Geometry of rare regions behind Griffiths singularities in random quantum magnets

In many-body systems with quenched disorder, dynamical observables can be singular not only at the critical point, but in an extended region of the paramagnetic phase as well. These Griffiths singularities are due to rare regions, which are locally in the ordered phase and contribute to a large susceptibility. Here, we study the geometrical properties of rare regions in the transverse Ising model with dilution or with random couplings and transverse fields. In diluted models, the rare regions are percolation clusters, while in random models the ground state consists of a set of spin clusters, which are calculated by the strong disorder renormalization method. We consider the so called energy cluster, which has the smallest excitation energy and calculate its mass and linear extension in one-, two- and three-dimensions. Both average quantities are found to grow logarithmically with the linear size of the sample. Consequently, the energy clusters are not compact: for the diluted model they are isotropic and tree-like, while for the random model they are quasi-one-dimensional.

where A > 0 is some constant.
Since the rare regions are well separated, one might assume that the related low-energy excitations are uncorrelated. In this case, the probability distribution function of the smallest gap in a finite system can be obtained from extreme statistics of independent and identically distributed (iid) random numbers having a parent distribution in Eq. (3). The result is the Fréchet distribution 13-15 : in terms of u = u 0 εL z , where u 0 is a constant. The validity of this relation has been studied numerically for the random transverse Ising model (TIM) and presumably exact agreement is found asymptotically in the L → ∞ limit in d = 1 16,17 . The numerically found agreement is also satisfactory for d = 2 17 . More recently, we have studied the finite size corrections to Eq. (4) for the d = 1 system 18 and systematic deviations are observed from the analytical results for iid random numbers. This fact shows that the weak correlations between low-energy excitations in random quantum magnets are relevant.
In the previous paragraphs we showed that the concept of rare regions for localized excitations is very useful in understanding the singular properties of Griffiths-phases in random quantum magnets. Here, we aim to visualize and understand the geometric characteristics of these rare regions. Specifically, we want to address the question whether the monotonous decrease of the excitation energy with L (see in Eq. (2)) goes hand in hand with the growth of these regions, or simply due to the appearance of stronger bonds in the same volume. In this paper, we carry out such an investigation on a paradigmatic system, the transverse Ising model, both with dilution (diluted model) and with random couplings and/or transverse fields (random model). For the diluted model, the low-energy clusters are the same as for classical percolation and we analyse the extreme statistics of these clusters. In the diluted model the largest cluster (containing the most sites) has the smallest excitation energy. For the random model, the calculation is more involved and the Strong Disorder Renormalization Group (SDRG) method 19,20 is used, in which the ground state of a given sample is represented by isolated sites and clusters of sites. To each cluster (and to each isolated site) an excitation energy is associated and the smallest one represents the lowest gap of the sample. The cluster with the smallest excitation energy (the so-called energy cluster) represents the rare region we are looking for.
In the actual calculation, we used a numerical implementation of the SDRG method 21,22 and for a large number of realizations we measured the excitation energy and different geometrical properties of the energy cluster (number of sites, linear extension, radius of gyration) and studied the dependence of their average value on the linear size of the system. We have most detailed calculations in d = 1 in which case we went up to L = 2 15 and used 10 6 realizations. In d = 2 and 3 the largest systems were L = 2 10 and 2 8 , respectively, and we have at least 10 4 realizations for the largest systems.

Models and the SDRG procedure
Here, we consider the zero temperature transverse Ising model in a d-dimensional hypercubic lattice and the Hamiltonian is defined by in terms of the σ x,z i Pauli matrices at site i and the first sum runs over nearest neighbours.

