Dicke quantum spin glass of atoms and photons
DDicke quantum spin glass of atoms and photons
Philipp Strack and Subir Sachdev
Department of Physics, Harvard University, Cambridge MA 02138 (Dated: October 31, 2018)Recent studies of strongly interacting atoms and photons in optical cavities have rekindled interest in theDicke model of atomic qubits coupled to discrete photon cavity modes. We study the multimode Dicke modelwith variable atom-photon couplings. We argue that a quantum spin glass phase can appear, with a randomlinear combination of the cavity modes superradiant. We compute atomic and photon spectral response functionsacross this quantum phase transition, both of which should be accessible in experiments.
PACS numbers: 37.30. + i, 42.50.-p, 05.30.Rt, 75.10.Nr, 11.30.Qc Introduction . Ultracold atoms in optical cavities haveemerged as attractive new systems for studying strongly-interacting quantum many body systems. Photon exchangecan mediate long-range interactions between the atomic de-grees of freedom, and this opens up rich possibilities for cor-related phases. In the celebrated atomic realizations of thesuperfluid-insulator quantum phase transition [1], the lightfield acts in a secular manner, creating a potential which trapsthe atoms in an optical lattice; consequently the atom-atominteractions are only on-site, and this limits the range of pos-sible phases. In contrast, the seminal recent experiments ofBaumann et al. [2, 3], realizing a supersolid phase, have long-range interactions mediated by active photon exchange [4].Baumann et al. argued that their experiments could be de-scribed by the Dicke model, as in the proposal of Nagy etal. [5]. The Dicke model couples photons in a single cav-ity mode uniformly to N atomic two-level systems (‘qubits’).In the limit N → ∞ , this model exhibits a phase transition[6–9] to a “superradiant” phase when the atom-photon cou-pling is strong enough. In terms of the qubits, the superra-diant phase is a ‘ferromagnet’ which spontaneously breaks aglobal Ising symmetry, and so we refer to it as FM SR . In theexperiments by Baumann et al. , the superradiance of the cav-ity photon mode is accompanied by ‘self-organization’ of theatoms into a density wave pattern [10–12].Here we study extensions of the Dicke model to multiplephoton cavity modes, and with non-uniform couplings be-tween the atomic qubits and the photon modes. Spatial modevariations for the single-mode Dicke model were consideredin Ref. 13. Multimode Dicke models have been studied earlier[6, 14–16], but were simplified by ignoring the variations inthe atom-photon couplings. We argue here that qualitativelynew physics emerges in the multimode case when the spatialvariation is treated seriously. We show that large variationsin the atom-photon couplings can give rise to a quantum spin-glass (QSG) phase. We will describe quantum-critical dynam-ics associated with the onset of this spin glass order.Dimer et al. [17] have discussed an experimental realiza-tion of the Dicke model using internal atomic degrees of free-dom, that is, Raman transitions between multiple atomic lev-els. We expect that such schemes can be generalized to a mul-timode Dicke model that respects a global Ising symmetry,which is then spontaneously broken in the FM SR and QSG phase, respectively. More specific realizations of the multi-mode Dicke model were described recently by Gopalakrish-nan et al. , in a paper [18] which appeared while our workwas being completed. The same authors had previously out-lined how Bose-Einstein condensates in multimode cavitiescan lead to frustration and glassy behavior [11, 12]. Suchexperiments on the multimode Dicke model would providea unique realization of a quantum spin glass with long-rangecouplings, and provide a long-awaited testing ground for theo-ries of quantum systems with strong interactions