Universal Renyi mutual information in classical systems: the case of kagome ice
UUniversal R´enyi mutual information in classical systems: the case of kagome ice
Armin Rahmani and Gia-Wei Chern Theoretical Division, T-4 and CNLS, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
We study the R´enyi mutual information of classical systems characterized by a transfer matrix. We firstestablish a general relationship between the R´enyi mutual information of such classical mixtures of configurationstates, and the R´enyi entropy of a corresponding Rokhsar-Kivelson–type quantum superposition. We then focuson chiral and nonchiral kagome-ice systems, classical spin liquids on the kagome lattice, which respectivelyhave critical and short-range correlations. Through a mapping of the chiral kagome ice to the quantum Liftshitzcritical field theory, we predict a universal subleading term in the R´enyi mutual information of this classicalspin liquid, which can be realized in the pyrochlore spin ice in a magnetic field. We verify our prediction withdirect numerical transfer-matrix computations, and further demonstrate that the nonchiral kagome ice (and thecorresponding quantum Rokhsar-Kivelson superposition) is a topologically trivial phase. Finally, we argue thatthe universal term in the mutual information of the chiral kagome ice is fragile against the presence of defects.
PACS numbers: 03.67.Mn, 05.20.-y,75.10.Hk
I. INTRODUCTION
Universal subleading terms in the entanglement entropy en-code valuable information about quantum systems. In gappedtopological phases, such as the Z spin-liquid ground state ofthe Kitaev toric-code model, there is a subleading correc-tion to the area law, known as the topological entanglemententropy, which encodes the total quantum dimension of thesystem. The ground state of the toric-code model can beviewed as an equal-weight superposition of loops. A very sim-ilar universal correction can appear in critical wave functionssuch as the ground state of the Rokhsar-Kivelson (RK) quan-tum dimer models; the so-called RK state is again an equal-weight superposition of dimer coverings. In this latter case,the universal term has a di ff erent interpretation: it encodesinformation about the sti ff ness constant of the underlying crit-ical field theory, and therefore the exponent of critical corre-lations. The concept of topological order in classical systems hasattracted considerable interest recently.
In particular, byusing the notion of mutual information (a generalization ofthe entanglement entropy to mixed states), it was shown inRef. 10 that the topological order can survive decoherenceand appear in classical systems. Moreover, it can be detectedthrough subleading corrections to the mutual information. Itwas argued in Ref. 10 that under fairly general conditions,the classical von Neumann mutual information is half of thecorresponding quantum coherent case. Primarily focusing oncritical systems, in this work we examine the universal termsin the
R´enyi mutual information of classical (thermal) mix-ture of configuration states, characterized by a transfer matrix,through a direct combinatorial approach. We obtain a generalrelationship between I cAB ( n ), the R´enyi mutual information ofthe thermal mixture (where n is the R´enyi index), and I qAB ( n ),of the corresponding quantum superposition: I cAB ( n ) = I qAB (cid:32) n + (cid:33) . (1)The expression above is valid for a long cylinder divided intotwo subsystems A and B is in Fig. 1(a), and consequently holds, generically, for the universal correction to the area law.This expression generalizes the finding of Ref. 10 to an arbi-trary R´enyi index and a wider class of systems. R´eny mutualinformation has been also applied recently to studying classi-cal phase transitions. A canonical example of classical thermal mixtures de-scribed above is provided by spin-ice systems, which are mas-sively degenerate magnetically disordered manifolds of spinconfigurations satisfying local constraints broadly referred toas ice rules. These systems can be thought of as the classi-cal analogs of quantum spin liquids. The entanglement prop-erties of quantum spin liquids (in particular their topologicalentanglement entropy) have been the subject of numerous re-cent studies.
Despite the prevalence of spin-ice systems innature, and the connection to quantum spin liquids, the entan-glement properties of spin-ice systems are largely unexplored.Here we focus on two spin-ice systems on the kagome lattice,known respectively as chiral and nonchiral kagome ice. Thelocal constraints, i.e., ice rules, characterizing these spin-icemanifolds allow for a transfer-matrix formulation of the prob-lem, which in turn makes the general relationship (1) applica-ble. By using a mapping of the chiral kagome ice to dimerson the honeycomb lattice, we relate the universal correctionto the R´enyi mutual information of this classical spin liquidto universal terms in the R´enyi entanglement entropy of theLiftshitz quantum critical point, and find γ cAB ( n ) = − n ) ln (cid:32) n + (cid:33) , (2)where n is the R´enyi index. As for the nonchiral kagome ice,through a numerical transfer-matrix calculation, we explicitlydemonstrate that γ cAB ( n ) vanishes.Chiral kagome ice provides an exactly solvable experimen-tally realized example, where subleading terms, analogousto the topological entanglement entropy, appear in the ab-sence of of quantum coherence or quantum fluctuations. Theadditional constraints in chiral kagome ice (with respect tothe nonchiral manifold) result in critical spin correlations, aswell as aforementioned subleading term γ cAB ( n ). In contrast,the nonchiral kagome ice has a finite correlation length, ex-tremely weak dependence of the R´enyi mutual information on a r X i v : . [ c ond - m a t . s t r- e l ] S e p the R´enyi index (which likely vanishes in the thermodynamiclimit), and a vanishing topological R´enyi mutual information.Interestingly, the RK quantum counterpart of the nonchiralkagome ice also has a vanishing topological entanglement en-tropy.Moreover, we study a continuous interpolation between thechiral and nonchiral kagome-ice systems, and argue that theuniversal terms (2) is a special property of the critical point,i.e., it is fragile against the presence of defects. A particularinterpolation is obtained by considering pyrochlore spin icein a [111] magnetic field: the chiral (nonchiral) kagome iceis then realized at temperature T = T = ∞ ). We arguethat in the thermodynamic limit, the subleading term in themutual information vanishes throughout the noncritical phase: γ cAB ( n ) = , T >
0, and jumps to the value given in Eq. (2)only at the critical point. Finally, our general formulation interms of a transfer matrix provides a tool for studying the en-tanglement properties of more complex classical systems suchas the three-dimensional pyrochlore spin ice, as well as theirRK-type quantum counterparts.The outline of our paper is as follows. In Sec. II, we presentthe general setup of the problem, and derive Eq. (1). We de-scribe in Sec. III the chiral and nonchiral kagome-ice systems,and discuss the relationship between the chiral manifold andthe Liftshitz quantum critical point. We predict the univer-sal subleading correction (2) using established results on theLiftshitz quantum critical point, and verify our predictionwith direct numerical calculations. We also explicitly verifythe vanishing of the topological R´enyi mutual information innonchiral kagome ice, and argue that the universal term (2) isfragile against thermal defects. We close the paper in Sec. IVwith a brief discussion.
