HHigh-degeneracy Potts coarsening
J. Denholm ∗ SUPA, Department of Physics, University of Strathclyde, Glasgow, G4 0NG, Scotland, UK (Dated: January 8, 2021)I examine the fate of a kinetic Potts ferromagnet with a high ground-state degeneracy thatundergoes a deep quench to zero-temperature. I consider single spin-flip dynamics on triangularlattices of linear dimension ≤ L ≤ and set the number of spin states q equal to the number oflattice sites L × L . The ground state is the most abundant final state, and is reached with probability ≈ . . Three-hexagon states occur with probability ≈ . , and hexagonal tessellations with morethan three clusters form with probabilities of O (10 − ) or less. Spanning stripe states—where thedomain walls run along one of the three lattice directions—appear with probability ≈ . . “Blinker”configurations, which contain perpetually flippable spins, also emerge, but with a probability that isvanishingly small with the system size. I. INTRODUCTION
When a kinetic ferromagnet with non-conserved mag-netisation undergoes a deep quench to zero-temperature,the final states are intriguingly diverse. After the ensuingcoarsening regime, surviving domain structures competefor dominion over the final state. Naïvley, one could as-sume that a single domain will ultimately prevail as theground state is necessarily reached—but this is far fromthe truth even in simple models.In the nearest-neighbour Ising model of linear dimension L , the ground state is always reached in one-dimension [1],yet never reached in three-dimensions [2, 3]. In two-dimensions, the situation is markedly richer: only of realisations proceed directly to the ground state ona timescale of O ( L ) [4, 5]. Surprisingly, of tra-jectories become trapped in frozen two-stripe states,which are infinitely long-lived and form on a timescaleof O ( L ln L ) [4, 6]. The remaining of instancesreach ephemeral diagonal winding configurations, whichultimately collapse to homogeneity on a timescale of O (cid:0) L . (cid:1) [4, 5].The explanation underpinning these final states is anapparent one-to-one mapping with the equivalent crossingprobabilities of critical continuum percolation [7–9]. Oncea percolating domain structure has formed, the fate of thezero-temperature Ising model is sealed [7–9]. Percolationand the domain growth in bi-dimensional coarsening havebeen readily studied [7–18].After understanding the fate of the zero-temperatureIsing model, it is natural to consider a system of greaterdegeneracy and to therefore study the dynamics of the q -state Potts model. Interest in the kinetics of Potts ferro-magnets has been motivated by their utility in understand-ing coarsening in soap froths [19–21], magnetic grains [22–25], natural tilings [26–28], superconductors [29] and mag-netic domains [30, 31].The domain growth in coarsening Potts systems hasbeen extensively studied and is well understood [32–44]. ∗ [email protected] The existence of non-ground final states after a zero-temperature quench was realised in the Potts model be-fore the equivalent finding in the Ising model [45–47].Nevertheless, there is an apparent lack of literature ex-amining the late-time configurations that persist after azero-temperature quench. In fact, there are seemingly onlytwo studies that explicitly focus on this problem—bothof which are in two dimensions [47, 48].On the square lattice with q = 3 , several odditiesemerge: the ground state probability is only ≈ . [47],and the most prevalent final states are “frozen” configu-rations with two or more surviving spin states [47]. Sur-prisingly, “blinker” configurations also emerge, where thesystem forever wanders at constant energy [47]. Thestrangest feature however is that of “pseudo-blinkers”:after exorbitant time periods strongly resembling blink-ers, single energy-lowering flips trigger sudden “energyavalanches”, where the system suddenly declines in en-ergy and macroscopically reorders [47]. Identifying these t = . × t = . × t = . × t = . × (a) t = . × t = . × t = . × t = . × (b) t = . × t = . × t = . × t = . × (c) FIG. 1. Snapshots of zero-temperature coarsening in atriangular-lattice Potts ferromagnet of q = L × L = 2500 for realizations reaching (a) ground, (b) two-stripe and (c)three-hexagon states. Distinct clusters are labelled by colour. a r X i v : . [ phy s i c s . c o m p - ph ] J a n configurations was a non-trivial numerical challenge [47].On the triangular lattice with q = 3 , the ground stateis reached approximately / of the time, and both three-hexagon and two-stripe states appear (FIG. 1 (a)–(c)) [48].Blinkers seem only to occur only in small system sizes with q > , and play a negligible role in the dynamics. Thedisparity between the fates of the square and triangularlattice Potts models at zero-temperature is surprisingconsidering the affinity in the equivalent Ising models.The origin of the contrasting fates of the square- andtriangular-lattice Potts models is rooted in the behaviourof so-called “T-junctions”. T-junctions are formed by themeeting of three domain structures comprised of differentspin-types [47]. At the centre of the junction, one findsspins which are trapped in local energy minima thatcannot flip. The square -lattice, by virtue of geometry,imposes the restriction that T-junctions are fixed in theirspatial location. Thus, the only way for the system toescape is through some macroscopic disruption of theconfiguration—which is not always possible. [47].However, on the triangular lattice, the geometricalconstraint on the location of T-junctions is lessened, andthe centre of the junctions can move. Consequently, thesystem can escape from these configurations [48]. Perhapsthe simplicity that emerges on the triangular lattice makesit a better candidate for one-day achieving the exactcomputation of the final states probabilities of the three-state Potts model at zero-temperature?Further study of the triangular lattice Potts model isinteresting for a number of reasons. In the Ising model,we see two categories of final state, both of which spanthe linear dimension of the system. When we move to thethree-state Potts model, a new topologically distinct finalstate emerges: the three-hexagon state. If we increase thedegeneracy to q = 5 , blinker configurations, which are an-other fundamentally different final state, appear. One cantherefore ask: are there other interesting features of thetriangular lattice Potts model that yet remain uncovered?The general behaviour of the final state probabilities athigh degeneracies is unknown.In this manuscript I examine the final state of a zero-temperature Potts ferromagnet with a high ground-statedegeneracy on the triangular lattice. I explore the specialcase where the number of spin states q is equal to thenumber of lattice sites L × L , which provides a naturalupper bound on the number of spin states with consider-ing. I detail the model and simulation method in Sec. II.In Sec. III, I introduce the final states that emerge andestimate the frequency with which they occur. In Sec-tion IV, I examine the number of clusters and survivingspins states as functions of time, before summarising myfindings in Section. V. II. ZERO-TEMPERATURE POTTS MODEL
I consider nearest-neighbour interactions on the trian-gular lattice geometry with periodic boundary conditions. I build the triangular lattice by taking a square lattice oflength L and adding diagonal bonds to the North-Westand South-East (see FIG. 2). FIG. 2. Equivalent triangular lattice geometries (a) and (b).
