magnetic charge and ordering in kagome spin...

Phil. Trans. R. Soc. A (2012) 370, 5718–5737doi:10.1098/rsta.2011.0388

Magnetic charge and ordering in kagomespin ice

BY GIA-WEI CHERN1,2 AND OLEG TCHERNYSHYOV3,*1Department of Physics, University of Wisconsin, Madison, WI 53706, USA

2Institute for Complex Adaptive Matter, University of California, Davis,CA 95616, USA

3Department of Physics and Astronomy, Johns Hopkins University,Baltimore, MD 21218, USA

We present a numerical study of magnetic ordering in spin ice on kagome, a two-dimensional lattice of corner-sharing triangles. The magnet has six ground states andthe ordering occurs in two stages, as one might expect for a six-state clock model. In spinice with short-range interactions up to second neighbours, there is an intermediate criticalphase separated from the paramagnetic and ordered phases by Kosterlitz–Thouless (KT)transitions. In dipolar spin ice, the intermediate phase has long-range order of staggeredmagnetic charges. The high- and low-temperature phase transitions are of the Ising and3-state Potts universality classes, respectively. Freeze-out of defects in the charge orderproduce a very large spin correlation length in the intermediate phase. As a result ofthat, the lower-temperature transition appears to be of the KT type.

Keywords: spin ice; magnetic ordering; magnetic charge

1. Introduction

Spin ice is a ferromagnet with peculiar properties [1]. Its discovery in the rare-earth pyrochlore Ho2Ti2O7 by Harris et al. [2] attracted the interest of physicistsbecause geometric frustration results in an enormous number of ground states inspin ice. The degeneracy is similar to that of proton positions in water ice [3]and manifests itself in a large residual entropy at low temperatures measuredby Ramirez et al. [4]. Later, it was shown that spin ice has another peculiarfeature: even though it remains disordered as the temperature goes to zero, spincorrelations decay with the distance algebraically rather than exponentially [5,6].Such critical behaviour is characteristic of systems with constraints exemplified bylattice models with hardcore dimers. In the case of spin ice, at low temperatures,each tetrahedron of rare-earth ions is forced to have two spins pointing towardsthe centre of the tetrahedron and two away from it.

The current resurgence of interest in spin ice stems from an unusual characterof magnetic excitations in this class of materials [7,8]. Reversing a single spin ina spin-ice state violates the above-mentioned constraint on the two tetrahedra*Author for correspondence ([email protected]).

One contribution of 8 to a Theo Murphy Meeting Issue ‘Emergent magnetic monopoles in frustratedmagnetic systems’.

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Kagome spin ice 5719

(a) (b) (c) (d)+

−− −

+−

−−

+

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+

−

−

−

++ +

+−

−

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−

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++ +

+−

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Figure 1. (a) Generic spin-ice state on kagome. (b) A spin-ice (micro)state with magneticcharges ordered in a staggered patter. (c) Expectation values of spins and magnetic charges ina charged-ordered intermediate state. Note that the expectation values for the Ising variables are〈si〉 = ± 1

3 in this state; so the charges on triangles are 〈Qa〉 = ±1. (d) A ground state of kagomespin ice with ordered charges and magnetic dipoles. The dashed line is the perimeter of a magneticunit cell. (Online version in colour.)

sharing that spin. One of them has three spins pointing in and one pointing out,the other one spin pointing in and three out. By reversing additional spins, thetwo defects can be moved around independently from each other. These two defecttetrahedra carry equal and opposite magnetic charges defined as sources and sinksof magnetic field H. A defect with magnetic charge Q in an external magnetic fieldHext experiences a Zeeman force m0QHext. Two defects with magnetic charges Q1and Q2 interact with one another via a Coulomb potential m0Q1Q2/(4pr). Variousproperties of spin ice can be most naturally described by focusing on the motionof these defects [9–12].

The concept of magnetic charge is not exactly novel. It can be found in earlierresearch articles [13] and even in textbooks [14]. Magnetic charges in spin ice areremarkable because they are mobile and represent a rare example of fractionalizedexcitations in three spatial dimensions: the underlying degrees of freedom,magnetic dipoles, must be split in half, so to speak, to create magnetic monopoles.

It is worth pointing out that magnetic charges in spin ice are fundamentallydifferent from the magnetic monopoles introduced by Dirac [15] to explainquantization of electric charge. They are sources and sinks of magnetic field H,rather than of magnetic induction B. For that reason, there are no restrictionson the possible values of magnetic charges in spin ice.

In this paper, we illustrate the utility of the concept of magnetic charge forthe problem of spin ice on a different lattice. A natural generalization of thepyrochlore network is a lattice consisting of corner-sharing simplexes [16]. In twodimensions, corner-sharing triangles form kagome, figure 1. A spin is shared bytwo triangles and points along the line connecting their centres. The easy axesof the three spins on a triangle make angles of 120◦ with each other. Like on thepyrochlore lattice, interactions of the antiferromagnetic kind are not frustrating:an isolated triangle has two ground states with all spins pointing in or all pointingout. A kagome lattice with nearest-neighbour antiferromagnetic interactions alsohas two ground states, in which spins point into triangles of the same orientation.On the other hand, ferromagnetic interactions are frustrating and yield six groundstates on a single triangle, with two spins pointing in (and one out) or two pointingout (and one in). The number of ground states grows exponentially with thenumber of spins when interactions are ferromagnetic and only include nearest

Phil. Trans. R. Soc. A (2012)



5720 G.-W. Chern and O. Tchernyshyov

neighbours. In that, spin ice on kagome is quite similar to its analogue on thepyrochlore lattice. However, the addition of longer-range interactions, especiallydipolar ones, reveals some important differences between the two systems.

The main difference between the pyrochlore network and kagome concernsthe allowed values of magnetic charge Q of a simplex defined as the fluxof magnetization M into the simplex. For a tetrahedron, Q is even (in theappropriate units): zero in a two-in-two-out state, ±2 in three-in-one-out orthree-out-one-in states and ±4 in the all-in or all-out states. The energy ofspin ice penalizes states with a large (in absolute value) charge Q so thatsimplexes are neutral, Q = 0, in low-energy states. The lack of magnetic chargeson tetrahedra explains a puzzling observation by Gingras & den Hertog: spin-ice states remain very nearly degenerate, even in the presence of long-rangedipolar interactions [17]. As Castelnovo et al. [8] showed, the energy of dipolarinteractions is well approximated by the Coulomb energy of magnetic chargesresiding on simplexes. Because these charges vanish in two-in-two-out states, theirCoulomb energy is the same. The residual interactions, neglected in the Coulombapproximation, force the system into a phase with long-range magnetic order attemperatures low compared with the strength of dipolar interactions [18].

