Charge avalanches and depinning in the Coulomb glass: The role of longrange interactions
Abstract
We explore the stability of farfromequilibrium metastable states of a threedimensional Coulomb glass at zero temperature by studying charge avalanches triggered by a slowly varying external electric field. Surprisingly, we identify a sharply defined dynamical (“depinning”) phase transition from stationary to nonstationary charge displacement at a critical value of the external electric field. Using particleconserving dynamics, scalefree systemspanning avalanches are observed only at the critical field. We show that the qualitative features of this depinning transition are completely different for an equivalent shortrange model, highlighting the key importance of longrange interactions for nonequilibrium dynamics of Coulomb glasses.
pacs:
75.50.Lk, 75.40.Mg, 05.50.+q, 64.60.iI Introduction
The longrange nature of the Coulomb interaction plays only a secondary role in metals, where it remains screened by mobile electrons down to atomic length scales. The situation is, however, far more interesting on the insulating side of disorderdriven metalinsulator transitions Dobrosavljević et al. (2012), where screening is suppressed due to charge localization. Here, the unscreened Coulomb interaction leads to the opening of the “Coulomb gap” in the electronic density of states, as first pointed out in pioneering works of Pollack Pollak (1970), as well as Efros and Shklovskii (ES). The ES theory Efros and Shklovskii (1975); Shklovskii and Efros (1988) predicts a universal form of the Coulomb gap, and explains how its existence modifies hopping transport Shklovskii and Efros (1988) in disordered insulators, consistent with numerous experiments Lee and Ramakrishnan (1985). Early work also revealed that Coulomb interactions in disordered insulators generally contribute to the formation of an extensive number of metastable states, i.e., the formation of the Coulomb glass (CG) Davies et al. (1982, 1984); Xue and Lee (1988). In subsequent work, various aspects of glassy behavior of the CG were explored theoretically Grannan and Yu (1993); Pastor and Dobrosavljević (1999); Pastor et al. (2002); Dobrosavljević et al. (2003); Müller and Ioffe (2004); Grempel (2004); Pankov and Dobrosavljević (2005); Kolton et al. (2005); Müller and Pankov (2007); Surer et al. (2009) and experimentally BenChorin et al. (1993); Ovadyahu and Pollak (1997); Vaknin et al. (2000); Bogdanovich and Popovic (2002); Vaknin et al. (2002); Orlyanchik and Ovadyahu (2004); Jaroszyński et al. (2004); Ovadyahu (2006); Jaroszyński and Popović (2006, 2007); Raičević et al. (2008, 2011); Lin et al. (2012).
More recent progress followed with the formulation of analytical theories of the CG Pastor and Dobrosavljević (1999); Pastor et al. (2002); Dobrosavljević et al. (2003); Müller and Ioffe (2004); Pankov and Dobrosavljević (2005); Müller and Pankov (2007) which adapted Parisi’s replica methods Parisi (1983); Rammal et al. (1986); Mézard et al. (1987); Young (1998) for spin glasses to disordered Coulomb systems. These theories find a Coulomb gap of the same universal form as predicted by the ES theory, but this behavior emerges only within the lowtemperature glassy phase (displaying replica symmetry breaking). Within this meanfield picture, the universality of the Coulomb gap, as well as the saturation of the appropriate stability bound, can be directly traced back to the “marginal stability” of the entire glassy phase Pastor and Dobrosavljević (1999). In physical terms, the marginal stability reflects the emergence of “replicons,” soft (gapless) collective excitations involving simultaneous rearrangements of many electrons. If such soft excitations indeed characterize the Coulomb glass, they should also govern the physical response to various weak perturbations (e.g.m the external electric fields), perhaps leading to largescale avalanches. Precisely such behavior has already been established Pázmándi et al. (1999); Andresen et al. (2013) for infiniterange spinglass models, leading to scalefree avalanches characterizing an entire manifold of metastable states. Despite the successes of the meanfield approach, its applicability to finite space dimensions remains the subject of much controversy and debate Young and Katzgraber (2004); Katzgraber and Young (2005); Boettcher et al. (2008); Jörg et al. (2008); Katzgraber et al. (2009); Larson et al. (2013). Furthermore, a computational search for a finitetemperature glass transition in the CG in two and three space dimensions has remained inconclusive Grempel (2004); Kolton et al. (2005); Surer et al. (2009). To shed additional light on the nature of excitations in the CG, and further test the meanfield ideas, it is therefore useful to examine the stability of the lowlying metastable states by external electric fields.
