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Θ CW trian- g sthe is < T ice to q=2 the sum on i, j runs over nearest-neighbour pairs and χ−1 h i non-generic q=3 the spins S are represented by classical vectors of unit length. J > 0 is the strength of the antiferromagnetic cooper- φ ative exchange. Let us first consider the case of a group of para- α q=4 para- q mutually interacting spins, for which the Hamiltonian magnet can be rewritten, up to a constant, as

J 2 HJ =J X Si Sj = L , (1) · 2 T q Θ T Θ where L Si is the total spin of the unit. CW F CW ≡ Pi=1 FIG. 2. Left: ’Susceptibility fingerprint’ of strongly frus- From this it can be read off that the ground states trated magnets. Top right: Units of q spins with total spin are those states in which the total spin L vanishes. The L = 0. The shaded pair of spins is rotated out of the plane appropriate configurations for Heisenberg spins are de- by an angle φ. Bottom right: The easy axes of a pyrochlore picted in Fig. 2. Note that, up to global rotations, the magnet. ground states for pairs and triplets of spins (q =2, 3) are non-degenerate, whereas there are two degrees of free- dom, α and φ, in the groundstate for a quartet of spins. main weak although the temperature is below the scale The origin of this difference is the following. For Heisen- set by the interactions. berg spins, the condition L 0 imposes three constraints This observation suggests a two step strategy for un- ≡ (Lx = Ly = Lz = 0), independent of q. The number of derstanding magnets in this class. For the cooperative degrees of freedom, however, increases with q. For q = 4, paramagnet, it should be sufficient to study a fairly sim- the constraints no longer suffice to determine the ground ple model system to capture the generic behaviour char- state uniquely, and the underconstraint shows up as the acterising this regime. Building on this, perturbations ground-state degrees of freedom. to the simple model Hamiltonian, appropriately chosen For a lattice built up of frustrated units, this argument for each compound, are introduced to describe the non- can be generalised: the dimension of the , D, generic regime. is given by the difference between the degrees of freedom, The remainder of this article adheres to this struc- F , and the ground-state constraints, K. Note that the ture in that we first identify and discuss an appropriate N 2 Hamiltonian can be written as H = J L , where class of model cooperative paramagnets and then con- J 2 Pα=1 α α runs over all N units. sider the effect of perturbations. In the process, we shall For spins with n components (n =2, 3 being XY and see that classical models of highly frustrated magnets Heisenberg spins, respectively), there are n ground-state have in common that, once the leading, frustrated ex- constraints per unit: the n-component vector L = 0. change interaction has been optimised energetically, a The number of degrees of freedom per unit is largest for large ground-state degeneracy remains. The collection lattices where the frustrated units share sites, since the of degenerate ground states (the ground-state manifold) degrees of freedom of each spin are only shared between provides no energy scale of its own and hence any per- two units in this case. The kagome lattice is thus made turbation has to be considered strong. Frustrated mag- up of corner-sharing triangles, whereas the pyrochlore nets are thus model strongly interacting systems. The lattice consists of corner-sharing tetrahedra. In addition, richness of their behaviour in the non-generic regime can more complicated lattices are possible, for example the be understood as a consequence of the non-perturbative SCGO lattice consisting of triangles and tetrahedra, or nature of any term added to the leading, frustrated ex- the GGG lattice made up of non-planar corner-sharing change Hamiltonian. triangles. For lattices of corner-sharing units of q spins with n components, one thus obtains D = F K = N[n(q 2) I. GROUND-STATE DEGENERACY OF − − − FRUSTRATED MAGNETS q]/2. The ground-state dimension D grows with n and q. For Heisenberg spins, it becomes extensive (D = N) at q = 4, i.e. the pyrochlore antiferromagnet has an exten- The main distinction between frustrated and unfrus- sive ground-state dimension. Since physically it is hard trated magnets appears to be the presence of a large to realise n> 3 or q> 4, Heisenberg magnets containing ground-state