ice-like configurations occur: A and A , or B and B , or C hedra of the opposite orientation are formed from the cor- and C′. They occur twice each. In the case shown in the ners of these (thin black lines). figure, once the configuration of tetrahedron at the origin is fixed as C, the tetrahedron at a3 must have the op- We now find the ground state (GS) for this system. For posite configuration (C′); and the tetrahedron at a2 may clarity we refer to the two different orientations of tetra- have either configuration (C or C′), but tetrahedron a1 hedra that occur as “up” and “down”: each “up” tetrahe- must have the opposite configuration to a2. With the pre- dron shares corners with only “down” tetrahedra, and vice viously quoted values for the dipole and superexchange versa. Consider a single tetrahedron: it has six possible interactions, the configurations in Figure 5 have an energy

3 7.5 K, and the next lower energy configurations have an system very well. − energy 6.9 K. This is exact except for one thing: we have ignored in- − If we note that each tetrahedron in its ground state has teractions between further-neighbour tetrahedra, so effec- a dipole moment, which is perpendicular to two possible tively confined our interaction range to the 5th neighbour, lattice translation vectors, the rule is this: in every GS which is the maximum separation of spins in two adja- only configuration of two opposite kinds occur (say C and cent tetrahedra. As long as only nearest-neighbour “up” C′), which therefore have antiparallel magnetic moments, tetrahedra interact, so that the whole system really can and two tetrahedra separated by a vector perpendicular to be decomposed into clusters as in Figure 5, our picture these moments must have opposite configurations. These is totally correct. In the presence of long ranged interac- are the twelve ground states allowed for this cluster, but we tions, luckily the nature of the system (with four different are making no theoretical argument for this; this is what local z directions) is such that the effects of the more dis- we learn from brute-force enumeration of states. In any of tant tetrahedra cancel heavily and have little effect on the these states, moreover, the included “down” tetrahedron energy of a single cluster. Thus, if one take a particular also has an ice-ruled configuration. “up” tetrahedron in the ground state, its energy turns out to be 21 K because of interactions with the immediately Now consider the entire fcc lattice of tetrahedra (that − is, the pyrochlore lattice). Once we fix the configuration neighbouring “up” tetrahedra (to which the cluster argument applies), but only 0.12 K because of interactions of a single “up” tetrahedron (say, as C) each of its neigh- − bouring tetrahedra, with which it forms part of a cluster with all other tetrahedra in the system. The effect is not of the above sort, must have the same or opposite con- only small, but tends to stabilise this order. (With a ran- figuration (say C or C′); and by extending further from dom ice-like configuration, this energy averages to zero, but each of these, this must be true for the entire lattice. We fluctuates considerably from site to site.) Recall further- can also see that if we travel in either of the two lattice more that the cost of disturbing a single four-tetrahedron directions perpendicular to the magnetic moment of these cluster from its ground state (Figure 5) is at least 0.6 K, tetrahedron configurations, we must have a perfectly alter- and in fact much more since each tetrahedron is shared by nating sequence (say C, C′, C, C′, . . . ); but when travel- four such clusters. ling in a third direction we have no ordering rule. So we So we can be satisfied that the long ranged interactions get a ground state ordering in two directions but not in will not affect the above conclusions. In fact, the simula- the third, and a degeneracy exponential in L (the system tions too don’t show much dependence on the range of the length) rather than L3 (the system volume), precisely as interaction, provided it extends to at least the third neigh- the simulations suggested. bour (Figure 6), as indeed we had argued in our earlier paper, where we cut off the interaction at the fifth-neighbour This is not the whole story, though. The system can be distance [2]. The suggestion [4] that ice-like behaviour is equally well described in terms of “down” tetrahedra, so restored by cutting off after the 10th neighbour seems un- the same sort of ordering should be evident if we describe tenable to us, and we do not observe it in our simulations a ground state configuration using “down” tetrahedra. So even on extending the interaction to the 12th neighbour only two configurations for these tetrahedra should be al- (which is halfway across our sample). Possibly the different lowed, each of which is the other with all spins reversed. results obtained in [4] is due to additional approximations But we note immediately that the two “down” tetrahedra involved, such as Ewald sums, and a somewhat smaller displayed as grey lines in the two clusters in Figure 5 do sample size; our simulations use no approximations in cal- not have opposite configurations. It follows that the two culating the energy except the cutoff, which as we have cluster configurations in that figure cannot both occur in seen, is quite justifiable. Longer simulations on bigger sys- the ground state: only one can. This immediately implies tems may throw more light on this question, but we now that the ordering sequence in a third direction is not ran- turn to some other interesting aspects of the problem. dom, but constant (say, C, C, C, . . . ). This true ground state ordering is more easily visualised So the true ground state for our model of holmium ti- (though less easily analysed) with the cubic unit cell rather tanate is only twelvefold degenerate, and viewed in terms than our parallelepiped; this is discussed in section 6. of configurations of either upward or downward tetrahedra, consists of alternating ordering of opposing configurations in two directions but a constant configuration in a third di- 3 Dysprosium titanate rection. However, the partially ordered states are also very low in energy, and moreover a system stuck in such a state Dysprosium titanate, which we earlier reported as show- can only get out by flipping entire planes of tetrahedron ing ice-like behaviour experimentally and in simulations, configurations, which is impossible in the thermodynamic appears to have a much weaker superexchange between limit. So simulations tend to get stuck in such states and nearest neighbours. With a nearest-neighbour-only model in other “domainised” states and the chances of a given of the superexchange, we find we need a superexchange simulation actually hitting a true ground state are very of around +1.1 K, that is, the nearest-neighbour interac- small—unless we use some sort of specialised “cluster” al- tion is around 1.25 K compared to the bare dipole-dipole gorithm which may not imitate the dynamics of the real value of 2.35− K. With these numbers, we get a reason- −