Diluted model.
In the bond diluted model, all the {h i } are equal to h > 0 and the couplings are equal to J = 1 with probability p or equal to J = 0 with probability 1 − p . We also consider the site diluted model, in which J ij = ǫ i ǫ j and h i = hǫ i and ǫ i = 1 with probability p and ǫ i = 0 with probability 1 − p . In d = 1 , for p < 1 , the system is in the paramagnetic phase: it is separated into finite segments. In higher dimensions, there is a percolation transition at p = p c and in the subcritical phase, δ = �p = p − p c < 0 , the system consists of isolated clusters 23 . In this subcritical region, having a sufficiently strong coupling, such that the pure system is in the ferromagnetic phase, h < h * , the finite clusters are trying to order, which results in a large susceptibility, a hallmark of the Griffiths-phase. We are going to study the system in this region, see in Fig. 1.
Random model. In the random model, the couplings are taken from the uniform distribution: and similarly for the transverse fields: www.nature.com/scientificreports/ Generally, the logarithmic variable, θ = ln(h 0 ) , is used as the control-parameter. The system is in the ferromagnetic and the paramagnetic phase for δ = �θ = θ − θ c < 0 and �θ > 0 , respectively. The properties of the system at the critical point, θ c , are controlled by infinite disorder fixed points, see in Refs. 21,22,[24][25][26][27][28][29][30][31][32] . Since the minimal value of the transverse field is h min = 0 , the Griffiths-phase extends to the complete paramagnetic phase, �θ > 0 , see in Fig SDRG procedure. We are going to study the properties of the random TIM by the SDRG procedure 19,20 , which has been introduced by Ma et al. 33,34 and further developed by Fisher 24,25 . The SDRG method provides analytical results in d = 1 24,25,35 and accurate numerical ones in higher dimensions 21,22,[26][27][28][29][30][31][32] . In this procedure, at each step of the renormalization, the largest parameter in the Hamiltonian (either coupling or transverse field) is eliminated and new terms are generated between remaining sites through second-order perturbation calculation. If a strong coupling, say J ij , is decimated, the two connected spins form a spin cluster having an additive moment, μ = µ i + µ j . (Initially, all spins have the same moment µ i = 1, ∀i .) The spin cluster is placed in an effective transverse field of strength: h = h i h j /J ij , which is obtained from the first gap of the cluster. If a large transverse field, say h i , is decimated, the actual spin is eliminated and new effective couplings are generated between each pair of spins, which are nearest neighbours to i, say j and k, having a value: J jk = J ji J ik /h i . In d = 1 the system remains a chain with fewer sites under decimation. In higher dimensions, however, the topology of the system is gradually changing. In this case it can happen, that at some step two parallel couplings appear between two neighboring sites, in which case the maximum of them is taken. The use of this "maximum rule" is asymptotically exact at an infinite disorder fixed point and results in simplifications in the renormalization algorithm. Using the optimized algorithm described in Refs. 21,22 the time to renormalize a cluster with N sites and E edges was t ∼ O(N ln N + E) , which is to be compared with the performance of a naïve implementation: The SDRG method is expected to provide asymptotically exact results at an infinite disorder fixed point, which has been checked in d = 1 by comparing the results with those of analytical and numerical calculations 12,36 . Also in d = 2 the SDRG results (obtained by the use of the maximum rule) are consistent with quantum Monte Carlo simulations 37,38 . Regarding the Griffiths-phase, the SDRG describes the Griffiths singularities asymptotically exactly in d = 1 , where the dynamical exponent is given by the positive root of the equation 35  www.nature.com/scientificreports/ The distribution of the smallest energy excitations obtained by the SDRG also agrees well with the precise numerical calculations, even for large but finite L values 18,41 . Therefore, we expect that the SDRG method provides valuable information about the statistics of energy clusters, too.
From a geometric perspective, the SDRG method decomposes the ground state into independent clusters in the GHZ-state. Already in d = 1 , the resulting clusters are generally disconnected, due to the generation of effective long-range couplings during the 'h-decimation' steps. Each time, when such an effective coupling is decimated in the 'J-decimation' step, a disconnected cluster is created in terms of the original sites. Such disconnected clusters can then form the building blocks of even larger disconnected clusters. As a result, at the critical point, the clusters are disconnected fractals with gaps emerging between the sites at all spatial scales 42 .
We mention that in d = 1 , the statistics of clusters has already been studied at the critical point 43 . In this case, the linear extension of the largest cluster scales with the length of the chains. Hence, density profiles are meaningful and -close to the open boundaries-are found to follow scaling predictions in numerical calculations. Critical energy clusters have also been studied at d = 2 and d = 3 , where they are found to span the finite system and have a quasi-one-dimensional structure 21,22 . In the Griffiths-phase, where dominantly transverse fields are decimated, the typical clusters have a finite extent, which is proportional to the correlation length. In the following sections, we are going to study the geometrical properties of the largest one, the energy cluster.

Results
Typical rare regions. The structure of the diluted model in d = 1 is trivial, it consists of separated connected segments, among which the energy cluster has the largest mass, which equals to its length. While in d = 1 the mass and the linear extension of a connected cluster are identical measures of the cluster size, in higher dimensions, the 'mass' stands for the number of sites in the cluster. Then, in higher dimensions, the ground state of the system is given by a set of percolation clusters, among which the one with the largest mass is the energy cluster. A typical energy cluster for the site-diluted TIM is shown in Fig. 3.
For illustration in the random model, a few energy clusters are shown in Fig. 4 for d = 1 and for d = 2 . The energy clusters are represented by the contributing original (non-decimated) sites, which are indicated by black squares in the figures. Note that the energy cluster in the random model generally consists of several disconnected parts.
From a geometrical point of view, the energy clusters are characterised by the linear extension, ℓ , the radius of gyration, R g , and the mass, µ . In d = 1 , ℓ is simply the length of the cluster. In higher dimensions, we draw a bounding box around the cluster (see in Figs   First, for a given dimension, we set the value of the control parameter in the Griffiths-phase to θ > θ c . For the uniform distribution, the critical point is known exactly in d = 1 , θ c = 0 , whereas in d = 2 and d = 3 these were determined previously as θ c = 1.6784 and θ c = 2.5305 , respectively, by the numerical application of the SDRG approach with the maximum rule 21,22 . Having a fixed value for θ , we considered a large set of random samples of different finite length, L, and performed the renormalization procedure up to the last site. In a given sample, during the decimation process, spin clusters are formed and then eliminated. In the end, we selected the energy cluster, which has the smallest excitation energy.     To verify that this result is not related to the relatively small sizes of the energy clusters, we have repeated the analysis closer to the percolation threshold, at p = −0.2 , for site-percolation in d = 2 . The obtained results in Fig. 7 show the same type of ln L scaling, although the energy clusters are comparatively large. On closer inspection of a typical energy cluster in Fig. 3, one can see an effective tree-like structure that corresponds to the observed linear mass-length relationship.