By contrast, the magnetic charge of a triangle is ±1 in a spin-ice state and ±3when its spins point all in or all out. Even if the interactions tend to suppressmagnetic charges, the lowest (in magnitude) values of charge on a triangle are±1. This has important consequences that we briefly outline below and considerin detail in the rest of the paper.

First, the presence of a charge degree of freedom on simplexes makes spin iceon kagome much more fluid than its pyrochlore counterpart. In pyrochlore spinice, spin dynamics slows down considerably at low temperatures. Single-spin flipsresult in the creation of two magnetic charges and thus become forbidden whenthe temperature falls below the energy cost of a monopole. Moves within the low-energy sector require flipping entire chains of alternating spins with a minimallength of six [18]. Alternatively, spin fluctuations can be mediated by the motionof a few remaining magnetic monopoles. Experimental studies confirm a very slowrelaxation in spin ice at liquid-helium temperatures [19,20], although the primarymechanism of low-T spin dynamics in the rare-earth pyrochlores remains to beclarified. No such impediment to spin fluctuations exists in kagome ice because asingle-spin flip does not necessarily take the magnet out of the low-energy sector.Such a local move is allowed if the magnetic charges of the two simplexes sharingthe spin change from +1 and −1 to −1 and +1. Although at present, thereare no materials realizing two-dimensional kagome spin ice, several experimentalgroups have made artificial magnetic arrays with this geometry [21]. Qi et al. [22]demonstrated that their artificial spin ice on kagome stayed strictly within thelow-energy sector in which the magnetic charges remain ±1. Ladak et al. [23]observed triple charges, but this is likely the result of strong quenched disorderin their samples [24].

Second, the presence of a charge degree of freedom opens a possibility ofnew kinds of ordered phases in spin ice. Pyrochlore spin ice is expected tohave only two thermodynamics phases. As the temperature is lowered from+∞, the paramagnetic state gradually crosses over to the spin-ice state, whichstill preserves all of the symmetries of the Hamiltonian and thus remains aparamagnet, albeit a correlated one, from the standpoint of the Landau theory. At





a sufficiently low temperature, the residual spin interactions not captured by theCoulomb model [8] induce a phase transition into an ordered state that breaks thetime-reversal and lattice symmetries [25]. On kagome ice, one has good reasonsto expect an intermediate ordered phase with staggered magnetic charges thatminimizes the Coulomb energy [26,27]. This phase is characterized by very strongspin fluctuations. At the lowest temperatures, the symmetry is broken further,and the system settles into a state with long-range magnetic order.

To test these heuristic considerations, we have performed extensive numericalstudies of spin ice on kagome, both with short-range interactions (nearest andnext-nearest neighbours) and with long-range dipolar interactions. The problemturned out to be subtle and the answers were found to be different for the modelswith short- and long-range interactions. In both cases, there is an intermediatephase. However, the nature of the intermediate phase is drastically different.The dipolar spin ice has the expected charge-ordered phase, whereas the short-range model has a critical intermediate phase without true long-range order.The determination of the universality classes of the phase transitions requiredrather large system sizes because some of the transitions were of the Kosterlitz–Thouless (KT) type. Simulating large systems is particularly difficult in thepresence of long-range interactions.

2. Kagome spin ice: the model

(a) The Hamiltonians

The model of spin ice on kagome was first introduced by Wills et al. [28]. It hasIsing spins Si = si ei living on the vertices of corner-sharing triangles of a kagomelattice. The unit vector ei points along the line connecting the centres of triangles,while the Ising variable si encodes the state of the spin. It is convenient to choosethe unit vectors ei in such a way that they point into triangles of one orientation(say, �). The Hamiltonian of the model with nearest-neighbour (nn) exchange J1can be written in two ways,

H1 = −J1

∑nn

Si · Sj = J1

2

∑nn

sisj , (2.1)

where we relied on the result ei · ej = − 12 for nearest neighbours. As

usual, a ferromagnetic exchange J1 > 0 for spins Si translates into an antiferro-magnetic interaction for the Ising variables si . It is well known that the Isingantiferromagnet on kagome remains paramagnetic down to zero temperature [29].It retains a finite entropy density in the ground state, 0.502 per spin [30].

To understand this residual spin degeneracy, we rewrite H1 in terms of magneticcharges of triangles

H1 = J1

4

∑a

Q2a. (2.2)

Here, we have neglected an irrelevant constant. The magnetic charge is defined as

Qa = ±∑i∈a

si , (2.3)





with the plus sign for one type of triangles and the minus for the other.As discussed earlier, the allowed values of magnetic charges are Qa = ±3 and±1. Equation (2.2) shows that triangles with triple charges are energeticallydisfavoured. Consequently, as temperature is lowered, the magnet gradually entersthe spin-ice phase consisting exclusively of triangles with Qa = ±1, correspondingto the two-in-one-out and one-in-two-out ice rules on kagome. The number of thespin-ice microstates grows exponentially with the number of spins. Invoking thefamous Pauling estimation gives an entropy density Sice = (1

3) ln(92) = 0.5014 per

spin, which agrees very well with the numerical results.Wills et al. [28] added interactions between next-nearest neighbours (nnn),

H2 = −J2

∑nnn

Si · Sj = J2

2

∑nnn

sisj . (2.4)

They considered both signs of J2, but we are interested in the antiferromagneticcase J2 < 0. Möller & Moessner [26] and Chern et al. [27] added long-rangedipolar interactions,

Hd = Dr3nn

2

∑i �=j

sisj(ei · ej) − 3(ei · rij)(ej · rij)

|ri − rj |3 , (2.5)

where D = m0m2/(4pr3nn) is the characteristic strength of dipolar coupling, ri are

spin locations, rij = (ri − rj)/|ri − rj |, m is the magnetic dipolar moment of a spinand rnn is the distance between nearest neighbours. For brevity, we shall refer tothe model of Wills et al. [28], with the Hamiltonian H1 + H2 and antiferromagneticJ2 < 0, as the short-range kagome ice and call the model with dipolar interactions,with the Hamiltonian H1 + Hd, the long-range kagome ice.

(b) The ground states

In both models, interactions between Ising variables si are antiferromagneticfor nearest neighbours and ferromagnetic for next-nearest neighbours. On thebasis of that, one might expect the same type of magnetic order at lowtemperatures. The ground states for the Ising model on kagome with first- andsecond-neighbour interactions are known exactly [31,32]. They have an enlarged(√

3 × √3) magnetic unit cell with nine spins. The corresponding ground states of

the ice model are shown in figure 2a. With the aid of a complex-order parameter,

M = 1N

∑i

si exp (iQ · ri), (2.6)

where Q = (4p/3, 0) and N is the number of spins, we can see that they aresimilar to those of the six-state clock model. The phase of the order parameterM = |M |eif takes on the six values f = np/3, where n is an integer (figure 2a).