4 3 10

9 3rd neighbour cutoff 2.5 Experiment 8 5th neighbour cutoff Simulation (nn superexchange) Simulation (uniformly scaled 7 12th neighbour cutoff interaction)

) 2 ) 2

6 −2

K −1 5 1.5

C/T (J/mol/K C/T (J mol (J C/T 3 1

0.5 1

0 0 0.5 1 1.5 2 2.5 3 3.5 4 0 T (K) 0 1 2 3 4 5 6 T (K)

Figure 6: The simulated specific heat when the interaction is cut off at the third (R 2r), fifth (R 2.646r), and Figure 7: A comparison with experimental data for dys- twelfth (R 4r) nearest neighbour≤ distances≤ (r =3.54 A˚ prosium titanate of a simulation using nearest-neighbour roughly).≤ The position of the phase transition and the superexchange and long-ranged dipole-dipole interactions, plot of the specific heat near the transition hardly change and a simulation using uniformly scaled down dipole-dipole at all on increasing the range of the interaction; but one interactions (as was done in ref. [3]). needs longer equilibriation times with increased interaction ranges. The simulations show a significant energy drop at chance to occur. In holmium titanate, on the other hand, the transition, suggesting that it is first order. the nearest-neighbour interaction is around 0.4 K, thus at 0.7 K the system is in no sense frozen, plenty of spin flips able agreement of the simulation with experiment (Fig- take place and the ordering transition occurs (Figure 8). ure 7). However, we get even better agreement by uni- So while our arguments show that the true ground state formly scaling down the long-ranged dipole-dipole interac- here is ordered and only twelve-fold degenerate, the system tion by roughly this factor. (This is what was reported in tends to get stuck in fairly generic ice-like states. We have our earlier paper.) This suggests that the superexchange is checked that the energy of the disordered low-temperature not strictly nearest-neighbour but extends over to further state of the system in our simulations is always slightly neighbours. but significantly higher (by around 0.5% to 1%) than the We need to understand why Ho2Ti2O7-like behaviour is energy of the fully ordered state if we include the full long- not observed in this case. With either of the two mod- ranged interactions in calculating the energy. els above (small nearest-neighbour-only superexchange, The low-temperature states we see are governed mainly or uniformly scaled-down dipole-dipole interaction) the by the ice rule (though evidence of some local ordering ground state for the cluster of four tetrahedra remains the of 4-clusters can be seen) and are probably macroscopic same as before, the difference in energy from the next- in number. This is why we observe an anomaly in the lowest state too remains roughly the same, and the above integrated entropy which we earlier attributed to a possi- arguments should still go through. ble ground state entropy. This low temperature “entropy” The difference is that, because of the stronger nearest- (as estimated from Figure 9) is around 10% lower than neighbour interactions here, this system undergoes a 1/2 log(3/2), which is itself an underestimate by around crossover from a paramagnetic phase to an ice-ruled phase 10%. Very much the same thing is likely to be true in at a considerably higher temperature (greater than 1 K); the real system (dysprosium titanate) too: its true ground and by the time it cools down to the temperature (< 0.7 K) state is ordered but the system can almost never access this where we expect a transition of the sort described here, ground state. The measured ground state entropy here is it is already stuck in a disordered ice-like state and (be- closer to 1/2 log(3/2), in fact a bit more; it is probably cause of the stronger nearest-neighbour interactions) can- a bit less than the true ground state entropy of nearest- not easily break out of this state to access other states. neighbour spin ice, though. It appears that the temperature of the crossover to the In the presence of a magnetic field, some interesting ice-like phase is dictated by the nearest-neighbour inter- things happen. As reported earlier [3], some of the ob- actions. In dysprosium titanate these have an energy of served ground state entropy is recovered experimentally; around 1.3 K, and hence the ice rule is already in place we see this also in simulations (Figure 9). Since only by the time we go down to 0.7 K and the spins are al- the dipole moment couples to the field, the quantitative most frozen, thus the ordering transition no longer has a agreement in this curve is an additional confirmation of