In this work, we investigate the outofequilibrium behavior of a threedimensional Coulomb glass at zero temperature and study the hopping and total charge displacement avalanches triggered by increasing an externallyapplied electric field. Previous work on avalanches in the CG in three space dimensions done by Palassini and Goethe Palassini and Goethe (2012), which trigger avalanches via dipole excitations or charge insertions, find scalefree behavior for longrange hopping dynamics, but when hopping is bounded by a finite fixed range they do not find any scalefree avalanches. Because physical electrons rearrange themselves by finiterange hopping it is of interest to search for a scalefree behavior in the CG for bounded hopping dynamics by other means.
Here we study the CG with particlenumberconserving shortrange hopping, by “adiabatically” increasing an external electric field up to a depinning electric field that separates the steady current state from just finite electron rearrangements as a reaction to the external field. We find that scalefree avalanches arise in the Coulomb glass when the electric field is close to . To emphasize the role played by the longrange Coulomb interactions we repeat our simulations for an equivalent shortrange interacting model. In this case we still find a sharply defined depinning transition, but a completely different form for the critical behavior. Here we do not find any scalefree avalanches, in dramatic contrast to the behavior of the CG model.
Ii Model
The Coulomb glass Hamiltonian (in dimensionless units) is given by Efros and Shklovskii (1975)
(1) 
where is the electron number at site , is the filling factor, is the coordinate of site , and a randomlydistributed onsite energy. For a charge neutral system, i.e., , in a constant external electric field in direction, Eq. (1) can be rewritten in an Ising spin formulation by setting Davies et al. (1982) ( an Ising spin variable)
(2) 
where the electric potential is and is the position of spin . This form of the Hamiltonian with is of a randomfield Ising model with longrange antiferromagnetic interactions given by
(3) 
The site energy is sampled from a Gaussian distribution with zero mean and standard deviation . To keep the dynamics of the two models identical it is necessary to constrain the Isinglike Hamiltonian in Eq. (2) to have a constant magnetization ( for ) at all times. This is accomplished by using magnetizationconserving Kawasaki dynamics Newman and Barkema (1999).
The corresponding shortrange model (SR) is given by the same Hamiltonian in Eq. (2), but with longrange interactions replaced by nearestneighbor interactions (on a cubic lattice) of the form
(4) 
ii.1 Determination of the initial configurations
In our simulations, we need to generate stable initial configurations of the system. In this context, “stable” refers to stable towards single nearestneighbor electron hopping. We implement this procedure for both the CG and the SR model. In order to have an initial configuration with a Coulomb gap and track its dependence on the electric field, we compute pseudogroundstate configurations using jaded extremal optimization (JEO) Middleton (2004).
The singleparticle density of states (DOS) of a classical Coulomb system is given by
(5) 
where the local singleparticle energy is given by
(6) 
and the average is performed both over thermal fluctuations and disorder instances. The ground state of the CG is well known to display a Coulomb gap Efros and Shklovskii (1975) in the DOS at the Fermi energy, which gradually fills up when temperature is increased Davies et al. (1982, 1984); Grannan and Yu (1993); Sarvestani et al. (1995); Surer et al. (2009).
For the CG, we can empirically check how “far” or “close” a given configuration is from the ground state by examining the form of the DOS. Depending on the depth of the Coulomb gap, we can argue whether the configurations are close or far from their respective ground state. The SR ground states do not have a Coulomb gap Boettcher et al. (2008), but have a “dip” at the Fermi energy that converges to a finite value in the thermodynamic limit. Again, we can empirically check if we have a good approximation of the ground state by studying at the DOS distribution. In Fig. 1(a), we show the DOS of the CG using the pseudo ground states for all simulated linear system sizes (the systems have spins). The occupation at is very close to zero, showing that the configurations found using JEO are not far from the true ground state. In Fig. 1(b), we show the DOS of the CG at electric fields . The data suggest that we are further away from a groundstate configuration, however, a pronounced gap in the DOS is still visible. The configurations for the SR model found by the JEO algorithm are likewise not far from the ground state (not shown).
