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Geometry of large-scale low-energy excitations in the one-dimensional Ising spin glass with power-law interactions

Helmut G. Katzgraber1 and A. P. Young2, ?

2

arXiv:cond-mat/0307583v2 [cond-mat.dis-nn] 8 Dec 2003

Theoretische Physik, ETH H¨nggerberg, CH-8093 Z¨rich, Switzerland o u Department of Physics, University of California, Santa Cruz, California 95064, USA (Dated: February 2, 2008)

1

Results are presented for the geometry of low-energy excitations in the one-dimensional Ising spin chain with power-law interactions, in which the model parameters are chosen to yield a ?nite spin-glass transition temperature. Both ?nite-temperature and ground-state studies are carried out. For the range of sizes studied the data cannot be ?tted to any of the standard spin-glass scenarios without including corrections to scaling. Incorporating such corrections we ?nd that the fractal dimension of the surface of the excitations, is either equal to the space dimension, consistent with replica symmetry breaking, or very slightly less than it. The latter case is consistent with the droplet and “trivial-nontrivial” pictures.

PACS numbers: 75.50.Lk, 75.40.Mg, 05.50.+q

I.

INTRODUCTION

There have been several numerical attempts at ?nite temperature1,2,3,4,5,6,7,8 and zero temperature9,10,11,12,13,14,15,16 to better understand the nature of the spin-glass state for short-range spin glasses. These results are generally interpreted in terms of the two main theories for the spin-glass phase: the replica symmetry-breaking (RSB) picture,17,18,19,20 and the “droplet picture”21,22,23,24 (DP). RSB predicts that excitations involving a ?nite fraction of the spins cost only a ?nite energy in the thermodynamic limit, and that the fractal dimension of the surface of these excitations ds is equal to the space dimension d. This is in contrast to DP where a low-energy excitation (droplet) has an energy proportional to ?θ , where ? is the characteristic length scale of the droplet and θ is a positive sti?ness exponent. In addition, the surface of these excitations is fractal with ds < d. More recently Krzakala and Martin,1 as well as Palassini and Young,2 suggest an intermediate picture (called “TNT” for trivial-nontrivial) in which droplets have a fractal surface with ds < d, and their energy is ?nite in the thermodynamic limit. Which of the above pictures describes the spin-glass state correctly is still widely debated. The RSB and TNT pictures require two sti?ness exponents for the energy of large-scale excitations. There is convincing numerical evidence that changing the boundary conditions (e.g., from periodic to antiperiodic), which induces a domain wall, costs an energy which increases as ?θ with θ > 0. On the other hand, in the RSB and TNT pictures, the energy of droplets, created by thermal noise or by applying a perturbation for a ?xed set of boundary ′ conditions, varies as ?θ with θ′ = 0. By contrast, the DP makes the reasonable ansatz that θ′ = θ (> 0). In a previous publication,25 we studied the onedimensional long-range Ising spin glass with power-law interactions. The model’s advantage is that large sys-

tem sizes can be studied, in contrast to the shortrange spin-glass models commonly used. The results of Ref. 25 showed that the sti?ness exponent θ for zero-temperature domain-wall excitations is positive and in fair agreement with analytical predictions from the droplet model.23,25,26 However, the sti?ness exponent θ′ for thermally induced droplet excitations is di?erent and consistent with zero. Hence, at least for the range of system sizes studied, L ≤ 512, the data of Ref. 25 are consistent with both the TNT and RSB scenarios since they have θ > θ′ = 0. The purpose of the present paper is to estimate ds , because this distinguishes between the RSB and TNT scenarios, since ds = d in RSB while ds < d in TNT. For short-range models, a droplet excitation forms a single connected piece, and so ds has to be zero in d = 1. However, for long-range interactions, a droplet may consist of disconnected pieces,23 so a nontrivial value of ds is possible in d = 1. We perform both ?nite-temperature Monte Carlo simulations and ground-state studies. Our results suggest that droplets are possibly compact in agreement with RSB, although the data are also consistent with a very small value of d?ds , which would be consistent with TNT. In Sec. II we introduce the model, observables, and details of the Monte Carlo technique. Results at zero temperature are presented in Sec. III, and those at ?nite temperature are presented in Sec. IV. Our conclusions are summarized in Sec. V.

II.

