Devil's Staircase Continuum in the Chiral Clock Spin Glass with Competing Ferromagnetic-Antiferromagnetic and Left-Right Chiral Interactions
aa r X i v : . [ c ond - m a t . d i s - nn ] M a y Devil’s Staircase Continuum in the Chiral Clock Spin Glass with CompetingFerromagnetic-Antiferromagnetic and Left-Right Chiral Interactions
Tolga C¸ a˘glar and A. Nihat Berker
1, 2, 3 Faculty of Engineering and Natural Sciences, Sabancı University, Tuzla, Istanbul 34956, Turkey Faculty of Engineering and Natural Sciences, Kadir Has University, Cibali, Istanbul 34083, Turkey Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
The chiral clock spin-glass model with q = 5 states, with both competing ferromagnetic-antiferromagnetic and left-right chiral frustrations, is studied in d = 3 spatial dimensions byrenormalization-group theory. The global phase diagram is calculated in temperature, antiferro-magnetic bond concentration p , random chirality strength, and right-chirality concentration c . Thesystem has a ferromagnetic phase, a multitude of different chiral phases, a chiral spin-glass phase,and a critical (algebraically) ordered phase. The ferromagnetic and chiral phases accumulate at thedisordered phase boundary and form a spectrum of devil’s staircases, where different ordered phasescharacteristically intercede at all scales of phase-diagram space. Shallow and deep reentrances of thedisordered phase, bordered by fragments of regular and temperature-inverted devil’s staircases, areseen. The extremely rich phase diagrams are presented as continuously and qualitatively changingvideos. PACS numbers: 75.10.Nr, 05.10.Cc, 64.60.De, 75.50.Lk
I. INTRODUCTION
The presence of chiral interactions, motivated by ex-perimental systems [1–5], can result in extremely richphase transition phenomena in otherwise simple systems[6]. In this respect, we study here a q = 5 state clockspin-glass model in d = 3 spatial dimensions, usingrenormalization-group theory. Our system has both com-peting ferromagnetic and antiferromagnetic interactions,as in the usually studied spin-glass models [8], and com-peting left-chiral and right-chiral interactions [6]. Wehave studied q = 5 states, because odd number of stateshave built-in entropy for antiferromagnetic interactions,even without quenched randomness and frustration.[7]The global phase diagram is calculated in temperature,antiferromagnetic bond concentration p , random chiral-ity strength, and right-chirality concentration c . We findan extremely rich phase diagram, with a ferromagneticphase, a multitude of different chiral phases, a chiral spin-glass phase, and a critical (algebraically) ordered phase[9, 10]. The ferromagnetic and chiral phases accumulateat the disordered phase boundary and form a devil’s stair-cases [11, 12], where different ordered phases characteris-tically intercede at all scales of phase-diagram space. Infact, a continuum of devil’s staircases is found. Shallowand deep reentrances of the disordered phase, borderedby fragments of regular and temperature-inverted devil’sstaircases, are seen. The extremely rich phase diagramsare presented as continuously and qualitatively changingvideos [13]. II. THE q − STATE CHIRAL CLOCKDOUBLE SPIN GLASS
The q − state clock spin glass is composed of unitspins that are confined to a plane and that can only point along q angularly equidistant directions, with Hamilto-nian − β H = X h ij i J ij ~s i .~s j = X h ij i J ij cos θ ij , (1)where β = 1 /k B T , θ ij = θ i − θ j , at each site i the spin angle θ i takes on the values (2 π/q ) σ i with σ i = 0 , , , . . . , ( q − h ij i denotes thatthe sum runs over all nearest-neighbor pairs of sites.As a ferromagnetic-antiferromagnetic spin-glass system[8], the bond strengths J ij , with quenched (frozen)ferromagnetic-antiferromagnetic randomness, are + J > − p and − J (antifer-romagnetic) with probability p , with 0 ≤ p ≤
1. Thus,the ferromagnetic and antiferromagnetic interactions lo-cally compete in frustration centers. Recent studies onferromagnetic-antiferromagnetic clock spin glasses are inRefs. [7, 14, 15].In the q − state chiral clock double spin glass intro-duced here, frustration also occurs via randomly frozenleft or right chirality [6]. The Hamiltonian in Eq. (1) isgeneralized to random local chirality, − β H = X h ij i [ J ij cos θ ij + ∆ δ ( θ ij + η ij πq )] . (2)In a cubic lattice, the x, y, or z coordinates increase assites along the respective coordinate direction are consid-ered. Bond-moving as in Fig. 1(a) is done transversely tothe bond directions, so that this sequencing is respected.Equivalently, in the corresponding hierarchical lattice,one can always define a direction along the connectiv-ity, for example from left to right in Fig. 1(b), and as-sign consecutive increasing number labels to the sites. InEq. (2), for each pair of nearest-neighbor sites h ij i thenumerical site label j is ahead of i , frozen (quenched) η ij = 1 (left chirality) or − δ ( x ) = 1 (0) for x = 0 ( x = 0). The overallconcentrations of left and right chirality are respectively1 − c and c , with 0 ≤ c ≤
1. The strength of the randomchiral interaction is ∆ /J , with temperature divided out.With no loss of generality, we take ∆ ≥
0. Thus, thesystem is chiral for ∆ >
0, chiral-symmetric for c = 0 . c = 0 .