We were able to show that the long-range model with dipolar interactionshas the same ground states. To do so, we treated the Hamiltonian H1 + Hd =siAijsj/2 as a quadratic form in the Ising variables. We replaced the constraintsi = ±1 with a less stringent normalization condition, sisi = N , where N is thenumber of sites, and minimized the quadratic form by finding the lowest (mostnegative) eigenvalue of the matrix Aij . Although this procedure minimizes the





(b)

ReMReM

ImM ImM

(a)

Figure 2. (a) The ground states of the dipolar spin ice on kagome have a complex-order parameterM = |M |eif with f = np/3. (b) A proposed intermediate phase [28,31] has f = (n + 1/2)p/3. Inboth cases, the magnetic unit cell contains nine sites. (Online version in colour.)

energy over an enlarged space of states, the method gives the right answer if theeigenstate with the lowest eigenvalue has si = ±1. That is indeed the case for theHamiltonian H1 + Hd. The corresponding eigenstates are the six ground states ofthe Ising model on kagome.

(c) Phase transitions: general considerations

Previous theoretical studies have revealed that the six-state clock model intwo dimensions may order in a number of scenarios [33]. In all of them, there is ahigh-temperature disordered phase with 〈M 〉 = 0 and a low-temperature orderedphase with 〈M 〉 �= 0.

(i) Two KT transitions with an intermediate critical phase in which 〈M 〉 = 0and spin correlations decay as a power of the inverse distance.

(ii) From the paramagnetic phase, the system enters a partially ordered phasevia an Ising transition. A second transition of the 3-state Potts universalityclass takes the system to the ordered state. The intermediate phase has anIsing-order parameter 〈M 3〉 �= 0, whereas 〈M 〉 = 0.

(iii) Similar to (ii) but with the Ising and Potts transitions exchanged. Theintermediate phase has a 3-state Potts-order parameter 〈M 2〉 �= 0, whereas〈M 〉 = 0.

(iv) A discontinuous phase transition between the paramagnetic and fullyordered phase.

Numerical studies [31,32] provide compelling evidence for scenario (i) with twoKT transitions in short-range Ising models on kagome. Takagi & Mekata [31]mistakenly ascribed a non-zero-order parameter 〈M 〉 to the intermediate phase,which is expected to have quasi-long-range order with power-law spin correlationsand 〈M 〉 = 0.





Wills et al. [28] also suggested an intermediate ordered phase with six possiblestates shown in figure 2b. They are equivalent to the intermediate states ofTakagi & Mekata and have a non-zero-order parameter 〈M 〉 with f = (n +1/2)p/3. Such an intermediate state is clearly inconsistent with any of the fourscenarios mentioned earlier.

The intermediate critical phase in scenario (i) is most simply understood inthe continuous version of the clock model, namely the XY model with a sixfoldanisotropy term a6 cos 6f in the free-energy functional. This term is irrelevantin the renormalization-group sense above the lower phase transition, so that theintermediate phase behaves as if the sixfold anisotropy were absent. In principle,there is a possibility that the sixfold anisotropy changes sign: at low temperatures,a6 < 0 yields the states shown in figure 2a, whereas at higher temperatures, a6 > 0stabilizes the states shown in figure 2b. However, if the intermediate phase isindeed ordered and fully breaks the Z6 symmetry, then the transition betweenthat phase and the fully symmetric paramagnet should follow one of the abovescenarios, which requires yet another intermediate phase or a discontinuous phasetransition. We see no evidence for either possibility.

Our proposal of the intermediate state with ordered magnetic chargescorresponds to scenario (ii). The charge-ordering transition is of the Ising type. Ittakes the system from the paramagnetic phase into a state where the Ising-orderparameter 〈M 3〉 has a non-zero expectation value. If 〈M 3〉 = 〈|M |3e3if〉 > 0, thestate can be thought of as a mixture of the clock states with f = 0, 2p/3 and−2p/3 (figure 2a). At a lower temperature, the system selects one of these statesin a transition of the 3-state Potts universality class.

To test which of the scenarios are realized in kagome spin ice, we haveperformed large-scale numerical simulations of both the short- and long-rangemodels with the Hamiltonians: H1 + H2 and H1 + Hd.

3. Numerical simulations

(a) Kagome ice with short-range interactions

Here we provide numerical evidence for scenario (i), namely two successive KTtransitions, in the short-range ice model on kagome. A standard Metropolisimportance sampling was used to simulate the behaviour of the model on L × Lunit cells with periodic boundary conditions; the number of spins N = 3L2. In asingle Monte Carlo step, each spin in the lattice is updated once with a probabilityp = min(1, exp(−DE/kBT )), where DE is the energy change of flipping a spin. It isworth noting that the single-spin update is sufficient for both the spin-ice regimeand even the intermediate critical phase; the corresponding acceptance rates areroughly 30 per cent and 15 per cent, respectively. About 10 000 initial MonteCarlo sweeps were discarded to allow the system to equilibrate. We then retainedresults from 105 to 106 Monte Carlo sweeps for computing averages. Our mostcomplete dataset was taken with parameters J2 = −J1/3, for which we describeour analysis in detail in the following.

Figure 3 shows the temperature variation of specific heat, staggered chargeorder and magnetic-order parameters for different lattice sizes. While theappearance of two peaks in the specific-heat curve suggests two phase transitions,the locations and amplitudes of the peaks vary only slightly with lattice size,





0.60

(a) (b) (c)

C Q

0.45

0.30

0.15

00.5 1.0 1.5

T/J1

1.00

0.75

0.50

0.25

00.5 1.0 1.5

T/J1

M

0.20

0.15

0.10

0.05

00.5 1.0 1.5

T/J1

L = 1224486072

L = 1224486072

L = 1224486072

Figure 3. Temperature dependence of (a) specific heat C = (〈E2〉 − 〈E〉2)/NkBT 2, (b) magneticcharge Q and (c) magnetic-order parameter M for varying lattice sizes obtained from Monte Carlosimulations of the short-range spin-ice model. The simulation was done with J2 = −J1/3. Theshaded area indicates the intermediate critical phase. The transition temperatures Tc1 = 0.735J1and Tc2 = 0.845J1 are determined using the finite-size scaling relation equation (3.4). (Onlineversion in colour.)

a feature characteristic of KT transitions. The intermediate critical phase,determined by finite-size scaling to be discussed below, is marked by the shadedregion in the figures. The charge-order parameter is defined as the difference ofmagnetic charges on the two different types of triangles,

Q = 12L2

∑a

(−1)aQa, (3.1)

where (−1)a = ±1 for up and down triangles, respectively, and 2L2 is the numberof triangles in the lattice. The charge order is proportional to the Ising-orderparameter, 〈Q〉 ∝ 〈M 3〉. As can be seen from figure 3b, the staggered charge orderdecreases rapidly with increasing L above the low-T transition, indicating thatmagnetic charges remain disordered in the intermediate phase.