degeneracy in the former. In the following, corner-sharing tetrahedra are the realistic systems where we first give a description of how the degeneracy arises, the effects of frustration are strongest. and then provide a general quantitative determination of This argument has relied on the constraints being in- the size of the ground-state degeneracy based on a simple 7 dependent and mutually compatible. For example, the Maxwellian counting argument. ground state in a very strong magnetic field is always non- Our starting point is the classical nearest neighbour an- degenerate (all spins aligned), although the Hamiltonian tiferromagnetic Hamiltonian, HJ =J Si Sj , where Phi,ji · can still be written as a sum of squares of n-component

2 n S η cause of thermal fluctuations around them, which give a 1 S 2 different entropic weighting to each ground state. The S4 η 4 S ground states 3 y softer the fluctuations around a particular ground state, the larger the region of phase space accessible near it, and the more time the system spends fluctuating around this x state. It can now happen that the fluctuations around a 3 special state (or set of states) are so soft that the systems phase space at low temperature effectively spends all its time nearby. ordered state For this to occur, the of this special set must dominate the total entropy of all the other states taken 2 together. In Fig. 3, this is represented as the area of the q shaded region near a special point becoming much larger at low temperature than the total shaded area elsewhere 3 4 5 6 FIG. 3. Schematic view of the barrier-less ground-state taken together. manifold (black line) inside phase space. The x are the coor- In practice, the states with the softest fluctuations dinates within the ground-state manifold, the y parametrise tend to incorporate some degree of long-range order (see the perpendicular directions. The regions accessible at low Fig. 3), so that their selection implies an ordering transi- temperature are shaded. Order by disorder occurs in the re- tion. This phenomenon is thus known as order by disor- gion of small q and n. The inset shows a ’soft mode’ for a der, since order is induced by thermal fluctuations, which unit of four spins: this fluctuation has zero energy cost, E(η) are normally associated with a disordering tendency (and (only) in the harmonic approximation, due to the collinearity indeed, upon raising the temperature, the increasingly of the spins: E(η)/J = L2/2 = 2(1 − cos η)2 = η4/2. violent fluctuations do destroy the order they initially stabilised).11 The concept of order by disorder was initially proposed vectors, the components of which, however, cannot all be by Villain12 and Shender13, and it has received a great made zero. Also, it turns out that the Kagome lattice deal of attention in connection with the selection of copla- Heisenberg antiferromagnet also has D N, because of 14,15 ∝ nar order in kagome Heisenberg magnets. Order by dependent constraints. The counting argument can thus disorder, although counterintuitive, has since been found not be applied blindly. However, with some physical in- to be almost ubiquitous – after a hard search, the first put, its simplicity can be very useful. For the Heisenberg magnet shown to avoid ordering to my knowledge was pyrochlore magnet, for example, D/N = 1 can be proven one on a Bethe lattice.16 to be correct by an explicit construction of all ground 8 In Ref. 8, a theory based on the Maxwellian mode states, and a careful application to the kagome magnet counting described above was worked out to determine can even be used to gain non-trivial insights into its be- 9 the presence of order by disorder for the general case of haviour under dilution. In addition, Ramirez et al. have n-component spins arranged in corner-sharing units of explored generalising the concept of underconstraint to a q. The result is that ordered states, provided they exist, more general set of problems, including for example the are selected for q and n both small, as depicted in Fig. 3. phenomenon of negative thermal expansion as a result of This region includes all the realistic cases (q 4, n 3), underconstrained lattice degrees of freedom (“frustrated ≤ ≤ 10 with the exception of the case q = 4, n = 3 which is soft mode”). marginal. This system, the Heisenberg pyrochlore anti- ferromagnet, is now universally agreed to remain disor- dered at all T ,17,18,8 a result first suggested by Villain.17 II. ENERGY BARRIERS AND ORDER BY This concludes the first part of our program, namely DISORDER the quest for a robust cooperative paramagnet. We can choose a classical magnet with sufficiently large q and Of course, the ground state manifold has many impor- n and expect it to reproduce the qualitative features of tant properties besides its dimensionality. An important that regime faithfully, as has been done for the Heisen- one, for example, is its topology: can one continuously berg pyrochlore antiferromagnet8 and subsequently for a go from one ground state to any other, or are there en- large-n kagome magnet.19 The former concentrated on ergy barriers separating different ground states? There the low-temperature statistical mechanics and dynam- is no general answer to this question. In the case of the ics of a cooperative paramagnet, whereas the latter con- pyrochlore magnets, it can be shown that there are in- tains a detailed treatment of the thermodynamics of this deed no such barriers, so that the ground-state manifold regime. ’comes in one piece’ as depicted in Fig. 3.8 However, at any nonzero temperature, the quantity that is minimised is not the internal energy but rather the free energy. Whereas the internal energy of different ground states is the same, their free energy can differ be-

3 III. DYNAMICS ρ(ω)

Having established that the cooperative paramagnet 30 can explore a vast region in phase space even down to the Νδ(0) ω lowest temperatures, the question naturally arises how it J 20 1 χ in fact does so. This question about its dynamics is one J 1/ of the most intriguing aspects of cooperative paramag- B netism, but one which has not received its fair share of 10 E/J attention over the years, despite the fact that experimen- 0 4 'liquid' 0 1 2 3 4 5 6 7 tal studies of this problem are not at all uncommon. 0 'gas' T 0 2 4 6 8 The semiclassical equations of motion for a spin pre- T/J cessing in the exchange field set up by its neighbours FIG. 4. Top left: The density of states of the pyrochlore can be written as dS /dt = J S (L + L ), with αβ αβ α β antiferromagnet in the harmonic approximation. The peak ¯h = 1, and S being the spin− shared× by tetrahedra α αβ at ω = 0 represents the ground-state degrees of freedom. and β. This leads to a simple equation of motion for the 8 Non-trivial dynamics along the ground state is generated by L: dLα/dt = J Pβ Sαβ Lβ. anharmonic interactions.8 Bottom left: The phase diagram A pioneering− numerical× study of this dynamics was 20 of in a magnetic field along the [100] direction (after undertaken by Keren, who contrasted the behaviour Ref.48). Right: Energy (inset) and susceptibility per spin of of Heisenberg spins on the square and kagome lattices Heisenberg pyrochlore antiferromagnet: single-unit approxi- and who found that kagome lattice correlations decayed mation (line) and Monte Carlo simulations (diamonds). qualitatively more rapidly. This is because, in ordered magnets, the dynamics can usually be described satis- factorily by considering excitations around the ordered IV. THE SINGLE-UNIT APPROXIMATION AND structure only, since overall changes in the ordered struc- THE SUSCEPTIBILITY FINGERPRINT ture (rotation of the ordering direction) occur at ex- 21 ponentially longer timescales. In cooperative param- We have seen that cooperative paramagnets have a agnets, however, the motion from one ground state to short-range correlations both in space and time, even in another (parametrised by the x-coordinates), typically a regime where the temperature is below the energy scale not related by symmetry, can occur on relatively short set by interactions. One can thus hope that there might timescales. be a description in terms of variables which behave ap- These ground-state modes have zero frequency in the proximately as if they were decoupled. A good candidate harmonic approximation (see Fig. 4). On top of the ‘spin- set of variables are the total magnetisation vectors Lα of waves’ (locally parametrised by the y-coordinates), one the basic units: these are the variables appearing in the therefore has to consider anharmonic effects (which are of Hamiltonian, and the equations of motion (see above) course also of relevance for many other properties). This can also be cast in a simple form with their aid. In the was done in Ref. 8, noticing that at low temperatures a low-temperature