MODEL AND NUMERICAL METHOD

The Hamiltonian of the one-dimensional long-range Ising spin chain with power-law interactions is given by H=?

i,j

Jij Si Sj ,

(1)

2 where the Ising spins Si = ±1 are evenly distributed on a circular ring of length L to ensure periodic boundary conditions. The sum is over all pairs of spins on the chain and the couplings Jij are given by Jij = c(σ) where25 rij = L sin π π|i ? j| L (3) ?ij σ , rij (2)

TABLE I: Parameters of the T = 0 simulations. The table shows the total number of Monte Carlo steps used for each value of ? and L. All data are computed with 104 disorder realizations. The lowest temperature used to calculate the ground states with parallel tempering Monte Carlo is T = 0.05, the highest 1.70. We use between 10 and 23 temperatures, depending on the system size, to ensure that the acceptance ratios of the parallel tempering moves are larger than ? 0.30. ? L = 16 L = 32 L = 64 L = 128 L = 256 L = 512 0.50 2 × 103 4 × 103 8 × 103 4.0 × 104 3.2 × 105 4.0 × 105 1.00 2 × 103 4 × 103 8 × 103 4.0 × 104 3.2 × 105 4.0 × 105 2.00 2 × 103 4 × 103 8 × 103 4.0 × 104 3.2 × 105 4.0 × 105

is the straight-line distance between sites i and j. The random part of the interactions ?ij is chosen according to a Gaussian distribution with zero mean and standard deviation unity, and the constant c(σ) in Eq. (2) is chosen25 MF to give a mean-?eld transition temperature Tc = 1. The one-dimensional long-range Ising spin chain has a very rich phase diagram23,25,26 in the d-σ plane. Spinglass behavior is controlled by the long-range part of the interaction if σ is su?ciently small, and by the shortrange part if σ is su?ciently large. In this work we focus on the long-range behavior at σ = 0.75 for which25 Tc > 0 and the critical exponents are non-mean-?eld like. Using the exact relation23,26 θ = d ? σ, we expect θ = 0.25 for d = 1, which is in moderate agreement with numerical results25 for domain walls induced by a change in boundary conditions at T = 0. By studying thermally induced droplet excitations Ref. 25 also estimated θ′ ≈ 0, consistent with RSB and TNT. In order to excite droplets at zero temperature we use the coupling-dependent ground-state perturbation method described elsewhere.2,27,28 First, we compute the (0) ground-state con?guration {Si }. Then we perturb the couplings Jij by the following amount: ?H(?) = 2? N

2 [Jij ]av (0) (0) S S Si Sj , MF (Tc )2 i j

(4)

i,j

where ? is a coupling constant and [· · ·]av represents a disorder average. The (total) energy of the unperturbed ground state then increases by exactly ?, whereas the energy of any other state α will increase by the lesser amount of ?ql , where ql is the link overlap between the unperturbed ground state and a state α: ql = 2 N

2 [Jij ]av (α) (α) (0) (0) S Sj Si Sj . MF (Tc )2 i

zero-temperature simulations the term “link overlap” will hereafter refer to the link overlap between the perturbed and unperturbed ground states. Ground states are calculated using the parallel tempering Monte Carlo method29,30 (at very low temperatures) as described in Refs. 25 and 31. The parameters used in the T = 0 simulations are shown in Table I. For each value of L and ? we compute 104 disorder realizations. We ?nd that for σ = 0.75, when the model is in the longrange phase, the e?ciency of the used algorithm to calculate ground states scales as Lz with z = 2.9 ± 0.3. For the current project this translates to a total CPU time of 70 CPU years. Curiously, for σ = 2.50, for which the interactions are e?ectively short range so frustration is minimal in the d = 1 model studied here, the algorithm performs poorly with the equilibration time varying as ? exp(aL), with a = 0.13 ± 0.02. It would be useful to understand intuitively the reasons for this. One quantity that we study at T = 0 is the link overlap, averaged over all samples. To see how this varies with size2 consider a large cluster of excited spins. This ′ has a characteristic energy of order ? Lθ , which is to be compared with the energy gained from the perturbation ?(1 ? ql ) ? ?L?(d?ds) . There is a distribution of cluster energies which we assume to have a ?nite weight at the origin, so the probability that the perturbation will cre′ ate the excitation is ? ?L?(θ +d?ds ) . When this occurs 1 ? ql ? L?(d?ds) , and so on average2,27,28 [1 ? ql ]av = ?L??l (a + bL?c ) , where ?l = θ′ + 2(d ? ds ) (7) (6)