and disorder.Condensed matter realizations of quantum spin glasses haveshorter-range couplings, and so do not directly map onto thetheoretically solvable systems analyzed in the present paper.Before describing our computations, we point out a keydistinction between the transitions involving onset of FM SR versus QSG order. In the single-mode Dicke model, all thequbits align in a common direction near the FM SR phase, andcan therefore be described by a collective spin of length N / N . Conse-quently, the dynamics near the phase transition can be de-scribed by classical equations of motion [19], and the single-mode Dicke model does not realize a quantum phase transi-tion in the conventional sense of condensed matter physics. Incontrast, we will argue here that the onset of QSG order in themultimode Dicke model has non-trivial quantum fluctuationseven in the limit of large N , and the critical properties cannotbe described by an e ff ective classical model. Experimentalstudies are therefore of great interest. Model . The Hamiltonian of the multimode Dicke model is H = M (cid:88) i = ω i a † i a i + ∆ N (cid:88) (cid:96) = σ z (cid:96) + N (cid:88) (cid:96) = M (cid:88) i = g i (cid:96) (cid:16) a i + a † i (cid:17) σ x (cid:96) . (1)This describes N two-level atomic qubits with level splitting ∆ / M photon modes with frequencies ω i coupled by anatom-photon coupling g i (cid:96) which depends on the photon ( i ) andatom ( (cid:96) ) number. a † i , a i are bosonic creation and annihila-tion operators, respectively, fulfilling canonical commutationrelations. σ x , z (cid:96) are spin-1 / ff erent stableground-state sublevels, | (cid:105) and | (cid:105) , of three-level Λ atoms. | (cid:105) and | (cid:105) are indirectly coupled through a pair of Raman transi-tions to an excited state | e (cid:105) which are driven by the classical a r X i v : . [ c ond - m a t . qu a n t - g a s ] N ov field of a pair of external lasers. Upon adiabatic eliminationof | e (cid:105) , one obtains Eq. (1) with σ z (cid:96) = | (cid:96) (cid:105)(cid:104) (cid:96) | − | (cid:96) (cid:105)(cid:104) (cid:96) | and σ x (cid:96) = | (cid:96) (cid:105)(cid:104) (cid:96) | + | (cid:96) (cid:105)(cid:104) (cid:96) | . The parameters ω i , ∆ , and g i (cid:96) can becontrolled through laser frequencies and intensities. This tun-ability enables access to the strong-coupling Dicke regime. Adispersive shift of the cavity frequencies ∼ a † i a j σ z does notmodify our results significantly, and so will be set to zerofor simplicity. A simple choice for a spatially varying atom-photon coupling is g i (cid:96) = g cos ( k i x (cid:96) ) with k i the wavevector ofthe photon mode, and x (cid:96) the coordinate of atom (cid:96) .In the single-mode, large photon wavelength case, we have M = ω i = ω , and g i (cid:96) = g / √ N and the model can besolved exactly in the N → ∞ limit [6, 7]. At zero temperature,there is a continuous phase transition between a paramagneticphase (PM) and a superradiant ferromagnetic phase (FM SR )at g = g c = √ ∆ ω / a , σ x ) → ( − a , − σ x ), is spontaneously broken.For the multimode Dicke model, it is useful to integrate outthe photon degrees of freedom in a path-integral representa-tion. Then the qubits are described by a Hamiltonian similarto the Ising model in a transverse field, H e ff = ∆ N (cid:88) (cid:96) = σ z (cid:96) − (cid:88) (cid:96) m J (cid:96) m σ x (cid:96) σ xm , (2)The exchange interactions J (cid:96) m are mediated by the photonsand have a frequency dependence associated with the photonfrequencies ω i ; thus Eq. (2) is to be understood as an actionappearing in an imaginary time path-integral summing overtime-histories of the qubits. The long-range exchanges J (cid:96) m ( Ω ) = M (cid:88) i = g i (cid:96) g im ω i Ω + ω i , (3)depend on Ω , the imaginary frequency of the qubits in thepath integral. Note