limit, they follow simple equipartition, separation of timescales occurs. The three timescales of while at high temperatures, they provide a Curie-Weiss relevance were determined to be the bandwidth of the law. These properties are shared in quantitative detail, excitation spectrum (1/J), a typical excitation lifetime upon inclusion of factors of two to account for the de- (1/√T J), and the decay time of the autocorrelation func- composition, by the magnet on full lattice. tion (τ = 1/T ). The dynamics looks ’diffusive’: the au- The partition function for isolated units of q spins were tocorrelation function decays like a simple exponential: obtained in Ref. 24, and the pyrochlore magnet was ap- S(t)S(0) = exp( cTt), where c is a constant and t is proximated by a set of isolated tetrahedra. In Fig. 4, h i − time. the resulting expressions for energy and susceptibility are The salient feature of this result is that the decay time compared against Monte Carlo simulations. The agree- τ grows only algebraically (and not, for example, accord- ment is very satisfactory, the error being below 5% ev- ing to an Arrhenius law) as the temperature is lowered. erywhere for the susceptibility and much less than that Note that this leads to a width, Γ T , in inelastic neu- for the energy. ∝ tron scattering linear in T , which is close to what is Note that the shape of the susceptibility is that of the 22 observed in SCGO, to which this theory should also susceptibility fingerprint pictured in Fig. 2 in that it fol- apply. There is no sign of a phase transition, the coop- lows the Curie-Weiss law down to temperatures well be- erative paramagnetic phase extends all the way down to low ΘCW , before bending upwards (in this case, towards zero temperature. Note also that the excitation lifetime the exact T = 0 result for the susceptibility). The single- 23 is much shorter than in an ordered magnet because unit approximation thus provides a simple model result- the dynamics along the ground-state manifold induces a ing in a closed-form expression for the susceptibility of 8 powerful scattering mechanism. the cooperative paramagnet at all temperatures. This model has been extended to describe quantum spins on

4 25 frustrated lattices by Garcia-Adeva and Huber. HP and HJ lead to a rugged energy landscape with barrier heights of order P .29 The existence of barriers is necessary to account for the glassiness seen in many V. PERTURBATIONS compounds.30 Recent XAFS experiments on lattice dis- 31 order in one such pyrochlore compound (Y2Mo2O7) do In real systems, the validity of the nearest-neighbour suggest a distribution of bond lengths wide enough to classical Heisenberg exchange Hamiltonian HJ can at give rise to a substantial degree of bond disorder. best be approximate. The real Hamiltonian will symboli- cally have the shape H(P )= HJ +HP where HP denotes one of many possible ‘perturbation’ such as anisotropies, VI. QUANTUM FRUSTRATION quenched disorder, further-neighbour exchange, dipolar interactions, coupling to lattice degrees of freedom... If Of all perturbations, the introduction of quantum fluc- P is the energy scale (perturbation strength) attached to tuations is probably the most interesting as the ground- HP , we expect the theory of the cooperative paramagnet state wavefunction can turn out to be any linear combi- to be useful in the temperature regime P

5 tion lies beyond the scope of this article, and we now The curious feature of this transition is that it takes move on to quite a different subject, the magnetic ver- place between two states not differing in symmetry. In sion of ice. particular, the line of first order transitions in the B T plane terminates in a critical point, very much the− same way as happens in a conventional liquid-gas phase VII. SPIN ICE diagram, allowing a continuous path from the liquid to the gas without encountering any transition (Fig. 4). Probably the most remarkable recent experimental de- Secondly, two groups are studying the microscopic de- velopment has been the discovery of ‘spin ice’, in experi- tails and the resulting behaviour of the two titanate 43 ments on the compound Ho2Ti2O7, which was supple- compounds mentioned above. Both are incorporating 44 mented by another titanate compound, Dy2Ti2O7. long-range dipolar interactions in addition to the near- 50,52,53 Harris et al. noticed that a strongly anisotropic ferro- est neighbour