II. GENERAL SETUPA. Preliminaries
The R´enyi entropy S X ( n ) of a subsystem X is a measureof entanglement with desired properties such as additivity andcontinuity: S X ( n ) = − n ln (cid:2) tr (cid:0) ρ nX (cid:1)(cid:3) . (3)Here the reduced density matrix ρ X of the subsystem X is de-fined through tracing out the degrees of freedom in the restof the system. The R´enyi entropy depends on a parameter n known as the R´enyi index, and the knowledge of the R´enyi en-tropies for all n potentially contains more information than thestandard von Neumann entropy, which is given by the n → n ) can yield the full spec-trum of the the reduced density matrix, known as the entangle-ment spectrum, thus providing a more complete characteriza-tion of entanglement properties. While it has been shownthat in a certain class of gapped topological phases, the uni-versal correction to the R´enyi entropy is independent of n , FIG. 1: (a) A convenient partitioning for studying universal entan-glement is obtained by dividing an infinite cylinder of circumference (cid:96) into two pieces. (b) Folding the B subsystem onto the A half allowsfor the application of boundary conformal field theory. for critical and also several gapped phases, the universal cor-rection generally has a nontrivial dependence on n .The R´enyi entropy is symmetric for a pure-state wave func-tion, i.e., S A ( n ) = S B ( n ) for a system comprised of two sub-systems A and B . On the other hand, for generic mixed stateswe have S A ( n ) (cid:44) S B ( n ). A natural generalization of the R´enyientropy to mixed states, known as the mutual information I AB ( n ), remedies this problem through an appropriate sym-metrization: I AB =
12 ( S A + S B − S A ∪ B ) , (4)where A ∪ B represents the whole system, and we have sup-pressed the explicit dependence on the R´enyi index n forbrevity. Since, for a pure state, S A ∪ B ( n ) = n , the mu-tual information reduces to the R´enyi entropy of entanglementin this case (the mutual information is often defined withoutthe factor of but this factor is convenient because of the con-nection with entanglement entropy). The mutual informationencodes the total amount of quantum and classical correla-tions between the two subsystems. For a system exhibiting area law with a universal order-onecorrection, we have I AB ∼ c (cid:96) + γ + · · · , (5)where c is a nonuniversal prefactor, (cid:96) is the ( d − d -dimensional subsystem, γ is the universal sub-leading term, and the ellipsis represents other subleading con-tributions, which vanish in the limit of (cid:96) → ∞ . Note that insome systems there could be logarithmic corrections to arealaw for subsystems with sharp corners or complex topology, but here we only consider subsystems with vanishing logarith-mic corrections.A convenient partitioning for extracting universal contribu-tions to the entanglement entropy [see Eq. (5)] is obtainedby dividing an infinite cylinder of circumference (cid:96) in half. The subsystems A and B then have simple topology and nosharp corners. We thus expect the area law to take the formof Eq. (5), with the subleading terms indicated by an ellipsisvanishing as e − (cid:96)/ξ for gapped phases (where ξ is the correla-tion length), and as a power-law of (cid:96) for critical systems. Inpractice, it is helpful to work with a finite cylinder of length 2 h (with free or fixed boundary conditions at the two endpoints),and take the thermodynamic limit h → ∞ at the end of thecalculation. B. Combinatorial argument
In order to derive Eq. (1), let us now consider a gen-eral set of (classical) configurations C satisfying some lo-cal constraints (which allows for a transfer-matrix formula-tion). A Hilbert space corresponding to this manifold can beconstructed as in the RK model: for each configuration weconstruct a state |C(cid:105) , and impose the orthogonality condition (cid:104)C|C (cid:48) (cid:105) = δ CC (cid:48) . If the degrees of freedom in a given configu-ration are classical Ising spins, then each state |C(cid:105) is a directproduct of single spin-up and -down states | ↑(cid:105) and | ↓(cid:105) . As-suming that there are N such configurations in a given (finite)system, an equal-weight quantum superposition and an equal-weight classical (thermal) mixture, respectively, have the fol-lowing density matrices: ρ q = N (cid:88) CC (cid:48) |C(cid:105)(cid:104)C (cid:48) | , ρ c = N (cid:88) C |C(cid:105)(cid:104)C| . (6)We start with the classical density matrix ρ c in Eq. (6). Rep-resenting the degrees of freedom (say Ising spins) in subsys-tems A and B with A and B , we can write S cA = − n ln (cid:32) N n (cid:88) C i A i B i (cid:104)A B |C (cid:105)(cid:104)C |A B (cid:105)(cid:104)A B |C (cid:105) . . . × (cid:104)A n B n |C n (cid:105)(cid:104)C n |A B