The Concept of Spin Ice Graphs and a Field Theory for their Topological Monopoles and Charges
TThe Concept of Spin Ice Graphs and Field Theory for their Topological Monopoles and Charges
The Concept of Spin Ice Graphs and Field Theory for theirTopological Monopoles and Charges
Cristiano Nisoli Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, 87545,USA (Dated: 1 October 2020)
Spin ices can now be fabricated in a variety of geometries which control their col-lective behavior and exotic properties. Therefore, their proper framework is graphtheory. We relate spin ice notions such as ice rule, ice manifold, Coulomb phases,charges and monopoles, to graph-theoretical notions, such as balance, in/out-degrees, and Eulerianicity. We then propose a field-theoretical treatment in whichtopological charges and monopoles are the degrees of freedom while the binary spinsare subsumed into entropic interaction among charges. We show that for a spin iceon a graph in a Gaussian approximation, the kernel of the entropic interaction isthe inverse of the graph Laplacian, and we compute screening functions from thegraph spectra as Green operators for the screened Poisson problem on a graph. Wethen apply the treatment on star graphs, tournaments, cycles, and regular spin icein different dimensions.
I. INTRODUCTION
Since the Bernal-Fowler ice rule was invoked by Pauling to explain the zero-point en-tropy of water ice the concept has come to describe a variety of other materials, such aspyroclore rare-earth spin ice antiferromagnets , artificial magnetic spin ice antiferromag-nets , or artificial particle-based ices . Often , but not always , frustrationimpedes ordering in these materials, and leads to degenerate states of constrained disorder,or ice manifolds , of interesting topological properties.Indeed, the ice rule is a topological concept related to the local minimization of a topo-logical charge . As such, it has a wide applicability, and artificial spin ice materialsare being designed for a variety of emergent behaviors not necessarily found in naturalmagnets . Violations of the ice manifold are topological excitations that, de-pending on the geometry and local degeneracy of the system, can be deconfined. Further,in magnetic materials these topological charges are also magnetic charges , and if deconfin-able, magnetic monopoles . As such, they interact via a Coulomb law , are sourcesand sinks of the (cid:126)H field, can pin superconductive vortices in spin ice/superconductors het-erostructures , and could perhaps exert a localized and mobile magnetic proximity effect in heterostructures that interface two-dimensional spin ices to transition metal dichalco-genides or Dirac materials. Finally, in artificial realizations, these topological objects canalso be read and written , and might therefore function as binary, mobile informationcarriers for spintronics for for neuromorphic computation .In non-magnetic spin ices—such as particle-based ones , which can be made of confinedcolloids , superconducting vortices , skyrmions in magnets or in liquid crys-tals —the mutual interaction among monopoles differs from a Coulomb law . Andyet, because they are topological charges, they always interact at least entropically , as weshall see.In this work, first we extend the concept of spin ice on general graphs, and relate the twofields and their jargons, because many concepts of spin ice physics have been investigatedunder different names and with different aims in graph theory.Then we propose here a unifying framework for degenerate spin ice on a general graph.We treat these charge excitations as degrees of freedom, subsuming the underlying spinvacuum into entropic interactions among charges. This work has many motivations.Firstly, a graph approach naturally separates geometry from topology. Graphs possessa metric yet are not necessarily embedded in a linear algebraic structure, each of them a r X i v : . [ c ond - m a t . d i s - nn ] S e p he Concept of Spin Ice Graphs and Field Theory for their Topological Monopoles and Charges 2essentially describing a topological class of various and different geometric realizations.Results obtained on a graph are ipso facto related only to the topology of the system, ratherthan to geometry, thus gathering under a unifying treatment many topological conceptsrelated to spin ice materials, some of which disseminated through many works pertainingto a variety of specific systems .Secondly, we also aim at guiding future artificial realizations of exotic topolo-gies , including finite size systems . For instance, interesting works on Penrosespin ices, based on finite size quasicrystals still lack a proper language.Thirdly, spin ice can now be realized in the quantum dots of a quantum annealer on awide variety of graphs.Finally, many common notions in spin ice physics posses a direct equivalent in graphtheory. For instance: spin ice charges are the degree-excess between indegrees and outde-grees ; the degeneracy of spin ice on a complete graph is simply the number of its regulartournaments ; an ice rule configuration realizes a graph that can be unraveled in an Eulerianpath, etcetera. A wealth of work has been done in graph theory, some of which could beeasily translated to spin ice physics. This work represent an invitation extended both tograph theorist, to contribute to the field of spin ice physics, and to physicists working inspin ice, to broaden their mathematical approaches to these systems. II. SPIN ICE GRAPHSA. Spin Ice
Ice comes in about eighteen crystalline forms , all of them involving oxygen atoms re-siding at the center of tetrahedra, sharing four hydrogen atoms with four nearest neighboroxygen atoms (Fig. 1). Two of such hydrogens are covalently bonded to the oxygen of theirmolecule, and two realize hydrogen bonds with oxygens of different molecules. Thus, twoare “in”, two are “out” of the tetrahedron, and this is the so called ice-rule introduced byBernal and Fowler . Each tetrahedron therefore realizes 6 admissible configurations outof the 2 = 16 ideally possible, and the collective degeneracy of the ice grows exponen-tially in the number of tetrahedra N as W N . This leads to a non-zero residual entropy pertetrahedron k B ln W . Pauling famously estimated W = 3 /