5. The global phase di-agram is in terms of temperature J − , antiferromagneticbond concentration p , random chirality strength ∆ /J ,and chiral symmetry-breaking concentration c . III. RENORMALIZATION-GROUP METHOD:MIGDAL-KADANOFF APPROXIMATION ANDEXACT HIERARCHICAL LATTICE SOLUTION
We solve the chiral clock double spin-glass modelwith q = 5 states by renormalization-group theory, in d = 3 spatial dimensions, with length rescaling factor b = 3. We use b = 3, as in previous position-spacerenormalization-group calculations of spin-glass systems,because it treats ferromagnetism and antiferromagnetismon equal footing. Our solution is, simultaneously, theMigdal-Kadanoff approximation [16, 17] for the cubic lat-tice and the exact solution [18–22] for the d = 3 hier-archical lattice based on the repeated self-imbedding ofleftmost graph of Fig. 1(b). Fig. 1(a) shows the Migdal-Kadanoff approximate renormalization-group transfor-mation for the cubic lattice, composed of the bond-moving followed by decimation steps. Fig. 1(b) showsthe exact renormalization-group transformation for thehierarchical lattice. The two procedures yield identicalrecursion relations.Exact calculations on hierarchical lattices are also cur-rently widely used on a variety of statistical mechanicsproblems.[23–39]. On the other hand, this approximationfor the cubic lattice is an uncontrolled approximation, asin fact are all renormalization-group theory calculationsin d = 3 and all mean-field theory calculations. However,as noted before [40], the local summation in position-space technique used here has been qualitatively, near-quantitatively, and predictively successful in a large va-riety of problems, such as arbitrary spin- s Ising models[41], global Blume-Emery-Griffiths model [42], first- andsecond-order Potts transitions [43, 44], antiferromagneticPotts critical phases [9, 10], ordering [45] and superflu-idity [46] on surfaces, multiply reentrant liquid crystalphases [47, 48], chaotic spin glasses [49], random-field[50, 51] and random-temperature [52, 53] magnets includ-ing the remarkably small d = 3 magnetization criticalexponent β of the random-field Ising model, and high-temperature superconductors [54].Under the renormalization-group transformation de-scribed below, the Hamiltonian of Eq. (2) maps ontothe more general form − β H = X h ij i V ij ( θ ij ) , (3) ( a )( b ) FIG. 1. (a) The Migdal-Kadanoff approximaterenormalization-group transformation for the cubic lat-tice, composed of the bond-moving followed by decimationsteps, with the length rescaling factor b = 3. The corre-sponding hierarchical lattice is obtained by the repeatedself-imbedding of the leftmost graph in (b). (b) The exactrenormalization-group transformation for this d = 3 hierar-chical lattice. The two procedures yield identical recursionrelations. where θ ij = θ i − θ j can take q different values, so thatfor each pair < ij > of nearest-neighbor sites, there areq different interaction constants { V ij ( θ ij ) } = { V ij (0) , V ij ( δ ) , V ij (2 δ ) , V ij (3 δ ) , V ij (4 δ ) } ≡ V ij , (4)which are in general different at each locality (quenchedrandomness). Here, δ ≡ π/ { V ij ( θ ij ) } at each locality < ij > is set to zero, by sub-tracting the same constant G from all q interaction con-stants, with no effect on the physics; thus, the q − e V ij ( θ ij ) − e G = b d − X k =1 V ( k ) ij ( θ ij ) , (5)and decimations e V ′ ( θ ) − G = X θ ,θ e e V ( θ )+ e V ( θ )+ e V ( θ ) , (6)where e G and G are the subtractive constants mentionedabove, and prime marks the interaction of the renormal-ized system.The starting double-bimodal quenched probability dis-tribution of the interactions, characterized by p and c asdescribed above, is not conserved under rescaling. Therenormalized quenched probability distribution of the in-teractions is obtained by the convolution [55] P ′ ( V ′ i ′ j ′ ) = Z i ′ j ′ Y ij d V ij P ( V ij ) δ ( V ′ i ′ j ′ − R ( { V ij } )) , (7) . . . . . . . S . . . . . . . Chirality Breaking Concentration c T e m p e r a t u r e / J ∆ /J = 3 .