n (cid:105) (cid:33) , (7)where {|AB(cid:105)} constitutes an orthonormal basis. We further as-sume that the configuration states |C(cid:105) belong to the set of basisstates (which is certainly the case for Ising spins or other localdegrees of freedom). We then find that the only contributionto the sum above comes from |C i (cid:105) = |A i B i (cid:105) = |A i + B i (cid:105) , whichgives A = A = . . . A n . In other words, the n replica config-urations C i are “stitched” together as shown in Fig. 2(a).Now if two configurations match in subsystem A , they mustalso match at the boundary between A and B . The sum inEq. (7) above can then be written as a sum over the configura-tions d at the boundary as follows: S cA = − n ln (cid:34) N n (cid:88) d N A ( d ) N nB ( d ) (cid:35) , (8)where N A ( d ) ( N B ( d )) denotes the number of configurationsin A ( B ) for a fixed configuration d on the boundary [seeFig. 2(a)]. The entropy S cB can be obtained simply by replac-ing A ↔ B in the above expression [see Fig. 2(b)]. It is alsoeasy to observe that S cA ∪ B = ln N for all n . A similar treatmentfor the quantum density matrix ρ q in Eq. (6) shows that the 2 n configurations ( C i and C (cid:48) i ) are “stitched” together as shown inFig. 2(c) in the quantum case, and we can write S qA = S qB = − n ln (cid:34) N n (cid:88) d N nA ( d ) N nB ( d ) (cid:35) . (9) FIG. 2: (Color online) (a) To calculate S cA , we need n replica states |C i (cid:105) , which have the same configurations in A subsystem (top panel).If we fold the B side onto the A side as in Fig. 1(b) (bottom panel),we get n + φ i in a field-theory representation) living on thehalf-cylinder. (b) To calculate S cC , we similarly identify the config-urations on the B side. (c) For the quantum case ( S qA = S qB ), theconfigurations are identified with a di ff erent pattern resulting in 2 n independent fields in the bulk of the half-cylinder. C. Transfer-matrix approach
As the configurations C are characterized by local con-straints, the combinatorial problem of computing the entropiesabove can be formulated in terms of a transfer matrix, whichhas proved useful in the calculation of entanglement entropyin both static and dynamical cases. Let us break up thesystem into rows parallel to the boundary between the twosubsystems, and indicate the configuration of the degrees offreedom on a row by a lower-case letter (the boundary config-uration d indicates the configuration of a row at the interfaceof the two subsystems). Some care must be taken in identify-ing the configuration d at the boundary with the configurationof a row, in terms of which the transfer matrix is constructed.Our assumption is that the subsystems A and B either satisfya symmetry with respect to the boundary, or there are hard lo-cal constraints. In the absence of these assumptions, one mayneed to resort to overlapping the subsystems as in Ref. 7.Considering two consecutive rows with configurations i and j , we can denote the number of allowed configurations of thedegrees of freedom between the two rows by T i j (if thereare no such degrees of freedom T i j is either 0 or 1). Wecan now represent the row configurations by states and write T i j = (cid:104) i | T | j (cid:105) , where T is the transfer matrix. Representing thestates at the left- and right-hand sides of the system, and theboundary between A and B [see Fig. 2(a)], respectively, by | l (cid:105) , | r (cid:105) , and | d (cid:105) , we can then write N A ( d ) = (cid:88) l (cid:104) l | T h | d (cid:105) , N B ( d ) = (cid:88) r (cid:104) d | T h | r (cid:105) , (10)where we have assumed free boundary conditions at the leftand right boundaries. Taking the thermodynamic limit, h →∞ , gives N A ( d ) = λ h (cid:88) l (cid:104) l | λ (cid:105)(cid:104) λ | d (cid:105) , N B ( d ) = λ h (cid:88) r (cid:104) d | λ (cid:105)(cid:104) λ | r (cid:105) , (11)where λ is the largest eigenvalue of T , and | λ (cid:105) is the corre-sponding eigenvector. Similarly, we have N = (cid:80) lr (cid:104) l | T h | r (cid:105) = λ h (cid:80) lr (cid:104) l | λ (cid:105)(cid:104) λ | r (cid:105) .Note that for fixed (Dirichlet) boundary conditions (insteadof free), we can use one particular | r (cid:105) or | l (cid:105) and omit thesummation over r and l . Our final result will not change asthe summation over the boundary conditions cancels out butwe may need to choose a di ff erent | λ (cid:105) : the largest-eigenvalueeigenstate of T , which has a nonvanishing overlap with bothfixed | r (cid:105) and | l (cid:105) . Also note that, more generally, the densitymatrices may not represent equal-weight mixtures (or super-positions in the quantum case). As discussed in the Appendix,however, as long as the configuration-dependent weights areultralocal, they