Dynamical Critical Properties of the Random Transverse-Field Ising Spin Chain

J. Kisker Institut für Theoretische Physik, Universität zu Köln, D-50937 Köln, Germany    A. P. Young Department of Physics, University of California, Santa Cruz, CA 95064
July 16, 2016

We study the dynamical properties of the random transverse-field Ising chain at criticality using a mapping to free fermions, with which we can obtain numerically exact results for system sizes, L𝐿L , as large as 256. The probability distribution of the local imaginary time correlation function S(τ)𝑆𝜏S(\tau) is investigated and found to be simply a function of α-logS(τ)/logτ𝛼𝑆𝜏𝜏\alpha\equiv-\log S(\tau)/\log\tau . This scaling behavior implies that the typical correlation function decays algebraically, Styp(τ)τ-αtypsimilar-tosubscript𝑆typ𝜏superscript𝜏subscript𝛼typS_{{\rm typ}}(\tau)\sim\tau^{-\alpha_{\rm typ}} , where the exponent αtypsubscript𝛼typ\alpha_{\rm typ} is determined from P(α)𝑃𝛼P(\alpha) , the distribution of α𝛼\alpha . The precise value for αtypsubscript𝛼typ\alpha_{\rm typ} depends on exactly how the “typical” correlation function is defined. The form of P(α)𝑃𝛼P(\alpha) for small α𝛼\alpha gives a contribution to the average correlation function, Sav(τ)subscript𝑆av𝜏S_{{\rm av}}(\tau) , namely Sav(τ)(logτ)-2xmsimilar-tosubscript𝑆av𝜏superscript𝜏2subscript𝑥𝑚S_{{\rm av}}(\tau)\sim(\log\tau)^{-2x_{m}} , where xmsubscript𝑥𝑚x_{m} is the bulk magnetization exponent, which was obtained recently in Europhys. Lett. 39 , 135 (1997). These results represent a type of “multiscaling” different from the well-known “multifractal” behavior.




I Introduction

Quantum phase transitions show a number of remarkable features and have attracted considerable interest in recent years. They occur at zero temperature and are driven by quantum, rather than thermal, fluctuations. Hence, they are induced by varying a parameter other than the temperature, such as an applied transverse magnetic field. In particular, systems with quenched disorder show surprising properties near a quantum critical point. For instance, it was found that in quantum Ising spin glasses (1); (2)and random transverse-field Ising ferromagnets, (3); (4)as well as in “Bose glass” systems (which have a continuous symmetry of the order parameter but lack “particle-hole” symmetry (5)), all or part of the disordered phase shows features which are usually characteristic of a critical point. More precisely, correlations in time decay algebraically and thus various susceptibilities may actually diverge. This behavior is due to Griffiths-McCoy singularities (6); (7), which arise from rare clusters which are more strongly coupled than the average, and it has become clear that their effect is much more pronounced near a quantum transition than near a classical critical point.

One of the simplest models exhibiting the characteristic features of a quantum phase transition is the random transverse-field Ising chain, defined by the Hamiltonian

H=-i=1L-1Jiσizσi+1z-i=1Lhiσix,𝐻superscriptsubscript𝑖1𝐿1subscript𝐽𝑖superscriptsubscript𝜎𝑖𝑧superscriptsubscript𝜎𝑖1𝑧superscriptsubscript𝑖1𝐿subscript𝑖superscriptsubscript𝜎𝑖𝑥H=-\sum_{i=1}^{L-1}J_{i}\sigma_{i}^{z}\sigma_{i+1}^{z}-\sum_{i=1}^{L}h_{i}% \sigma_{i}^{x}, (1)

where the {σiα}superscriptsubscript𝜎𝑖𝛼\{\sigma_{i}^{\alpha}\} are Pauli spin matrices at site i𝑖i and the interactions Jisubscript𝐽𝑖J_{i} and the transverse fields hisubscript𝑖h_{i} are random variables with distributions π(J)𝜋𝐽\pi(J) and ρ(h)𝜌\rho(h) , respectively.