(5)

i,j

In previous work ql has been de?ned for nearest-neighbor models in which the sum is over nearest-neighbor pairs. Here we have generalized the link overlap to long-range models in a natural way. Because the coupling constant ? is of order unity and not of order L only low-energy excitations can be generated. We compute the new ground state of the perturbed system and record the link overlap between the old and new ground states. In the context of

and we have added a correction to scaling term bL?c . In RSB we have ?l = 0 so Eq. (6) tends to a constant for L → ∞, whereas in DP and TNT ?l > 0 so Eq. (6) tends to zero in this limit. In addition, we consider averages over only those samples in which a large excitation is generated,27,28 comprising a ?nite fraction of spins. The criterion we take is |q| ≤ 0.50. Averaging just over these samples gives27,28 [1 ? ql ]′ = L?(d?ds) (a + bL?c ) , av (8)

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TABLE II: Parameters of the ?nite-T simulations. Nsamp is the number of samples, Nsweep is the total number of Monte Carlo sweeps for each of the 2NT replicas for a single sample, and NT is the number of temperatures used in the parallel tempering method. L 16 32 64 128 256 512 Nsamp 2.0 × 104 2.0 × 104 2.0 × 104 2.0 × 104 1.0 × 104 5.0 × 103 Nsweep 2.0 × 103 4.0 × 103 8.0 × 103 4.0 × 104 2.0 × 105 8.0 × 105 NT 10 10 12 14 17 24

the prime representing the restricted average. Equation (8) follows trivially from the arguments presented in the derivation of Eq. (6) with the probability factor ′ ?L?(θ +d?ds ) replaced by unity. We expect that [1 ? ql ]′ av will be independent of ?. In order to study droplet geometries at ?nite temperatures, we compute the distribution of the link overlap ql between two replicas α and β of the system with the same disorder: ql = 2 N

2 [Jij ]av (α) (α) (β) (β) [ S Sj Si Sj ]av . MF (Tc )2 i

FIG. 1: Zero-temperature data for [1 ? ql ]′ as a function of av system size L for di?erent values of the coupling constant ?. Note that the data only depend slightly on ?, thus indicating only small deviations from the scaling form. The dashed lines correspond to a three-parameter ?t to a + bL?c as expected in RSB. TABLE III: Fits of zero-temperature data for [1 ? ql ]′ to av L?(d?ds ) (a + bL?c ), appropriate for DP/TNT, for di?erent coupling constants ?. The last column is χ2 per degree of freedom, where for this data with six points and four ?tting parameters, the number of degrees of freedom (ndf) is two. ? 0.50 1.00 2.00 d ? ds 0.043(14) 0.003(28) 0.019(19) a 0.81(8) 0.59(14) 0.67(10) b 1.95(64) 1.29(7) 1.30(8) c 0.83(19) 0.51(9) 0.54(8) χ2 /ndf 0.64 0.31 1.63

(9)

i,j

Here · · · represents a thermal average, and [· · ·]av represents a disorder average. From the ?nite-size scaling arguments4 used to derive Eq. (6) we expect that the variance of the distribution of the link overlap scales as Var(ql ) = L??l a + bL?c . (10)

Note that in RSB ?l = 0 so Var(ql ) tends to a constant for L → ∞. However, ?l > 0 in DP (since θ′ = θ > 0 and ds < d) and in TNT (since ds < d). The bL?c term is a correction to scaling, which turns out to be necessary since the data cannot be ?tted without it. To speed up equilibration of the ?nite-T simulations we use the parallel tempering Monte Carlo method.29,30 We test for equilibration using the criterion developed earlier,4 now generalized25 for the Hamiltonian in Eq. (1). For all sizes, the lowest temperature used is T = 0.05, well below Tc ? 0.63.25,32 The highest temperature is 1.70 which is well above the mean-?eld critical temMF perature ( Tc = 1) and so the spins equilibrate fast there. We choose the spacing between the temperatures such that the acceptance ratios for the global moves are around 0.30. Parameters of the ?nite-T simulations are summarized in Table II. To summarize, for L → ∞ all the quantities that we calculate, [1 ? ql ]av in Eq. (6), [1 ? ql ]′ in Eq. (8), and av Var(ql ) in Eq. (10) tend to a non-zero constant in RSB, whereas they tend to zero with a power of L in TNT and DP.

III.