01 2 3 4 ∆ /J = 2 . /J = 1 . /J = 0 . FD ∆ /J = 0 . /J = 1 . /J = 3 . /J = 2 . S ∆ /J = 1 . F ∆ /J = 0 .
13 1 4 2 A ∆ /J = 0 . D ∆ /J = 1 . p = 0), on the left side of the figure,and antiferromagnetic ( p = 1), on the right side, systems with quenched random left- and right-chiral interactions. Thehorizontal axis c is the concentration of right-chiral interactions. Phase diagrams for different random chirality strengths∆ /J are shown. The system exhibits ferromagnetic (F), a multitude of different chiral, and spin-glass (S) ordered phases.On some of the chiral phases, the δ multiplicity of the asymptotically dominant interaction is indicated. The ferromag-netic and chiral phases accumulate as different devil’s staircases at their boundary with the disordered (D) phase. Theantiferromagnetic system also exhibits an algebraically ordered (A) phase. The full richness of the continuum of widelyvarying devil’s staircase phase diagrams can also be seen in video form, four of which are accessible as SupplementalMaterial [13]. These four videos are also accessible at http:// web.mit.edu/physics/berker/temperatureDeltac0scanp.avi,web.mit.edu/physics/berker/temperatureDeltac05scanp.avi, web.mit.edu/physics/berker/temperaturecp1scanDelta.avi,web.mit.edu/physics/berker/temperaturecp0scanDelta.avi where V ij ≡ { V ij ( θ ij ) } as in Eq. (4), R ( { V ij } ) repre-sents the bond moving and bond decimation given in Eqs.(5) and (6), and primes refer to the renormalized system.Similar previous studies, on other spin-glass systems, arein Refs. [7, 14, 56–63]. For numerical practicality thebond moving and decimation of Eqs. (5) and (6) areachieved by a sequential pairwise combination of interac-tions, each pairwise combination leading to an interme-diate probability distribution resulting from a pairwiseconvolution as in Eq. (7).We effect this procedure numerically, first starting withthe initial double delta distribution of Eq. (2) giving4 possible interactions quenched randomly distributedthroughout the system, and generating 1000 interactionsthat embody the quenched probability distribution re-sulting from the pairwise combination. Each of the gen-erated 1000 interactions is described by q interaction con-stants, as explained above [Eq. (4)]. At each subsequentpairwise convolution as in Eq. (7), 1000 randomly cho-sen pairs, representing quenched random neighbors in thelattice, are matched by (5) or (6), and a new set of 1000 interactions is produced. As a control, we have also calcu-lated phase diagrams given below using 1500 interactionsand the phase diagrams did not change.Our calculation simply consists in following therecursion relations, Eqs.(5-7) to the various fixed pointsand thereby mapping the initial conditions that arethe basins of attraction of the various fixed points.This map is the phase diagram: The different ther-modynamic phases of the system are identified by thedifferent asymptotic renormalization-group flows ofthe quenched probability distribution P ( V ij ). Tworenormalization-group trajectories starting at each sideof a phase boundary point diverge from each other,flowing towards the phase sinks (completely stable fixedpoints) of their respective phases. Thus, the phaseboundary point between two phases is readily obtainedto the accuracy of the figures. We are therefore able tocalculate the global phase diagram of the chiral clockdouble spin-glass model. − − − − − − − Random Chirality Strength ∆ /J T e m p e r a t u r e / J p = 0 . F p = 0 . D p = 0 . p = 0 . p = 0 . S p = 0 . p = 1 .
