Critical and Griffiths-McCoy singularities in quantum Ising spin-glasses on d-dimensional hypercubic lattices: A series expansion study
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] N ov Critical and Griffiths-McCoy singularities in quantum Ising spin-glasses ond-dimensional hypercubic lattices: A series expansion study
R. R. P. Singh
University of California Davis, CA 95616, USA
A. P. Young
University of California Santa Cruz, CA 95064, USA (Dated: October 12, 2018)We study the ± J transverse-field Ising spin glass model at zero temperature on d-dimensionalhypercubic lattices and in the Sherrington-Kirkpatrick (SK) model, by series expansions around thestrong field limit. In the SK model and in high-dimensions our calculated critical properties are inexcellent agreement with the exact mean-field results, surprisingly even down to dimension d = 6which is below the upper critical dimension of d = 8. In contrast, in lower dimensions we find arich singular behavior consisting of critical and Griffiths-McCoy singularities. The divergence ofthe equal-time structure factor allows us to locate the critical coupling where the correlation lengthdiverges, implying the onset of a thermodynamic phase transition. We find that the spin-glasssusceptibility as well as various power-moments of the local susceptibility become singular in theparamagnetic phase before the critical point. Griffiths-McCoy singularities are very strong in two-dimensions but decrease rapidly as the dimension increases. We present evidence that high enoughpowers of the local susceptibility may become singular at the pure-system critical point. The combination of quantum mechanics and disorderleads to rich behavior at and near zero temperature quan-tum critical points (QCPs). For example, the one dimen-sional random transverse field Ising model has a QCP inwhich average and typical correlation functions have dif-ferent critical exponents [1, 2], and the time-dependenceis described by activated dynamical scaling, in which thelog of the relaxation time is proportional to a power of thecorrelation length ξ , rather than conventional dynamicalscaling in which the relaxation time itself is proportionalto ξ z , where z is dynamical exponent. Distributions ofseveral quantities are very broad at the quantum criticalpoint (QCP) so QCP’s with these features are said tobe of the “infinite-randomness” type. It has been pro-posed [3] that infinite-randomness QCP’s can occur indimension higher than 1. It is also proposed [3] that theinfinite-randomness QCP can occur in spin glasses, onthe grounds that frustration is irrelevant since the dis-tribution of renormalized interactions (as one performrenormalization group transformations) is so broad thatonly the largest one matters.In addition, singularities can occur in the paramagnetphase in the region where the corresponding non-random system would be ordered. This was first pointed out forclassical systems by Griffiths [4], though the singularitiesturn out to be unobservably weak in that case [5]. How-ever, these singularities are much stronger in the quan-tum case, as first shown by McCoy [6, 7], and can leadto power-law singularities in local quantities in part ofthe paramagnetic phase. For quantum problems we willrefer to these effects as Griffiths-McCoy (GM) singular-ities. For a review see Ref. [8], and for recent experi-mental observations of GM singularities see Ref. [9]. Inthe quantum paramagnetic phase in the limit as T → z ′ which varies as the QCP is approached. Forinfinite-randomness QCP’s, z ′ → ∞ as the QCP is ap-proached [3, 8].In this paper we study the QCP and GM singularitiesin quantum spin glasses. The infinite-range version, theSherrington-Kirkpatrick (SK) [10] in a transverse field,has been studied in detail [11–13], and the mean field be-havior determined. There have also been quantum MonteCarlo (QMC) studies in dimension d equal to two [14] andthree [15]. In this paper we study quantum spin glassesusing series expansions at T = 0, in which we expandaway from the high transverse-field limit. We feel thatthe series expansion method is complementary to QMCsimulations and has certain advantages including: (i) westudy the whole range of dimensions from d = 2 to theSK model (which is effectively infinite- d ), (ii) we work atstrictly zero temperature whereas in QMC one has to ex-trapolate to T = 0 using a