that although we have formally integratedout the photons, we demonstrate below that the photon-photoncorrelation function is directly related to the atom-atom cor-relation function as obtained by solving Eq. (2).If we ignore the frequency dependence in Eq. (3), the J (cid:96) m have a structure similar to the Hopfield model of associativememory [20], with M ‘patterns’ g i (cid:96) . For M small, it is ex-pected that such a model can have M possible superradiantground states with FM SR order (cid:68) σ x (cid:96) (cid:69) ∝ g i (cid:96) , i = . . . M . Inthe spin-glass literature, these are the Mattis states which are“good” memories of the patterns g [20]. The critical proper-ties of the onset of any of these FM SR states should be similarto those of the single mode Dicke model.Our interest in the present paper is focussed on larger valuesof M , where the summation in Eq. (3) can be viewed as a sumover M random numbers. Then, by the central limit theorem,the distribution of J (cid:96) m ( Ω ) is Gaussian. Alternatively, the ran-domness of J (cid:96) m ( Ω ) can be enhanced by passing the trappinglaser beams through di ff users so that atomic positions x (cid:96) arerandomly distributed inside the cavity [18]. In either case, weassume a Gaussian distribution characterized by its mean and FM SR q (cid:185) Ψ(cid:185) Ψ =0 QSGq (cid:185) Ψ =0 t g FIG. 1: (Color online) Zero-temperature phase diagram for ω = ∆ = SR super-radiant ferromagnet, and QSG quantum spin glass. q is the Edwards-Anderson order parameter and ψ is the atomic population inversionor ferromagnetic order parameter. variance J (cid:96) m ( Ω ) = J ( Ω ) / N (4) δ J (cid:96) m ( Ω ) δ J (cid:96) (cid:48) m (cid:48) ( Ω (cid:48) ) = ( δ (cid:96)(cid:96) (cid:48) δ mm (cid:48) + δ m (cid:96) (cid:48) δ (cid:96) m (cid:48) ) K ( Ω , Ω (cid:48) ) / N , where the line represents a disorder average, and δ J (cid:96) m is thevariation from the mean value. We have assumed couplingsbetween di ff erent sites are uncorrelated, and this will allowan exact solution in the N → ∞ limit, modulo an innocuoussoftening of the fixed length constraint on the Ising variable[21, 22]. We will allow arbitrary frequency dependencies in J ( Ω ) and K ( Ω , Ω (cid:48) ). The factors of N ensure an interesting N → ∞ limit [23]. Especially for finite M , one could also usethe methods of Ref. [20] to extend our analysis to models inwhich the g i (cid:96) rather than the J (cid:96) m ( Ω ) are taken as independentrandom variables. However, as long as the photon modes canbe chosen so that the J (cid:96) m ( Ω ) vary in sign and magnitude, ouranalysis should remain qualitatively correct also for smallervalues of M . Key results . We will show below that, in the limit of largeatom number N , the results depend only upon J ( Ω =
0) and K ( Ω , − Ω ). Here, we will display the phase diagram and spec-tral response functions for the simple choices J (0) = g /ω and K ( Ω , − Ω ) ≡ J ( Ω ) with J ( Ω ) = t ω / ( Ω + ω ) . (5)In Fig. 1, we depict the ground state phase diagram; a re-lated phase diagram in a condensed matter context was ob-tained in Ref. 24. All phase transitions are continuous andthe respective phase boundaries merge in a bicritical point at( t = . , g = t ).The intersection of the PM-FM SR phase boundary with thevertical axis at t = FM SR Ω = 1.3 Ω = 1.015 Ω (cid:61) (cid:87) (cid:45) I m Q aa FIG. 2: (Color online) rf spectral response function of the atomicqubits in the FM SR phase for various photon frequencies and t = . g = . ∆ = Ω is a real measurement frequency. Thered arrow at Ω = q ∼ ψ from Eqs. (12,13). The value of the gap is givenabove Eq. (13). For the Dicke model without disorder ( t = Ω =