2, remarkably close to both theexperimental and the numerically obtained value ( W = 1 . ± . ).One can associate to the ice rule a spin model (Fig 1c) where binary spins are assignedto the bonds between molecules pointing toward the proton. Then the ice rule dictatesthat two spins point in, two point out, as is the case of magnetic spin ices . In rare earthtitanates such as Ho Ti O and Dy Ti O , the magnetic cations Ho + and Dy + carry avery large magnetic moment, µ ∼ µ B . At low temperature they can be considered binary,classical Ising spins constrained to point along the directions of the lattice bonds which forma pyrochlore lattice, and are thus expected to interact as magnetic dipoles obeying the icerule .While this ice manifold represents an interesting manifestation of a Coulomb phase whichhas been amply studied and which can be considered a prototype for classical topolog-ical order , the exotic behaviors of spin ices proceed not so much from said topologicalstructure, but rather from how it is broken , e.g. via fractionalization into monopoles ,and how much of it is instead retained, e.g. via spin fragmentation . B. Spins on a Graph
We consider the most general case of a spin ice on a connected, undirected, simple graph G . The phase space for G is the set of all directed graphs, o digraphs, that can be built on G by assigning an orientation to its edges.he Concept of Spin Ice Graphs and Field Theory for their Topological Monopoles and Charges 3 a b c FIG. 1. a) The crystalline structure of ice shows proton disorder in the allocation of hydrogenatoms among oxygen centered tetrahedra (image from ref ). b) In water ice oxygen atoms sitat the center of tetrahedra, connected to each other by a hydrogen atom. Two of such protonsare close (covalently bonded) to the oxygen at the center, two are further away. c) One mightreplace this picture with spins pointing in or out depending on whether the proton is close or faraway. Then two spins point in, two point out. This corresponds to the disposition of magnetic mo-ments on pyrochlore spin ices, rare earth titanates whose magnetic ensemble does not order at lowtemperature, because of frustration, and, much like water ice, provides non-zero low temperatureentropy density. (Figures b, c from ref ). Consider an undirected, simple graph G of a number N l of edges labeled by l , connectinga number N v of vertices labeled by v and of various degree of coordination z v . We callhe Concept of Spin Ice Graphs and Field Theory for their Topological Monopoles and Charges 4 FIG. 2. Left: undirected graph. Center: directed graph. Right: directed graph obeying the icerule. { vv (cid:48) } an edge l among vertices v, v (cid:48) . For such graph, the adjacency matrix is the matrix A vv (cid:48) , such that A vv (cid:48) = 1 if v, v (cid:48) are connected and A vv (cid:48) = 0 otherwise. It contains all theinformation of the graph. Obviously, A vv (cid:48) is symmetric and z v = (cid:80) v (cid:48) A vv (cid:48) .We can define binary variables or Ising spins S l on each edge l , as in Fig. 2, via anantisymmetric matrix S vv (cid:48) such that S vv (cid:48) = 0 if v and v (cid:48) do not share an edge, S vv (cid:48) = 1 ifthey do and the spin points toward v (cid:48) , and S vv (cid:48) = − v . In the language of graph theory, A vv (cid:48) defines an undirected graph, while an Ising spinstructure S vv (cid:48) defines a digraph. The S matrix is related to the non-symmetric adjacencymatrix of the corresponding digraph, A dir vv (cid:48) , whose elements have value one if and only if vv (cid:48) are connected by an edge pointing toward v (cid:48) (on simple graphs). Then S is the anti-symmetrization of A dir , or S vv (cid:48) = A dir vv (cid:48) − A dir v (cid:48) v (and of course A vv (cid:48) = A dir vv (cid:48) + A dir v (cid:48) v ).For an undirected graph G , we call its phase space P , the set of the digraphs that can bespecified on it. Clearly, the cardinality of the phase space is |P| = 2 N l .
1. Ice Manifolds An ice manifold is a proper subset of the phase space P that minimizes the topologicalcharge, defined as follows.Given S vv (cid:48) , its topological charge distribution is the vector Q v defined for each vertex v as the difference between the edges pointing in and out of v , or Q v [ S ] = (cid:88) v (cid:48) S v (cid:48) v . (1)In graph theory, Q v is thus the difference between indegrees and outdegrees of the directedgraph that corresponds to a particular spin configuration on G . Q v can have the values z v , z v − , . . . , − z v , − z v , and thus only vertices of even coordination can have zero charge.To relate Eq. (1) to the more physical picture , we can introduce the divergence operatoron a graph as div[ S ] v = (cid:88) vv (cid:48) S vv (cid:48) (2)from which we immediately have Q v [ S ] = − div[ S ] v . (3)A digraph S vv (cid:48) (i.e. a spin configuration) for which | Q v | is minimal on each vertex, i.e.has zero charge on all vertices of even coordination and ± obey the ice-rule .he Concept of Spin Ice Graphs and Field Theory for their Topological Monopoles and Charges 5Then, given a graph G , we call the ice manifold of G the subsets I ⊂ P of its spin phasespace P made of all directed graphs that obey the ice rule. Then, |I| is the ice-degeneracyof the graph and s = N − l ln |I| is its Pauling entropy per spin.In the language of graph theory, an ice rule configuration corresponds to a balanced or quasi-balanced directed graph. By a famous theorem, if all vertices have even degrees(i.e. even coordination) a balanced digraph is Eulerian . That means that an Eulerianpath exists (a walker can follow the arrows and walk the entire graph without passing thesame edge twice ). Moreover, if the graph is complete and has odd vertices, a spinconfiguration is called a tournament because it describes winners and losers in a round-robin tournament . Thus, an ice rule configuration on a complete graph describes a regular tournament, one where all players score the same. Therefore, the ice-degeneracy ofa complete graph of odd number of vertices counts all the possible outcomes of a round-robin tournament where every player beats the same number of other players (which is stillan open problem).We see therefore that spin ice physics is a case of the theory of balanced or Euleriangraphs. The latter is greatly consequential, with applications in biology , computer sci-ence , logistics , and sociology . In complex networks, digraphs can describe the sharingof a token among agents, and deviation from the ice-rule can quantify the agent satisfactionor frustration, and give measures of fairness .Depending on the topology of the graph it is not obvious if and when the cardinality ofthe ice manifold scales exponentially with the size of the graph, thus leading to a non-zeroPauling entropy . Indeed, the one dimensional ferromagnetic Ising model can be mappedinto a spin ice on a path graph, or a cycle, and its ice manifold has cardinality two, regardlessof the number of vertices. We will show elsewhere that the path graph is the only such case.