On the other hand, the magnetic-order parameter shows quite different finite-size behaviour across the three different regimes (figure 3c). Above the high-Tpeak, the rapid decrease of M with increasing L is consistent with a magneticallydisordered phase, whereas a negligible finite-size effect implies an ordered statebelow the lower transition at Tc1. In the intermediate regime, the order parameterM falls off rather slowly with increasing system size. These observations aresummarized in figure 4, which shows magnetic-order parameter M as a functionof L at different temperatures. The magnetic-order parameter extrapolates tozero at high temperatures, whereas it levels out to a constant at low T . In theintermediate regime, the linear behaviour in the double-logarithmic plot indicatesa power-law dependence

M ∝ L−h/2, (3.2)





0.18

M

0.16

0.14

0.12 0.24

0.20

0.16

0.120.76 0.80 0.84

0.10

15 30 60L

T/J1

h

120 240

0.90 0.890.88

0.87

0.86

0.850.840.830.820.810.800.790.780.770.760.750.740.730.720.71

0.70

0.69T = 0.68

Figure 4. The log–log plots of order parameter M as a function of system size L at varioustemperatures. The solid curves indicate the linear behaviour that corresponds to a power-lawdependence, M ∝ L−h/2, in the intermediate critical phase. The inset shows the extracted criticalexponent h as a function of temperature in the critical regime. The two data points representedby symbol � are obtained using the finite-size scaling equation (3.4). (Online version in colour.)

which is consistent with an infinite correlation length in a critical phase. Theextracted value of critical exponent h as a function of temperature is shown inthe inset of figure 4.

As discussed in §2c, the sixfold anisotropy a6 cos 6f in the Landau expansionof the free-energy functional is irrelevant in the intermediate critical phase.To confirm this, we show in figure 5 the distribution of instantaneous-orderparameter M = |M | eif in the complex (ReM , ImM ) plane at various tempera-tures. As can be seen in figure 5c, which corresponds to a temperature in theintermediate regime, the order parameter has a finite amplitude |M | �= 0 owingto a finite system size (L = 180). More importantly, a continuous O(2) symmetryemerges in this critical phase. In the ordered phase below Tc1, the anisotropy termreasserts itself and restores the Z6 symmetry, as demonstrated in figure 5a,b.

To further corroborate the proposed scenario of two KT transitions, we resortto the method of finite-size scaling [34]. For a KT transition, the correlation lengthnear the critical temperature Tc diverges as [35]

x ∝ exp(at−1/2), (3.3)

where a is a non-universal constant and t = |T − Tc|/Tc is the reducedtemperature. The order parameter and its susceptibility exhibit power-law





0.2(a) (b) (c) (d)

0.1

0

–0.1

–0.20.20.10–0.1–0.2 0.20.10–0.1–0.2 0.20.10–0.1–0.2 0.20.10–0.1–0.2

Figure 5. Distribution of instantaneous magnetic-order parameter M in the complex plane(ReM , ImM ) for different temperatures. (a) T = 0.5J1, (b) T = 0.7J1 (c) T = 0.8J1 and (d) T = J1.(Online version in colour.)

behaviour, M ∝ x−h/2 and cM ∝ x2−h, in a KT transition. For finite systems, thesingular part of the free energy is expected to depend only on x/L. The orderparameter and susceptibility are assumed to have the functional forms

M = L−h/2M(

x

L

)and cM = L2−hX

(x

L

), (3.4)

where M and X are unknown universal functions. At the critical point,substituting x = ∞ into equation (3.4) gives the power-law relation (3.2). For anextended regime consisting of a line of critical points, the power law M ∼ L−h/2

should hold over the entire temperature range from Tc1 to Tc2, which is indeedobserved in our simulation, figure 4.

The finite-size scaling relations (3.4) can also be used to determine the upperand lower critical temperatures of the intermediate KT phase. For example, withan appropriately chosen set of parameters a, b and Tc1, the plot of MLb versusL−1 exp(a/

√Tc1 − T ) should collapse on a universal curve for different system

sizes. The same should hold for cML−c versus L−1 exp(a/√

T − Tc2). The twoconstants, b and c, are related to the critical exponent: b = h/2 and c = 2 − h.As shown in figure 6, excellent data collapse was obtained using the following setof parameters: a = 1.22, b = 0.0615, c = 1.746, Tc1 = 0.735J1 and Tc2 = 0.845J1.From this, we estimated the critical exponent at the two transition temperatures:h(Tc1) = 2b = 0.123 and h(Tc2) = 2 − c = 0.255. These values agree reasonablywell with those extracted by linear fitting of the power-law relation (3.2) withinthe critical phase (see the inset of figure 4). Both values are close to the theoreticalpredictions 1

9 and 14 for the six-state clock model [36], and the slight overestimation

could be due to the finite-size effects on KT transitions.

(b) Kagome ice with dipolar interactions

Although the single-spin Metropolis algorithm is quite efficient for simulatingshort-range kagome ice, it suffers from a dynamical freezing in the charge-orderedphase. A similar problem arises in the low-temperature simulation of pyrochlorespin ice [18] in which a single-spin flip violates the ice rules, leading to a largeenergy cost and low acceptance rate of the updates. To overcome this problem,we added the non-local loop moves first introduced by Barkema & Newman [37]for square ice models. In a non-local update, a loop is first formed by randomly





ML

b

c ML

–C

0.28

(a) (b)

0.26

0.24

0.22

0.05L = 1218243048607290

120150180

L = 243048607290

120150180

0.04

0.03

0.02

0.01

010–1 102 103 1041 10 10–1 102 103 1041 10

L–1 exp(a/÷Tc1 – T ) L–1 exp(a/÷T – Tc2)

Figure 6. (a) Finite-size scaling of the order parameter M for the lower-temperature transition atTc1. (b) Finite-size scaling of susceptibility cM = (〈M 2〉 − 〈M 〉2)/NkBT for the higher-temperaturetransition at Tc2. The following values of the parameters are used: a = 1.22, b = 0.0615, c = 1.746,Tc1 = 0.735J1 and Tc2 = 0.845J1. (Online version in colour.)

tracing a path through triangles satisfying the ice rules, alternating between spinspointing in and spins pointing out of the triangles. The process is completed whenthe path closes upon the starting spin or encounters any spin already included inthe loop. Flipping all the spins in such a loop leaves the magnetic charges {Qa}intact and conserves the nearest-neighbour energy H1. The loop move results ina small gain or loss of the dipolar energy DEd, and the update is accepted withthe probability p = min(1, exp(−DEd/kBT )).