Analysis of the
We repeated the previous analyzes for bond-diluted TIMs, where the energy clusters were selected from the extremal clusters of bond percolation and the geometric properties were qualitatively the same as for site percolation.
Random model. For the random TIMs, the cluster structure of the ground state is calculated through the numerical application of the SDRG method. In d = 2 , we considered the point θ = 2.6784 ( �θ = 1 ) and calcu-   Fig. 8.
Repeating the calculations at other points of the Griffiths-phase, we obtained the same qualitative picture: the size-dependence of the parameters of the energy clusters are logarithmic in L, just the prefactors are θ dependent. Consequently, we obtained the same result as for the diluted model: the energy clusters are quasi-one-dimensional, having a tree-like topology even in d = 2 and d = 3 . We note that this observation was already made at the critical point 21,22 , which explains why the critical exponent ψ in Eq. (1) is almost independent of dimensions. Our numerical results show that this behavior of the energy clusters remains valid even in the Griffiths-phase.

Discussion
One peculiarity of random many-body systems is that dynamical observables can be singular not only at the critical point, but in extended regions of the paramagnetic and ferromagnetic phases as well. This singular behavior is due to rare regions, which occur with very low probability, but have a very large relaxation time, which actually diverges in the thermodynamic limit. In this paper, we studied the geometrical properties of these rare regions in the paramagnetic phase of ferromagnetic quantum magnets, such as the transverse Ising model, which are either diluted or contain random parameters (couplings and/or transverse fields).  www.nature.com/scientificreports/ In the diluted model, a fraction of sites or bonds are missing and the existing spins form well-defined clusters, the structure of which is described by percolation theory. On the contrary, in the random model, clusters are formed during the SDRG procedure when sites become decimated when the energy-scale is lowered below a characteristic local value. In this way, clusters in the diluted model are defined sharply, while in the random model they depend on the energy-scale and the final cluster structure is obtained only at the lowest energy scale of the SDRG procedure.
In the Griffiths-phase, these clusters are locally in the ordered phase and the rare region responsible for Griffiths singularities is associated to such cluster, which has the smallest excitation energy. In the diluted model, the energy cluster also has the largest mass, while in the random model it is the cluster decimated at the lowest energy scale. From a topological point of view, the energy cluster is connected for the diluted model, while it contains several disconnected components for the random model.
Specifically, we considered the transverse Ising model in one-, two-and three-dimensional hypercubic lattices and calculated the energy clusters in diluted and random samples of varying lengths, L. We have determined the mass (number of sites), µ , the linear extension, ℓ , and the radius of gyration, R g , of the energy clusters. The average mass is found to scale as ln L for large systems. For the diluted model, this follows from extreme statistics and from the result of percolation theory that the distribution of the mass of the clusters is exponential.
More surprisingly, the average linear parameters of the energy clusters, [ℓ] av and [R g ] av are found to scale with ln L , also in higher dimensions. Thus, from the numerical results, we observe a linear relation between mass and linear extension, [µ] av ∼ [ℓ] av , both for diluted and random models. Regarding the diluted model, the typical (not extreme) percolation clusters in the subcritical region, p < p c , have a finite extent, ∼ ξ , and these are compact and isotropic. One could naïvely expect that the same properties hold for the extreme clusters, too, which is not the case, however. As seen in Fig. 3, the extreme clusters have a large linear extent, ℓ ≫ ξ , are isotropic, but not compact. As seen in Fig. 3, the extreme energy clusters have a tree-like structure from building blocks of the size ξ.
In higher dimensional random models, the energy clusters are not compact either. Previous studies performed at the critical point showed that the energy clusters are worm-like, quasi-one-dimensional 21,22 , disconnected objects. This is in agreement with the observation that the critical exponent ψ , defined in Eq. (1) is close to 0.5 in any dimensions. Our present investigations revealed that the quasi-one-dimensional form of the energy clusters is likely in the Griffiths-phase, too.
In our numerical work, we were deep in the paramagnetic phase, so that the relation L ≫ ξ is well satisfied. Being closer to the critical point, a cross-over from critical dynamics to Griffiths dynamics should take place, which is accompanied with the change of the structure and symmetry of the energy cluster.
Our results about the geometric properties of the energy cluster are expected to be valid for other models having infinite disorder scaling and strong Griffiths effects. For diluted models, the cluster structure evidently does not depend on the actual form of the dynamical variable. For random models, the TIM fixed point is expected to govern the critical properties of models having a discrete symmetry (q-state Potts model 48,49 , Ashkin-Teller model [50][51][52] , etc.), as well as random stochastic models, such as the contact process [53][54][55] . For these models, the form of the Griffiths singularities is very similar, which should hold for the basic geometrical properties of the energy clusters, too 42 .

Data availibility
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