2. Coulomb Phases
The concept of a Coulomb phase appeared first in gauge field theories and was thenintroduced in the theory of pyrochlore spin ice . In simple terms, it corresponds toa disordered spin texture that can be coarse-grained to a solenoidal magnetization field.It can be considered a case of classical topological order where no order parameterexists but instead the disordered states are labeled by gauge fields whose solenoidal natureexpresses the constraint over the disorder. Excitations are then violations of the solenoidalrequirement.We can generalize this notion to a graph spin ice by expressing it without coarse grainingor gauge fields. We say that two spin assignations S, S (cid:48) ∈ P are charge equivalent if andonly if the difference ( S vv (cid:48) − S (cid:48) vv (cid:48) ) / zero topological charge on every vertex. Charge-equivalence is an equivalence relation and thus induces a partition on the phase space P .We call each class of equivalence in that partition a Coulomb class . Each Coulomb class islabeled by a distribution of charge Q v .A trivial example: a graph made of only two vertices connected by one edge has a spinphase space of cardinality 2, corresponding to the two orientations of its only spin. Its icemanifold coincides with the spin phase space, and it contains 2 Coulomb classes, each ofcardinality 1. For a less trivial example, the ice manifold of pyrochlore ice is a Coulombclass corresponding to charge Q v = 0 everywhere. The same is true by definition for everygraph that has only even coordinations.Clearly, not all ice manifolds are Coulomb classes. For instance the ice manifold of thehoneycomb graph, so-called kagome ice, is not: for any spin configuration there is alwaysat least one spin, and in fact an extensive number of spins, that can be flipped individually,thus changing charge configurations (and Coulomb class) without violating the ice rule. Andyet, the kagome ice manifold can be partitioned into Coulomb classes. Crucially, its Ice IIphase corresponds to two Coulomb classes, each of charge alternating in sign ± Coulomb phase . Thus, the ice manifold of spin ices of even coordination, includingtherefore pyrochlores, is a Coulomb class and also a Coulomb phase. In kagome ice, eachof the two Coulomb classes of the Ice-II phase are Coulomb phases, which might explainwhy it is so hard to observe it experimentally , whereas the Ice-I phase is foundeasily .Indeed, a Coulomb class imposes topological constraints on the kinetics within the class:it prohibits single spin flips and requires collaborative flips (corresponding to Euler trails)that might be extremely unlikely in a realistic dynamics. Thus, in a Coulomb phase, allkinetics must happen above that phase, by breaking the topology of the Coulomb class,that is by changing its defining distribution of charges. In pyrochlore ices or in degenerateartificial square ices, where the entire ice manifold is a Coulomb phase, the breakage oftopological protection consists in the appearance of monopoles, that is ± et al. , kinetics consists in the breaking ofthe ± III. FIELD THEORY OF EMERGENT CHARGES
We need a Hamiltonian whose ground state is the ice manifold, and thus allows us todeal with breakages of the topological protection in terms of energy costs. The following isminimal H [ S ] = (cid:15) (cid:88) v Q v , (4)where (cid:15) > H more generally measures how “unbalanced” a digraphis. Note also that for every digraph (cid:80) v Q v is larger than or equal to the number of verticeswith odd coordination.The corresponding partition function reads Z [ H, V ] = (cid:88) { S vv (cid:48) } e − β ( H [ Q ] − (cid:80) vv (cid:48) S vv (cid:48) H vv (cid:48) − (cid:80) v V v Q v ) , (5)where β = 1 /T , H vv (cid:48) is an antisymmetric matrix modeling an external “magnetic” fielddefined on the edges with respect to the orientation l = vv (cid:48) , so that H vv (cid:48) (cid:54) = 0 if and onlyif v and v (cid:48) share an edge, and H vv (cid:48) = − H v (cid:48) v . H has dimensions of an energy (the Zeemanenergy), and V v is an external potential acting on the charges.Clearly then (cid:104) S l (cid:105) = T δ ln Z [ H, V ] δH l (cid:104) S l S l (cid:48) (cid:105) = T δ ln Z [ H, V ] δH l δH l (cid:48) (cid:104) Q v (cid:105) = T δ ln Z [ H, V ] δV v (cid:104) Q v Q v (cid:48) (cid:105) = T δ ln Z [ H, V ] δV v δV v (cid:48) (6)are the one- and two-point correlation functions for spins and charges.he Concept of Spin Ice Graphs and Field Theory for their Topological Monopoles and Charges 7To produce a field theory we need to remove the discrete variables S l . We do so by a com-mon trick. We insert in the sum of (5) the tautological expression 1 = (2 π ) − N v (cid:81) v (cid:82) dq v dφ v exp [ iφ v ( q v − Q v )]and then we sum over the spins. We introduceΩ[ φ, H ] = (cid:88) { S vv (cid:48) } e − i (cid:80) v φ v Q v + β (cid:80) vv (cid:48) S vv (cid:48) H vv (cid:48) (7)= (cid:89) (cid:104) vv (cid:48) (cid:105) (cid:88) S = ± e [ − i ( φ v (cid:48) − φ v )+ βH vv (cid:48) ] S (8)where (cid:104) vv (cid:48) (cid:105) are edges (counted once, so that if (cid:104) vv (cid:48) (cid:105) is counted (cid:104) v (cid:48) v (cid:105) is not). From it weobtain Ω[ φ, H ] = 2 N l (cid:89) (cid:104) vv (cid:48) (cid:105) cos ( ∇ vv (cid:48) φ + iβH vv (cid:48) ) , (9)where the matrix ∇ φ is defined on the edges only as ∇ vv (cid:48) φ = φ v (cid:48) − φ v (10)and is called the gradient matrix of φ . Note that if the graph is embedded in a linear spaceand (cid:126)vv (cid:48) is the vector pointing from v to v (cid:48) then ∇ vv (cid:48) φ = (cid:126)vv (cid:48) · (cid:126) ∇ φ + O ( | vv (cid:48) | ). Z [ H, V ] from (5) can now be rewritten as Z [ H, V ] = (cid:90) [ dq ] ˜Ω[ q, H ] e − β H [ q ]+ β (cid:80) v q v V v (11)where [ dq ] = (cid:81) v dq v , H [ q ] is given by Eq. (4), and ˜Ω[ q, H ] is the functional Fourier transformof Ω[ φ, H ], or ˜Ω[ q, H ] = 1(2 π ) N v (cid:90) [ dφ ]Ω[ φ, H ] e (cid:80) v iq v φ v . (12)The averages and correlations of q are the same as for Q and therefore, for instance (cid:104) q v (cid:105) = (cid:104) Q v (cid:105) and also (cid:104) div[ S ] v div[ S ] v (cid:48) (cid:105) = (cid:104) q v q v (cid:48) (cid:105) . (13)We have thus replaced the binary spin variables S l defined on the edges with a continuumfield q v defined on the discrete set of vertices, but we have gained a term ˜Ω[ q ]. It representsa generalized degeneracy or density of states for the charge distribution q v , emergent fromthe many possible underlying spin ensembles compatible with q v . It also constraints q v toproper, discretized values.We can call S = ln ˜Ω[ q ] the generalized entropy for the charge distribution q . Then, theeffective free energy at zero loop for q v is given by the quadratic part of H [ q ] − T S [ q ]as we shall see. Also, in absence of a field, from (12), (6) and the parity of the functionsinvolved, (cid:104) q v (cid:105) = 0 for every v and (cid:104) S l (cid:105) = 0 for every l , as one would indeed expect fromtrivial considerations on the model.To see the same formalism from a different angle we can rewrite the partition function as Z [ H, V ] = 1(2 π ) N v (cid:90) [ dqdφ ] e − β F [ q,φ,H ] (14)with F [ q, φ, H ] = H [ q ] − (cid:88) v q v ( iT φ v + V v ) + F [ φ, H ] (15)and F [ φ, H ] = − T ln Ω[ φ, H ] . (16)he Concept of Spin Ice Graphs and Field Theory for their Topological Monopoles and Charges 8Equations (14), (15) look familiar in the language of quantum field theory. They correspondto a charge field q v , for which iT φ v acts as a bosonic field (of “Lagrangian” F [ φ ]) mediatingan interaction between charges. Again, the interaction is not real but comes from theunderlying binary ensemble from which the charge field is an emergent observable. Bothpictures come useful in different scenarios, as we shall see.Integrating (14), over dq v we obtain Z [ H, V ] = (cid:18) T π(cid:15) (cid:19) Nv (cid:90) [ dφ ] Ω[ φ, H ] e − T (cid:15) (cid:80) v ( φ v − iβV v ) (17)which shows that in general (cid:104) iφ (cid:105) is real. Indeed from Eqs. (6,17) we have (cid:15) (cid:104) q v (cid:105) = iT (cid:104) φ v (cid:105) + V v . (18)To further clarify the intuitive meaning of iφ as an emergent, entropic field translatingthe effect of the spin correlations, from Eq. (17) we can immediately deduce (cid:104) S vv (cid:48) (cid:105) = (cid:104) tanh ( βH vv (cid:48) + i ∇ vv (cid:48) φ ) (cid:105) . (19)It is then natural to introduce an entropic “magnetic field” B evv (cid:48) = i ∇ vv (cid:48) φ , and then (cid:104) S vv (cid:48) (cid:105) (cid:39) βH vv (cid:48) + (cid:104) B evv (cid:48) (cid:105) (20)when (cid:104) S vv (cid:48) (cid:105) is small (which requires H vv (cid:48) small). Note than in general, from Eq. (18), theentropic magnetic field is related to the gradient of the charges, or (cid:15) ∇ vv (cid:48) (cid:104) q (cid:105) = (cid:104) B evv (cid:48) (cid:105) + E vv (cid:48) (21)where E is the gradient of V .It is time to confess that V v is superabundant, though useful. Indeed, nothing changesby incorporating V v into H vv (cid:48) by replacing in our equations V → H vv (cid:48) → H vv (cid:48) + ∇ vv (cid:48) V .Similarly, if H can be divided into a gradient plus a term irreducible to a gradient, or H vv (cid:48) = H (cid:48) vv (cid:48) + ∇ V (cid:48) vv (cid:48) , the equations above are still valid with the substitution V → V + V (cid:48) , H → H (cid:48) . In the following we will assume V = 0.Finally, taking the divergence of Eq. (19), we obtain (cid:104) q (cid:105) = − div [ (cid:104) tanh ( βH + i ∇ φ ) (cid:105) ] . (22)In the linearized limit in which Eq. (20) is valid, we obtain (cid:104) q (cid:105) = − βµ q ext + i ˆ Lφ. (23)In the previous equation the external charge is defined as µq ext = div[ H ], where µ is anenergy, and the Laplace operator ˆ L is defined via the Laplacian matrix L vv (cid:48) = z v δ vv (cid:48) − A vv (cid:48) , (24)or ˆ L = ˆ D − ˆ A where D vv (cid:48) = z v δ vv (cid:48) is the degree matrix and z v is the degree or coordinationof the vertex v . The Laplacian matrix is the generalization on a graph of the discretizedLaplacian operator on a lattice. The reader can easily verify that on a square lattice of edgelength a when one takes the usual continuum limit for a → L → − a ∇ . Also,the reader can verify that for a generic ζ v defined on the nodes v div[ ∇ ζ ] v = − (cid:88) v (cid:48) L vv (cid:48) ζ v (cid:48) (25)as one would expect as the generalization of the notorious (cid:126) ∇· (cid:126) ∇ = ∇ , valid in linear spaces,and which we have used to deduce Eq. (23).Thus, Eq. (23) tells us that charges are the sources and sinks of the entropic potential,and represent a generalization of the at least in a linear approximation. There, the entropicfield iφ obeys a generalized Poisson equation and is therefore the Coulomb potential of thecharges on a graph. We shall see in the last section that indeed when the graph can beproperly embedded in a linear space, the entropic interaction among charges is indeed thestandard Coulomb interaction in the proper dimension of the space.he Concept of Spin Ice Graphs and Field Theory for their Topological Monopoles and Charges 9
IV. HIGH T APPROXIMATION