We used a combination of single-spin flips and loop moves to simulate the long-range ice model H1 + Hd. With periodic boundary conditions, dipolar interactionswere summed over periodic copies up to a distance of 500L. Most of the resultspresented here are obtained with a dipolar coupling D = 2J1. Figure 7 shows thetemperature dependence of specific-heat-, charge- and magnetic-order parametersfor various system sizes. Similar to the case of short-range kagome ice, two peakscan be seen in the specific-heat curves, indicating two phase transitions. While thehigh-temperature peak becomes sharper with increasing L, the low-T transitionbarely shows any finite-size dependence. The behaviours of charge- and magnetic-order parameters shown in figure 7b,c are consistent with scenario (ii) discussedin §2. We discuss both transitions below.

The staggered charge order corresponds to the partially ordered phase inscenario (ii) that is characterized by an Ising-order parameter 〈Q〉 ∝ 〈M 3〉. Thecharge-ordering transition is thus expected to be in the universality class of theIsing model. To verify this conjecture, we performed finite-size scaling analysisand found excellent data collapse, figure 8, using the critical exponents of thetwo-dimensional Ising universality class. The critical temperature Tc2 = 0.29Dis determined from the crossing of Binder’s fourth-order cumulant for differentlattice sizes.





C

(a) (b) (c)

1.40

1.05L = 9

12152430 L = 9

1215243036

L = 91215243036

0.70

0.35

0

Q

1.00

M

0.20

0.15

0.10

0.05

0

0.75

0.50

0.25

00.15 0.20 0.25

T/D0.30 0.35 0.15 0.20 0.25

T/D0.30 0.35 0.15 0.20 0.25

T/D0.30 0.35

Figure 7. Temperature dependence of (a) specific heat C = (〈E2〉 − 〈E〉2)/NkBT 2, (b) charge Qand (c) magnetic M order parameters for varying system sizes. The simulation was done withD = 2J1. (Online version in colour.)

The origin of an intermediate phase with ordered charges can be understoodby expressing the energy of the dipolar ice in terms of magnetic charges. This isachieved through the so-called dumbbell approximations [8], in which the dipolesare stretched into bar magnets of length a = (2/

√3)rnn such that their poles meet

at the centres of triangles,

E({Qa}) =∑

a

v0

2Q2

a + m0q2m

8p

∑a�=b

QaQb

|ra − rb| . (3.5)

Here, qm = m/a is the magnetic charge of the dumbbell, m is the moment ofthe spin. The self-energy v0 = J /2 + (11 + 3

√3)D/8 for kagome ice. It can be

shown that the dipolar energy H1 + Hd is well approximated by equation (3.5),with corrections that vanish with distance at least as fast as 1/r5 for eachdipole pair.

The dumbbell approximation also illustrates an important difference betweenpyrochlore and kagome ices. Because the allowed charges are even on atetrahedron and odd on a triangle, minimization of the self-energy term resultsin Qa = 0 on the pyrochlore lattice and ±1 on kagome. The conditions of minimalallowed charges on simplexes correspond to the ice rules on the respective lattices.For pyrochlore ice, the vanishing of magnetic charge also makes the Coulombinteraction, the second term in equation (3.5), strictly zero. This observationexplains why all states satisfying the ice rules are essentially degenerate inpyrochlore spin ice over a wide range of temperatures [17]. The degeneracy islifted by the residual interactions neglected in the Coulomb approximation anda long-range magnetic order with wavevector Q = (0, 0, 2p) sets in at a lowertemperature compared with the dipolar energy scale [18].

The situation on kagome is quite different as non-zero charges generate amagnetic field. This results in substantial energy differences between statesobeying the ice rule Qa = ±1. Consequently, interactions between uncompensated





B4Q

CL

–a/n

QL

b/n

c QL

–g/n

0.6

(a) (b)

(c) (d)

L = 91215243036

L = 9121524303642

L = 9121524303642

L = 121524303642

1.2

0.8

0.4

0.6

0.4

0.2

0

0

0.40.6

0.5

0.4

0.26 0.28 0.30 0.32T/D

0.2

1.2

0.8

0.4

0

0

–2 –1 0(T – Tc)L

1/n1 2 –2 –1 0

(T – Tc)L1/n

1 2

–2 –1 0 1 2 –2 –1 0 1 2

Figure 8. Finite-size scaling of (a) Binder’s fourth-order cumulant B4Q = 1 − 〈Q4〉/3〈Q2〉2,(b) specific heat, (c) staggered charge-order parameter and (d) charge susceptibility cQ = (〈Q2〉 −〈Q〉2)/NkBT for the high-temperature transition. The inset in (a) shows Binder’s cumulant ofdifferent lattice sizes crossing at the critical temperature Tc2 = 0.29D. The critical exponentsof the two-dimensional Ising model a = 0, b = 1

8 , g = 74 and n = 1 are used. (Online version

in colour.)

charges induce an Ising transition into a state with staggered charge order thatminimizes the Coulomb energy. This partially ordered phase is closely related tothe ice states of pyrochlore spin ice in a 〈111〉 magnetic field [38,39]. Triangleswith positive (negative) charges on kagome correspond to three-in-one-out(three-out-one-in) tetrahedra of the pyrochlore lattice.

Spins remain disordered in the charge-ordered ice phase, as evidenced bythe loop moves discussed at the beginning of this section. Such non-local loopupdates flip spins on a closed path while maintaining the charge configuration.To quantify this residual degeneracy, we note that each triangle in a state withperfect charge order has two majority spins pointing into (or out of) the triangle





C

0.14

(a) (b)

0.12

0.08

0.04

0.2

0.1M

02 4

0

0.12

0.10

L = 912152436

L = 12306090

150

0.08

0.060.12 0.16 1 2 3

T/D T/u2

0.20

Figure 9. Specific heat as a function of temperature for (a) the dipolar ice model on kagome and(b) the dimer model with next-nearest-neighbour attraction on the hexagonal lattice. The insetin (b) shows the temperature variation of the order parameter characterizing the

√3 × √

3 dimerorder. (Online version in colour.)

and a minority spin pointing the other way. Such states can be mapped to dimercoverings on the honeycomb lattice by identifying the minority spins with thedimers. The number of dimer coverings on a honeycomb grows exponentiallywith the lattice size, giving rise to a residual entropy density S = 0.108 perspin [38].

The remaining entropy of the charge-ordered phase is completely removedby a magnetic phase transition corresponding to the low-T peak of thespecific-heat curve (figure 7a). The magnetic order is shown in figure 1d andis again characterized by order parameter M ; it has an enlarged

√3 × √

3unit cell containing nine spins. Similar to pyrochlore spin ice, the magneticordering is induced by residual interactions beyond the dumbbell approximationequation (3.5).

As the Z6 symmetry of the spin-ice Hamiltonian is reduced to a threefoldsymmetry in the charge-ordered phase, the magnetic transition is expected to bein the universality class of 3-state Potts model. However, Monte Carlo simulationson systems up to L = 36 fail to turn up any signature of the Potts criticality.Instead, the lack of a singularity in the specific heat, figure 9a, seems to beconsistent with a KT transition.

