## Abstract

The formulation of a complete theory of classical electromagnetism by Maxwell is one of the milestones of science. The capacity of many-body systems to provide emergent mini-universes with vacua quite distinct from the one we inhabit was only recognized much later. Here, we provide an account of how simple systems of localized spins manage to emulate Maxwell electromagnetism in their low-energy behaviour. They are much less constrained by symmetry considerations than the relativistically invariant electromagnetic vacuum, as their substrate provides a non-relativistic background with even translational invariance broken. They can exhibit rich behaviour not encountered in conventional electromagnetism. This includes the existence of magnetic monopole excitations arising from fractionalization of magnetic dipoles; as well as the capacity of disorder, by generating defects on the lattice scale, to produce novel physics, as exemplified by topological spin glassiness or random Coulomb magnetism.

This article is part of the themed issue ‘Unifying physics and technology in light of Maxwell's equations’.

## 1. Introduction

The tremendous importance of Maxwell’s equations is hard to state, let alone overstate. They underpin a large fraction of our sensory perceptions, those related to light, are at the root of literally innumerable applications of electromagnetism, and provide the rung above gravity in the ladder of fundamental interactions, which climbing up further has brought forth the spectacular successes of high-energy physics, in particular the standard model, and led us to string theory.

It was only in the twentieth century that it was fully recognized that much is to be gained from moving in the opposite direction of energy. While particle accelerators got larger and more powerful in order to push back the frontier of high-energy experiments, it was by cooling that many phenomena in solid-state physics were discovered which account for much of the interest in that field. The liquefaction of helium, and the resulting discoveries of superfluidity and superconductivity a century ago, are representatives for a string of many important developments.

We are now used to, as a matter of course that hardly requires comment, the fact that low-energy degrees of freedom of a system can differ completely from those of its high-energy constituents. This phenomenon is now known under the name emergence and was formulated most crisply in Anderson’s manifesto entitled ‘More is different’ [1]. This independence of degrees of freedom across scales is crisply noticed in the case of crystals, whose low-energy excitations—the phononic lattice vibrations—exhibit bosonic quantum statistics, regardless of the fermionic or bosonic nature of the atoms making up the crystal.

The capacity of phase transitions—such as the translational symmetry breaking leading to the phonons in the above example—to generate various types of emergent excitations being well established, it perhaps comes as a surprise that there are not all that many experimental instances where these degrees of freedom are not a fermionic or bosonic field but instead gauge fields. This experimental scarcity is not due to theoretical neglect, there being no shortage of models where their existence is, if not established, at least fervently desired.

This article is devoted to an account of the emergence of Maxwell electromagnetism in a class of magnetic systems known as highly frustrated magnets. In these systems, competing interactions yield a large number of nearly degenerate low-energy states; fluctuations between these lead to new collective behaviour which in some cases turns out to be naturally captured in a description based on an emergent gauge field.

In §2, we account for how this comes about in some detail; and then focus on observable consequences, placing some emphasis on aspects of the emergent physics which are not known to have a counterpart in conventional electromagnetism. These include the existence of magnetically charged quasi-particles known as emergent magnetic monopoles. In addition, a selection of disorder-induced phenomena is covered in §3, where the interplay of lattice scale defects to the ‘ether’ of emergent electromagnetism and its long-wavelength Coulomb field is discussed: disorder can nucleate gauge-charged defects, the interactions between which lead to new collective disorder physics. We conclude with an outlook in §4.

The material in this article has been selected rather idiosyncratically in order to provide a flavour of this field to the interested outsider; it is neither comprehensive in its choice of topics nor a complete historical account. For a more detailed pedagogical introduction, we mention review articles on geometrically frustrated magnets [2] and their Coulomb phase [3] and spin ice [4].

## 2. Emergent gauge fields

Antiferromagnets present an impressive variety of possible ground state configurations, crucially depending on the geometry of the lattice where the spins reside. This is related to the fact that interactions are frustrated once the lattice contains odd length loops and the ground state configurations become highly degenerate. This high degeneracy can usually be traced back to an underconstraint in the system of equations found by minimizing the Hamiltonian [5]. This idea, amusingly enough, can also be traced back to Maxwell [6].