By taking the high T , high H limit but keeping H/T finite, the Gaussians in (17) tendto delta functions in φ and we obtain Z [ H ] = 2 N l (cid:89) l cosh ( βH l ) . (26)Unsurprisingly, the above is the standard “paramagnetic” partition function for an uncor-related system. It leads, via Eqs. (6) to the familiar magnetization law for a paramagnet (cid:104) S l (cid:105) = tanh( βH l ) . (27)When H = 0, we get from (26) the correct entropy per spin at high temperature, or s = ln 2.Thus under assumption of disorder, the high T expansion corresponds to an expansion inthe small entropic field φ . In doing so we lose the constraint to discrete values of q v imposedby Ω[ φ ], which is however not relevant at high T . A. Free Energy and Entropic Interactions
We can expand Eq. (16) at lowest order and find β F H [ φ ] = − (cid:88) vv (cid:48) A vv (cid:48) ln cos ( ∇ vv (cid:48) φ + iβH vv (cid:48) ) (cid:39) (cid:88) vv (cid:48) A vv (cid:48) ( ∇ vv (cid:48) φ + iβH vv (cid:48) ) = 12 (cid:88) vv (cid:48) φ v ( z v δ vv (cid:48) − A vv (cid:48) ) φ v (cid:48) − iβ (cid:88) vv (cid:48) φ v A vv (cid:48) H vv (cid:48) − β (cid:88) vv (cid:48) A vv (cid:48) H vv (cid:48) , (28)and thus write F in Eq. (15) at the second order as β F [ q, φ ] = β(cid:15) q + 12 φ ˆ Lφ − iφ · (cid:16) q + β div[ ˆ H ] (cid:17) − β | H | (29)where matrices are expressed as operators acting on q , φ , which are vectors of dimension N v , q = q · q = (cid:80) v q v , and | H | = (cid:80) vv (cid:48) A vv (cid:48) H vv (cid:48) / only because the divergence of H is hereby defined discretelywith respect to vertices by Eq. (2). One might at first suspect that it is incorrect, andimmagine that, e.g. on a square lattice a continuous solenoidal H would not couple to thespin ice. However, a moment thought shows that even for an uniform H , its divergence, asdefined on vertices in Eq. (2), is necessarily non-zero at the boundaries of the lattice evenfor a uniform field. In other words, the definition of divergence on a graph already accountsfor boundary conditions (inclusive of “internal” boundaries ).By integrating over q v we can express the partition function in Eq. (14) in the high T approximation as coming from a free energy in the entropic field only, or β F [ φ ] = 12 T(cid:15) φ + 12 φ ˆ Lφ − iφ · (cid:16) β div[ ˆ H ] (cid:17) − β | H | . (30)In a similar way, to find an effective free energy for the charges, using Eqs (14), (29), wecan integrate instead over the entropic fields φ . However, to do so, we must first considerthe spectrum of ˆ L .he Concept of Spin Ice Graphs and Field Theory for their Topological Monopoles and Charges 10The following is well known from the spectral theory of graphs. ˆ L is symmetric andthus has real eigenvalues { γ ( k ) } k =0 ...k max with k max ≤ N v −
1, and corresponding N v eigenvectors ψ αv ( k ) (where α counts the eigenvalue degeneracy). It is immediate to verifythat γ (0) = 0 for the uniform eigenvector ψ v (0) = 1 / √ N v . In a simple and connectedgraph all other eigenvalues are strictly positive.We can go to the new basis, defining ˜ q α ( k ) = ψ α ∗ ( k ) · q and ˜ φ α ( k ) = ψ α ∗ ( k ) · φ . Thenin Eqs (14), (29), the integration over d ˜ φ (0) merely returns a δ (˜ q (0)), which in “real space”corresponds to δ ( (cid:80) v q v ). This ensures that in the new free energy we sum only overcharge configurations of zero net charge–as it should be, since a system of dipoles is chargeneutral. All other charge modes have zero net charge. Indeed for any eigenvector of L except the one of zero eigenvalue it is true that (cid:80) v ψ v ( k ) = 0. This follows immediatelyfrom (cid:80) v L vv (cid:48) = z v (cid:48) − z v (cid:48) = 0 and ψ v ( k ) = (cid:80) v (cid:48) L vv (cid:48) ψ v (cid:48) ( k ) /γ ( k ) .From Gaussian integration of F in Eq. (29) in the space orthogonal to ψ (0) (where ˆ L can be inverted) we obtain, in absence of field H , the effective free energy for q in the form β F [ q ] = β(cid:15) q + 12 q ˆ L − q, (31)which can be interpreted both in real or spectral space.The first term is the usual energy cost to produce charges. The second term tells usthat, at quadratic order, the effect of the underlying spin manifold can be subsumed intoa pairwise, entropic interaction that corresponds to T L − , i.e. the Green operator of theLaplacian.In regular lattices embedded in a linear space (see below), Eq. (60) implies that in threedimensions (3D) charges interact entropically via a 1 /x law, as indeed found numerically.In two dimensions (2D) one expects instead a logarithmic interaction. In both case we haveCoulomb potentials in the proper dimension, as anticipated in our discussion of Eq. (23).This, however, also implies a mismatch in systems of reduced dimensionality, for instancein square ice, whose entropic interaction is the Green function of the 2D laplacian whilethe real monopole interaction (not considered in this work) is the Green function of the 3Dlaplacian. Such mismatch leads to lack of screening, as we show elsewhere. B. Charge Correlations
We define ˆ W q as the inverse of the kernel of the free energy for the charge of Eq. (60), orˆ W q − = ˆ1 β(cid:15) + ˆ L − , (32)and then, from equipartition we have (cid:104) q v q v (cid:48) (cid:105) = W qvv (cid:48) (33)We note that ˆ W q can be written in various ways, includingˆ W q = ˆ L ( ξ ˆ L + 1) − = ξ − − ξ − ˆ G, (34)where we call ξ = (cid:112) (cid:15)/T (35)the correlation length at high temperature , as it was already appreciated in more specificsystems, via other means (see next section). In Eq. (34) we have introduced the “Greenfunction” (actually, a matrix) of the screened Poisson equation on a graph, orˆ G = (cid:16) ˆ L + ξ − (cid:17) − , (36)he Concept of Spin Ice Graphs and Field Theory for their Topological Monopoles and Charges 11which, as we show in the next subsection, controls the screening from an external charge.Note that, from Eq. (30) ˆ G is also the correlation for the entropic field φ , or (cid:104) φ v