(c) Dimers on a honeycomb lattice

To shed light on this puzzling result, we consider a similar transition inthe honeycomb dimer model. As discussed above, the charge-ordered statescan be uniquely mapped to dimer coverings on the honeycomb. To inducethe corresponding

√3 × √

3 ordering of dimers, we introduce a next-nearest-neighbour attractive interaction between the dimers. Note that nearest-neighbour





dimers are precluded by the hardcore constraints that each lattice site isattached to exactly one dimer. The partition function of the modified dimermodel reads

Z =∑

Cexp

(v2N2(C)

kBT

), (3.6)

where the sum is over all dimer coverings of the hexagonal lattice and N2(C)counts the number of next-nearest-neighbour pairs in a given covering C.An attractive interaction between next-nearest-neighbour dimers correspondsto v2 > 0.

We used the direct-loop Monte Carlo algorithm [40] to simulate the dimermodel (3.6). The non-local update is performed by initially breaking up anarbitrary dimer into a pair of monomers and then moving one monomer across thelattice by flipping a sequence of dimers along a path. The process is completed,and a new dimer configuration is created once the two monomers meet andrecombine with each other. Because detailed balance is maintained at each stepof the monomer’s movement, a large number of dimers can be efficiently updatedwithout rejection. This method significantly reduces the autocorrelation time inthe Monte Carlo process and allows us to simulate large lattices up to L = 150.Figure 9b shows the specific heat versus temperature for various system sizes;also shown in the inset is the temperature dependence of the order parametercharacterizing the

√3 × √

3 dimer order. The rather weak finite-size dependenceand a lack of singularity in specific heat again shows that the dimer orderingis characterized by a KT transition similar to the case of the dimer model withaligning interactions on the square lattice [41].

The appearance of a KT transition indicates the critical nature of thedisordered hardcore dimer phase, or equivalently the charge-ordered ice phase.Indeed, by further mapping the dimer covering to a ‘height’ field [42], the dimermodel is described by a sine–Gordon model in the coarse-grained approximation.It is well known that the height model undergoes a KT transition into a roughphase at high temperatures [43]. As the confining cosine potential term isirrelevant at high T , the rough phase is described by a Gaussian field theorywith critical correlation functions in two dimensions.

The existence of such a critical phase relies on the absence of charge defects.At finite temperatures, however, thermally excited defects spoil the mappingto hardcore dimer coverings. There are two types of defects in the charge-ordered phase: triangles with ±3 charges (ice defects) and triangles that violatethe charge order (charge defects). In the dimer model, the triply charged sitesbecome monomers, whereas defects in charge order become sites with two dimers,figure 10. We ignore the triply charged defects in the intermediate phase onaccount of their high energy cost and focus on the charge defects. To that end,we modified the honeycomb model (3.6) by allowing dimer pairs with fugacityz = exp(−3/kBT ). The hardcore dimer model is recovered in the limit 3 → ∞.Figure 11 shows the temperature variation of the

√3 × √

3 order parameter,susceptibility and density of dimer pairs for 3 = 2, 10 and ∞ in units of v2.The model with larger fugacity for dimer pairs shows a dramatically differentbehaviour from that of the hardcore dimer covering (figure 11).





(e)(a)(i)

(ii)

(b) (c) (d)

Figure 10. (i) Spin configurations. (ii) Respective dimer configurations. (a) Reference state withcharges Qa = ±3 on the two sublattices. (b) A charge-ordered state with Qa = ±1. (c–e) Two chargedefects (DQa = ±2) are created and pulled apart. Arrows denote the minority spins. (Online versionin colour.)

M cM

rd

0.20

(a) (b) (c)

0.15

e = 2

•10

0.10

0.05

0.20 0.30

0.15

0

0.15

0.10

0.05

01 2 3

T/u2

4 5 6 1 2 3T/u

2

4 5 6 1 2 3T/u

2

4 5

e = 2

•10

e = 2

•10

Figure 11. Temperature variation of (a) order parameter M , (b) susceptibility cM = (〈M 2〉 −〈M 〉2)/NkBT and (c) density of the dimer pair (charge defect) for phase transitions to a long-range

√3 × √

3 dimer order. The three curves correspond to different fugacities z = exp(−3/kBT )for the charge defects. The value 3 = ∞ corresponds to models with hardcore dimer covering. Thesimulations were done on a honeycomb lattice with L = 12. (Online version in colour.)

For 3 < +∞, a finite density of dimer pairs sets a bound on the dimercorrelation length, thus altering the criticality of magnetic ordering to theuniversality class of 3-state Potts model. To confirm this, we conducted a finite-size scaling study on the dimer model with 3 = 2v2. As can be seen from figure 12,one obtains excellent data collapse with the critical exponents of the two-dimensional 3-state Potts model. However, when the average separation betweencharge defects exceeds the lattice size, we are back to hardcore dimers with power-law spatial correlations for all distances. This explains the KT-like behaviourobserved in spin ice at small lattice sizes. The critical behaviour characteristic ofthe 3-state Potts universality only reveals itself for sufficiently large systems [44].





B4M

ML

b/n

c ML

–g/n

CL

–a/n

0.6

(a) (b)

(c) (d)

0.5

0.66

L = 120.4

0.3

0.2

0.1

182436

L = 12182436

0.60

0.54

1.2 1.3T/u2

0.4

0.3

0.15

0.6

0.4

0.2

0

0.10

0.05

–6 –4 –2 0 2 4 6 –8 –4 0 4 8

–8 –4 0 4 8–6 –4 –2 0 2 4 6

(T – Tc)L1/n (T – Tc)L

1/n

L = 12182436

L = 12182436

Figure 12. Finite-size scaling of (a) Binder’s fourth-order cumulant B4M = 1 − 〈M 4〉/3〈M 2〉2, (b)specific heat, (c) order parameter M and (d) susceptibility cM = (〈M 2〉 − 〈M 〉2)/NkBT for thetransition into the

√3 × √

3 dimer order. The inset in panel (a) shows Binder’s cumulant of differentlattice sizes crossing at the critical temperature TM = 1.226v2. The critical exponents of the two-dimensional 3-state Potts model a = 1

3 , b = 19 , g = 13

9 and n = 56 are used. (Online version in colour.)

4. Discussion

Magnetic charges in spin ice are an example of an emergent phenomenon.Although the fundamental degrees of freedom in these materials are magneticdipoles, the behaviour of the system at low energies is often more convenientlyexpressed in the language of magnetic charges. In spin ice on kagome, magneticcharge of a simplex can take on odd values ±1 and ±3 in natural units.Even though magnetostatic interactions tend to minimize magnetic charge,low-energy spin-ice states have non-vanishing magnetic charges on simplexes.Such a magnet may have, in addition to the paramagnetic and fully ordered states,a distinct intermediate phase with ordered magnetic charges and disordered spins.