Ω = g √ ∆ /ω (plotted for ω = . (11), in agreement with the earlier work. The critical atom-photon coupling is g c = ∆ ω / σ x (cid:96) spin suscep-tibility in the FM SR phase is (for imaginary frequencies) Q aa (cid:96) ( Ω ) (cid:12)(cid:12)(cid:12) t = = ∆Ω + ∆ g /ω + ψ πδ ( Ω ) . (6)The corresponding radiofrequency (rf) spectral responsefunction of the atomic qubits for real frequencies, − Im [ Q aa ( i Ω → Ω + i + )], is depicted in Fig. 2. The su-perradiance is encoded in the zero frequency delta functioncontribution, whose weight is proportional to the atomic pop-ulation inversion ψ . However, away from the zero frequencydelta function, there is a spectral gap, and the remainingspectral weight is a delta function at frequency (cid:112) ∆ g /ω .The superradiance also appears as a photon condensate (cid:104) a i (cid:105) = − (cid:80) (cid:96) ( g i (cid:96) / (2 ω i )) (cid:104) σ x (cid:96) (cid:105) . We have computed the atomicpopulation inversion, (cid:104) σ x (cid:96) (cid:105) = ψ , and the Edwards-Andersonorder parameter (cid:104) σ x (cid:96) (cid:105) = q QSG in Eqs. (13,14). Both of theseare related to (cid:104) a i (cid:105) , but computation of the latter requires morespecific knowledge of the g i (cid:96) . For Ω (cid:44)
0, the photon correla-tion function follows from Eq. (6) (cid:104) a † i ( Ω ) a j ( Ω ) (cid:105) = ( i Ω − ω i ) δ i j + N (cid:88) (cid:96) = g i (cid:96) g j (cid:96) Q aa (cid:96) ( Ω ) − , (7)where the right-hand-side is a matrix inverse, as can be ob-tained from integrating out the atomic fields from the path-integral representation of Eq. (1).Upon introducing small disorder (with t (cid:44) SR phase, the zero frequency delta functionand spectral gap survive, although the higher frequency spec-tral weight changes, as shown in Fig. 2. This spectral gap is Ω (cid:61) Ω = 1.59 Ω (cid:61) (cid:87) (cid:45) I m Q aa FIG. 3: (Color online) rf spectral response function of the atomicqubits in the QSG phase for various photon frequencies and t = . g = . ∆ =
1. The red arrow at
Ω = q ∼ q QSG from Eqs. (12,14). present across the phase transition from the FM SR phase to thePM phase. Thus all the low energy fluctuations in the criticaltheory for this transition are restricted to the zero frequencydelta function, which can be described in classical theory forthe spins: this is the reason this transition is more properlyconsidered as a classical phase transition.For a su ffi ciently large value of t , the system undergoes aquantum phase transition to the QSG ground state. In con-trast to the PM-FM SR transition, at the QSG transition, and inthe entire QSG phase, there is spectral weight at a continuumof frequencies reaching zero (see Fig. 3). Thus the onset ofQSG order from the PM phase is a genuine quantum phasetransition, whose universality class was described in Ref. 22.The PM phase is clearly delineated from both, the QSGand the FM SR phases: the PM phase has a gapped spectralresponse and no superradiant photon condensates.We also note that in all phases, while the spectral functionhas a universal form at low frequencies, its high frequencybehavior is strongly dependent upon the forms of J ( Ω ) and J ( Ω ). For the forms in Eq. (5), the spectral function is sup-pressed to zero at Ω = ω . Experimental signatures . The rf spectral response functionof the atomic qubits presented in Figs. 2,3 should be observ-able via radiofrequency spectroscopy [25, 26].Measuring the spectrum of photons leaving the cavitythrough its imperfect mirrors at loss rate κ allows for an in-situ measurement of our phase diagram, Fig. 1. Our predic-tion for the spectrum of intra-cavity photons, Eq. (7), can berelated to the extra-cavity photons via the input-output formal-ism [17, 27, 28]. For this case of a dissipative Dicke model,we note a similarity of the decay e ff ects to those in theoriesof metallic spin glasses [29], in which the spin qubits arecoupled to a “reservoir” of continuum spin excitations nearthe Fermi surface. This coupling leads to a damping term