can be absorbed into the transfer matrix be-tween consecutive rows (in this case the number of configura-tions with given boundary conditions transforms into a parti-tion function with those boundary conditions).Putting all these together, and assuming the overlaps be-tween | λ (cid:105) and the boundary states above are real, we then ob-tain S cA = − n ln λ h ( n + (cid:16)(cid:80) d (cid:104) λ | d (cid:105) n + (cid:17) λ hn (cid:0)(cid:80) l (cid:104) λ | l (cid:105) (cid:1) n − . (12)For this particular geometry, the expression for S cB isgiven by changing the dummy variable l to r (and yieldsan identical result). Similarly, we can write S cA ∪ B = − − n ln (cid:104) λ h ( n − (cid:0)(cid:80) l (cid:104) λ | l (cid:105) (cid:1) n − (cid:105) , which leads to the general ex-pression I cAB ( n ) = − n ln (cid:88) d (cid:104) λ | d (cid:105) n + , (13)where the R´enyi index n is written explicitly. The entangle-ment entropy of the quantum case was studied in Ref. 7 usingthis transfer matrix method. For completeness, we restate thatresult in our notation: by inserting Eq. (11) into Eq. (9), weimmediately obtain I qAB ( n ) = − n ln (cid:88) d (cid:104) λ | d (cid:105) n . (14)Notice that the row configurations d form an orthonormalbasis and the normalization of the eigenvector | λ (cid:105) implies that (cid:80) d (cid:104) λ | d (cid:105) =
1. In other words, p d ≡ (cid:104) λ | d (cid:105) is a probabil-ity, and the above expression is identical to that of Ref. 7.Comparing the two equations above, we finally arrive at thegeneral relationship I cAB ( n ) = I qAB (cid:16) n + (cid:17) , which is one of themain results of this paper [see Eq. (1)], and provides a non-trivial generalization of the prediction of Ref. 10 to arbitrary n . It is also worth mentioning that subtle issues (such as theinter-dependence of the compactification radii of fields) makeit di ffi cult to apply the field-theoretical approach of Ref. 10 tocertain problems such as RK dimer models. Despite this,the result of our combinatorial approach, which directly ap-plies to dimer models, is in perfect agreement with Ref. 10 for n → FIG. 3: (Color online) A kagome layer in the pyrochlore lattice.
III. KAGOME ICE AND QUANTUM LIFTSHITZ MODELA. Kagome-ice manifolds
Highly frustrated magnets are canonical examples of clas-sical spin liquids with nontrivial correlations. In particular,the so-called spin-ice materials in which Ising-type momentsreside on a three-dimensional network of corner-sharing tetra-hedra (the pyrochlore lattice shown in Fig. 3) exhibit a crit-ical dipolar-like spin correlations in an emergent Coulombphase. The origin of this critical phase can be traced to thenontrivial local constraints known as the two-in-two-out icerules that govern the ordering of magnetic moments in indi-vidual tetrahedra. In this work, we apply the above analysis to two-dimensional analogs of the pyrochlore spin ice: (i) chiraland (ii) nonchiral kagome ice. The kagome lattice shownin Fig. 4(a) consists of corner-sharing triangles with two dif-ferent orientations. The projections of the easy axes on thekagome plane form 120-degree ordering as in Fig. 4(a). Inchiral kagome ice, every up triangle has a two-in-one-out spinconfiguration with a net positive magnetic charge, while everydown triangle has a one-in-two-out configuration with a netnegative charge [see Fig. 4(a)]; the entropy density of this icephase is 0 . k B per spin. Since the two types of trianglescontain opposite magnetic charges, hence breaking the sublat-tice symmetry, this phase is referred to as the chiral kagomeice. The nonchiral kagome ice is a less constrained manifold:every triangle (whether up or down) is in either the two-in-one-out or the one-in-two-out state [see Fig. 4(c)]; the entropydensity of this ice phase is 0 . k B per spin. .The nonchiral degenerate manifold minimizes the nearest-neighbor spin interaction energy. This ice phase is noncriticalwith short-range spin-spin correlations. The chiral kagomeice appears in the intermediate plateau regime of the py-rochlore spin-ice in a [111] magnetic field, where thespins align themselves with the field on the triangular layers,and, consequently, form the degenerate chiral manifold on thekagome layers (see Fig. 3). An artificial version of the kagomeice is also realized in arrays of single-domain ferromagneticnanoisland arranged in a honeycomb network. In artifi-cial kagome ice, the chiral ice phase is induced by further-neighbor spin-spin interactions.
FIG. 4: (Color online) (a) A chiral kagome ice configuration. (b)The mapping to dimer coverings. The configurations of the bondscut across by the solid black line represent a boundary state. (c) Anonchiral kagome-ice configuration. (d) The mapping to dimers withone type of defect.