5 7

Dysprosium titanate (experiment) 6

7 ) Holmium titanate (experiment) −1 K 5 Rln2

−1 6 Holmium titanate (simulation) R(ln2 − 1/2ln3/2)

5 4

4 3

C/T (J/mol/K)

3 2 H = 0 (Expt)

Integrated entropy (J mol H = 0.5 T (Expt) 2 1 * H = 0 (Sim) 1 H = 0.5 T (Sim) 0 0 2 4 6 8 10 12 0 0 0.5 1 1.5 2 2.5 3 3.5 4 T (K) T (Kelvin)

Figure 9: The integrated specific heat per unit temper- Figure 8: Specific heat curves of dysprosium titanate (ex- ature in simulations without a magnetic field and with a perimental, and simulations give excellent agreement) and half-tesla magnetic field. This shows the entropy gained holmium titanate (experimental, and a simulation retain- over the ground state entropy. The entropy is expected ing dipole-dipole interactions to 12 nearest-neighbour dis- to be R log 2 (dotted line) at high temperatures; the inte- tances). By the time the ordering temperature is reached grated value falls short of this, indicating a ground state (which is expected to be the same for both systems) dys- entropy, but the ground state entropy is reduced in the prosium titanate is already frozen into an ice-like configu- presence of a magnetic field. Both the experimental data, ration from which it would find it hard to locate a lower- taken from our earlier paper [3], and the simulation results energy state. are plotted here for easy comparison. Based on simulations, we suggest that a magnetic field of around 3 Tesla should recover all or nearly all the ground state entropy. our model of co-existing dipole-dipole interactions and superexchange. The curves are similar in features and the amount of entropy recovered is also roughly the same. In and [111] direction. The system seems to go through sev- stronger fields, sharp spike-like features start to show up in eral magnetic transitions before reaching its fully polarised the experimental specific heat curves at low temperatures. state. Here, too, we find reasonable qualitative and quantitative All these features would be averaged over in the experi- agreement between simulations and experiment. All this ments on the powder samples, and single crystals of these confirms that our calculation of the dipole moment of the materials could turn out to be worth studying in their own f electrons and our supposition that the reduced energy right. scales are due to another interaction (superexchange) are correct, since the interaction with a magnetic field is purely magnetostatic. The experiments were done using powder 4 An analogous six-vertex model samples, and the simulations show that the behaviour is strongly dependent on the direction of the field. To com- If we examine the nature of states just above the transition pare with powder averaged experimental results, we would in simulations of our holmium titanate model, we find that need a very large number of simulations in random direc- a large fraction of the tetrahedra are already in the ice- tions, which we have not done to our satisfaction. ruled state. This suggests that the sharp transition here is The nature of the ground state also depends on the field not the transition from paramagnetism to an ice-like phase, direction, and with a sufficiently strong field and a suitable but a phase transition from a disordered ice-like state to field direction the ground state may not even satisfy the ice an ordered or partially ordered state. rule. For instance, with a field along the z axis in Figure 3 To check that this is the case, we can try putting the ice (which corresponds to the [111] direction with the conven- constraint in by hand, and check that the phase transition tional unit cell) the ground state consists of three spins is still reproduced at the same temperature. The model we pointing into each upward tetrahedron and one pointing get then is a form of six-vertex model on the “diamond” out of it. lattice, whose sites are the centres of the tetrahedra in the It is interesting to ask how the transition to a different pyrochlore lattice. The six-vertex model has been widely ground state occurs as one slowly turns on a magnetic field studied on the square lattice; the diamond lattice, like the at low temperatures. Figure 11 shows the result of doing square lattice, has a coordination number of four and can this in a simulation at 0.2 Kelvin, for a field in the [11√2] be divided in two sublattices, but is three-dimensional. So