01 2 3 4 A − . . . . . . − p = 0 . F − p = 0 . D p = 0 . p = 0 . − p = 0 . S p = 0 . − p = 1 . A c = 0), on the upper side of the figure,and quenched random left- and right-chiral ( c = 0 . /J . The consecutive phase diagrams arefor different concentrations of antiferromagnetic interactions p . The system exhibits ferromagnetic (F), a multitude of differentchiral, and spin-glass (S), and critical (algebraically) ordered (A) phases. On some of the chiral phases, the δ multiplicity of theasymptotically dominant interaction is indicated. The ferromagnetic and chiral phases accumulate as different devil’s staircasesat their boundary with the disordered (D) phase. Note shallow and deep reentrances of the disordered phase at p = 0 . p = 0 .
7, respectively, surrounded by regular and temperature-inverted devil’s staircases. The full richness of the continuum ofwidely varying devil’s staircase phase diagrams can also be seen in video form, four of which are accessible as SupplementalMaterial [13]. These four videos are also accessible at http:// web.mit.edu/physics/berker/temperatureDeltac0scanp.avi,web.mit.edu/physics/berker/temperatureDeltac05scanp.avi, web.mit.edu/physics/berker/temperaturecp1scanDelta.avi,web.mit.edu/physics/berker/temperaturecp0scanDelta.avi
IV. GLOBAL PHASE DIAGRAM OF THE q = 5 STATE CHIRAL CLOCK DOUBLE SPIN GLASS
The global phase diagram of the q = 5 state chiralclock double spin-glass model in d = 3 spatial dimen-sions, in temperature J − , antiferromagnetic bond con-centration p , random chirality strength ∆ /J , and right-chirality concentration c , is a four-dimensional object, sothat only the cross-sections of the global phase diagramare exhibited.Figs. 2 show the calculated sequence of phase diagramsfor the ferromagnetic ( p = 0), on the left side of the fig-ure, and antiferromagnetic ( p = 1), on the right side,systems with quenched random left- and right-chiral in-teractions. The horizontal axis c is the concentration of right-chiral interactions. Phase diagrams for differentrandom chirality strengths ∆ /J are shown. The systemexhibits ferromagnetic (F), a multitude of different chiral,and spin-glass (S) ordered phases. The antiferromagneticsystem also shows an algebraically (A) ordered (critical)phase, in which every point is a critical point with diver-gent correlation length [9, 10]. In all cases, the ferromag-netic and different chiral phases accumulate as differentdevil’s staircases [11, 12] at their boundary with the dis-ordered (D) phase. The definition of the devil’s staircaseis that this accumulation is seen at every expanded scaleof the phase diagram variables. This accumulation atevery expanded phase diagram scale is indeed revealedfrom our calculations, as seen further below.Figs. 3 show the calculated sequence of phase dia- − . . . . . − .
01 0 0 . . . . − .
001 0 0 . Random Chirality Strength ∆ /J T e m p e r a t u r e / J FIG. 4. (Color online) The phase diagram cross-section in the upper left of Fig. 3, with a calculated 10-fold zoom and with100-fold zoom. The devil’s staircase structure appears at each zoom level. grams for the left-chiral ( c = 0), on the upper side, andquenched random left- and right-chiral ( c = 0 . /J .The consecutive phase diagrams are for different concen-trations of antiferromagnetic interactions p . The sys-tem exhibits ferromagnetic (F), a multitude of differ-ent chiral, spin-glass (S), and algebraically ordered (A)phases. The ferromagnetic and different chiral phasesaccumulate as different devil’s staircases [11, 12] at theirboundary with the disordered (D) phase. Note shallowand deep reentrances of disorder [48, 64–67] at p = 0 . p = 0 .