The intricate structure of correlations within the degenerate set of ground states will give the excitations special properties.

For some systems the ground state correlations can be interpreted as arising from constraints defining emerging conservation laws, which are then resolved by a gauge field. This mapping from the original spin variables to the emergent gauge field leads not only to an understanding of the correlations within the set of ground state configurations, but also allows prediction of the behaviour of the local excitations.

On the particular class of lattices consisting of corner sharing frustrated units containing *q* spins (where *q*≥3, figure 1), an *n*-component spin model can be rewritten in a particularly intuitive form:
2.1where is the total spin on a single frustrated unit (e.g. a tetrahedron in the case of a pyrochlore lattice, or a triangle for a kagome lattice, figure 1), and the variables are *n*-component vectors. The local constraints characterizing the ground state configurations are then explicitly
2.2This set of equations can be turned into a conservation law of charges living on the dual lattice (e.g. in the case of pyrochlore, on the diamond lattice). For that, the original spin variables are mapped onto fields *B*^{i} residing on the links of the dual lattice [8], in such a way that each component *i* of *L*_{α} is described by the Gauss law: The factor is a sublattice-dependent staggering factor: it equals +1 (−1) if the frustrated unit occupies the A (B) sublattice. It is therefore implicitly assumed that the frustrated units occupy a bipartite lattice. In fact, only in this case is it known how to define the map to the ** B** fields [3] giving the Gauss law required above.

The constraints, equation (2.2), can then be resolved by an emergent gauge field *B*^{i}=**∇**×*A*^{i}. Furthermore, the spin correlations are recovered by the *B*^{i} fields if one imposes a magnetostatic action for each of them as the simplest ansatz for a coarse-grained theory:
2.3This is the prominent *Coulomb phase* [3,8] action. It has the form of an emergent Maxwell magnetostatics advertised above, stabilized by fluctuations between classical ground states. Here, the gauge theory nature of Maxwell’s equations comes about for purely energetic reasons—it follows from imposing the ground state constraint. Adding quantum fluctuations to such a system is one way to generate the conjugate electric potential and to produce Maxwell electrodynamics also hosting, for example, emergent photons [9,10].

Demanding that the field ** B** be divergenceless implies for its Fourier modes the condition

**⋅**

*q***=0, and this together with the quadratic action gives immediately that pair correlations of the Fourier modes must have the form 2.4which in real space translates into a dipolar form. This power-law decay translates into characteristic features in the**

*B**T*=0 structure factor of the original spin model, that hallmark all the systems presenting a Coulomb phase, the so-called

*pinch points*(figure 2

*a*).

Owing to Gauss’s law, the gauge charges themselves are , and these correspond to excitations on the original spin system, which can be induced thermally or, in a quenched manner, by diluting the original spin system as described below.

Tracing the magnetostatic action, equation (2.3), with the sole constraint of fixing two charges at positions *r*_{1,2} one obtains at zero temperature (when the ** B** field in all other places is divergenceless) that these charges are subject to an action for two Coulomb charges
2.5

Furthermore, a simple consequence of the staggering factor in the gauge charges definition is that the following condition must hold: 2.6akin to global charge neutrality of the universe as a whole.

Up to now we had in mind systems with discrete Ising or continuous Heisenberg *O*(3) vector spins leading to one or three flavours for the gauge charges; the latter are therefore continuous variables. These charges represent gapless excitations on the original spin system.

The Ising case leads to gapped, discrete charges. This occurs in nature as a result of a local easy-axis anisotropy combined to ferromagnetism [11] as occurs in the spin ice materials Ho_{2}Ti_{2}O_{7}, Dy_{2}Ti_{2}O_{7} [12]. In these materials, the spin variables can be described as , with *σ*_{iα} being Ising variables, and describing the local easy axis, defined along the bonds of the diamond lattice, dual to the original pyrochlore lattice where spins reside. The Hamiltonian has a nearest-neighbour contribution, as well as a long-ranged, dipolar one [13] (borrowing notation from [14]):
2.7and
2.8with *a* the nearest-neighbour distance, the pyrochlore adjacency matrix and the interaction matrix of the magnetostatic dipolar interactions.