Remarkably, spin-spin correlations in chiral kagome ice de-cay algebraically. The critical correlations of this phase canbe understood by mapping the chiral ice configurations todimer coverings on the dual honeycomb lattice obtained byconnecting the centers of the corner-sharing triangles of thekagome lattice [see Fig. 4(b)]. Since every triangle hasexactly one negative spin (lying on a bond of the honeycomblattice), each site of the honeycomb lattice is visited by exactlyone negative spin, which we identify with a dimer. The dimer-covering problem on honeycomb is exactly solvable using thewell-known Pfa ffi an method for computing the partition func-tion. Exact calculations showed a power-law dimer-dimercorrelation decaying as 1 / r . In the language of dimers, thenonchiral kagome ice then corresponds to dimer coverings onthe honeycomb lattice with one type of defect: a site visitedby exactly two dimers [see Fig. 4(d)].
B. Chiral kagome ice and Liftshitz quantum model
In order to obtain the universal correction term to the R´enyimutual information for the chiral kagome ice [Eq. (2)], wefirst consider the R´enyi entropy of the corresponding quan-tum RK state and then invoke the classical-quantum corre-spondence (1). Universal entanglement entropy in RK wavefunctions of dimer coverings is a well-studied problem, butthe connection to spin-ice systems has not been exploited tostudy the the entanglement properties of these classical spinliquids. Let us briefly review the field-theoretical approachto the quantum RK case. The RK wave function | Ψ (cid:105) is anequal weight superposition of all dimer coverings on the hon-eycomb lattice, or equivalently an equal weight superpositionof all chiral kagome-ice states.Dimer models on a bipartite lattice in turn have a height-model representation: scalar heights are defined on every hexagonal plaquette, when a small circular path around asite (which crosses three bonds) goes across an empty bond(dimer), we decrease (increase) the height by 1 (2). Thepath is directed clockwise for sites on one sublattice andcounterclockwise for sites on the other. We then obtain aunique height field modulo the height of one reference pla-quette. A periodic structure is imposed on the heights toavoid overcounting configurations. Upon coarse-graining,the height-model representation then leads to a critical fieldtheory of compactified noninteracting bosons (coarse-grainedheight field) with e ff ective action: S [ φ ] = g π (cid:90) d x ( ∂ µ φ ) , φ ∼ φ + π R , (15)which represents a universality class characterized by the dy-namical exponent z =
2, sti ff ness g , and the compactificationradius R . By rescaling the bosonic fields φ , we observe thatthe only free parameter of the theory is R √ g , which can becomputed by comparing the dimer correlation functions ob-tained from the above continuum quadratic filed theory withexact results based on the Pfa ffi an method. The projection of the RK wave function to a particularheight field configuration and the quantum density matrix [seeEq. (6)] are given by (cid:104) φ | Ψ (cid:105) ∝ e − S [ φ ] , ρ q = | Ψ (cid:105)(cid:104) Ψ | . (16)The calculation of the entanglement entropy in the cylindergeometry then involves folding half of the cylinder (subsystem B ) onto the other half. The identification of Fig. 2(c) resultsin 2 n ( n form each subsystem) independent (in the bulk) fieldswith say Dirichlet boundary condition on the left-hand side ofthe folded system (the far ends of both A and B subsystems),and a special boundary condition φ = φ = · · · = φ n on theright-hand side of the folded system (the boundary betweenthe A and B subsystems).By explicitly constructing the boundary state correspond-ing to this boundary condition, it was shown in Ref. 9 (seealso Refs. 7,8) that the universal R´enyi entropy should go as − n ln (cid:20) ( (cid:112) gR ) − ( N / − (cid:113) N (cid:21) , where N is the number of inde-pendent fields, which in the quantum case equals N = n .Exactly the same argument also holds for the classical caseexcept that, as demonstrated in Figs. 2(a) and 2(b), the num-ber of independent fields is N = n + n from one sub-system and 1 from the other) with the boundary condition φ = φ = · · · = φ n + on the right-hand side of the foldedsystem. This is consistent with our Eq. (13) found throughan explicit combinatorial approach. The actual sti ff ness ofthe dimer model corresponds to (cid:112) gR = n with a critical R´enyi index n c = Thus, our results for the classical case should onlyhold for n < × − = A and B subsystems cuts across (cid:96) bonds ofthe honeycomb lattice [see Fig. 4(b)]. The transfer matrix isthen a (2 (cid:96) × (cid:96) )-dimensional matrix, which can be constructedusing the local constraints (counting the number of dimer cov-erings by transfer matrices was pioneered by Lieb. ). Thetransfer matrix on the honeycomb lattice has a block-diagonalstructure: if the transfer matrix has a nonzero matrix element (cid:104) i | T | j (cid:105) , the states | i (cid:105) and | j (cid:105) must have the same number ofdimers. A mapping from dimers to fermions yields elegantanalytical results for the transfer matrix both on the square and the honeycomb lattices. In the case of free boundary conditions at infinity, we needoverlaps (cid:104) λ | i (cid:105) for the eigenstate with the largest eigenvalue andall of the 2 (cid:96) row states. As the transfer matrix is block diag-onal, we can consider only the states in the block with thelargest eigenvalue. It was shown in Ref. 7 that the number ofdimers in this block is (cid:96)/
3. Denoting the positions of dimersin state | i (cid:105) as α i with 0 (cid:54) α < α < · · · < α (cid:96)/ (cid:54) (cid:96) −
1, themapping to fermions allows us to compute |(cid:104) λ | i (cid:105)| as a Van-dermonde determinant: |(cid:104) λ | i (cid:105)| = (cid:96) (cid:96)/ (cid:89) (cid:54) j < j (cid:48) (cid:54) (cid:96)/ (cid:20) π(cid:96) ( α j − α j (cid:48) ) (cid:21) . (17)The above expression provides an e ffi cient way to computethe R´enyi entropy for large (cid:96) (here we went up to (cid:96) = (cid:96) up to 18. The results areshown in Fig. 5(a) for n = . . .