A lot of results on the critical and off-critical properties of this model have been obtained, both analytically and numerically. The ground state properties of the Hamiltonian in Eq. ( 1) are closely related to a two-dimensional classical Ising model where the disorder is perfectly correlated along one direction, the latter model first being studied by McCoy and Wu (8). Subsequently, the quantum model was studied by Shankar and Murthy (9), and recently the critical properties have been worked out in great detail by D. S. Fisher, using a real space renormalization group approach (3); (10). The quantum model, Eq. ( 1), has also been investigated numerically (10); (11); (12); (13); (14)using a mapping to free fermions by means of a Jordan-Wigner transformation.

We now briefly summarize some of the surprising features of this model. Distributions of equal time correlation functions are found to be very broad which leads to (3): (i) different critical exponents for the divergence of the “typical” (15)and average correlation lengths, and (ii) the typical equal time correlation function at criticality falls off as a stretched exponential function of distance, quite different from the power law variation of the average. Recently, it has been shown that these main results are not restricted to one dimension, but also seem to hold in the two-dimensional random Ising ferromagnet (16); (17).

A number of results for dynamics have also been found. For example, at the critical point the dynamical exponent z𝑧z is infinite (3). Away from the critical point the distribution of local (imaginary) time dependent correlation functions is very broad (12). The average varies as a (continuously varying) power of imaginary time τ𝜏\tau involving an exponent (18)z(δ)superscript𝑧𝛿z^{\prime}(\delta) , where δ𝛿\delta is the deviation from criticality. By contrast, the typical correlation function varies as a stretched exponential function of time. Quite detailed information on the whole distribution of time dependent correlation functions in the paramagnetic phase has also been found (12). At the critical point the average correlation function is found to decay with an inverse power of the log of the time (13), corresponding to the result z=𝑧z=\infty mentioned above.

However, the distribution of time dependent correlation functions at the critical point has not yet been determined and is the focus of this study. We find that the distribution is very broad and can be expressed in terms of a single (logarithmic) scaling variable, defined in Eq. ( 6) below. This result implies that there is a continuous range of exponents α𝛼\alpha , somewhat analogous to (but also with important differences from) “multifractal” behavior (19); (20).

We present our results in the following section, and we conclude with a discussion in Sec. III.

II Results

Throughout the paper we assume the following rectangular distributions for the couplings Jisubscript𝐽𝑖J_{i} and the transverse fields hisubscript𝑖h_{i}

π(J)𝜋𝐽\displaystyle\pi(J) =\displaystyle= {1for0<J<10otherwisecases1for0𝐽10otherwise\displaystyle\left\{\begin{array}[]{ll}1&\mbox{for}\quad 0<J<1\\ 0&\mbox{otherwise}\end{array}\right. (2)
ρ(h)𝜌\displaystyle\rho(h) =\displaystyle= {h0-1for0<h<h00otherwise,casessuperscriptsubscript01for0subscript00otherwise,\displaystyle\left\{\begin{array}[]{ll}h_{0}^{-1}&\mbox{for}\quad 0<h<h_{0}\\ 0&\mbox{otherwise,}\end{array}\right. (3)

which are characterized by a single control parameter h0subscript0h_{0} . The system possesses a critical point at δ=[lnJ]av-[lnh]av=0𝛿subscriptdelimited-[]𝐽avsubscriptdelimited-[]av0\delta=[\ln J]_{{\rm av}}-[\ln h]_{{\rm av}}=0 , i.e. h0=1subscript01h_{0}=1 , at which the distributions of the bonds and fields are equal. The lattice size is L𝐿L and we impose free rather than the more conventional periodic boundary conditions (11); (12).

For the numerical work we make use of the mapping of Hamiltonian ( 1) onto a model of free fermions (21); (22); (23). Since the transformation has been used in previous work, we only give a brief summary here and refer to Refs. (11); (12); (14)for further details. For free boundary conditions, which we shall assume here, the most convenient representation is given in Refs. (11); (24), necessitating only the diagonalization of a 2L×2L2𝐿2𝐿2L\times 2L real, tridiagonal matrix. The spin operators occurring in the expectation value of the correlation functions can then be expressed as a product of fermion operators, which is evaluated using Wick’s theorem. The resulting Pfaffian is then given by the square root of the determinant of a matrix, where the matrix elements can be calculated from the eigenvectors and eigenvalues of the Hamiltonian in the free fermion representation. The imaginary time correlation functions are always positive, so there is no ambiguity in sign when taking the square root. For L128𝐿128L\leq 128 we average over 30000 realizations of the disorder, while for the largest size, L=256𝐿256L=256 , we average over 10000 realizations.