I initialise the system by placing each spin in a uniquestate, giving q = L × L states in total. The spin statesare denoted by the integers S i ∈ { , , . . . q } . Like spinsare said to be aligned and unlike spins misaligned. Thetotal energy of the system is given by the Hamiltonian H = − J (cid:88) i,j (cid:2) δ ( S i , S j ) − (cid:3) , (1)where J > is a ferromagnetic coupling constant, δ ( S i , S j ) is the Kronecker delta and j indexes the nearest-neighbours of each S i . Thus, each misaligned neighbourprovides an energy contribution of +2 J .To implement the dynamics, I employ continuous-timerejection-free kinetic Monte Carlo—where each spin isallowed to flip once, on average, in a single Monte Carlotime unit [34, 49–51]. This method is equivalent to thestandard discrete time Monte Carlo method, where oneallows N = L × L randomly selected spins the chanceto flip once in a single time step [49–51]. I endow theHamiltonian with zero-temperature metropolis dynamics:energy lowering and energy conserving moves are acceptedwith probability , while energy raising moves are forbid-den [34, 51]. The choice of dynamics is relatively flexibleso long as one adheres to the principle of detailled balance,therefore one might also use Glauber dynamics. [51].The total rate, r i , of spin S i is simply the sum ofthe transition probabilities over each of the ( q − ori-entations it may flip to. Since I use zero-temperatureMetropolis dynamics—where the transition probabilitiesare or —the rate r i is simply a count of the numberof transitions permitted by the dynamics. Let the totalrate in the system be R = (cid:80) r i . To flip a spin, I select asite with probability r i /R , draw randomly from its list of r i permissible transitions, and then flip the spin. Timeadvances as ∆ t = − log( u ) × ( q − /R , where u ∈ (0 , isa uniform random number and (cid:104)− log( u ) (cid:105) = 1 . I includea brief note on the simulation time in Appendix. A.An important simplifying feature at zero-temperatureis that spins with no aligned neighbours may flip to ( q − other spin states, whereas spins with at least one alignedneighbour may only flip to align with other neighbouringspins. This consideration is useful when computing r i ,particularly at large q . . . . . . . / ln L . . . . . . . . P r ob a b ilit y × − (a) Ground 0 . . . . . . / ln L . . . . . . . . × − (b) Three-hexagonTwo-stripe 0 . . . . . . / ln L × − (c) Five-hexagon 0 . . . . . . / ln L × − (d) Twelve-hexagonEight-hexagon
FIG. 3. Probability of reaching: (a) the ground state; (b) a three-hexagon and two-stripe state; (c) a five-hexagon state; (d) atwelve- and eight-hexagon state. The data are based on × realisations. III. FINAL STATES
I begin my examination of the final state probabilitieswith the ground state case, which I plot in FIG. 3 (a). Theground state is the most abundant final state, and when L = 128 , it is reached with a probability of ≈ . . Theground state probability varies non-monotonically with L , making it difficult to obtain an asymptotic estimate.The non-monotonicity in the ground state probability isnot unique to this system: it is also present in both theIsing model and small q Potts models on the square andtriangular lattice geometries [4, 5, 47, 48].The next most common final states are three-hexagonand two-stripe states, which are reached with probabilitiesof ≈ . and ≈ . respectively (FIG. 3 (b)). As wellas three-hexagon states, I also find rarer subspecies ofhexagonal tilings containing five, eight and twelve clus-ters—examples of which are shown in FIG. 4 (a)–(c).Hexagonal configurations with more than three clustersare much rarer than the three-cluster case; I plot the prob-ability of finding five-, eight- and twelve-hexagon statesin FIG. 3 (c)–(d), showing they are orders of magnitudeless abundant than their three-cluster counterparts. I alsofound a single realisation which reached a sixteen -hexagonstate (see FIG. 4 (d)), making it the rarest of its kind.The energy of an n -hexagon state depends only on thenumber of clusters, and not the individual cluster arrange-ment. Consider the three hexagon state in FIG. 1 (c).The total length of interface between the domains is L .There are spins on either side of these interfaces, giving L boundary spins. Each interface spin has two misalignedneighbours giving an energy contribution of +4 J . Con-sequently, the total energy of any three-hexagon stateis L . I extend this reasoning to hexagon states withmore than three