4. As expected, for each n , themutual information I cAB ( n , (cid:96) ) satisfies the area law, i.e., scalesas ∝ (cid:96) for large (cid:96) . To extract the universal subleading term γ , we fit these data to I = c (cid:96) + γ + c (cid:48) /(cid:96) , and obtained γ as afunction of n . The agreement with Eq. (2) is excellent as seenin Fig. 5(b). C. Nonchiral kagome ice and defects
It is also illuminating to consider the case of nonchiralkagome ice, which has less constrained ice rules than the chi-ral manifold. The analogous quantum wave function then hasa finite correlation length. The subleading correction to thearea law in this case can then be interpreted as topologicalentanglement entropy. We find through numerical transfer-matrix calculations that the mutual information satisfies thearea law, and a fit to c (cid:96) + γ gives γ of order 10 − , i.e., thenonchiral kagome ice (and the corresponding quantum RKwave function) is topologically trivial. This provides an in-teresting example where a quantum wave function, with a su-perficial resemblance to a quantum spin liquid, is actually atopologically trivial paramagnet. Notice that the transfer ma-trix does not have a block-diagonal structure in the nonchiralcase, and we do not have an exact analytical solution for theoverlaps (cid:104) λ | i (cid:105) so we are limited to smaller systems in the nu-merics. However, we do not need to go to larger systems be-cause the finite-size corrections are exponentially small. Theuniversal terms can then be easily extracted from a simple lin-ear fit for very small system sizes.
10 15 20 25 30234567 2 4 6 8 10−0.25−0.2−0.15−0.10 2 4 6 800.511.522.53
FIG. 5: (Color online) (a) Numerically computed mutual informationusing Eqs. (17) and (13) exhibits the area law. (b) Fitting the data inpanel (a) gives the universal piece γ (blue stars), which are in perfectagreement with the analytical prediction Eq. (2) (red line). (c) Forthe nonchiral kagome ice, the mutual information has a very weakdependence on the R´enyi index n , and the linear fit to data gives γ cAB ≈ One comment is in order before proceeding. In the transfer-matrix calculation above, we used a 2 (cid:96) × (cid:96) transfer matrix forthe configuration of a row shown in Fig. 4(b). The subsystemsare constructed such that all the degrees of freedom betweenthe the last row of A and the first row of B belong to one ofthe two subsystems, say A . In the chiral kagome-ice case, thefirst row of B is uniquely determined by these degrees of free-dom and the last row of A . In the nonchiral case, however, theconstraints are less rigid and identifying the boundary config-uration d [see Eq. (8)] with a row as in Fig. 4(b) is rathersubtle. We have checked (for smaller systems), however, thatusing a 2 (cid:96) × (cid:96) transfer matrix (so that there are no degreesof freedom between two consecutive rows and the two subsys-tems are symmetric with respect to the boundary) leads to thesame result. Such microscopic considerations do not appearin the continuum field-theoretic approach, and seem unlikelyto play an important role in general.An interesting question is the behavior of γ when we in-terpolate between these two manifolds. If we consider thepyrochlore spin ice in a magnetic field H in the [111] direc-tion, we have a density matrix that only depends on the ratio H / T at temperature T . In terms of the dimer model, there isan energy cost proportional to H for any defect (two dimerstouching on one site). At zero temperature (and H > H / T diverges and no defects are allowed, which results inthe critical chiral kagome-ice manifold. On the other hand,the nonchiral ice corresponds to H / T =
0, where there is nocost for such defects. At low temperatures, any infinitesimalfinite temperature destroys the critical phase, resulting in afinite density of defects and a correlation length that is expo-nentially large in H / T . This instability follows from the factthat in the height representation, a defect is like a vortex (thecompactified height field winds once when going around suchdefect) with an operator proportional to cos (cid:16) √ θ (cid:17) , where θ is the dual field to the height field φ , which is a relevant (inrenormalization-group sense) perturbation. . If we compute γ in a finite system (smaller than the correlation length) atsmall finite temperatures, our results changes continuouslyfrom the T = γ to change within anoncritical phase as ξ/(cid:96) vanishes if we take the limit of (cid:96) → ∞ before any other limit. We are thus lead to conjecture that theuniversal term (2) is fragile against any density of defects inthe thermodynamic limit.For finite H / T , where we have a small correlation length,we have performed numerical transfer-matrix calculations,and verified the presence of a γ = H / T =