The emergent gauge charges on these systems have a fascinating physics of their own, and we devote the next two subsections to them.

### (a) Magnetic monopoles

The ground state local constraints on spin ice systems receive usually the special name of *ice rules*, coming from a natural mapping between these systems, and common water ice, where the ice rules were first formulated. Local violations of the ice rules occur by single-spin flips which leads to gapped *magnetic monopoles* [7]. These are emergent gauge charges, and on account of the discussion in the previous section also experience a Coulomb entropic force.

Only a finite amount of free energy is necessary to separate monopoles infinitely, and therefore these are *deconfined* [4,7] objects. This constitutes a ‘fractionalization’ of the original magnetic dipole moment into magnetic charges. They can be separated by further spin flips, thereby leaving behind an observable ‘Dirac string’ [15–17] (figure 1). Indeed, an easy way to mathematically see the genesis of magnetic charges is to observe that inverting a string of dipoles sets up a potential equivalent to that of two equal and opposite charges, ±*Q*, at its ends. In a continuum approximation, the dipole moment density along the string corresponds to one atomic magnetic moment *μ* per bond of the dual diamond lattice, which has length *a*; since a string is flipped, rather than considered in isolation, there is an additional factor of 2: *Q*=2*μ*/*a*.

The Dirac strings themselves are loosely defined objects, as a given configuration does not define them uniquely. Nevertheless, it is possible to define them stochastically, and study the statistics of their lengths *l*, which present a broad distribution, with two power-law regimes: short loops with a probability distribution scaling as ∼*L*^{3}*l*^{−2.50(1)} and long loops with a scaling of ∼*l*^{−0.98(3)} (figure 2) [18]. The effective low-temperature system can therefore be seen as a soup of magnetic monopoles wandering around, with the ‘spaghetti’ of Dirac strings fluctuating between them. The power-law form of their length distribution reflects their thermodynamic tensionlessness, and underpins the deconfinement of the magnetic monopoles.

A perhaps illuminating approach to understand the origin of the Coulomb interaction from the dipolar one is obtained through the simplified ‘dumbbell model’ [3,7] (figures 1 and 3). Within this model, the original spins having magnetic moment *μ* are replaced by a dipole of charges ±*Q*/2 separated by the diamond lattice bond spacing *a*, so that the effective dipole moment of this dumbbell coincides with the original spin magnetic moment, which fixes the charges to be *Q*=2*μ*/*a*, the value derived above.

The dumbbell model is useful, since monopole charges are directly given by the sum of dumbbell charges on a tetrahedron. The original dipolar interactions between spins are translated into effective Coulomb interactions between these monopoles. This interaction is therefore of an energetic origin, contrary to its entropic counterpart originating from an averaging over the many degenerate configurations compatible with defects placed at fixed positions.

### (b) Intrinsic versus emergent gauge charge

The dumbbell model has led us to the important distinction between two contributions to the monopoles’ interactions. In fact, these two contributions correspond to two kinds of charges in our system.

The emergent gauge charges exist due to the cooperative behaviour of the whole system, which on account of the non-trivial ground state correlations, leads to entropically interacting magnetic monopoles.

On the other hand, the intrinsic gauge charges interact on energetic grounds alone, as a result of the original dipolar interactions for the magnetic dipoles constituting the system. The ground state constraints let these original dipolar interactions, decaying as , be described by effective Coulomb interactions between the intrinsic gauge charges, thus with a slower decay, decaying as 1/*r*_{ij}.

## 3. Disorder in a spin liquid

The most unavoidable way for the system to disobey Gauss’s law is on account of thermal fluctuations. Nonetheless, the original microscopic model provides another manner for violating these constraints, robust, down to the zero temperature limit. This is achieved on diluting the original system.