7, respectively, surrounded by regular andtemperature-inverted devil’s staircases.Fig. 4 shows the phase diagram cross-section in theupper left of Fig. 3, with a calculated 10-fold zoom andwith 100-fold zoom. The devil’s staircase structure ap-pears at each zoom level.The full richness of the continuum of widely varyingdevil’s staircase phase diagrams can best be seen in videoform, four of which are accessible as Supplemental Ma-terial [13]. These four videos are also accessible at http://web.mit.edu/physics/berker/temperatureDeltac0scanp.avi,web.mit.edu/physics/berker/temperatureDeltac05scanp.avi,web.mit.edu/physics/berker/temperaturecp1scanDelta.avi,web.mit.edu/physics/berker/temperaturecp0scanDelta.avi.These videos effectively exhibit a very large number ofcalculated phase diagram cross-sections.
V. ENTIRE-PHASE CRITICALITY,DIFFERENTIATED CHAOS IN THESPIN-GLASS AND AT ITS BOUNDARY
The renormalization-group mechanism for thealgebraically ordered (critical) phase is that, allrenormalization-group trajectories originating inside this phase flow to a completely stable fixed point(sink) that occurs at finite temperature (finite couplingstrength).[9, 10, 68–76] In all other ordered phases, thetrajectories flow to strong (infinite) coupling.In the ferromagnetic phase, the interaction V ij (0) be-comes asymptotically dominant. In the chiral phases,in the renormalization-group trajectories, one of thechiral interactions from the right-hand side of Eq.(4), { V ij ( δ ) , V ij (2 δ ) , V ij (3 δ ) , V ij (4 δ ) } , becomes asymp-totically dominant. However, in each of the sep-arate phases, it takes a characteristic number n ofrenormalization-group transformations, namely a lengthscale of 3 n , to reach the dominance of one chiral inter-action. This distinct number of iterations, namely scalechanges, determines, by tracing back to the periodic se-quence in the original lattice, the pitch of the chiral phasein the original unrenormalized system. Thus, the chi-ral phases in the original unrenormalized system, withdistinct chiral pitches, are distinct phases. After thedominance of one chiral interaction, the renormalization-group trajectory follows the periodic sequence V ij ( δ ) → V ij (3 δ ) → V ij (4 δ ) → V ij (2 δ ) → V ij ( δ ) resulting frommatching q = 5 and b = 3.Our calculation is exact for the hierarchical lattice pic-tured in Fig. 1(b) therefore for which the phase diagramsin Fig. 2 and 3 are exactly applicable. However, our cal-culation is approximate for the cubic lattice, as picturedin Fig. 1(a). Thus, one could speculate that in the cubiclattice, the multitude of chiral phases would appear asa single chiral phase with a continuously varying pitch:Fig. 5 shows all the chiral phases merged into a singlephase. It is seen that a quite unusual phase diagram stillappears, with the interlacing of the ferromagnetic phasewith the chiral phase, throughout the bulk of the phaseregion.The renormalization-group trajectories starting in thechiral spin-glass phase, unlike those in the ferromagnetic . . . . . . . S − − − T e m p e r a t u r e / J Chirality Breaking Concentration c Random Chirality Strength ∆ /J ∆ /J = 3 .
01 2 3 4 ∆ /J = 2 . /J = 1 . /J = 0 . FD ∆ /J = 0 . /J = 1 . p = 0 . F p = 0 . D p = 0 . p = 0 . S p = 0 . p = 1 .