RESULTS AT ZERO TEMPERATURE

We ?rst discuss the results for the constrained average of 1 ? ql , including only samples |q| ≤ 0.5, since this yields d ? ds independent of θ′ , see Eq. (8). The results are shown in Fig. 1. Note that the data only depend slightly on ?, indicating only small deviations from the expected scaling form. The results of a DP/TNT ?t to [1 ? ql ]′ = av ?(d?ds ) L (a + bL?c ) are presented in Table III, while the corresponding RSB ?ts to [1 ? ql ]′ = a+ bL?c are shown av in Table IV. In the DP/TNT ?ts we ?nd that d ? ds is close to zero. Both DP/TNT and RSB ?ts are acceptable (χ2 /ndf ? 1). However, in ?ts to a nonlinear model, one cannot convert χ2 /ndf to a con?dence limit33 even if the data have a normal distribution. Similarly, unlike for the case for ?ts to a linear model, the error bars do not correspond to a 68% con?dence. We are particularly interested to get a con?dence limit on the value of d ? ds in the DP/TNT

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TABLE IV: Fits of zero-temperature data for [1 ? ql ]′ = av a + bL?c , appropriate for RSB, for di?erent values of the coupling constant ?. The number of degrees of freedom (ndf) here is three. ? a b c χ2 /ndf 0.50 0.580(7) 1.35(8) 0.53(3) 1.74 1.00 0.575(6) 1.29(5) 0.50(2) 0.21 2.00 0.569(5) 1.28(4) 0.474(14) 1.36

FIG. 3: Zero-temperature data for [1 ? ql ]av as a function of system size L for di?erent values of the coupling constant ?. The dashed lines represent ?ts according to [1 ? ql ]av = a + bL?c (RSB).

TABLE V: Fits of the zero-temperature data for [1 ? ql ]av = L??l (a + bL?c ), which assume the DP/TNT picture, for different coupling constants ?. FIG. 2: The cumulative probability for d ? ds from the ?ts to [1 ? ql ]′ as discussed in the text. The inner pair of dashed av horizontal lines show 68% con?dence levels and the outer pair show 95.5% con?dence levels. ? ?l 0.50 0.065(62) 1.00 ?0.15(18) 2.00 0.018(7) a 0.28(1) 0.07(12) 0.44(25) b 1.0(8) 1.07(7) 1.25(8) c 0.81(46) 0.51(6) 0.49(13) χ2 /ndf 0.12 0.68 0.64

?ts. We do this by computing χ2 as a function of d ? ds , minimizing with respect to the other parameters (a, b, and c). The probability of the ?t P is proportional to exp(?χ2 /2) which we numerically integrate to get the cumulative probability for x = d ? ds :

x

Q(x) =

P (x′ )dx′ .

(11)

indicative that corrections to scaling have to be included. The DP/TNT ?ts, Eq. (6), are shown in Table V, while the RSB ?ts (which ?x ?l to zero) are shown in Table VI. Both ?ts have acceptable χ2 . The cumulative probabilities shown in Fig. 4 give a lot of weight to unphysical negative values of ?l , especially for ? = 1. For all values of ? the weight is small for ?l greater than about 0.10 so we conclude that 0 ≤ ?l < 0.10. ? (13)

The results are shown in Fig. 2. The data for ? = 0.5 and 2.0 constrain d ? ds to zero or a small positive value. The data for ? = 1.0 constrain d ? ds less and allow a range of negative values which are unphysical. At a 68% con?dence level the data are consistent with 0 ≤ d ? ds < 0.05, ? (12)

We should, perhaps, be cautious about this statement in view of the large weight at negative values of ?l in Fig. 4. However, Eq. (13) is consistent with Eq. (12) and the result of Ref. 25 that θ′ ? 0.

apart from the ? = 0.5 data which would exclude zero at the 68% level but, from Fig. 2, are consistent with it at the 86% level. We take Eq. (12) to be our estimate for d ? ds . It is consistent with the RSB prediction of zero and also consistent with a small non-zero value in the DP/TNT scenarios. Data for the average of 1?ql over all samples are shown in Fig. 3 along with RSB ?ts. The data show curvature

TABLE VI: RSB ?ts of zero-temperature data for [1?ql ]av = a + bL?c for di?erent values of the coupling constant ?. ? 0.50 1.00 2.00 a 0.168(8) 0.280(8) 0.378(10) b 0.73(11) 1.15(11) 1.25(6) c 0.56(7) 0.57(5) 0.46(3) χ2 /ndf 0.25 0.90 0.45

5

FIG. 4: The cumulative probability for d ? ds from the ?ts to [1 ? ql ]av as discussed in the text. The inner pair of dashed horizontal lines show 68% con?dence levels and the outer pair show 95.5% con?dence levels.

FIG. 6: χ2 as a function of ?l , optimized with respect to the other parameters (a, b, and c) in Eq. (10), for the variance of the link overlap. The arrows mark the minima.