In contrast to our conventional Maxwell vacuum, consideration of defects on the underlying space in a condensed matter system is quite natural, because the materials, which we are ultimately interested in describing by our simplified models, very often present unavoidable, or deliberately introduced, non-magnetic impurities.

Two important cases must be distinguished here, according to whether we dilute a lattice presenting discrete or continuous spins.

### (a) Topological spin glass

The ground states of spin ice systems are characterized by the ice rules which can only be violated by a finite energy gap, Δ, e.g. by flipping a single spin. This corresponds to creation of two nearest-neighbour monopoles with opposite charges. Interestingly, if instead of flipping a spin, a single spin in a configuration satisfying the ice rules is removed, the same effect is produced in the alternative monopole picture—two nearest-neighbour monopoles with opposite (half-)charges *Q*/2 are created. Still an important difference here applies, namely, the ice rules where the monopoles were created cannot be restored by further spin flips, as in the disorder-free case, since each of these frustrated units contains now an odd number of discrete spins and with this, *L*=0 is impossible. At low temperatures, when thermally activated monopoles become exponentially suppressed, only the quenched monopoles due to disorder are relevant. A quenched pair of the nearest-neighbour oppositely charged monopoles corresponds therefore to an emerging *ghost spin* [19], which becomes the relevant degree of freedom, while all the original spins can be integrated out in this limit (figure 3).

The randomly placed ghost spins mediate through the correlated background dipolar interactions, which have two contributions as for the usual monopoles: one entropic, due to the fluctuations among the degenerate ground states, and one energetic, due to long ranged dipolar interaction present on spin ice compounds. The latter survives even down to the zero temperature limit, while the entropic contribution vanishes linearly with *T*.

This low-temperature description for spin ice in terms of ghost spins turns out to present a spin glass phase transition at non-zero temperatures (dipolar spin glass), where the ghost spins freeze, while the remaining bulk is still able to fluctuate [19]. The freezing transition is found to depend linearly on the dilution, *T*_{c}(*x*)∝*x*. This can be simply understood in terms of the typical energy scale of the dipolar interaction, scaling as , while *r*^{3}_{typ}∼1/*x*.

### (b) Orphan spins

The consideration of dilution in a Coulomb phase system with vector charges has a long history [20] with the first systematic experimental exploration being made by Schiffer & Daruka [21] on the material SrCr_{9p}Ga_{12−9p}O_{19} (SCGO), even before the Coulomb phase was properly identified. They proposed a two-population model on explaining the observed uniform magnetic susceptibility a population of correlated spins coexisting with uncorrelated ones, leading to a Curie tail *T*^{−1} on the susceptibility, the so-called *orphans*.

The single-unit approximation proposed [22] shortly after these experimental findings led to the conclusion that dilution in these systems can lead to orphans on those frustrated units where all but one spin is left alone (*L*=0 is impossible only for *q*=1). These are still correlated with the rest of the system through the remaining frustrated unit to which it belongs.

This correlation of the orphan with its spin liquid ‘bath’ has important consequences [23,24] as a texture emerges around each orphan, and partially screens it, effectively reproducing an orphan + texture fractionalized object, with a fractional moment of of the original spin moments (figure 4).

In the magnetostatic picture, this texture is a natural consequence of Gauss’s law: due to the defect charge (orphan), , an emergent magnetic field originates around it. In *d* dimensions, since , the field decays as 1/*r*^{d−1} with distance, and this is precisely the scaling found for the texture at *T*=0. Despite this slow decay, the texture forms a total moment conspiring to cancel only half of the orphan moment, and this is due to the oscillations coming from the staggered definition of the local fields (figure 4).

### (c) Interactions between orphan spins

The orphans are disorder nucleated gauge charges and as such are subject to mutual interactions mediated by the background Coulomb phase in which they are embedded. These are obtained in the same way as considered before, by integrating out the remaining spins, which thus serve as ‘bath’ to the orphans, leaving an effective description for the orphans alone, which, not surprisingly, interact as vector charges
3.1with the sublattice dependent staggering factors *η*_{i} explicitly shown. This staggering is removable by a trick due to Mattis [26]: define new spin variables on one of the two sublattices to be equal to the reverse of the old spin variables. This Mattis transformation removes the sublattice dependence at the cost that certain observables (such as the magnetization) obtain a modified meaning (staggered magnetization).