6 1200

B || [1 1 1] [1 −1 1.414] 1000

H=0 ) B || [1 −1 1.414]

−2 800

K −1 1 600

M (arb. units) C/T (J mol (J C/T 400

200

0.1 0.1 1 10 0 0 0.5 1 1.5 2 2.5 3 T (K) B (Tesla)

Figure 10: Specific heat in the presence of a 2 Tesla Figure 11: The growth of magnetisation in the simu- magnetic field, for various directions of the field. lation sample, for magnetic field in the [11√2] and [111] directions, at a temperature of 0.2 K. we study a system where one assigns arrows to bonds on the diamond lattice, such that each site (or “vertex”) has spin interactions: we therefore also insert a factor of half. two arrows pointing in at it and two pointing away: so six Formally, we can write kinds of vertices are possible. But unlike conventional six- vertex models, we don’t assign different weights to these H = Eij E(Si, Sj , ri rj ) (7) ≡ − six vertices, since all of them are really equivalent here; {Xi,j} {Xi,j} instead, the thermodynamics comes from interactions between different vertices. In other words, we have a Hamil- and then break this sum up into nearest-neighbour terms, tonian of the sort next-neighbour terms, and so on; throw out the nearest- neighbour terms because we have used them in enforcing H = J(c(i),c(j), ri rj ), (6) the ice constraint; and group the next few terms into pairs ≡ − i,j i,j where one spin belongs to one “up” tetrahedron, the other X X spin belongs to an adjacent “up” tetrahedron. where ci is the configuration (a six-valued variable) of the i- th vertex, and J is the interaction energy of vertices i and j, 4 4 which depends not only on their configurations but on the H = E(Sαm, Sβn, Rα Rβ + xm xn) − − m=1 n=1 vector joining them (thanks to the underlying direction- {Xα,β} X X dependent dipole-dipole interaction). We have to calculate +other terms (8) the pairwise J’s appropriately. What we do is the following: we note that the sites on where the sum is over neighbouring “up” tetrahedra at the diamond lattice fall into two sublattices, correspond- sites α and β, and we sum the interaction of each of the ing to up and down tetrahedra. First consider adjacent four spins at α with each of the four spins at β. Rα is the vertices (adjacent corner-sharing tetrahedra). The inter- position of spin 1 on tetrahedron α, likewise Rβ, and xm nal interactions between these spins can be separated into are as in equation (2). All the “other terms” involve spins nearest-neighbour interactions, which we can assume has separated by three times the nearest-neighbour distance already been taken into account via the ice rule, and next- or more, so we drop them. We can do precisely the same neighbour interactions, which we can equally include by grouping of terms for the “down” tetrahedra; we do both, considering only next-neighbour vertices (that is, nearest- and insert a factor of half. Thus our Ising spin Hamilto- neighbour tetrahedra of like orientation). We thus ignore nian is reduced to the vertex Hamiltonian (7) with appro- interactions between nearest-neighbour vertices and con- priately chosen interaction energies J, extending only to sider interactions only between next-neighbour vertices, or the nearest neighbour on the same sublattice. nearest-neighbour vertices on a single sublattice. The in- As before, we simulate this Hamiltonian. First, a word teraction between two such vertices is the energy of inter- on how we do this. Flipping a single bond will not do: action between the two corresponding tetrahedra, as given it will destroy the ice constraint on both adjoining ver- by the sum of interaction energies of all pairs of spins. We tices. We must find a closed loop, a set of bonds whose ignore all further-range interactions. But the interactions arrows lead from vertex to vertex and return to the start- already included, if carried out over all vertices over both ing vertex, and flip the whole loop at one go. Such “loop sublattices, will actually double-count the pairwise spin- algorithms” have been discussed previously [7] and it has