rather complicated anisotropicfinite-size scaling procedure [14, 15], and (iii) averagingover bond disorder is done exactly. Like QMC we cansee GM singularities (we think our work is the first timethese singularities have been seen using series methods),and also go beyond simple averages of local quantities, inour case by computing moments up to high order.Our main conclusions are as follows. We show system-atically how the strength of GM singularities diminishrapidly as the dimension increases above two, vanishing,as expected, for the SK model. In two dimensions, whereGM singularities are strongest, our results make plausi-ble the expectation that GM singularities persist in theparamagnetic phase all the way to the critical point of thepure system. We find that critical behavior close to thatof mean field theory persists below the upper critical di-mension, d u = 8 [13], down to d = 6, which is surprisingsince renormalization group finds no perturbative fixedpoint below d = 8 [13]. Our results in two and threedimensions agree very well with earlier work [14, 15].We consider the Hamiltonian H = − h T N X i =1 σ xi − X h i,j i J ij σ zi σ zj , (1)where the σ αi are Pauli spin operators and h T is the trans-verse field. The interactions J ij are quenched randomvariables with a bimodal distribution. The N spins ei-ther lie on a hypercubic lattice, in which case the interac-tions are between nearest-neighbors and take values ± J with equal probability, or correspond to the Sherrington-Kirkpatrick (SK) [10] model in which case there is nolattice structure, every spin interacts with every otherspin, and J ij = ± J/ √ N . We choose a bimodal distri-bution because the series can be worked out much moreefficiently for this case than for a general distribution [16].We will also add longitudinal fields h i , coupling to σ zi todefine the spin glass susceptibility, and set them to zeroafterwards, see Eqs. (3) and (4) below.The zero temperature quantities we calculate are: • The zero-wavevector, equal-time structure factordefined as S (0) = 1 N X i,j [ h | σ zi σ zj | i ] av , (2)where the state | i is the ground state of the sys-tem, and the average [ · · · ] av refers to disorder av-erage over the quenched random bonds. • The spin-glass susceptibility defined as χ SG = 1 N X i,j [ χ ij ] av (3)where the ground state energy E ( { h i } ) in the pres-ence of infinitesimal local longitudinal fields h i de-fines the local susceptibilities χ ij by the relation E ( { h i } ) = E − X i,j χ ij h i h j . (4) • The moments of the local susceptibility defined as χ m = 1 N X i [ χ mii ] av . (5)We use the linked cluster method to generate the se-ries [19] and discuss the details of the computationalmethod elsewhere [16]. We expand away from the trivialparamagnetic state with J = 0, so the expansion param-eter is x = ( J/h T ) . (6) TABLE I: Estimates of points of singularity and exponentsin various dimensions and the SK model. Note that 1 /x c ≡ ( h Tc /J ) . We anticipate that the singularity found for theequal time structure factor S (0) is the critical singularityand so has exponent γ − zν . If the QCP is of the infinite-randomness type, then z and γ are infinite but the combi-nation γ − zν is presumably finite. For χ SG the criticalexponent is γ , but for low dimensions the susceptibility sin-gularity is clearly at a larger value of 1 /x c than the criticalsingularity determined from S (0), i.e. it is in the paramagneticphase. Consequently, the series is finding a GM singularityfor χ SG rather than the critical singularity, so we denote theexponent by λ rather than γ . For the SK model GM sin-gularities do not occur, so λ = γ and the exact values are γ = 1 / , γ − zν = − / χ SG as discussed in the text. The value of ( h Tc /J ) is estimated to be 2 .
268 in Ref. [17] and 2 . ± .
03 in Ref. [18].In d = 6 and 8, GM singularities are very weak, since the twovalues of x c are almost the same, we expect that the seriesfor χ SG gives the critical singularity in those cases too. Forthe SK model, and for d = 6 and 8, we show results for χ SG both with and without the mean-field log correction. No logcorrection is applied to S (0) so the results for this quantityare the same in both rows. It is curious that the differencebetween the exponents for χ SG and S (0) is close to 1 for allthe models studied.Model χ SG S (0)1 /x c λ /x c γ − zν SK 2 .
267 0.578 2 .