This phase possesses considerable residual entropy (0.108 per spin) because agiven pattern of magnetic charges can be realized by many configurations ofmagnetic dipoles.

Our numerical simulations provide evidence for an intermediate phase withstaggered magnetic charges in spin ice with dipolar interactions on kagome.As the system is cooled down from the paramagnetic state, it first graduallyenters the spin-ice regime in which magnetic charges of triangles are restrictedto the smallest (in magnitude) values of ±1. A phase transition of the Isinguniversality class takes the magnet into the intermediate phase with staggeredmagnetic charges.

At a lower temperature, the system enters a magnetically ordered phase withsix ground states shown in figure 2a. Because only three of them are accessiblefrom each of the two states of the charge-ordered phase, this transition is expectedto be in the universality class of the 3-state Potts ferromagnet. However, observingthe corresponding critical behaviour proved challenging because the transitiontakes place in the background of nearly-perfect magnetic charge order. When thecharge order has no defects at all, magnetic configurations can be mapped ontostates of hardcore dimers with attraction. This mapping establishes that spinsin the background of perfectly ordered magnetic charges are in a quasi-orderedphase with algebraic spatial correlations. The transition to the fully orderedphase is then of the KT type. Thermal fluctuations generate defects in chargeorder, thus violating the hardcore constraint for dimers. The spin correlationlength is then set by the typical distance between isolated charge defects. Asfigure 7b shows, the charge-order parameter quickly approaches saturation uponcooling below Tc2 = 0.29D. At T = 0.24D, i.e. considerably above the onset ofmagnetic order at Tc1, the concentration of charge defects is about 0.01 pertriangle. Furthermore, most of these defects come in pairs with zero net chargeand minimal separation (resulting from single-spin flips); so they just renormalizethe spin correlation function without shortening the correlation length. Notsurprisingly, the intermediate phase exhibits power-law spin correlations up tovery large distances and the transition to the magnetically ordered phase appearsto be of the KT type. Observing the true critical behaviour of the 3-state Pottsuniversality class would require extremely large system sizes inaccessible to us.

The authors thank Paula Mellado, Gunnar Möller and Roderich Moessner for useful discussions.G.W.C. was supported by the US National Science Foundation grant no. DMR-0844115. O.T. wassupported in part by the US Department of Energy, Office of Basic Energy Sciences, Division ofMaterials Sciences and Engineering under award no. DE-FG02-08ER46544.

References

1 Gingras, M. J. P. 2011 Spin ice. In Introduction to frustrated magnetism (eds C. Lacroix,P. Mendels & F. Mila). Berlin, Germany: Springer.

2 Harris, M. J., Bramwell, S. T., McMorrow, D. F., Zeiske, T. & Godfrey, K. W. 1997Geometrical frustration in the ferromagnetic pyrochlore Ho2Ti2O7. Phys. Rev. Lett. 79,2554–2557. (doi:10.1103/PhysRevLett.79.2554)

3 Petrenko, V. F. & Whitworth, R. W. 1999 Physics of ice. New York, NY: Oxford UniversityPress.

4 Ramirez, A. P., Hayashi, A., Cava, R. J., Siddharthan, R. & Shastry, B. S. 1999 Zero-pointentropy in ‘spin ice’. Nature 399, 333–335. (doi:10.1038/20619)



http://dx.doi.org/doi:10.1103/PhysRevLett.79.2554

http://dx.doi.org/doi:10.1038/20619



5 Isakov, S. V., Gregor, K., Moessner, R. & Sondhi, S. L. 2004 Dipolar spin correlations in classicalpyrochlore magnets. Phys. Rev. Lett. 93, 167204. (doi:10.1103/PhysRevLett.93.167204)

6 Henley, C. L. 2005 Power-law spin correlations in pyrochlore antiferromagnets. Phys. Rev. B71, 014424. (doi:10.1103/PhysRevB.71.014424)

7 Ryzhkin, I. A. 2005 Magnetic relaxation in rare-earth oxide pyrochlores. J. Exp. Theor. Phys.101, 481–486. (doi:10.1134/1.2103216)

8 Castelnovo, C., Moessner, R. & Sondhi, S. L. 2008 Magnetic monopoles in spin ice. Nature 451,42–45. (doi:10.1038/nature06433)

9 Fennell, T., Deen, P. P., Wildes, A. R., Schmalzl, K., Prabhakaran, D., Boothroyd, A. T.,Aldus, R. J., McMorrow, D. F. & Bramwell, S. T. 2009 Magnetic Coulomb phase in the spinice HO2Ti2O7. Science 326, 415–417. (doi:10.1126/science.1177582)

10 Jaubert, L. D. C. & Holdsworth, P. C. W. 2009 Signature of magnetic monopole and Diracstring dynamics in spin ice. Nat. Phys. 5, 258–261. (doi:10.1038/nphys1227)

11 Morris, D. J. P. et al. 2009 Dirac strings and magnetic monopoles in spin ice Dy2Ti2O7. Science326, 411–414. (doi:10.1126/science.1178868)

12 Ryzhkin, I. A. & Ryzhkin, M. I. 2011 Screening of the magnetic field by magnetic monopolesin spin ice. JETP Lett. 93, 384–387. (doi:10.1134/S0021364011070095)

13 Saitoh, E., Miyajima, H., Yamaoka, T. & Tatara, G. 2004 Current-induced resonance andmass determination of a single magnetic domain wall. Nature 432, 203–206. (doi:10.1038/nature03009)

14 Landau, L. D., Lifshitz, E. M. & Pitaevskii, L. P. 1984 Electrodynamics of continuous media,vol. 8, ch. 44, pp. 153–157, 2nd edn. Amsterdam, The Netherlands: Elsevier.

15 Dirac, P. A. M. 1931 Quantised singularities in the electromagnetic field. Proc. R. Soc. Lond.A 133, 60–72. (doi:10.1098/rspa.1931.0130)

16 Chalker, J. T. 2011 Geometrically frustrated antiferromagnets: statistical mechanics anddynamics. In Introduction to frustrated magnetism (eds C. Lacroix, P. Mendels & F. Mila),pp. 3–22. Berlin, Germany: Springer.