We calculate the probability distribution P(lnS(τ))𝑃𝑆𝜏P(\ln S(\tau)) of the single site imaginary time correlation function

Sii(τ)=σiz(τ)σiz(0)subscript𝑆𝑖𝑖𝜏delimited-⟨⟩superscriptsubscript𝜎𝑖𝑧𝜏superscriptsubscript𝜎𝑖𝑧0S_{ii}(\tau)=\langle\sigma_{i}^{z}(\tau)\sigma_{i}^{z}(0)\rangle (4)

at the critical point, i.e. δ=0𝛿0\delta=0 . For convenience, we will denote Sii(τ)subscript𝑆𝑖𝑖𝜏S_{ii}(\tau) by S(τ)𝑆𝜏S(\tau) from now on. To obtain better statistics, we determine the correlation function for every second site from L/4𝐿4L/4 to L/2𝐿2L/2 , making a total of L/8𝐿8L/8 sites. All sites are far from the boundary so we do not expect the results to be affected by boundary effects. The average correlation function is then given by

Sav(τ)=8Li[Sii(τ)]av,subscript𝑆av𝜏8𝐿subscript𝑖subscriptdelimited-[]subscript𝑆𝑖𝑖𝜏avS_{\rm av}(\tau)=\frac{8}{L}\sum_{i}[S_{ii}(\tau)]_{\rm av}, (5)

where []avsubscriptdelimited-[]av[\cdots]_{\rm av} denotes an average over samples.

Figure 1: The probability distribution, PSsubscript𝑃𝑆P_{S} , of -lnS(τ)𝑆𝜏-\ln S(\tau) for L=128𝐿128L=128 and different values of the imaginary time τ𝜏\tau at the critical point δ=0𝛿0\delta=0 . The data is averaged over 300003000030000 samples.
Figure 2: The probability distribution of -lnS(τ)𝑆𝜏-\ln S(\tau) for L=256𝐿256L=256 at the critical point δ=0𝛿0\delta=0 . The disorder average is over 10000 samples.

Since we expect strong finite size effects at the critical point, we calculated data for different system sizes. Data for the distribution of -lnS(τ)𝑆𝜏-\ln S(\tau) for L=128𝐿128L=128 and L=256𝐿256L=256 is shown in Figs. 1and 2. One observes that the distributions are broad and that for larger times the probability distribution gains more weight in the tail, indicating that correlations decrease for larger times, as expected.

Since the curves for fixed L𝐿L and different times τ𝜏\tau appear to be shifted by a roughly constant amount on the (logarithmic) x-axis, we attempt a scaling plot of the data with the parameter free scaling variable

α=-lnS(τ)lnτ.𝛼𝑆𝜏𝜏\alpha=-\frac{\ln S(\tau)}{\ln\tau}\,. (6)

Note that in Ref. (12), which investigated the dynamics in the paramagnetic phase, the scaling variable was found to be -lnS(τ)/τ1/μ𝑆𝜏superscript𝜏1𝜇-\ln S(\tau)/\tau^{1/\mu} , where μ𝜇\mu is expected to diverge at the critical point. Hence α𝛼\alpha in Eq. ( 6) is a natural scaling variable at the critical point.

Figure 3: Scaling plot of the probability distribution in Fig. 1( L=128𝐿128L=128 ). The scaling variable α𝛼\alpha is that given in Eq. ( 6). For larger values of α𝛼\alpha systematic deviations from scaling occur. This comes from data for small times τ𝜏\tau which is presumably not in the scaling region.
Figure 4: Scaling plot of the probability distribution in Fig. 2( L=256𝐿256L=256 ). The scaling variable α𝛼\alpha is that given in Eq. ( 6). Note, by comparison with Fig. 3that the range where the data scale well increases with increasing system size.

The corresponding scaling plots for P(α)𝑃𝛼P(\alpha) against α𝛼\alpha are shown in Figs. 3and 4for L=128𝐿128L=128 and L=256𝐿256L=256 . One sees that the data collapse is good for α𝛼\alpha not too large and that the range of α𝛼\alpha where scaling works increases with increasing L𝐿L . The data for large α𝛼\alpha where scaling breaks down corresponds to small τ𝜏\tau , and it is reasonable to expect deviations from scaling in this region.