265 -0.509SK (log) 2 .
268 0.486 2 .
265 -0.509SK [11–13, 17] 2 .
268 1 / . − / . . .
92 0.475 32 .
77 -0.5158d 32 .
83 0.606 32 .
77 -0.5156d (log) 23 .
75 0.489 23 .
60 -0.5136d 23 .
68 0.628 23 .
60 -0.5134d 14 .
46 0.697 14 .
23 -0.4413d 9 .
791 0.796 9 .
411 -0.3002d 5 .
045 1.281 4 .
521 0.284
The series are obtained to order 14 for the SK model andfor d = 2 and 3, and to order 10 in higher dimensions.The series coefficients can be found in the source materialon the arXiv. Throughout this paper we generically referto the coefficients of the series expansions as a n , meaningthat the series is of the form Q = X n a n x n . (7)We find that, in contrast to classical spin-glasses [20, 21],the series for the quantum systems are surprisingly wellbehaved. Most of our analysis is based on the simpleratio method, although, we have checked that d-log Padeanalysis gives answers consistent with them. If the serieshas a simple power-law variation, Q ∝ ( x − x c ) − λ , thenthe ratios satisfy r n ≡ a n a n − = 1 x c (cid:18) λ − n (cid:19) . (8) a n / a n - (a) SK χ SG S(0) 2 2.1 2.2 0 0.2 0.4 a n / a n - χ SG
18 20 22 24 26 28 30 32 34 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 a n / a n - (b) 8d χ SG S(0) 30 32 0 0.2 0.4 a n / a n - χ SG
14 16 18 20 22 24 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 a n / a n - (c) 6d χ SG S(0) 22 23 24 0 0.2 0.4 a n / a n - χ SG FIG. 1: Ratio plots for (a) the SK model, (b) d = 8, and (c) d = 6. In all cases the differences in the intercept, i.e. the valuesof 1 /x c , between results for S (0) and χ SG , is very small or zero indicating that GM singularities are very small or non-existent.Thus it is plausible that the singularity exponent for χ SG is the critical exponent γ for these figures. The fits are quadraticexcept for χ SG for the SK model where we used a linear fit. The parameters of the fits are given in Table I. In all three figures,the series for χ SG had the mean-field log factor [11–13] incorporated. The insets show the ratios for χ SG series without the logfactor. The greater curvature is apparent. Hence, in a plot of the ratio r n against 1 /n , the interceptgives 1 /x c and the slope for 1 /n → λ − /x c .We will do linear and quadratic fits to our data to extract x c and the exponent.Griffiths-McCoy (GM) singularities occur in a quan-tum disordered system at low- T in the region betweenthe critical point of the system and the critical pointof the corresponding pure system. In this range, thereare regions of the sample which are non-disordered andso are “locally in the ordered, symmetry-broken state”.The very slow tunneling between between the symmetry-broken states leads to power-law singularities [8, 22–24]in the paramagnetic phase, coming from purely localphysics, namely a distribution of local relaxation timeswhich extends up to very high values.Since χ SG is the divergent response function for thisproblem, its critical exponent is defined to be [13] γ , i.e. χ SG ∝ ( x c − x ) − γ . (9)However, it is important to stress that, because of GMsingularities, the exponent determined in the series isusually different from γ , and shall generally call it λ ,see Table I. The equal time structure factor S (0) doesnot have the two time-integrals present in χ SG . Sincethe dynamic exponent is z and the correlation exponentis ν , the critical behavior of S (0) is S (0) ∝ ( x c − x ) − ( γ − zν ) . (10)Because there are no time integrals in S (0) we expectthat GM singularities will not occur for this quantity,and any classical-like Griffiths [4] singularities will beunobservably weak [5]. If the QCP is of the infinite-randomness type, then γ and z will be infinite, thoughpresumably the combination γ − zν will be finite since itdescribes the critical behavior of an equal time quantity S (0).We now discuss our results. Figures 1 and 2 showthe ratios r n for each model. The reader should also refer to Table I for values of the points of singularity andexponents.We begin with the results for the SK model shown inFig. 1(a). The critical point obtained, 1 /x c = 2 . γ = 1 / − (1 /t ) log(1 − t )] / , where t = x/x c . Ratios of theresulting series give γ = 0 .