17 Gingras, M. J. P. & den Hertog, B. C. 2001 Origin of spin-ice behavior in Ising pyrochloremagnets with long-range dipole interactions: an insight from mean-field theory. Can. J. Phys.79, 1339–1351. (doi:10.1139/p01-099)

18 Melko, R. G., den Hertog, B. C. & Gingras, M. J. P. 2001 Long-range order at low temperaturesin dipolar spin ice. Phys. Rev. Lett. 87, 067203. (doi:10.1103/PhysRevLett.87.067203)

19 Snyder, J., Slusky, J. S., Cava, R. J. & Schiffer, P. 2001 How ‘spin ice’ freezes. Nature 413,48–51. (doi:10.1038/35092516)

20 Giblin, S. R., Bramwell, S. T., Holdsworth, P. C. W., Prabhakaran, D. & Terry, I. 2011 Creationand measurement of long-lived magnetic monopole currents in spin ice. Nat. Phys. 7, 252–258.(doi:10.1038/nphys1896)

21 Tanaka, M., Saitoh, E., Miyajima, H., Yamaoka, T. & Iye, Y. 2006 Magnetic interactionsin a ferromagnetic honeycomb nanoscale network. Phys. Rev. B 73, 052411. (doi:10.1103/PhysRevB.73.052411)

22 Qi, Y., Brintlinger, T. & Cumings, J. 2008 Direct observation of the ice rule in artificial kagomespin ice. Phys. Rev. B 77, 094418. (doi:10.1103/PhysRevB.77.094418)

23 Ladak, S., Read, D. E., Perkins, G. K., Cohen, L. F. & Branford, W. R. 2010 Directobservation of magnetic monopole defects in an artificial spin-ice system. Nat. Phys. 6, 359–363.(doi:10.1038/nphys1628)

24 Ladak, S., Read, D. E., Branford, W. R. & Cohen, L. F. 2011 Direct observation and controlof magnetic monopole defects in an artificial spin-ice material New J. Phys. 13, 063032.(doi:10.1088/1367-2630/13/6/063032)

25 Melko, R. G. & Gingras, M. J. P. 2004 Monte Carlo studies of the dipolar spin ice model.J. Phys.: Condens. Matter 16, R1277. (doi:10.1088/0953-8984/16/43/R02)

26 Möller, G. & Moessner, R. 2009 Magnetic multipole analysis of kagome and artificial spin-icedipolar arrays. Phys. Rev. B 80, 140409. (doi:10.1103/PhysRevB.80.140409)

27 Chern, G.-W., Mellado, P. & Tchernyshyov, O. 2011 Two-stage ordering of spins in dipolarspin ice on kagome. Phys. Rev. Lett. 106, 207202. (doi:10.1103/PhysRevLett.106.207202)




http://dx.doi.org/doi:10.1103/PhysRevB.71.014424

http://dx.doi.org/doi:10.1134/1.2103216

http://dx.doi.org/doi:10.1038/nature06433

http://dx.doi.org/doi:10.1126/science.1177582

http://dx.doi.org/doi:10.1038/nphys1227

http://dx.doi.org/doi:10.1126/science.1178868

http://dx.doi.org/doi:10.1134/S0021364011070095



http://dx.doi.org/doi:10.1098/rspa.1931.0130

http://dx.doi.org/doi:10.1139/p01-099


http://dx.doi.org/doi:10.1038/35092516






http://dx.doi.org/doi:10.1088/1367-2630/13/6/063032

http://dx.doi.org/doi:10.1088/0953-8984/16/43/R02





28 Wills, A. S., Ballou, R. & Lacroix, C. 2002 Model of localized highly frustrated ferromagnetism:the kagomé spin ice. Phys. Rev. B 66, 144407. (doi:10.1103/PhysRevB.66.144407)

29 Syôzi, I. 1951 Statistics of kagomé lattice. Prog. Theor. Phys. 6, 306–308.30 Kano, K. & Naya, S. 1953 Antiferromagnetism. The kagomé Ising net. Prog. Theor. Phys. 10,

158–172. (doi:10.1143/PTP.10.158)31 Takagi, T. & Mekata, M. 1993 Magnetic ordering of Ising spins on kagome lattice with the 1st

and the 2nd neighbor interactions. J. Phys. Soc. Jpn 62, 3943–3953. (doi:10.1143/JPSJ.62.3943)32 Wolf, M. & Schotte, K. D. 1988 Ising model with competing next-nearest-neighbour interactions

on the kagome lattice. J. Phys. A Math. Gen. 21, 2195–2209. (doi:10.1088/0305-4470/21/9/032)33 Cardy, J. L. 1980 General discrete planar models in two dimensions: duality properties and

phase diagrams. J. Phys. A Math. Gen. 13, 1507–1515. (doi:10.1088/0305-4470/13/4/037)34 Challa, M. S. S. & Landau, D. P. 1986 Critical behavior of the six-state clock model in two

dimensions. Phys. Rev. B 33, 437–443. (doi:10.1103/PhysRevB.33.437)35 Kosterlitz, J. M. 1974 The critical properties of the two-dimensional xy model. J. Phys. C 7,

1046–1060.36 José, J. V., Kadanoff, L. P., Kirkpatrick, S. & Nelson, D. R. 1977 Renormalization, vortices,

and symmetry-breaking perturbations in the two-dimensional planar model. Phys. Rev. B 16,1217–1241. (doi:10.1103/PhysRevB.16.1217)

37 Barkema, G. T. & Newman, M. E. J. 1998 Monte Carlo simulation of ice models. Phys. Rev.E 57, 1155–1166. (doi:10.1103/PhysRevE.57.1155)

38 Udagawa, M., Ogata, M. & Hiroi, Z. 2002 Exact result of ground-state entropy for Isingpyrochlore magnets under a magnetic field along [111] axis. J. Phys. Soc. Jpn 71, 2365–2368.(doi:10.1143/JPSJ.71.2365)

39 Moessner, R. & Sondhi, S. L. 2003 Theory of the [111] magnetization plateau in spin ice. Phys.Rev. B 68, 064411. (doi:10.1103/PhysRevB.68.064411)

40 Sandvik, A. W. & Moessner, R. 2006 Correlations and confinement in nonplanar two-dimensional dimer models. Phys. Rev. B 73, 144504. (doi:10.1103/PhysRevB.73.144504)

41 Alet, F., Ikhlef, Y., Jacobsen, J. L., Misguich, G. & Pasquier, V. 2006 Classical dimers withaligning interactions on the square lattice. Phys. Rev. E 74, 041124. (doi:10.1103/PhysRevE.74.041124)

42 Blöte, H. W. J. & Hilhorst, H. J. 1982 Roughening transitions and the zero-temperaturetriangular Ising antiferromagnet. J. Phys. A Math. Gen. 15, 631–637. (doi:10.1088/0305-4470/15/11/011)

43 Kardar, M. 2007 Statistical physics of fields, ch. 8, Cambridge, UK: Cambridge University Press.44 Otsuka, H. 2011 Monomer-dimer mixture on a honeycomb lattice. Phys. Rev. Lett. 106, 227204.

(doi:10.1103/PhysRevLett.106.227204)




http://dx.doi.org/doi:10.1143/PTP.10.158

http://dx.doi.org/doi:10.1143/JPSJ.62.3943





http://dx.doi.org/doi:10.1103/PhysRevE.57.1155

http://dx.doi.org/doi:10.1143/JPSJ.71.2365









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