inthe dynamics of each spin, but does not significantly modifythe spin-spin interactions responsible for the spin glass phase.Similarly, for the dissipative Dicke model, decay into photonsoutside the cavity will introduce various damping terms e.g. a κ | Ω | term in the denominator of Eq. (3). As in the previousanalyses [29], we expect that the quantum spin glass transi-tion will survive in the presence of damping, although therewill be some changes to the critical properties [30].As in other glasses, we expect slow relaxational dynamics,along with memory and aging e ff ects in the QSG phase whichshould be observable via local spin addressing protocols andmeasuring the spin relaxation time scale [18]. Conclusion . Observations of these e ff ects in quantum op-tic systems would be remarkable. Moreover, the spin glassphysics is driven by long-range interactions, and this makesthe theoretical models analytically tractable. We thereforehave prospects for a quantitative confrontation between the-ory and experiment in a glassy regime, something which haseluded other experimental realizations of spin glasses. Details of the calculation . As discussed in Refs. 21, 22,each Ising qubit, with on-site gap ∆ /
2, is conveniently rep-resented by fluctuations of a non-linear oscillator φ (cid:96) ( τ ) ( τ isimaginary time) which obeys a unit-length constraint. Theiraction at temperature T is then S [ φ, λ ] = ∆ N (cid:88) (cid:96) = (cid:90) / T d τ (cid:104) ( ∂ τ φ (cid:96) ) + i λ (cid:96) ( φ (cid:96) − (cid:105) (8)where τ is imaginary time, and the λ (cid:96) are Lagrange multiplierswhich impose the constraints. The only approximation of thispaper is to replace the λ (cid:96) by their saddle-point value, i λ (cid:96) = λ , and to ignore their fluctuations. For decoupled oscillators,this saddle-point value is λ = ∆ /
4, the φ susceptibility is ∆ / ( Ω + ∆ / ∆ /
2, has been matchedto that of the Ising spin.The interactions between the qubits are accounted for asbefore [21]: we introduce replicas a = . . . n , average over the J (cid:96) m using Eq. (4), decouple the resulting two- φ coupling byHubbard-Stratonovich transformation using a ferromagneticorder parameter Ψ a ( Ω ), and the four- φ coupling by the bilocalfield Q ab ( Ω , Ω ) [22] (the Ω are Matsubara frequencies). Thecomplete action is S = (cid:88) a S [ φ a , λ a ] + T (cid:88) a , Ω J ( Ω ) (cid:34) N | Ψ a ( Ω ) | − Ψ a ( − Ω ) N (cid:88) (cid:96) = φ a (cid:96) ( Ω ) (cid:35) + T (cid:88) a , b , Ω , Ω (cid:48) K ( Ω , Ω (cid:48) ) (cid:34) N | Q ab ( Ω , Ω (cid:48) ) | − Q ab ( − Ω , − Ω (cid:48) ) N (cid:88) (cid:96) = φ a (cid:96) ( Ω ) φ b (cid:96) ( Ω (cid:48) ) (cid:35) . (9)Now we perform the Gaussian integral over the φ (cid:96) : the re-sulting action has a prefactor of N , and so can be replaced byits saddle-point value. By time-translational invariance, thesaddle-point values of the fields can only have the followingfrequency dependence Ψ a ( Ω ) = (cid:0) δ Ω , / T (cid:1) ψ Q ab ( Ω , Ω (cid:48) ) = (cid:0) δ Ω+Ω (cid:48) , / T (cid:1) (cid:104) χ ( Ω ) δ ab + (cid:0) δ Ω , / T (cid:1) q (cid:105) , (10) and we take λ a = λ . We have assumed replica symmetry forthe Edwards-Anderson order parameter q because our inter-est will be limited here to T = ψ , the spin susceptibility χ ( Ω ), q , and λ have to bedetermined by optimizing the free energy. The latter is ob-tained by inserting Eq. (10) in Eq. (9); after taking the replicalimit n →