01 2 3 4 A FIG. 5. (Color online) Our calculation is exact for the hierarchical lattice pictured in Fig. 1(b), therefore for which the phasediagrams in Fig. 2 and 3 are exactly applicable. However, our calculation is approximate for the cubic lattice, as pictured inFig. 1(a). Thus, it could be speculated that in the cubic lattice, the multitude of chiral phases would appear as a single chiralphase with a continuously varying pitch: This Fig. 5 shows all the chiral phases merged into a single phase. It is seen that aquite unusual phase diagram still appears, with the interlacing of the ferromagnetic phase with the chiral phase, throughoutthe bulk of the phase region. The left side of this figure is derived from the left portion of Fig. 2; the right side is derived fromthe top portion of Fig. 3 or chiral phases, do not have the asymptotic behaviorwhere at any scale a single potential V(theta) is domi-nant. These trajectories of the spin-glass phase asymp-totically go to a strong-coupling fixed probability distri-bution P ( V ij ) which assigns non-zero probabilities to adistribution of V ij values, with no single V ij ( θ ) beingdominant. Projections of this distribution (a functionof five variables) are shown in Fig. 6. This situation isa direct generalization of the asymptotic trajectories ofthe ± J Ising spin-glass phase, where a fixed probabilitydistribution over positive and negative values of the in-teraction J is obtained, with no single value of J beingdominant [14].Since, at each locality, the largest interaction in { V ij (0) , V ij ( δ ) , V ij (2 δ ) , V ij (3 δ ) , V ij (4 δ ) } is set to zero andthe four other interactions are thus made negative, bysubtracting the same constant from all five interactionswithout affecting the physics, the quenched probabilitydistribution P ( V ij ), a function of five variables, is actu-ally composed of five functions P σ ( V ij ) of four variables,each such function corresponding to one of the interac-tions being zero and the other four, arguments of thefunction, being negative. Fig. 6 shows one of the latterfunctions: The part of the fixed distribution, P ( V ij ),for the interactions V ij in which V ij (3 δ ) is maximum and therefore 0 (and the other four interactions are neg-ative) is shown in this figure. The projections of P ( V ij )onto two of its four arguments are shown in each panel ofthis figure. The other four P σ ( V ij ) have the same fixeddistribution. Thus, chirality is broken locally, but notglobally.Another distinctive mechanism, that of chaos un-der scale change [49, 77, 78] or, equivalently, underspatial translation [14], occurs within the spin-glassphase and differently at the spin-glass phase bound-ary [14], in systems with competing ferromagnetic andantiferromagnetic interactions [14, 49, 62, 77–105] and,more recently, with competing left- and right-chiral in-teractions [6]. The physical hierarchical lattice thatwe solve here is an infinite system, where 1000 quin-tuplets { V ij (0) , V ij ( δ ) , V ij (2 δ ) , V ij (3 δ ) , V ij (4 δ ) } are ran-domly distributed over the lattice bond positions. Thus,as we can fix our attention to one lattice position andmonitor how the quintuplet at that position evolves un-der renormalization-group transformation, as it mergeswith its neighbors through bond moving [Eq. (5)] anddecimation [Eq. (6)], and thereby calculate the Lyapunovexponent [14, 62], which when positive is the measure ofthe strength of chaos.Fig. 7 gives the asymptotic chaotic renormalization- − − − − − − v i j ( ) v ij ( δ ) − − v ij (2 δ ) − − v i j ( δ ) . P ( v i j ) / m a x ( P ( v i j )) FIG. 6. (Color online) Asymptotic fixed distribution of thespin-glass phase. The part of the fixed distribution, P ( V ij )for the interactions V ij in which V ij (3 δ ) is maximum andtherefore 0 (and the other four interactions are negative) isshown in this figure, with v ij ( σδ ) = V ij ( σδ ) / < | V ij ( σδ ) | > .The projections of P ( V ij ) onto two of its four arguments areshown in each panel of this figure. The other four P σ ( V ij )have the same fixed distribution. Thus, chirality is brokenlocally, but not globally. group trajectories of the spin-glass phase and, dis-tinctly, of the phase boundary between the spin-glassand disordered phases. The chaotic trajectories foundhere are similar to those found in traditional (Ising)spin-glasses [14, 62], with of