This leads at zero temperature to the effective random Coulomb antiferromagnet model describing the system of orphans, which has been studied in detail in [25].

The limit to zero temperature is necessary as only on this limit do the interactions become truly long-ranged, as they are at finite temperatures thermally screened. More precisely one encounters in this limit orphans interacting with:
3.2with *J*_{ij} the Coulomb potential between orphans *i* and *j*, and the coupling strength *A* being fixed by this limit and the original microscopic model parameters.

### (d) Random Coulomb magnets

The new models describing effective charges, induced by dilution on a Coulomb phase with continuous charges, are thus of the form (at *T*=0)
3.3with the properties

—

*J*_{ij}>0,∀*i*,*j*(after a Mattis transformation).—

*J*_{ij}∼1/|*r*_{i}−*r*_{j}| in three, or in two dimensions.— The spin positions,

*r*_{i}, are quenched random variables.

These models we call the *random Coulomb magnets* (RCMs) a new type of frustrated disordered magnetic model which naturally arises in the context of diluted Coulomb phases.

The interaction matrices on these effective models belong to the class of Euclidean random matrices, which are generally characterized by the dependence of *J*_{ij} on the Euclidean distance *J*_{ij}=*J*(|*r*_{ij}|), while the randomness lies alone on the independently distributed random variables *r*_{i} [27,28]. The properties of this ensemble are comparatively not as intensively studied in the literature of spin glass models as the commonly considered random matrices there, which usually involve the important different assumption of independence between the different matrix elements. Also, in equation (3.3), all interactions are antiferromagnetic—we have a random Coulomb antiferromagnet, unlike the usual spin glass Hamiltonian with random signs of the magnetic interaction.

These models resemble plasma models, with the fluctuating degrees of freedom being, instead of the charged particle positions, the ‘signs’ of the charges themselves, which can then be allowed to take on values given by *O*(*N*) symmetry. The case *N*=3 corresponds thus to the original microscopic model giving birth to these models. The same model in two dimensions for the case *N*=1 has been found as the effective description of disorder-induced defects on a completely different microscopic model many years ago by Villain [29].

The global charge neutrality constraint on the microscopic theory (equation (2.6)) translates into (after the Mattis transformation) 3.4This is quite a natural constraint, which even without explicit imposition in the original microscopic model, alone on energetic grounds of the long ranged Coulomb model described by equation (3.3) would be very nearly satisfied.

The RCMs have been studied in some detail in [25], both for the cases *N*=3 and , the latter also serving as a semi-analytical proofing of the (necessarily) purely numerical results on *N*=3.

With the aim of understanding the general phase diagram of these models, Rehn *et al*. [25] allow the coupling strength *A* to vary, while the density of orphans *x* is fixed but kept small (the behaviour at some high enough values of *x* leads to trivially ordered, striped or Neel phases), in the expectation of possibly detecting a spin glass ordering as one increases *A*. This quest is inspired by a proposal already made by Schiffer and Daruka in their original work on the orphans, that these might actually be degrees of freedom undergoing freezing known to occur on SCGO at low temperatures, while the bulk of spins could well keep their own dynamics, and not necessarily freeze at the same temperature as the orphans. These experimental observations on SCGO and similar highly frustrated magnetic materials constitute a very old puzzle waiting for explanation already for several decades [30–32]. Furthermore, the idea that a spin glass phase might arise out of emergent degrees of freedom originating on an insulating matrix provided by a highly frustrated magnetic system goes much further back in time, as argued by Villain [33].