exchange. Siddharthan et al. find 50 magnet on the pyrochlore lattice is frustrated, whereas an Ho2Ti2O7 to be in a partially ordered state, whereas antiferromagnet is not. The reason for this lies in the ori- den Hertog et al. conclude it to be a bona fide spin ice 52 entation of the easy axes, depicted in Fig. 2, which the compound. The origins of this disagreement are not spins are constrained to point along by the anisotropy. entirely clear and may be due to problems involved in The exchange Hamiltonian (Eq. 1) seeks to mimimise approximating the long-range nature of the dipolar in- (maximise) the total spin of the tetrahedron in case of an- teractions. tiferromagnetic (ferromagnetic) exchange. The two con- figurations with the spins pointing all in and all out on VIII. WHAT’S NOT HERE alternating tetrahedra have Lα = 0 everywhere. The an- tiferromagnetic ground state is thus only trivially degen- erate and appears unfrustrated. By contrast, there are In this article, I have tried to give a non-technical intro- exponentially many configurations maximising the total duction to and a short overview of the theory of strongly spin on each tetrahedron, namely all those with two spins frustrated magnets. I hope to have conveyed to the pointing in and two pointing out on each tetrahedron. reader the idea that this field is a rich one, and, if nothing This is equivalent to the ice model, as the ice rules state else, that a review article not constrained by size limits is that each has two protons sitting near and two far by now overdue. I had to skip many exciting topics; these from it in the ice structure. In this model, a spin point- include the unconventional heavy fermion behaviour in 56 ing out/in is taken to represent a proton sitting near/far LiV2O4, interactions of orbital and spin degrees of from an imaginary oxygen placed at the centre of the freedom,57–59 rigorous results on ground states of frus- tetrahedron.43 These ice states are in fact the ground trated quantum magnets60 and the many facets of quan- states of an on the pyrochlore lattice, which tum itinerant magnetism61,56 to name just a few. I have curiously had already been noticed by Anderson45 in the completely omitted any mention of one-dimensional sys- first discussion of frustration on the pyrochlore lattice in tems. Other overview articles can be found in Refs. 62,63. 1956. Thus we now have a magnetic compound which For magnets on the triangular lattice, there exists a cannot only be used to study ice physics but which also very thorough review by Collins and Petrenko.64 Clas- turns out to have a large number of other interesting sical frustrated Ising magnets on a wide range of lattices aspects.43–52 are reviewed in Ref. 35. Theoretical work has so far thrust in two directions. Firstly, unusual properties of spin ice as a model many- body system are being explored. For example, the spin ACKNOWLEDGEMENTS ice ground states are massively (discretely) degenerate, 35 giving the system an extensive zero-point entropy. Har- I am immensely grateful to John Chalker, who intro- 48 ris et al. noted that a magnetic field, B, applied in the duced me to this field. Many of the ideas presented here [100] direction can lift this degeneracy completely as a have originated with him. Other theorists who have gen- result of the orientations of the easy axes. The contribu- erously shared their knowledge with me over the years are tions of both the field and the entropy to the free energy John Berlinsky, Premi Chandra, Michel Gingras, Chris are extensive, the latter being weighted by the temper- Henley, Peter Holdsworth, David Huse, David Sherring- ature. At T = 0, the magnetic field energy thus dom- ton and Shivaji Sondhi. I am also very grateful for inates, and the ground state (termed an entropy-poor, the large number of discussions with members, past and “liquid” state) has the maximal magnetisation compati- present, of the experimental groups radiating outwards ble with the easy axis constraints. However, as T is in- from Bell, Edinburgh, Grenoble, McMaster, RAL and creased, the entropic contribution eventually dominates, Vancouver. Thanks are due for figures to Mark Harris and a first order transition to the entropy-rich (“gas”) (single tetrahedron), Oleg Petrenko (GGG) and Martin state ensues, with a discontinuous drop in the magneti- Zinkin (pyrochlore lattice). sation.

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