φ v (cid:48) (cid:105) = G vv (cid:48) . (37)Finally, going to the spectrum of L we find (cid:104) ˜ q α ∗ ( k )˜ q α (cid:48) ( k (cid:48) ) (cid:105) = δ αα (cid:48) δ kk (cid:48) w ( k )= δ αα (cid:48) δ kk (cid:48) γ ( k ) γ ( k ) ξ + 1 . (38)In the infinite temperature limit the correlation length becomes zero, and from Eq. (38) weobtain (cid:104) q v q v (cid:48) (cid:105) → L vv (cid:48) for T → ∞ . (39)From Eq. (39) we have (cid:104) q v (cid:105) → z v for T → ∞ (40)which indeed corresponds to the average square charge of uncorrelated vertices, or q defined as the square charge obtained from counting arguments. Indeed, considering vertexmultiplicities only, and computing the average square charge for a vertex of degree z witheach charge 2 n − z weighted merely by its vertex multiplicity (cid:0) zn (cid:1) , one obtains q = 2 − z z (cid:88) n =0 ( z − n ) (cid:18) zn (cid:19) = z. (41)Note also that Eq. (39) implies zero correlations among vertices that are not nearest neigh-bors, but a correlation of − (cid:104) q v q v (cid:48) (cid:105) = (cid:34) L ∞ (cid:88) n =0 (cid:0) − ξ L (cid:1) n (cid:35) vv (cid:48) . (42)Note now that L n can be written as sums of products of n ˆ A and ˆ D matrices (e.g. ADDAADAD . . . ). A moment’s thought should convince that if v , v (cid:48) are separated bymore edges than the number of A matrices in such product, then the vv (cid:48) element of theproduct is zero. An obvious notion of distance between two vertices on a graph is givenby the number of edges in the shortest path (aka graph geodesic) connecting them . Itfollows that if v and v (cid:48) are at a distance d vv (cid:48) > (cid:104) q v q v (cid:48) (cid:105) = (cid:0) − ξ (cid:1) ( d vv (cid:48) − (cid:2) A d vv (cid:48) (cid:3) vv (cid:48) + O (cid:0) ξ d vv (cid:48) (cid:1) . (43)Interestingly, A kvv (cid:48) is known to be the number of walks of length k between the two vertices v and v (cid:48) . Thus, the coefficient (cid:2) A d vv (cid:48) (cid:3) vv (cid:48) in Eq. (43) is the number of walks between thetwo vertices v and v (cid:48) of length equal to the distance d vv (cid:48) . It is therefore the number ofgeodesics connecting the two vertices. Note that this considerations are conditional to thepossibility of the expansion of Eq. (42), which might not hold in the thermodynamic limitof certain systems.By construction our approximation doesn’t work for T → ξ ( T ) → ξ r ( T ) where ξ r ( T ) is the real correla-tion length at low T . For instance in pyrochlore ice, correlations become screened-algebraicat low T , suggesting that a quadratic free energy might work with proper renormalizationsof the parameters.In the limit ξ → ∞ From Eq. (38) we find (cid:104) q v q v (cid:48) (cid:105) = ξ − δ vv (cid:48) − ξ − (cid:2) L − (cid:3) vv (cid:48) + O ( ξ ) . (44)Then, for graphs of even coordination we obtain ξ ∼ / (cid:104) q (cid:105) for ξ → ∞ , (45)Note that the equation above corresponds to the Debye screening length for a Coulombpotential whose coupling constant is proportional to T , which is indeed the case of ourentropic field. At high temperature, the situation is much different, with ξ = (cid:15)/T . TheDebye-H¨uckel approach applies to strong electrolytes that are fully dissociated, and thedisorder brought by higher temperature prevents charges from screening, thus increasingthe screening radius. In spin ice, instead, charges carry an energy cost and at highertemperature there are more charges available for the screening.Furthermore, Eq. (45) corresponds in pyrochlore spin ice to the experimentally found exponentially divergent behavior of the correlation length in the proximity of the ice mani-fold. And because for v (cid:54) = v (cid:48) the correlations tend to ∝ ˆ L − , we call them Coulomb in thislimit.
C. Entropic Screening of External Charges
Since we have no real interaction among charges, all screening is entropic. One canconsider two cases: screening of an external charge and screening of a pinned charge.One can define external charges as sources and sinks of the H field, which interact withemergent spin ice charges via the coupling of the H field to the spins S . To understand thisformally, consider the term (cid:80) vv (cid:48) S vv (cid:48) H vv (cid:48) in Eq. (5). Imagine that we can write a Helmholtzdecomposition on the graph, so that H can be represented as H vv (cid:48) = ∇ vv (cid:48) Ψ + H ⊥ vv (cid:48) whereΨ v is a field and the second term has no divergence. Then we have12 (cid:88) vv (cid:48) S vv (cid:48) H vv (cid:48) = − (cid:88) v Q v Ψ v + 12 (cid:88) vv (cid:48) S vv (cid:48) H ⊥ vv (cid:48) , (46)that is, the potential responsible for the divergence-full part of the magnetic field couplesto the emergent spin ice charges, thus generalizing on a graph the notion of magneticfragmentation in spin ice .Let us then call q ext = µ div[ H ] the “external charge” (here µ has dimension of an energy)and assume (cid:80) v q ext ,v = 0. Then from Eq. (29), integrating over φ one has (cid:104) q (cid:105) = − µT ˆ W q ˆ L − q ext = − µ(cid:15) ˆ Gq ext . (47)Thus, an external charge is screened by the screened Green function of the Laplacian. Notethat because ˆ L Ψ = − q ext we can also write (cid:104) q v (cid:105) = − µT (cid:104) q v q v (cid:48) (cid:105) Ψ v (cid:48) . (48)From Eq. (47) we obtain the two correlation limits (cid:104) q v (cid:105) = − µ(cid:15) ξ δ vv (cid:48) q ext + O ( ξ ) for ξ → (cid:104) q v (cid:105) = − µ(cid:15) (cid:2) L − (cid:3) vv (cid:48) q ext ,v (cid:48) + O ( ξ − ) for ξ → ∞ . (49)he Concept of Spin Ice Graphs and Field Theory for their Topological Monopoles and Charges 13 D. Entropic Screening of Pinned Charges