The simulations by Rehn *et al*. [25] found a broad paramagnetic regime for the RCMs and up to very large values of *A*, no spin glass phase transition for a finite value of *A* was found both in two and three dimensions in systems with *N*=3. Furthermore, in two dimensions, the system with indicated the critical glassy behavior occurring for , although no conclusive statements have so far been obtained for three dimensions.

Although numerics in these models cannot explain orphan freezing, one particularly interesting property on the two-dimensional case is that at very low orphan densities *x*, a global scaling transformation of the orphan distances is innocuous on account of the logarithmic form of the interactions, and the global charge neutrality (3.4) thereby leaving the partition function unchanged up to an irrelevant constant factor:
3.5with *M* the total number of orphans. This implies that in the limit of low densities *x*, any critical coupling *A*_{c} will be independent of *x*: the partition function itself is a scaling function.

The analogy of these models to plasma physics can be pushed further and a general screening theory in same spirit of Debye–Hückel theory is developed by Rehn *et al*. [25], with the result that due to spin fluctuations alone (note that the spins are quenched on random lattice positions) a thermal screening length originates, scaling in a similar way to the Debye–Hückel screening length as . The necessity for screening in these models can be seen directly in a high-temperature expansion (HTE), as on account of the long rangedness of the interactions, the series of the HTE must be resummed on each term to remove a divergence.

In summary, RCMs present a very rich physics, with connections to several areas of physics, such as the physics of plasmas and Debye–Hückel screening, or the physics of spin glass models and the relatively poorly explored ensemble of Euclidean random matrices.

## 4. Summary and outlook

In this article, we have showcased how Maxwellian physics arises in particular, geometrically frustrated, condensed matter systems. We have given some examples of how it behaves like conventional electromagnetism, and how it can appear to be richer, such as in the case of the emergence of freely mobile magnetic monopoles from the fractionalization of spin moments, which in addition to their magnetic charge carry an emergent Coulomb gauge charge.

We have also discussed how new degrees of freedom can emerge when disorder is added to the system, and how the Coulomb spin liquid which hosts them mediates interactions between these. This leads to the appearance of unusual ghost spin degrees of freedom, which encode the missing spins in much the same way as holes in a band insulator represent missing electrons. These ghost spins themselves undergo a freezing transition. We have also introduced the rich physics of RCMs, which is only beginning to be explored.

The study of quenched disorder in the lattice is a particularly attractive feature of this class of models, as it has a natural interpretation in terms of defects in the ‘ether’ underpinning the Coulomb phase.

Varying the lattice then allows for the emergence of different types of disorder physics. For instance, in all the cases currently known, where the map from spin variables to an emerging field is described by a magnetostatic action, one important requirement is that the frustrated units occupy a *bipartite lattice* [3]. This has an important consequence for the gauge charges, as their lattice scale dependence is completely removable, and their long wavelength description is purely the one of emerging Coulomb charges on the continuum.

In this sense, it is very interesting that at least one system has been recently recognized where the emerging gauge charges interact as emerging Coulomb charges [34], but the frustrated units occupy a non-bipartite lattice. Here, the fractionalization of the spin moments is into orphan spins carrying an odd-denominator fraction, , of the microscopic magnetic moments, the first time such fractionalization has been observed in a classical spin system.

We hope that the study of Coulomb spin liquids will continue to produce further such surprises, and continue to allow condensed matter physics to provide novel aspects to add to the venerable field of Maxwell electromagnetism.

## Authors' contributions

Both authors contributed extensively to the manuscript.

## Competing interests

The authors declare they have no competing interests.

## Funding

This work was supported by the Deutsche Forschungsgemeinschaft under grant no. SFB 1143.

## Acknowledgements

We thank our collaborators on the various topics covered here: Alex Andreanov, Claudio Castelnovo, John Chalker, Kedar Damle, Karol Gregor, Masud Haque, Peter Holdsworth, Sergei Isakov, Ludovic Jaubert, Anto Scardicchio, Arnab Sen, Shivaji Sondhi and Peter Young.

## Footnotes

One contribution of 7 to a discussion meeting issue ‘Unifying physics and technology in light of Maxwell's equations’.

- Accepted April 25, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.