49, in excellent agreement withthe exact result. Without taking logarithms into account,the exponent γ is estimated too high as 0 .
58. Other pre-dictions are [13] z = 2 , ν = 1 /
4, so γ − zν = − / S (0)). Our result for this is − .
51 againin good agreement. The critical points for χ SG and S (0)agree with each other to high precision indicating thatthere are no GM singularities in the SK mode, as ex-pected.Results of the ratio analysis of the series in d = 8 and d = 6 are shown in Fig. 1(b)-(c). Curiously the resultsfor χ SG work better, in the sense that the ratio plot iscloser to a straight line, if one includes the same mean-field log-correction as for the SK model. Of course, evenif there are log corrections there is no a priori reason toassume that they have the same form as in mean fieldtheory. Including this correction the exponent for χ SG isvery close to the mean field value of 1 /
2. The exponentfor S (0) (for which no log-correction is performed) is alsoclose to the mean field prediction of − /
2. There is verylittle difference in the critical points for χ SG and S (0) in-dicating that GM singularities, if present, are very weak.It is surprising that the same near-mean-field-like behav-ior is found in d = 6 as well as d = 8 since d = 8 is theupper critical dimension for this problem and no pertur-bative fixed point is found [13] below d = 8. One mighttherefore expect a dramatic change in critical behaviorin going below d = 8, but this is not what we find.In d = 4, see Fig. 2(a), there are clear deviations frommean field exponents, and a clear, though small, differ- a n / a n - (a) 4d χ SG S(0) 6.5 7 7.5 8 8.5 9 9.5 10 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 a n / a n - (b) 3d χ SG S(0) 3.5 4 4.5 5 5.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 a n / a n - (c) 2d χ SG S(0)
FIG. 2: Ratio plots for (a) d = 4, (b) d = 3, and (c) d = 2. In all cases there is a difference in the intercept, i.e. the value of1 /x c , between results for S (0) and χ SG , indicating that χ SG diverges in the paramagnetic phase before the QCP is reached,i.e. the singularity in χ SG corresponds to GM singularities, not critical singularities. Note that the difference in x c values isvery large in 2d but rapidly decreases with increasing dimension. The fits are linear for d = 2 and quadratic for d = 3 and 4.The parameters of the fits are given in Table I. ence between the critical points for χ SG and S (0) indi-cating the presence of rather weak GM singularities.Comparing the results for d = 4 with those for d = 3and 2 in Fig. 2(b) and (c), we see that the strength ofGM singularites increases considerably with decreasingdimension. The same conclusion follows from comparingQMC results in d = 3 [23] with those in d = 2 [24].For the critical singularity of S (0) in d = 3 we find anexponent γ − zν = − .
30 which is in excellent agreementwith the QMC calculations of Ref. [15] who obtain − . z ≃ . , /ν ≃ . , η + z ≃ . γ/ν = 2 − η . Thereis also good agreement in d = 2 between our value of0 .