Since the data is only a function of the scaling variable α𝛼\alpha , it follows that typically the correlation function falls off with a power of τ𝜏\tau . We shall now see that the precise value of the power depends in detail on how the typical correlation function is defined. For example, if we define “typical” to be the exponential of the average of the log\log , i.e.

Savlog(τ)=exp([lnS(τ)]av),subscript𝑆avlog𝜏subscriptdelimited-[]𝑆𝜏avS_{\rm avlog}(\tau)=\exp([\ln S(\tau)]_{\rm av})\,, (7)

we obtain

[lnS(τ)]av=-0P(α)αlnτdα=-αlnτ,subscriptdelimited-[]𝑆𝜏avsuperscriptsubscript0𝑃𝛼𝛼𝜏𝑑𝛼delimited-⟨⟩𝛼𝜏[\ln S(\tau)]_{{\rm av}}=-\int_{0}^{\infty}P(\alpha)\alpha\ln\tau\,d\alpha=-% \langle\alpha\rangle\ln\tau\,, (8)

which yields, the algebraic decay

Savlog(τ)=τ-α,subscript𝑆avlog𝜏superscript𝜏delimited-⟨⟩𝛼S_{\rm avlog}(\tau)=\tau^{-\langle\alpha\rangle}\,, (9)

where delimited-⟨⟩\langle\ldots\rangle denotes an average with respect to the distribution P(α)𝑃𝛼P(\alpha) . From our data we get α0.7similar-to-or-equalsdelimited-⟨⟩𝛼0.7\langle\alpha\rangle\simeq 0.7 .

On the other hand if we define “typical” to be the median of the distribution, then one easily sees that

Smedian(τ)=τ-αmed,subscript𝑆median𝜏superscript𝜏subscript𝛼medS_{\rm median}(\tau)=\tau^{-\alpha_{\rm med}}, (10)

where αmedsubscript𝛼med\alpha_{\rm med} is the median of P(α)𝑃𝛼P(\alpha) , i.e. it is defined implicitly by

12=0αmedP(α)𝑑α.12superscriptsubscript0subscript𝛼med𝑃𝛼differential-d𝛼{1\over 2}=\int_{0}^{\alpha_{\rm med}}P(\alpha)\,d\alpha. (11)

Any reasonable definition of “typical” will give a power law, in contrast to the average which has a much slower logarithmic variation, which we discuss next.

Contributions to the average correlation function can come both from the scaling function and from non-scaling contributions (14); (10). The scaling part comes from the small α𝛼\alpha part of the scaling function where there is an upturn in the data, similar to that found for the scaling function of the distribution of the static spin-spin correlations C(r)=σiz(0)σi+rz(0)𝐶𝑟delimited-⟨⟩superscriptsubscript𝜎𝑖𝑧0superscriptsubscript𝜎𝑖𝑟𝑧0C(r)=\langle\sigma_{i}^{z}(0)\sigma_{i+r}^{z}(0)\rangle at the critical point (14). If we assume an algebraic relation P(α)α-λsimilar-to𝑃𝛼superscript𝛼𝜆P(\alpha)\sim\alpha^{-\lambda} for small α𝛼\alpha , we can calculate the scaling contribution to the average correlation Sav(τ)subscript𝑆av𝜏S_{\rm av}(\tau) from

Sav(τ)=0P(α)S(τ)𝑑α.subscript𝑆av𝜏superscriptsubscript0𝑃𝛼𝑆𝜏differential-d𝛼S_{\rm av}(\tau)=\int_{0}^{\infty}P(\alpha)S(\tau)\,d\alpha. (12)

Noting that S(τ)=exp(-αlnτ)𝑆𝜏𝛼𝜏S(\tau)=\exp(-\alpha\ln\tau) one obtains

Sav(τ)subscript𝑆av𝜏\displaystyle S_{\rm av}(\tau) similar-to\displaystyle\sim 0α-λexp(-αlnτ)𝑑αsuperscriptsubscript0superscript𝛼𝜆𝛼𝜏differential-d𝛼\displaystyle\int_{0}^{\infty}\alpha^{-\lambda}\exp(-\alpha\ln\tau)\,d\alpha (13)
similar-to\displaystyle\sim (lnτ)-(1-λ).superscript𝜏1𝜆\displaystyle(\ln\tau)^{-(1-\lambda)}.