TABLE VII: DP/TNT ?ts of Var(ql ) to L??l (a + bL?c ) for di?erent temperatures. T ?l 0.05 ?0.21(54) 0.10 0.10(11) 0.16 0.16(6) 0.23 0.13(8) a 0.002(11) 0.047(42) 0.079(39) 0.050(34) b 0.073(8) 0.16(3) 0.29(6) 0.41(2) c 0.52(35) 0.55(24) 0.60(17) 0.52(2) χ2 /ndf 0.05 0.52 0.83 1.18

FIG. 5: Log-log plot of ?nite-T data for the variance of the link overlap Var(ql ) as a function of system size L for several low temperatures. In all three cases we see strong curvature in the data suggesting corrections to scaling. The dashed lines represent ?ts according to a + bL?c (RSB) with the ?tting parameters shown in Table VIII.

the form aL??l is improbable and corrections to scaling must be included. Fits to Eq. (10) (DP/TNT picture) are shown in Table VII, and ?ts to the RSB picture (in which ?l is ?xed to be zero) are shown in Table VIII. The quality of the ?ts is acceptable. However, the DP/TNT ?t for T = 0.05 gives an unphysical negative value for ?l with a very small amplitude a. To clarify this situation, we plot, in Fig. 6, χ2 as a function of ?l , optimizing with respect to the other parameters (a, b, and c). For T = 0.05, χ2 is quite small out to very large negative values of ?l (not shown) and increases rapidly for ?l greater than about 0.12. Since physically ?l cannot be negative, the only conclusion we can deduce from the T = 0.05 data is that ?l lies between

IV.

RESULTS AT FINITE TEMPERATURE

TABLE VIII: RSB ?ts of Var(ql ) to a + bL?c for di?erent temperatures. T 0.05 0.10 0.16 0.23 a 0.015(2) 0.020(2) 0.021(1) 0.017(1) b 0.073(10) 0.155(11) 0.261(11) 0.391(9) c 0.47(7) 0.47(3) 0.50(2) 0.55(1) χ2 /ndf 0.10 0.47 1.32 1.35

In this section we study the model at temperatures well below25 Tc ≈ 0.63. Figure 5 shows data for the variance of the link overlap for several low temperatures. The data show strong curvature indicative that a simple ?t of

6 zero and about 0.12, consistent with the result from the T = 0 data in Eq. (13). The data for χ2 for T = 0.10 in Fig. 6 has a minimum at ?l = 0.10 but it is shallow and ?l = 0 has only a slightly greater χ2 value. The T = 0.10 data are therefore also consistent with Eq. (13). The data at higher temperatures, T = 0.16 and 0.23 have a deeper minimum at nonzero χ2 , suggesting that ?l = 0 is somewhat unlikely, but experience from short-range systems4 suggests that estimates of ?l at ?nite T are e?ective exponents which need to be extrapolated to T = 0 to get close to the asymptotic value. Hence we do not feel that the results at T = 0.16 and 0.23 rule out ?l = 0. Overall, the ?nite-T data are consistent with ?l in the range given by Eq. (13) which came from the T = 0 data, and do not constrain ?l any further.

V. CONCLUSIONS Acknowledgments

of d ? ds . Substantial corrections to scaling had to be incorporated into all the ?ts. We have also estimated the exponent ?l = θ′ +2(d?ds ), where θ′ characterizes the dependence of the energy of droplet excitations on their length scale. We ?nd it to be in the range 0 ≤ ?l < 0.10, which is consistent with the ? value for d ? ds in Eq. (12) and our earlier result25 that θ′ ? 0. Note that this result for θ′ , if also valid in the thermodynamic limit, is inconsistent with the DP. By studying a one-dimensional model, we have been able to study a much larger range of sizes, 16 ≤ L ≤ 512, than is generally possible in spin glasses. However, in the absence of a good understanding of corrections to scaling in spin glasses, we still cannot rule out the possibility that di?erent behavior may occur in the thermodynamic limit.

We have studied the geometry of the large-scale, lowenergy excitations in a one-dimensional Ising spin glass where the interactions fall o? as r?σ with σ = 0.75, both at T = 0 and at temperatures well below the spin-glass transition temperature. We ?nd that the fractal dimension of the surface of these excitations, ds , lies in the range 0 ≤ d ? ds < 0.05. This is consistent with the ? RSB picture (d ? ds = 0). It is also consistent with the DP/TNT picture (d ? ds > 0) but with a small value

We would like to thank K. Tran for carefully reading the manuscript. The simulations were performed on the Asgard cluster at ETH Z¨ rich. We are indebted to u M. Troyer and G. Sigut for allowing us to use the idle time on the Asgard cluster. The work of APY is supported by the National Science Foundation under Grant No. DMR 0086287.

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