clusters to obtain the energies shown inTable I.The scarcest final states are blinkers, which are config-urations that contain perpetually active sites. “Blinking”spins flip eternally as they only have energy conservingtransitions available to them. Consider the zoom-in on ablinker configuration in FIG. 5. When the spin is aligned (a) (b)(c) (d) FIG. 4. Frozen (a) five-, (b) eight-, (c) twelve- and (d) sixteen-hexagon states. Each cluster is labelled by colour and thelattice size is L = 90 . with its North and North-west neighbours, it can onlyflip to align with its South and South-east neighbours,and vice versa. It can never align with its East or West n Interface length Interface spins Energy L L L L L L L L L
12 6 L L L
16 7 L L L TABLE I. Energies of n -hexagon final states. Note: each bondis counted twice. FIG. 5. Zoom-in on a blinker spin (B) which can align with itsNorth and North-west neighbours ( × ) or its South and South-east neighbours ( (cid:13) ), but never its East or West neighbours. neighbours without raising its energy, which is forbiddenat zero-temperature.In small q triangular-lattice Potts models, blinkers seem-ingly only occur when q > with a probability that isvanishingly small with increasing L [48]. Here I find threemain categories of blinkers: five-, eight- and twelve-clusterconfigurations. Each contain O (1) active sites which arepinned in the same way as the blinker spin in FIG. 5. Theprobability of reaching blinkers with five, eight and twelveclusters is O (10 − ) , O (10 − ) and O (10 − ) respectively.I also found a single realisation that reached a blinkerstate with sixteen clusters with L = 22 . The probabilityof reaching a blinker configuration—of any kind—on thetriangular lattice is vanishingly small with increasing L . IV. TIME EVOLUTION
Two natural observables to consider when a high- q Potts system is quenched are the number of clusters, N c ,and the number of extant spins states, E q . A clusteris simply a group of aligned spins that are connectedthrough nearest-neighbour contact, and the number ofextant q is a count of the distinct spin states that remainpresent in the system.I compare the time evolution of these quantities forPotts systems with q = 3 , q = 60 and q = L in FIG. 6.As the number of spin states increases, the departure fromthe unmagnetised initial state slows—both N c and E q areincreasingly stagnant at early times with greater q .The explanation for this slow initial evolution is simple:consider the unmagnetised initial condition where no spinhas any aligned neighbours. Each spin may freely undergoone of ( q − possible transitions. In a small q Pottssystem, the probability that a spin should flip to alignwith a neighbour is relatively high. However, when q islarge, the probability that a single flip should result in twoneighbours aligning is small. Consequently, at large q , ittakes longer for the spins to align, so the departure fromthe unmagnetised initial condition becomes increasinglyslow. log ( time ) l og ( N c ) (a) q = L q = q = log ( time ) l og (cid:0) E q (cid:1) (b) q = L q = q = FIG. 6. Time dependence of (a) the number of clusters N c and (b) the number of extant spin states E q . The data arebased on realisations with L = 60 . V. DISCUSSION
I investigated the final state of a zero-temperature Pottsferromagnet where the ground-state degeneracy was equalto the number of lattice sites. With respect to q = 3 ,the geometric and topological nature of the final states isnot materially different, but their relative abundance is different.The ground state is the most prevalent final state, andis reached with probability ≈ . when L = 128 . Three-hexagon states appear more frequently with increasing L , and are reached with probability ≈ . when L =128 . I also found hexagon states with five, eight, andtwelve clusters with probabilities of O (10 − ) , O (10 − ) and O (10 − ) respectively. On-axis stripe states, wherethe domain walls run along one of the three lattices axes,also appeared. The