7 been pointed out that, in a six-vertex model, every line lines in Figure 13). We ignore nearest-neighbour interac- of arrows if followed must return to the starting vertex tions for the same reason as earlier, ie that is taken care of and every configuration is accessible via loop-flips alone, by assigning an ice rule. so a random loop-flip algorithm is ergodic. The earlier The possible interactions are shown in Figure 14; for algorithms have several improvements and optimizations; symmetry reasons, we need have only three interaction pa- however, they are concerned with conventional vertex mod- rameters, all other non-zero interactions can be obtained els where different vertices have different weights but don’t from these by rotation, reflection, or inversion of one or interact, and cannot be completely translated to this sit- both vertices (inverting a single vertex will simply change uation. We found it sufficient to merely pick up random the sign of the interaction energy). If we calculate the starting sites, form loops randomly, calculate the energy interaction parameters from an assumed dipole-dipole in- difference, and flip them according to the Metropolis algo- teraction between magnets aligned along the edges con- rithm. necting the respective vertices, we obtain A = 8.6678, − The results are shown in Figure 12. It exhibits a phase B = 10.753, C = 10.345 (arbitrary units). The ground − transition at exactly the point where both the real sys- state then looks as shown on the left of Figure 15; but tem, and our model for it, do. This is a distinctly first a very slight alteration in the choice of A, B and C will order phase transition. Thus we have verified that the give a ground state as shown on the right of Figure 15, phenomenon driving this phase transition is not the for- and the choice A B = 2C may lead to rather interest- − mation of ice-like tetrahedra, but the further ordering of ing results. The model is probably worth studying both tetrahedra that have already attained ice-rule configura- in its own right and because of the long historical inter- tions; and we have displayed a fairly simple vertex model est square-lattice vertex models have held; but since it is which shows the same features as our Ising pyrochlore. not really related to the rest of this article, we postpone The ground state of this system would be expected to further discussion of it to a future work. be fully ordered, but typically only partially ordered states are accessible. The argument is similar to that in the case of holmium titanate, and the simulations bear this out.

9 Holmium titanate (experiment)

Holmium titanate (simulation) 8 Six−vertex model (simulation) 7 Six−state fcc spin model (simulation) 6

C (J /mol/K) (J C 4

1 Figure 13: An ice model on the square lattice, with similar

0 properties to the earlier, diamond-lattice model. Interac- 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 T (K) tions are between diagonally-opposite vertices (grey lines).

Figure 12: Comparison of speciﬁc heat curves for holmium titanate (the real system); our model of holmium titanate; the six-vertex model of section 4; and the six-state spin model of section 6

A BC 5 A square lattice vertex model Figure 14: The three possible interactions between neigh- The sort of physics involved can perhaps be better seen in bouring vertices. All other possibilities are symmetry re- a square-lattice vertex model. Such models have been ex- lated, or zero. In particular, reversing all arrows on a single tensively studied [8], but the thermodynamics has typically vertex will simply change the sign of the interaction. arisen from assigning diﬀerent weights to diﬀerent vertices; instead we give the same weight to all vertices, but consider interactions between diagonally-opposite vertices (the grey