The case of a pinned charge is obtained by summing the partition function only overspin configurations corresponding to a fixed charge q pin on a vertex ¯ v . This correspondsto inserting a δ ( q ¯ v − q pin ) in the functional integral. We leave to the reader the simplecalculation, whose result is (cid:104) q v (cid:105) = W v ¯ v W ¯ v ¯ v q pin = (cid:104) q v q ¯ v (cid:105)(cid:104) q v (cid:105) q pin , (50)and correctly yields (cid:104) q ¯ v (cid:105) = q pin . Note also, from Eq. (34), (cid:104) q v (cid:105) = L v ¯ v q pin z ¯ v + O ( T − ) for T → ∞ . (51)The difference between the two screenings should not surprise. In the first case an externalfield interacts locally [see Eq. (46)] with the spin ice, inducing local effects that then prop-agate via the charge-charge correlation, while the second case is due to correlations of thefree charge with the pinned one. V. EXAMPLESA. Star Graph
A star graph is made of n nodes v i each connected only a central node w . Thus, N l = n, N v = n + 1. The number of ice rule configurations are W = (cid:0) nn/ (cid:1) when n is even and W = 2 (cid:0) n ( n +1) / (cid:1) when n is odd. This leads to a Pauling entropy per spin s = ln(2) in thelimit of large n . The fact that the Pauling entropy coincides with the entropy at infinitetemperature is a feature of spin ices with infinite spins per vertex. In particular, to have afinite energy per spin we need to scale the energy coupling as (cid:15) = J/n .To find the charge correlations, note immediately that by symmetry it must be (cid:104) q v (cid:105) = 0, (cid:104) q w (cid:105) = 0, and (cid:104) q v (cid:105) = 1 for every v . The only possible correlations are (cid:104) q v (cid:105) = 1, (cid:104) q v q w (cid:105) , (cid:104) q w (cid:105) and (cid:104) q v q v (cid:48) (cid:105) . Using tricks such as (cid:104) q v q w (cid:105) = n − (cid:80) v (cid:104) q v q w (cid:105) = n − (cid:104) (cid:80) v q v q w (cid:105) and chargeneutrality one readily finds (cid:104) q v q w (cid:105) = −(cid:104) q w (cid:105) /n (cid:104) q v q v (cid:48) (cid:105) = (cid:104) q w (cid:105) / [ n ( n − − / ( n + 1) , (52)and we need to find only (cid:104) q w (cid:105) . Note that Z = exp( βJ/ Z w , where the latter is given by Z w = n (cid:88) k =0 (cid:18) nk (cid:19) e − J n (2 k − n ) ∼ (cid:90) − dρe − βnf ( ρ ) (53)in the limit of large n [ ρ = (2 k − /n , f ( ρ ) = Jρ / − T s ( ρ ) with s ( ρ ) = σ (1 / ρ/ σ ( x ) = − x ln( x ) − (1 − x ) ln(1 − x ) is the usual binomial entropy]. In the limit of large n we can perform the quadratic expansion around the minimum of f (which is at ρ = 0)obtaining f ( ρ ) ∼ (1 / βJ + 1) ρ , and thus (cid:104) ρ (cid:105) = n − ( βJ + 1) − . Then from q w = nρ andfrom the Eqs. (52) we have (cid:104) q w (cid:105) = n βJ (cid:104) q v q w (cid:105) = −
11 + βJ (cid:104) q v (cid:105) =1 (cid:104) q v q v (cid:48) (cid:105) = − n βJ βJ for v (cid:54) = v (cid:48) . (54)he Concept of Spin Ice Graphs and Field Theory for their Topological Monopoles and Charges 14Note that (cid:104) q w (cid:105) → z w at large T and (cid:104) q w (cid:105) , (cid:104) q v q w (cid:105) → T , as expected. Note alsothat the Eq. (43) is valid: from Eq. (54), (cid:104) q v q (cid:48) v (cid:105) = − ξ + O ( ξ ) and indeed the number ofgeodesics between v and v (cid:48) is one.What are the correlations deduced from the field theory at high T ? We leave to thereader the simple task of computing the matrix ˆ W q and verifying that W qww = n n + ( n + 1) βJW qvw = − nn + ( n + 1) βJW qvv = 1 + βJ n +1 n βJ + n +1 n ( βJ ) W qvv (cid:48) = − βJn + ( n + 1) βJ + n +1 n ( βJ ) for v (cid:54) = v (cid:48) . (55)The equations above correctly reduce to the Eqs. (54) in the limit of large n in whichthe latter were derived. Therefore, the field theory is exact at any temperature for the stargraph in the thermodynamic limit. That should not surprise. We know that field theoriesbecome exact for Ising models of binary spins when each spin interact with each other inthe same way, that is when the graph connecting interacting spins is a complete graph,which is the case for the star graph.Finally, pinning a charge Q p on w elicits, from Eq. (50) a screening charge (cid:104) q v (cid:105) = − Q p /n ,which can also be obviously deduced from charge cancellation. More interesting is whenthe charge is pinned on v . Then it must be Q p = ±
1, and the screening is: (cid:104) q w (cid:105) = − Q p (1 + βJ ) − , (cid:104) q v (cid:48) (cid:54) = v (cid:105) = − ( Q p /n ) βJ/ (1 + βJ ), going to zero in the thermodynamic limit. B. Complete Graph
In a complete graph K n all nodes are connected to all others. We have N v = n, N l = n ( n − /
2. As already explained, a spin configuration configuration is a digraph called atournament, and there are |P| = 2 n ( n − / tournaments. When n is odd, vertex coordinationis even and an ice-rule obeying tournament is called a regular tournament. Thus |I| is thenumber of regular tournaments with n players, which is still an open problem. Thereare, however, asymptotic formulas for large n . From McKay’s formula we find that thePauling entropy per spin in the thermodynamic limit is s = ln(2). As in the star graph iscoincides to the entropy at infinite temperature.By the symmetry of the problem there are only two charge correlations, (cid:104) q v q v (cid:48) (cid:105) (when v (cid:54) = v (cid:48) ) and (cid:104) q v (cid:105) , and they are related: (cid:104) q v q v (cid:48) (cid:105) = ( n − − (cid:80) v (cid:48) (cid:54) = v (cid:104) q v q v (cid:48) (cid:105) = −(cid:104) q v (cid:105) / ( n − n . Indeed, consider K n − , then K n can be former by adding a vertex v n and n − v . . . v n − vertices of K n − . Its partition function is therefore Z n = n (cid:88) k =0 (cid:20)(cid:18) nk (cid:19) e − β(cid:15) (2 k − n ) × (cid:88) { S }∈ K n − e − β(cid:15) [ (cid:80) kj =1 ( q j − + (cid:80) n − j = k +1 ( q j +1) ] (cid:21) = Z n − n (cid:88) k =0 (cid:18) nk (cid:19) e − β(cid:15) (2 k − n ) + j k , (56)where j k = ln (cid:104) e β(cid:15) (cid:80) kj =1 q j (cid:105) n − , and the suffix indicates that the average is performed onthe K n − graph. Notice than, again, we must assume (cid:15) = J/n in order to have an extensivehe Concept of Spin Ice Graphs and Field Theory for their Topological Monopoles and Charges 15energy. Then, with the same transformation as for the star graph, β(cid:15) (2 k − n ) = nβJρ scalesas n , whereas j k is subextensive in n and can be neglected. Thus, for large n , Z n = Z n − Z w ,with Z w given by Eq. (53). We obtain therefore, as before (cid:104) q v (cid:105) = n βJ , (57)and from (cid:104) q v q v (cid:48) (cid:105) = −(cid:104) q v (cid:105) / ( n −
1) we have (cid:104) q v q v (cid:48) (cid:105) = −
11 + βJ . (58)Note in particular that while the entropy is extensive in the number of spins, energyscales linearly in the number of vertices. Indeed E = (cid:104) (cid:15) (cid:80) v q v (cid:105) / n/ JT / ( T + J ): theenergy per vertex is finite and non zero (except for T = 0) while the energy per spin isalways zero.From Eq. (50) we see that by pinning a charge Q p in a vertex, the charge elicited byentropic screening in all other vertices is (cid:104) q (cid:105) = − Q p /n , which is obvious since there must becharge cancellation. The elicited charge is zero in the thermodynamic limit if Q p is finite,and non-zero if Q p instead scales linearly in n .Finally, the adiacency matrix for the complete graph K n is ˆ A = ˆ J − ˆ1, and thus theLaplacian is ˆ L = ( n − − ˆ J , where ˆ J is the matrix of ones. From that it is immediate tofind ˆ W q as W qvv = n −
11 + βJW qvv (cid:48) = −
11 + βJ for v (cid:54) = v (cid:48) . (59)which corresponds to the correlations of Eqs. (57, 58) for n large. Again, we see that thefield theory is exact at any temperature. C. Spin Ice on a Lattice
We will treat elsewhere the cases of spin ice on a lattice in full. Here we report generalconsiderations that are common to spin ices that can be properly embedded in a linearspace of dimension d , such that in the thermodynamic limit it can be homogenized to acontinuum ( v → (cid:126)x ), and distances are measured in units of the lattice constant such thatEq. (60) becomes F [ q ] = (cid:15) (cid:90) q ( x ) d d x + T (cid:90) d d x d d y q ( x ) V e ( x − y ) q ( y ) . (60)Because the Laplacian operator of the graph coincides with that of the linear space, itseigenvectors are plane waves and at small momentum (or large distances) and the eigenval-ues are γ ( k ) (cid:39) (cid:126)k (we measure space in units of the lattice constant). Then we have forthe entropic interaction at large distances V e ( x ) = − T | x | d = 1 − πT ln | x | d = 2 T πξ x d = 3 , (61)which are all Coulomb potentials in their proper dimension, as already anticipated in sectionIII. The third line coincides with the numerically verified entropic interaction for pyrochlorespin ice .he Concept of Spin Ice Graphs and Field Theory for their Topological Monopoles and Charges 16From Eq. (38) we obtain (cid:104)| q ( k ) | (cid:105) = k ξ k = ξ − − ξ − k + ξ − , (62)from which we have, via Fourier transform, (cid:104) q ( x ) q (0) (cid:105) = − ξ exp ( −| x | /ξ ) d = 1 − πξ K ( | x | /ξ ) d = 2 − πξ x exp ( −| x | /ξ ) d = 3 , (63)valid at large distances. The first line corresponds to the exponential form of the screeningin a 1D Ising system, where domain walls correspond to charges. We note that ξ thus knownexactly as ξ = − / ln tanh( β(cid:15) ) and does not follow the ∼ (cid:112) (cid:15)/T at high T . Thoughone can verify that at low T we have ξ ∼ / (cid:104)| q |(cid:105) which of course is different from ξ ∼ / (cid:104) q (cid:105) . In general, Eq. (35) underestimates the correlation in the one-dimensional casebecause one-dimensional spin ice has an ordered ice manifold. As we show in future work,it enjoys a unique position in such regard.The second line in Eq. (63) was recently experimentally verified in square spin ice realizedon a quantum annealer by pinning a charge and relaxing the system . VI. CONCLUSIONS
We have proposed the concept of spin ice on a general graph, and related it to well knowngraph-theoretical concepts of balance and Eulerian paths. We have then developed a fieldtheoretical framework for its excitations, monopoles or charges. We have obtained a seriesof results that are independent of geometry. The partition function of general Graph SpinIce can be reformulated exactly as a functional integral over the charge distribution and itsentropic field, the latter subsuming the effect of the underlying spin ensemble on emergentcharges [Eqs (11)-(16)]. The high T behavior is described by a quadratic free energy of theaverage charges and contains informations on the graph via the graph Laplacian [Eq. (60)].In absence of charge interaction and external fields and in the limit of high T , the en-tropic interaction among charges corresponds to the Green operator of the graph Laplacian.Thus, correlations correspond to the screened Green operator of the graph [Eqs (33)-(38)].The correlation length is ξ = (cid:15)/T at high T and ξ = 1 / (cid:104) q (cid:105) at low T , results alreadyappreciated in various special spin ice systems .Beside generalizing condensed matter notions to graphs where they can provide intuitiveinsight or inspiration for broader problems in complex networks, we have shown that manyof the properties of spin ice systems follow directly from the graph structure, which isessentially topological. VII. ACKNOWLEDGEMENTS
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