0. Asthe correlation length keeps increasing with H / T , however,the calculation becomes less reliable in the vicinity of the thechiral phase (note that we can not use the exact solution of thechiral kagome ice, which allowed for large system sizes in thepresence of defects). Nevertheless, by excluding smaller sys-tem sizes in fitting the data, we observe a very suggestive trendshown in Fig. 6: the dramatic change from γ is shifted towardsmaller temperatures, supporting our conjecture of fragility.Similar behavior has been proposed in the topological Kitaevtoric code model. IV. DISCUSSION
Motivated by the predictions of Ref. 10 on the appear-ance of topological entanglement entropy in classical sys-tems, which arise through the decoherence of topologicallyordered quantum wave functions, we obtained a generalizedrelationship [Eq. (1)] between the R´enyi mutual informationof generic classical systems (which can be described by atransfer matrix), and their quantum-coherent RK-type coun-terparts. The classical systems above can be similarly viewed
FIG. 6: (Color online) The von Neuman mutual information as afunction of e − H / T , where H is the energy cost of one defect. By ex-cluding smaller systems in the fitting, we observe that the dramaticchange away from the γ = as decohered versions of these RK quantum wave functions.We examined the universal entanglement properties of twospin-ice manifolds (chiral and nonchiral) on the kagome lat-tice, and found a nonvanishing universal subleading term inthe mutual information of the chiral state (a highly constrainedclassical spin liquid with critical correlations). The experi-mentally relevant chiral kagome ice, which appears in kagomelayers of pyrochlore spin ice in the presence of a [111]-direction magnetic field, readily maps onto a dimer model onthe honeycomb lattice. The RK quantum dimer model, andits well-known corresponding quantum Liftshitz universalityclass, thus describe the quantum-coherent counterpart of thechiral kagome ice. This mapping, and the general relation-ship (1) thus leads to our prediction of the presence of a uni-versal subleading term in chiral kagome ice, which bears astriking resemblance to the topological entanglement entropyin gapped topologically ordered phases.The universal mutual information in this critical classicalspin liquid, has a nontrivial dependence on the R´enyi in-dex. Moreover, we verified that for the short-range correlatednonchiral kagome ice, the dependence of the mutual infor-mation on the R´enyi index is extremely weak (likely only inthe subleading terms), and the topological mutual informationvanishes. The quantum counterpart of the nonchiral kagomeice provides an interesting example of a trivial paramagnet,which, on the surface, resembles a topological quantum spinliquid. The topologically trivial phase extends even when Z chirality is explicitly broken by a magnetic fields. In otherwords, in the thermodynamic limit, the universal term ob-tained at the critical point is fragile to any infinitesimal densityof defects. Acknowledgments
We thank C. Batista, C. Castelnovo, C. Chamon, E. Frad-kin, B. Hsu, and G. Misguich for helpful discussions and com-ments. We are specially grateful to I. Martin for collaborationin the early stages of this work and for several helpful sug-gestions. This work was supported by U.S. DOE under theLANL / LDRD program.
Appendix A: DENSITY MATRICES WITH NONEQUALWEIGHTS
In this appendix, we outline how the derivation of Eq. (1)changes when the density matrices have nonequal (but ultralo-cal) weights. The quantum and classical density matrices canbe written as ρ q = Z ( β ) (cid:88) CC (cid:48) e − β ( E C + E C(cid:48) ) |C(cid:105)(cid:104)C (cid:48) | , ρ c = Z ( β ) (cid:88) C e − β E C |C(cid:105)(cid:104)C| , (A1)where the partition function is given by Z ( β ) = (cid:80) C e − β E C .Computing S cA gives S cA = − n ln Z n ( β ) (cid:88) AB i e − β ( E AB + E AB + ··· E AB n ) . (A2)Once again all configurations AB i must match at the bound-ary between A and B represented by the row configuration d so we can write the generalized version of Eq. (8) as S cA = − n ln Z n ( β ) (cid:88) d Z A ( β, d ) Z nB ( β, d ) , (A3) where Z A ( β, d ) and Z B ( β, d ) represent partition functions