probability of reaching two-stripestates decays with increasing L , and was ≈ . when L = 128 . Blinker configurations with five, eight and twelveclusters also emerged. The probability of finding blinkerconfigurations is O (10 − ) or less, and is vanishingly smallwith increasing L .The time evolution at high q is inherently slow, meaningmy probability estimates as L → ∞ are necessarily crude.I illustrated this slow evolution by comparing the numberof clusters and extant spin states as functions of time inPotts models with q = 3 , q = 60 and q = L = 3600 .There are a number of open questions concerning thefate of kinetic Potts ferromagnets at zero-temperature.The exact computation of the final state probabilitieswith q = 3 has not yet been achieved. The connectionwith two-colour percolation that emerged in the Isingmodel enabled the precise conjecture of the final stateprobabilities—perhaps a similar connection exists betweenthe less-well-understood three-colour percolation and thethree-state Potts model? Nevertheless, the affinity be-tween the final states of the square- and triangular-latticeIsing models at zero temperature is not present in theequivalent Potts models, so the universality of a connec-tion to three-colour percolation is unclear.Furthermore, even if three-colour percolation does ap-ply to a Potts ferromagnet with q = 3 , the question of q > remains; the general dependence of the final stateprobabilities on the number of spin states is still unknown.It is concevable that the high- q limit will one day play arole in an analytical solution for the final state probabili-ties of the zero-temperature Potts model on the infinitetriangular lattice geometry. In such a case, knowledgeof how the final state probabilities behave in finite ge-ometries, and of what kinds of final state to expect, willbe important. The fact the final state probabilities for q = L are different to the q = 3 case is an interestingfinding, and suggests the ground-state degeneracy playsan important role in determining the final state.Additionally, the fate of the zero-temperature Potts fer-romagnet on the simple cubic lattice remains unexplored.On the triangular lattice, where the spins have six nearest neighbours, the final states become materially simplerand more tractable. Perhaps the three-state Potts modelon the simple cubic lattice, which also has a coordinationnumber of six, will exhibit a similar simplicity? ACKNOWLEDGEMENTS
I thank Sid Redner for an interesting suggestion thatsparked this work and Leticia Cugliandolo for encourag-ing comments. I acknowledge the ARCHIE-WeSt HighPerformance Computer based at the University of Strath-clyde and grant EP/P015719/1 for computer resources.I also acknowledge EPSRC DTA5 grant EP/N509760/1for financial support. [1] P. L. Krapivsky, S. Redner, and E. Ben-Naim,
A KineticView of Statistical Physics (Cambridge University Press,2010).[2] J. Olejarz, P. L. Krapivsky, and S. Redner, Phys. Rev.E , 030104(R) (2011).[3] J. Olejarz, P. L. Krapivsky, and S. Redner, Phys. Rev.E , 051104 (2011).[4] V. Spirin, P. L. Krapivsky, and S. Redner, Phys. Rev. E , 036118 (2001).[5] V. Spirin, P. L. Krapivsky, and S. Redner, Phys. Rev. E , 016119 (2001).[6] J. Denholm and B. Hourahine, Journal of Statistical Me-chanics: Theory and Experiment , 093205 (2020).[7] J. J. Arenzon, A. J. Bray, L. F. Cugliandolo, and A. Si-cilia, Phys. Rev. Lett. , 145701 (2007).[8] K. Barros, P. L. Krapivsky, and S. Redner, Phys. Rev.E , 040101(R) (2009).[9] J. Olejarz, P. L. Krapivsky, and S. Redner, Phys. Rev.Lett. , 195702 (2012).[10] T. Blanchard and M. Picco, Phys. Rev. E , 032131(2013).[11] L. F. Cugliandolo, Journal of Statistical Mechanics: The-ory and Experiment , 114001 (2016).[12] T. Blanchard, L. F. Cugliandolo, M. Picco, andA. Tartaglia, Journal of Statistical Mechanics: Theoryand Experiment , 113201 (2017).[13] F. Corberi, L. F. Cugliandolo, F. Insalata, and M. Picco,Phys. Rev. E , 022101 (2017).