Iii Numerical Details
iii.1 Algorithm
For the description of the algorithm, we introduce a stability criterion, which for an electron () or a vacancy () at a given site is given by
(7)  
(8) 
For each pseudoground state generated via JEO [see Fig. 1(a)], we proceed as follows.

Select the least stable electron with one nearestneighbor hole in the opposite direction of the electric field.

Apply an electric field just strong enough to destabilize the selected electron, such that it will hop to the neighboring hole.

Perform the electronhole hopping that minimizes the energy; go to step 3.
The careful reader will have noticed that the above procedure is in fact an infinite loop stuck between steps 3 and 4 when a certain electric field threshold is reached. This electric field threshold is the depinning field of the system, which separates two regions: Below there are only short charge displacement pulses due to the rearrangement of the electrons as a response to the external electric field, and above it there is a steady current. A sketch of the different scenarios is shown in Fig. 2. The infinite loop between step 3 and step 4 is the steady current flowing through the system. Since we are interested in the number of times step 3 and step 4 are repeated at each field (this, in turn, yields the avalanche size ) before we reach the depinning field, we artificially stop the process if the avalanche size surpasses a given number , where is the total number of sites of the system. Note that is much larger than the maximal avalanche size measured for for a given system size .
To cope with the longrange Coulomb interactions between the electrons we use the Ewald summation method Wang and Holm (2001). Furthermore, the applied electric field is periodic to avoid an electron pileup at the edge of the system. The simulation parameters are listed in Tab. 1.
model  

CG  
CG  
CG  
CG  
CG  
CG  
CG  
CG  
SR  
SR  
SR  
SR  
SR  
SR  
SR 
iii.2 Measured observables and statistical data analysis
At each increase of we count the number of electrons that hopped and the total charge displacement in the direction of the applied electric field. Using these data, we compute their distributions and , respectively (see, for example, Fig. 3). To determine the depinning field we compute the cumulative distribution function of the depinning distributions which gives the probability whether a randomly picked sample is in the pinned or depinned state for a given system size and at a given field. We perform a finitesize scaling assuming that the function has a universal form Dong et al. (1993); Newman and Ziff (2000); Xi et al. (2015)
(9) 
[see Fig. 5(b) and Fig. 8(b) for the CG model and the SR model, respectively], which gives us an estimate of the depinning field. Note that the depinning field is defined as the typical electric field necessary to induce a continuous current for a given system size, i.e, for , the system just rearranges its electron configuration by electron hopping, whereas for , the field induces a steady current.
In addition, we define the characteristic avalanche size of the system by fitting the exponential tail of the avalanche distributions to an exponential function . For each system size , we thus obtain a characteristic avalanche size . To estimate the value of in the thermodynamic limit, we do an extrapolation of by using the following functional ansatz:
(10) 
where , , and are fitting parameters.
Finally, we also monitor the DOS as a function of the applied electric field . For example, Fig. 1(b) shows the density of states at an electric field range of .
Different finitesize scaling Ansätze have been attempted Pázmándi et al. (1999) to scale the and data without yielding any satisfactory results. We therefore empirically resized the avalanche curves without making any a priori assumptions. Interestingly, the following scaling ansatz showed good results:
(11)  
(12) 
where and in Eqs. (11) and (12), respectively, are universal functions.