withboundary configuration d for the two subsystems.We can similarly write S qA as S qA = − n ln Z n ( β ) (cid:88) A i B i e − β ( E A B + E A B + E A B ··· E A B n ) , (A4)which yields a generalized version of Eq. (9), i.e., S cA = − n ln (cid:104) Z n ( β ) (cid:80) d Z A ( β, d ) n Z nB ( β, d ) (cid:105) (see Ref. 7 for closely re-lated derivation based on Schmidt decomposition). Note thatthe energy of each replica subsystem A i and B i appears twicecanceling the factor of in β . If the weights are ultralocal, wecan then write E AB i = (cid:15) AB i a a + (cid:15) AB i a a + · · · (cid:15) AB i a h d + (cid:15) AB i db i + · · · (cid:15) AB i b ih − b ih , (A5)where a i and b i represent configurations of rows parallel tothe boundary. We then see that the weights can be absorbedinto the transfer matrix T , and all steps of our derivation gothrough. A. Y. Kitaev, Ann. Phys. , 2 (2003). A. Hamma, R. Ionicioiu, and P. Zanardi, Phys. Rev. A , 022315(2005). A. Kitaev and J. Preskill, Phys. Rev. Lett. , 110404 (2006). M. Levin and X.-G. Wen, Phys. Rev. Lett. , 110405 (2006). D. S. Rokhsar and S. A. Kivelson, Phys. Rev. Lett. , 2376(1988). B. Hsu, M. Mulligan, E. Fradkin, and E.-A. Kim, Phys. Rev. B ,115421 (2009). J.-M. St´ephan, S. Furukawa, G. Misguich, and V. Pasquier, Phys.Rev. B , 184421 (2009). B. Hsu and E. Fradkin, J. Stat. Mech: Theor. Exp. p. P09004(2010). M. Oshikawa, arXiv:1007.3739 (2010). C. Castelnovo and C. Chamon, Phys. Rev. B , 174416 (2007). C. L. Henley, J. Phys.: Condens. Matter , 164212 (2011). M. Hastings, Phys. Rev. Lett. , 210501 (2011). A. J. MacDonald, P. C. W. Holdsworth, and R. G. Melko, J. Phys.:Condens. Matter , 164208 (2011). L. D. C. Jaubert, M. J. Harris, T. Fennell, R. G. Melko, S. T.Bramwell, and P. C. W. Holdsworth, Phys. Rev. X , 011014(2013). R. Z. Lamberty, S. Papanikolaou, and C. L. Henley,arXiv:1210.8134 (2012). J. Wilms, M. Troyer, and F. Verstraete, J. Stat. Mech. p. P10011(2011). J. Iaconis, S. Inglis, A. B. Kallin, and R. G. Melko, Phys. Rev. B , 195134 (2013). S. T. Bramwell and M. J. P. Gingras, Science , 1495 (2001). S. V. Isakov, M. B. Hastings, and R. G. Melko, Nat. Phys. , 772(2011). Y. Zhang, T. Grover, and A. Vishwanath, Phys. Rev. B , 075128(2011). J.-M. St´ephan, G. Misguich, and V. Pasquier, J. Stat. Mech: Theor.Exp. p. P02003 (2012). H.-C. Jiang, Z. Wang, and L. Balents, Nat. Phys. , 902 (2012). D. Poilblanc, N. Schuch, D. P´erez-Garc´ıa, and J. I. Cirac, Phys.Rev. B , 014404 (2012). T. Grover, Y. Zhang, and A. Vishwanath, New J. of Phys. ,025002 (2013). H. Li and F. D. M. Haldane, Phys. Rev. Lett. , 010504 (2008). S. Dong, E. Fradkin, R. G. Leigh, and S. Nowlin, JHEP , 16(2008). S. T. Flammia, A. Hamma, T. L. Hughes, and X.-G. Wen, Phys.Rev. Lett. , 261601 (2009). H.-C. Jiang, R. R. P. Singh, and L. Balents, arXiv:1304.0780(2013). B. Groisman, S. Popescu, and A. Winter, Phys. Rev. A , 032317(2005). M. M. Wolf, F. Verstraete, M. B. Hastings, and J. I. Cirac, Phys.Rev. Lett. , 070502 (2008). E. Fradkin and J. E. Moore, Phys. Rev. Lett. , 050404 (2006). A. Rahmani and C. Chamon, Phys. Rev. B , 134303 (2010). C. L. Henley, Annu. Rev. Condens. Matter Phys. , 179 (2010). M. Udagawa, M. Ogata, and Z. Hiroi, J. Phys. Soc. Jpn. , 2365(2002). R. Moessner and S. Sondhi, Phys. Rev. B , 064411 (2003). A. S. Wills, R. Ballou, and C. Lacroix, Phys. Rev. B , 144407(2002). K. Matsuhira, Z. Hiroi, T. Tayama, S. Takagi, and T. Sakakibara,Journal of Physics: Condensed Matter , L559 (2002). M. Tanaka, E. Saitoh, H. Miyajima, T. Yamaoka, and Y. Iye, Phys.
Rev. B , 052411 (2006). Y. Qi, T. Brintlinger, and J. Cumings, Phys. Rev. B , 094418(2008). G.-W. Chern, P. Mellado, and O. Tchernyshyov, Phys. Rev. Lett. , 207202 (2011). G. M¨oller and R. Moessner, Phys. Rev. B , 140409 (2009). R. Moessner and S. L. Sondhi, Phys. Rev. B , 224401 (2001). P. W. Kasteleyn, Physica , 1209 (1961). C. S. O. Yokoi, J. F. Nagle, and S. R. Salinas, J. Stat. Phys. ,729 (1986). E. Ardonne, P. Fendley, and E. Fradkin, Ann. of Phys. , 493(2004). C. L. Henley, J. Stat. Phys. , 483 (1997). C. Zeng and C. L. Henley, Phys. Rev. B , 14935 (1997). J.-M. St´ephan, G. Misguich, and V. Pasquier, Phys. Rev. B ,195128 (2011). E. H. Lieb, J. Math. Phys. , 2339 (1967). F. Alet, Y. Ikhlef, J. L. Jacobsen, G. Misguich, and V. Pasquier,Phys. Rev. E , 041124 (2006). H. Otsuka, Phys. Rev. Lett. , 227204 (2011). C. Castelnovo and C. Chamon, Phys. Rev. B , 184442 (2007). Z. Nussinov and G. Ortiz, Proc. Natl. Acad. Sci. U.S.A. ,16944 (2009). Z. Nussinov and G. Ortiz, Ann. Phys.324