0, we have the free energy per site F = J (0) ψ + T (cid:88) Ω K ( Ω , − Ω ) | χ ( Ω ) | + K (0 , χ (0) q + T (cid:88) Ω ln (cid:32) ( Ω + λ ) ∆ − K ( Ω , − Ω ) χ ( Ω ) (cid:33) − λ ∆ − K (0 , q + J (0) ψ λ/ ∆ − K (0 , χ (0) . (11)Note that this free energy depends only upon J (0) and K ( Ω , − Ω ), as claimed earlier. Our results described in Eq. (6)and Figs. 1-3 are derived from a set of coupled saddle-pointequations obtained from varying Eq. (11) with respect to χ ( Ω ), q , ψ , and λ for every Ω . Subsequently we let T → K ( Ω , − Ω ) and J (0) of Eq. (5), the rfspectral response function of the atomic qubits plotted in fig-ures 2,3 is given by the expression: − Im (cid:2) Q aa ( i Ω → Ω + i + ) (cid:3) = (12) (cid:12)(cid:12)(cid:12) ω − Ω (cid:12)(cid:12)(cid:12) (cid:113) ∆ t ω − (cid:0) λ − Ω (cid:1) (cid:16) ω − Ω (cid:17) ∆ t ω + q πδ ( Ω ) . The first term is non-zero only for frequencies Ω so that theexpression underneath the square-root is positive. The valueof the Lagrange multiplier in the FM SR is pinned to λ FM =∆ ( J (0) + K (0 , / J (0)). The value of the gap in Fig. 2 is (cid:115) (cid:32) λ FM + ω − (cid:113) ∆ t ω + (cid:16) λ FM − ω (cid:17) (cid:33) . This expressionequates to zero in the gapless QSG phase shown in Fig. 3,where λ QSG = ∆ √ K (0 , t − t c ) when approaching the QSG phase boundarydue to the square-root behavior of the spectral weight [21, 31].The ferromagnetic moment obtains as ψ = J (0) − K (0 , J (0) (cid:32) − (cid:90) ∞−∞ d Ω π χ ( Ω ) (cid:12)(cid:12)(cid:12)(cid:12) λ = λ FM (cid:33) , (13)and ψ vanishes continuously at the FM SR -QSG phase bound-ary (at which J (0) = √ K (0 , β FM = . (cid:104) a i (cid:105) vanishes with the same exponent. Notethat the Edwards-Anderson order parameter q is continuousacross this transition and in the QSG phase given by: q QSG = − (cid:90) ∞−∞ d Ω π χ ( Ω ) (cid:12)(cid:12)(cid:12)(cid:12) λ = λ QSG . (14)As expected, one obtains numerically β QSG = . = β FM . Acknowledgments . We thank A. Amir, J. Bhaseen,T. Esslinger, J. Keeling, B. Lev, M. Punk, J. Ye, P. Zoller,and especially J. Simon for useful discussions. This researchwas supported by the DFG under grant Str 1176 / [1] M. Greiner, O. Mandel, T. Esslinger, T. W. H¨ansch, andI. Bloch, Nature , 39 (2002).[2] K. Baumann, C. Guerlin, F. Brennecke, and T. Esslinger, Nature , 1301 (2010).[3] K. Baumann, R. Mottl, F. Brennecke, and T. Esslinger, Phys.Rev. Lett. , 140402 (2011).[4] C. Maschler and H. Ritsch, Phys. Rev. Lett. , 130401 (2010).[6] K. Hepp, and E. H. Lieb, Annals of Physics , 360, (1973).[7] Y. K. Wang and F. T. Hioe, Phys. Rev. A , 831 (1973).[8] C. Emary and T. Brandes, Phys. Rev. E , 066203 (2003).[9] J. Ye and C. Zhang, Phys. Rev. A , 023840 (2011).[10] P. Domokos and H. Ritsch, Phys. Rev. Lett. , 253003 (2002).[11] S. Gopalakrishnan, B. L. Lev, and P. M Goldbart, Nat. Phys. ,845 (2009).[12] S. Gopalakrishnan, B. L. Lev, and P. M Goldbart, Phys. Rev. A , 043612 (2010).[13] J. Larson and M. Lewenstein, New. J. Phys. , 063027 (2009).[14] V. I. Emeljanov and Y. L. Klimontovich, Phys. Lett. , 366 (1976).[15] B. V. Thompson, J. Phys. A , 89 (1977); ibid. , L179(1977).[16] D. Tolkunov and D. Solenov, Phys. Rev. B , 024402 (2007).[17] F. Dimer, B. Estienne, A. S. Parkins, and H. J. Carmichael,Phys. Rev. A , 013804 (2007).[18] S. Gopalakrishnan, B. L. Lev, and P. M Goldbart,arXiv:1108.1400 (2011).[19] J. Keeling, M. J. Bhaseen, and B. D. Simons, Phys. Rev. Lett. , 043001 (2010).[20] D. J. Amit, H. Gutfreund, and H. Sompolinsky, Phys. Rev. Lett. , 1530 (1985).[21] J. Ye, S. Sachdev, and N. Read, Phys. Rev. Lett. , 4011(1993).[22] N. Read, S. Sachdev, and J. Ye, Phys. Rev. B , 384 (1995).[23] K. H. Fischer, and J. A. Hertz, Spin Glasses (Cambridge Univ.Press, Cambridge, 1991).[24] D. Dalidovich, and P. Phillips, Phys. Rev. B , 11925 (1999).[25] J. T. Stewart, J. P. Gaebler, and D. S. Jin, Nature , 744(2008).[26] R. Haussmann, M. Punk, and W. Zwerger, Phys. Rev. ,063612 (2009).[27] C. W. Gardiner, and M. J. Collett, Phys. Rev. A , 3761 (1985).[28] D. F. Walls, G. J. Milburn, Quantum Optics (Springer-Verlag,Berlin, 2008).[29] S. Sachdev, N. Read, and R. Oppermann, Phys. Rev. B ,10286 (1995).[30] P. Strack and S. Sachdev, in preparation.[31] J. Miller and D. A. Huse, Phys. Rev. Lett.70