[14] A. Tartaglia, L. F. Cugliandolo, and M. Picco, Journalof Statistical Mechanics: Theory and Experiment ,083202 (2018).[15] F. Insalata, F. Corberi, L. F. Cugliandolo, and M. Picco,Journal of Physics: Conference Series , 012018 (2018).[16] K. Humayun and A. J. Bray, Journal of Physics A: Math-ematical and General , 1915 (1991).[17] A. Bray, Physica A: Statistical Mechanics and its Appli-cations , 41 (1993).[18] A. Sicilia, J. J. Arenzon, A. J. Bray, and L. F. Cuglian-dolo, Phys. Rev. E , 061116 (2007).[19] J. A. Glazier, M. P. Anderson, and G. S. Grest, Philo-sophical Magazine B , 615 (1990).[20] G. L. Thomas, R. M. C. de Almeida, and F. Graner,Phys. Rev. E , 021407 (2006). [21] D.Weaire and N. Rivier, Contemporary Physics , 199(2009).[22] D. Srolovitz, M. Anderson, P. Sahni, and G. Grest, ActaMetallurgica , 793 (1984).[23] V. Fradkov and D. Udler, Advances in Physics , 739(1994).[24] D. Raabe, Acta Materialia , 1617 (2000).[25] D. Zöllner, Computational Materials Science , 2712(2011).[26] J. C. M. Mombach, M. A. Z. Vasconcellos, and R. M. C.de Almeida, Journal of Physics D: Applied Physics ,600 (1990).[27] J. C. M. Mombach, R. M. C. de Almeida, and J. R.Iglesias, Phys. Rev. E , 598 (1993).[28] A. Hočevar, S. El Shawish, and P. Ziherl, The EuropeanPhysical Journal E , 369 (2010).[29] R. Prozorov, A. F. Fidler, J. R. Hoberg, and P. C. Can-field, Nature Physics , 327 (2008).[30] K. L. Babcock, R. Seshadri, and R. M. Westervelt, Phys.Rev. A , 1952 (1990).[31] E. A. Jagla, Phys. Rev. E , 046204 (2004).[32] S. A. Safran, P. S. Sahni, and G. S. Grest, Phys. Rev. B , 466 (1982).[33] S. A. Safran, P. S. Sahni, and G. S. Grest, Phys. Rev. B , 2693 (1983).[34] P. S. Sahni, D. J. Srolovitz, G. S. Grest, M. P. Anderson,and S. A. Safran, Phys. Rev. B , 2705 (1983).[35] P. S. Sahni, G. S. Grest, M. P. Anderson, and D. J.Srolovitz, Phys. Rev. Lett. , 263 (1983).[36] G. S. Grest, M. P. Anderson, and D. J. Srolovitz, Phys.Rev. B , 4752 (1988).[37] E. E. Ferrero and S. A. Cannas, Phys. Rev. E , 031108(2007).[38] M. J. de Oliveira, Computer Physics Communications , 480 (2009).[39] D. J. Srolovitz and G. S. Grest, Phys. Rev. B , 3021(1985).[40] E. A. Holm, J. A. Glazier, D. J. Srolovitz, and G. S.Grest, Phys. Rev. A , 2662 (1991).[41] A. Petri, M. de Berganza, and V. Loreto, PhilosophicalMagazine , 3931 (2008).[42] M. P. O. Loureiro, J. J. Arenzon, L. F. Cugliandolo, andA. Sicilia, Phys. Rev. E , 021129 (2010). [43] M. P. O. Loureiro, J. J. Arenzon, and L. F. Cugliandolo,Phys. Rev. E , 021135 (2012).[44] F. Corberi, L. F. Cugliandolo, M. Esposito, and M. Picco,Journal of Physics: Conference Series , 012009(2019).[45] M. J. de Oliveira, A. Petri, and T. Tomé, Physica A:Statistical Mechanics and its Applications , 97 (2004).[46] M. J. de Oliveira, A. Petri, and T. Tomé, EPL (Euro-physics Letters) , 20 (2004).[47] J. Olejarz, P. L. Krapivsky, and S. Redner, Journalof Statistical Mechanics: Theory and Experiment ,P06018 (2013).[48] J. Denholm and S. Redner, Phys. Rev. E , 062142(2019).[49] A. Bortz, M. Kalos, and J. Lebowitz, Journal of Compu-tational Physics , 10 (1975).[50] P. Landau and K. Binder, A guide to Monte–Carlo Simu-lations in Statistical Physics (Cambridge University Press,2009).[51] G. N. Hassold and E. A. Holm, Computers in Physics ,97 (1993). Appendix A: Note on simulation time
Consider a zero-temperature Potts system with onlya single active site (see FIG 7). If each site is allowed to flip once, on average, in a single time step, and siteswith no aligned neighbours have q − possible transitionsavailable to them, the configuration in FIG. 7 reaches theground state in ( q − Monte Carlo time steps. When q = 2 , the ground state is reached in a single Monte Carlotime step, and when q = 3 , the ground state is reachedin two Monte Carlo time steps, and so on. This featuresignificantly encumbers my simulations: say L = 100 and q = L ; the active spin in FIG. 7 will flip O (10 ) timesbefore aligning with its neighbours. FIG. 7. A realisation reaching the ground state at time (cid:104) T G (cid:105)(cid:105)