This agrees with the results of Rieger and Iglói (13)who found

Sav(τ)(lnτ)-2xm,similar-tosubscript𝑆av𝜏superscript𝜏2subscript𝑥𝑚S_{\rm av}(\tau)\sim(\ln\tau)^{-2x_{m}}\,, (14)

(where xm=(1-ϕ/2)0.191subscript𝑥𝑚1italic-ϕ2similar-to-or-equals0.191x_{m}=(1-\phi/2)\simeq 0.191 is the bulk magnetization exponent with ϕ=(1+5)/2italic-ϕ152\phi=(1+\sqrt{5})/2 ), provided λ=1-2xm0.618𝜆12subscript𝑥𝑚similar-to-or-equals0.618\lambda=1-2x_{m}\simeq 0.618 .

Figure 5: An enlarged plot of the data in Fig. 4(L=256). The full line is α-0.67similar-toabsentsuperscript𝛼0.67\sim\alpha^{-0.67} .

To check this we show in Fig. 5an enlarged plot of the data in Fig. 4( L=256𝐿256L=256 ) for small α𝛼\alpha , together with the function α-0.67similar-toabsentsuperscript𝛼0.67\sim\alpha^{-0.67} , which is the best power law fit to the data. The curve fits the data fairly well and the exponent of 0.670.670.67 is reasonably close to the value of λ0.618similar-to-or-equals𝜆0.618\lambda\simeq 0.618 calculated above. Note again that there may be additional non-scaling contributions to the average correlation function as in Ref. (10).

III Conclusions

We have studied numerically the distribution of local (on-site) correlations in imaginary time for the random transverse-field Ising chain at the critical point. The distribution was found to be logarithmically broad and the scaling variable α=logS(τ)/logτ𝛼𝑆𝜏𝜏\alpha=\log S(\tau)/\log\tau was established. This means that while the correlations typically decay with a power of τ𝜏\tau , there is a range of exponents α𝛼\alpha with a distribution P(α)𝑃𝛼P(\alpha) . The small α𝛼\alpha part of the scaling function, which dominates the average correlations, can be fitted by P(α)α-0.67similar-to𝑃𝛼superscript𝛼0.67P(\alpha)\sim\alpha^{-0.67} , which gives (close to) the correct exponent in Eq. ( 13) for the logarithmic decay of the average correlation function. Note that all positive moments are determined by the small α𝛼\alpha region and so fall off with the same (25)decay given in Eq. ( 13).

The behavior of the distribution of S(τ)𝑆𝜏S(\tau) found here is somewhat analogous to “multifractal” behavior (19)predicted for the decay of spatial correlations in the classical two-dimensional Potts model at the critical point (20), since both have a distribution of scaling exponents. Beyond that, however, there are significant differences. Whereas in our case the probability of having a scaling exponent α𝛼\alpha is P(α)𝑃𝛼P(\alpha) , which does not explicitly depend on τ𝜏\tau , for the corresponding multifractal behavior, the probability would be τ-f(α)superscript𝜏𝑓𝛼\tau^{-f(\alpha)} . As a result, for multifractal behavior, the typical correlation function has an exponent αminsubscript𝛼min\alpha_{\rm min} , the value of α𝛼\alpha at the minimum of f(α)𝑓𝛼f(\alpha) , whereas here we find that the exponent for the typical correlation function depends on exactly how “typical” is defined. Averages of the n𝑛n -th moment of the correlation function for positive n𝑛n are also quite different. In our case, for all n>0𝑛0n>0 , the behavior is dominated by the small α𝛼\alpha region of the distribution and the exponent is independent of n𝑛n (25), whereas for multifractal behavior, the moments depend on n𝑛n in a non-trivial way and are given by (19); (20)the Legendre transform of f(α)𝑓𝛼f(\alpha) . It would be interesting to see if there are other systems which have a “multiscaling” behavior of the type found here.

IV Acknowledgments

J.K. is indebted to H. Rieger for valuable hints on the numerics and useful discussions. He thanks the Department of Physics of UCSC for its kind hospitality and the Deutsche Forschungsgemeinschaft (DFG) for financial support. This work is supported by NSF grant DMR 9713977.


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