8 in one of two opposite directions; each plane perpendicular to this direction is antiferromagnetically ordered; and perpendicular to this plane, the ordering is random. It is tempting to use the total dipole moments of the tetrahedra as the site variables, and for the interaction simply to use their mutual magnetic interactions, since we know that the dipole-dipole interaction favours antiferromagnetic ordering in planes perpendicular to the spins; but it turns out that this is not the ground state of such a sys- Figure 15: With interaction energies A, B, C between tem. Only by using the actual interaction energies of the vertices calculated from dipole-dipole interactions between tetrahedra do we obtain such a ground state. However, spins aligned along their edges, we obtain a ground state there is an obvious connection between this system and a as on the left. But a slightly diﬀerent choice of weights well studied-problem, which we turn to in the next section. will yield a ground state as on the right, and there is the possibility of a “level crossing” between the ground states for a choice A B =2C. − 7 Ising model on the fcc lattice 6 Multistate spin model on an fcc The nearest-neighbour antiferromagnetic Ising model on the fcc geometry has been studied by several authors, and lattice its ground state is known to have exactly the sort of ordering we are considering. In our vertex model earlier, we had two sublattices, and in- The ordering is easy to understand if there is a bit of teractions only within a single sublattice. Apart from the anisotropy in the system: consider Figure 16, where we ice-rule constraint, the two sublattice could just as well have an fcc crystal, and within the x–y plane and planes be noninteracting. So the next logical step is to separate parallel to it there is an antiferromagnetic interaction J be- the two sublattices. We consider an fcc lattice, with a six- tween nearest neighbours, but out of the plane there is an valued variable at each site. Only nearest-neighbour inter- interaction J ′

9 which have a second order transition at the Onsager tem- [5] See eg. N. D. Mackenzie and A. P. Young, J. Phys. C (Solid perature. In fact, simulations suggest that for J ′ close to State Phys) 14, 3927 (1981); Phani et al. Phys. Rev. Lett. J, the transition is first order, but for J ′ somewhat less it 42, 577 (1979), Phys. Rev. B 21, 4027 (1980); R. Kikuchi is second order, and for J ′ = 0.6J the transition temper- and H. Sato, Acta Metall. 22, 1099 (1974) ature is almost exactly the Onsager temperature. Since [6] G. Wannier, Phys. Rev. 79, 357 (1950) the planes have a zero interaction energy in the ground [7] H. G. Evertz, G. Lana and M. Marcu, Phys. Rev. Lett. 70, state and a very small interaction energy at low tempra- 875 (1993); H. G. Evertz, cond-mat/9707221 (revised Jan tures, they behave like almost uncoupled 2D Ising systems. 2000) Thus this system seems to exhibit either a first order tran- [8] R. J. Baxter, Exactly Solved Models in Statistical Mechan- sition or a second order transition with the same sort of ics (Academic, 1989) ground state, depending on the parameters. [9] R. Siddharthan, Ph. D. Thesis submitted to the Indian Institute of Science, Bangalore, July 2000. 8 Conclusion [10] B. C. den Hertog, R. G. Melko, M. J. P. Gingras, cond- mat/0009225 We have clarified the true nature of the ground states of the Ising pyrochlores holmium titanate and dysprosium titanate. We have pointed out that the ice-like behaviour of dysprosium titanate seems to arise not from a macroscopic degeneracy of the true ground state, but from its inaccessibility in practice, and consequently the tendency of the system to fall into one of a large number of slightly excited ice-like states. In holmium titanate, the ordering temperature is higher than the expected ice-formation temperature; here, too, the system gets stuck into excited states, but these are partially ordered states and the model system shows a clear phase transition. By rigidly enforcing the ice constraint, we show that this transition exists independently of the broad ice-state crossover in spin ice, and we exhibit several models, including the well-known fcc Ising model and a diamond lattice vertex model, which undergo a similar phase transition. Analogous to this vertex model we also exhibit a square lattice vertex model which has differently ordered ground states depending on what interaction parameters we choose, and which we hope to examine further sometime in the future. In addition, we have looked at what happens to dysprosium titanate when a magnetic field is applied, and compared our conclusions to available experimental data; we have reproduced earlier experimental data for a weak field and see similarities in our simulations and the strong-field data, further confirming our underlying model. We see strongly anisotropic behaviour in these systems and predict some interesting results for possible experiments on single-crystal samples. This work appears as part of the Ph. D. thesis of R. S.[9] Note. After this manuscript was prepared, we learned that the ordering discussed by us has recently been observed in simulations by den Hertog et al.[10] References

[1] A. P. Ramirez, Annu. Rev. Mater. Sci. 24, 453 (1994) [2] R. Siddharthan et al., Phys. Rev. Lett. 83, 1854(1999) [3] A. P. Ramirez et al., Nature 399, 333 (1999) [4] B. C. den Hertog and M. J. P. Gingras, Phys. Rev. Lett. 84, 3430 (2000)