28 for the exponenent for S (0) and the QMC value [14]of 0 . ± . γ − zν ) /ν = 0 . ± . ν = 1 . ± . S (0) are summarized inFig. 3.We now study the GM singularities in more detail, fo-cussing on the local susceptibility χ ii . According to thestandard picture [8, 22–24], GM singularities occur be-cause χ ii is a random quantity, with a broad distributionextending out to very large values. Although we can’tcompute the distribution of χ ii directly we can get in-formation on it indirectly by computing the series formoments of it, up to high order. The results are sum-marized in Fig. 4. The y -axis is defined such that thecritical value of x for the spin glass problem is at y = 1and the critical value for the pure ferromagnet (i.e. allinteractions equal to J ) is at y = 0.Consider first the results for d = 2 shown in Fig. 4. Wesee that the higher moments have a singularity furtherand further away from the spin glass critical point (whichcorresponds to y = 1). For large values of the order ofthe moment m , the 14-th order series lies below the 10-thorder series. It is therefore plausible that, for an infinitelylong series, the singularity approaches the pure systemcritical point (i.e. y = 0) for m → ∞ .Also shown in Fig. 4, by circles, are the locations of -0.6-0.4-0.2 0 0.2 0.4 1 2 3 4 5 6 7 8 9 γ - z ν d seriesQMCMF FIG. 3: The critical exponent γ − zν for S (0) as a functionof dimension. It is seen that no apparent change occurs at theupper critical dimension of d = 8 [13], and that our resultsin d = 2 and 3 agree well with QMC simulations of Ref. [14]and [15] respectively. Note: this figure is not included in thepublished version. the divergence of the spin-glass susceptibility. This quan-tity has two time integrals and so we put these points at m = 2. In d = 2 the χ SG singularities agree very wellwith the singularities in the local- χ , confirming that thesingularity found in χ SG is a (local) GM singularity, notthe critical singularity.In d = 3, there seems to be a difference in Fig. 4 be-tween the local- χ and χ SG results, but notice the oppositetrends in the data between the 14-term and 10-term se-ries, so it is plausible that the two quantities would besingular at the same point in an infinitely-long series.In d = 4, the GM singularities are sufficiently weakthat the local- χ does not show a singularity at the QCPor in the paramagnetic phase, at least with a 10-term se- ( x c - x p ) / ( x S G - x p )
2d moments (order 14)2d χ SG (order 14)2d moments (order 10)2d χ SG (order 10)3d moments (order 14)3d χ SG (order 14)3d moments (order 10)3d χ SG (order 10)4d moments (order 10)4d χ SG (order 10) FIG. 4: Location of the singularity for the m -th moment ofthe local susceptibility in d = 2 , /m . Here x p is the critical point of the pure system, and x SG is the spin glass critical point as determined from equal-time structure factor, so the y -axis is scaled such that thepure system critical point corresponds to y = 0 and the spinglass critical point to y = 1. The spin glass phase correspondsto y > < y < ries. We expect that the singularity would be at the samelocation as that of χ SG (which is just in the paramagneticphase) for an infinitely long series.To conclude, we have shown systematically how thestrength of GM singularities diminish rapidly as the di-mension increases above two, vanishing, as expected, forthe SK model. In two dimensions, where GM singulari-ties are strongest, our results make plausible the expec-tation that GM singularities persist in the paramagneticphase all the way to the critical point of the pure sys-tem. We find that critical behavior close to that of meanfield theory persists below the upper critical dimension, d u = 8 [13], down to d = 6, which is surprising sincerenormalization group finds no perturbative fixed pointbelow d = 8 [13]. Our results in two and three dimensionsagree very well with earlier work [14, 15].Since the series for χ SG sees GM singularities ratherthan critical singularities we can not determine whetheror not the QCP is of the infinite-randomness type (forwhich γ and z are infinite, though γ − zν is finite). Re-cent numerical simulations in two dimensions [26] are ar-gued to support the infinite-randomness scenario, thoughit seems to us that conventional critical behavior fits thedata about as well. As the dimension increases, the ef-fects of GM singularities become much weaker than in d = 2, so we conclude that if the infinite-randomnessscenario occurs at all for d >
2, it must manifest itselfonly over a very small region around the quantum criticalpoint.
Acknowledgments:
One of us (APY) would like tothank the hospitality of the Indian Institute of Science,Bangalore and the support of a DST Centenary Chair Professorship. He is particularly grateful for stimulatingdiscussions with H. Krishnamurthy which initiated thisproject. We would like to thank Thomas Vojta for in-forming us of recent experimental work on GM singular-ities and for an informative correspondence on infinite-randomness quantum critical points. We would like tothank D. A. Matoz-Fernandez for bringing Ref. [26] toour attention. The work of RRPS is supported in partby US NSF grant number DMR-1306048. [1] D. S. Fisher,
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