course different Lya-punov exponents seen below. The five interactions V ij (0) , V ij ( δ ) , V ij (2 δ ) , V ij (3 δ ) , V ij (4 δ ) at a given location < ij > , under consecutive renormalization-group trans-formations, are shown in Fig. 7. As noted, chaos is mea-sured by the Lyapunov exponent [14, 62, 96, 106, 107],which we here generalize, by the matrix form, to ourmulti-interaction case: λ = lim n →∞ n ln (cid:12)(cid:12)(cid:12) E (cid:16) n − Y k =0 d v k +1 d v k (cid:17)(cid:12)(cid:12)(cid:12) , (8)where the function E ( M ) gives the largest eigenvalue ofits matrix argument M and the vector v k is v k = { v ij (0) , v ij ( δ ) , v ij (2 δ ) , v ij (3 δ ) , v ij (4 δ ) } , (9)with v ij ( σδ ) = V ij ( σδ ) / < | V ij ( σδ ) | > , at step k of therenormalization-group trajectory. The product in Eq.(8) is to be taken within the asymptotic chaotic band,which is renormalization-group stable or unstable for thespin-glass phase or its boundary, respectively. Thus, wethrow out the first 100 renormalization-group iterations − − Chiral Spin-Glass Phase Boundary λ = 1 . θ ij = 0 − π − π − π − π − −
40 0 250 500 750 1000
Chiral Spin-Glass Phase λ = 2 . θ ij = 0 − π − π − π − π I n t e r a c t i o n V i j ( θ i j ) / h | V i j ( θ i j ) | i Renormalization-group iteration number n FIG. 7. Chaotic renormalization-group trajectories of thespin-glass phase (bottom) and of the phase boundary betweenthe spin-glass and disordered phases (top). The five interac-tions V ij (0) , V ij ( δ ) , V ij (2 δ ) , V ij (3 δ ) , V ij (4 δ ) at a given location < ij > , under consecutive renormalization-group transforma-tions, are shown. The θ ij = σδ angular value of each interac-tion V ij ( θ ij ) is indicated in the figure panels. Bottom panel:Inside the spin-glass phase. The corresponding Lyapunov ex-ponent is λ = 2 .
01 and the average interaction diverges as < | V | > ∼ b y R n , where n is the number of renormalization-group iterations and y R = 0 .
26 is the runaway exponent. Toppanel: At the phase boundary between the spin-glass anddisordered phases. The corresponding Lyapunov exponent is λ = 1 .
70 and the average non-zero interaction remains fixedat < V > = − .
99. As indicated by the Lyapunov exponents,chaos is stronger inside the spin-glass phase than at its phaseboundary. to eliminate the transient points outside of, but leadingto the chaotic band. Subsequently, typically using 1,000renormalization-group iterations in the product in Eq.(8) assures the convergence of the Lyapunov exponentvalue λ , which is thus accurate to the number of signifi-cant figures given. Spin-glass chaos occurs for λ > λ , the stronger is chaos, as seen forexample in the progressions in Figs. 6 and 7 of Ref. [62].In the spin-glass phase of the currently studied system,the Lyapunov exponent is λ = 2 .
01 and the average inter-action diverges as < | V | > ∼ b y R n , where n is the numberof renormalization-group iterations and y R = 0 .
26 is therunaway exponent. At the phase boundary between thespin-glass and disordered phases, the Lyapunov exponentis λ = 1 .
70 and the average non-zero interaction remainsfixed at < V > = − .
99. As indicated by the Lyapunovexponents, chaos is stronger inside the spin-glass phasethan at its phase boundary.
VI. CONCLUSION
It is thus seen that chirality and chiral quenchedrandomness provides, in a simple model, remarkably rich phase transition phenomena. These include a multitudeof chiral phases, a continuum of widely varying devil’sstaircases, shallow and deep reentrances of the disorderedphase surrounded by regular and temperature-inverteddevil’s staircases, a critical phase, and a chiral spin-glassphase with chaotic rescaling behavior inside and differ-ently at its boundary. The widely varying continuum ofdevil’s staircase phase diagrams are best seen in videoform, four of which are accessible as Supplemental Ma-terial [13]. These four videos are also accessible at http://web.mit.edu/physics/berker/temperatureDeltac0scanp.avi,web.mit.edu/physics/berker/temperatureDeltac05scanp.avi,web.mit.edu/physics/berker/temperaturecp1scanDelta.avi,web.mit.edu/physics/berker/temperaturecp0scanDelta.avi.Finally, the study of an even number of q states, whichdo not have a built-in entropy as mentioned above,should yield equally rich, but qualitatively differentphase diagrams. ACKNOWLEDGMENTS
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