Iv Results
Figure 3 shows electron hop, as well as total charge displacement avalanche distributions for the CG for different ranges of the electric field . The field is increased in the different panels from top to bottom. Figures 3(a) – 3(c) show how the avalanche sizes progressively become system spanning, i.e., when [as is the case in Fig. 3(c)] avalanche size distributions become power laws. As the field reaches a hunch in the curves emerges separating a powerlaw region from an exponential cutoff, for the measured avalanches distribution . Figure 6 shows the dependence of the inverse of the characteristic avalanche size as a function of the electric field . We can extract from the figure that the depinning field lies somewhere around . A precise estimate of the depinning field can be obtained by analyzing the cumulative distribution function as shown in Fig. 5(b). For the CG model we obtain . In Fig. 7 we show an example of the estimation of using Eq. (10) for a given field window . Similar qualitative results are obtained for the charge displacement distribution , as shown in Fig. 3(d). We attempt to scale the data for the distributions and in Fig. 4. The data scale well with no adjustable parameters (especially for the larger system sizes) according to Eqs. (11) and (12).
In addition, we study the total charge displacement distribution and electron hop distribution as a function of the applied field for the SR model, where the estimated depinning field is as seen in Fig. 8. Electron avalanche distributions are shown in Fig. 9. For low fields, i.e. , the characteristic avalanche size can be estimated analogously as for the CG model, i.e., fitting the tail to an exponential function and using Eq. (10) to extrapolate to the thermodynamic limit. As for the CG model at low fields, no systemspanning avalanches were found, moreover no emergent avalanche size dependence is observed [Fig. 9(a)(b)]. For fields closer to the depinning field, i.e. , the exponential fitting function [Eq. (10)] gives unsatisfactory fitting results, therefore we additionally fitted the distribution to a stretched exponential function
(13) 
The characteristic avalanche size defined through the stretched exponential function is bounded in the thermodynamic limit for all fields, especially close to the depinning field: the inset of Fig. 9(d) shows the values of for the field window . The stretched exponential exponent has a strong field dependence as seen in Fig. 10. At low fields and as the field increases it monotonically decreases to com ().
We observe that the CG model and the SR model have a well defined depinning field transition, but that they differ in the way they behave close to . The CG model total charge displacement and electron hop avalanche distributions close to the depinning field have a powerlaw shape (with powerlaw exponent ) with a systemsize dependent exponential cutoff. This finitesize effect vanishes in the thermodynamic limit, revealing its scalefree behavior at . In clear contrast the SR model total charge displacement and electron hop avalanche distributions show no signs of scalefree avalanche behavior (powerlaw shape) close to and are best described by a stretched exponential function, which is defined by the exponent and the parameter . The exponent shows a strong field dependence; it decreases monotonically as the field is increased, while does not show any systematic systemsize dependence at any field, not even close to the depinning field. The different avalanche distributions in the SR and CG models hint towards a different mechanism behind the depinning transition.
V Conclusions
Our largescale computational study of the Coulomb glass has demonstrated that, under external electric fields and nearestneighbor particleconserving hopping dynamics, scalefree avalanches only occur in the vicinity of a characteristic depinning field . For small external electric fields, no large avalanches are present, in agreement with the results of Palassini and Goethe Palassini and Goethe (2012). For a shortrange variation of the Coulomb glass model we do not find any sign of scalefree avalanches, not even close to the depinning electric field. Furthermore, we find that the initial Coulomb gap vanishes as the field is ramped up, suggesting that it is not a generic feature on the hysteresis loop formed in an external electric field. We empirically find a simple scaling ansatz to collapse the avalanche and charge displacement distributions, reinforcing the notion that the scalefree behavior of the CG emerges close to the depinning electric field.
The scalefree behavior found in the CG is not a selforganized critical (SOC) state, because an external parameter has to be tuned Bak et al. (1987); Drossel and Schwabl (1992); Schenk et al. (2002); Andresen et al. (2013), namely the electric field . Nevertheless, it is interesting to note the difference between the CG and the SR model: in the former the combination of the diverging number of neighbors and disorder results in powerlaw distributions, which is not the case in the latter. This behavior is very similar to that found for the threedimensional randomfield Ising model Sethna et al. (1993); Perkovic et al. (1995, 1999); Kuntz et al. (1998); Sethna et al. (2004), where scalefree avalanches have been observed at a critical field strength . These unexpected results for the Coulomb glass show that a diverging number of neighbors is necessary but not sufficient in a model Hamiltonian to show SOC behavior, and that the dynamics of a model might play an important role for showing SOC (i.e., the orderparameter conserving Kawasaki dynamics used here vs singlespin flip dynamics used for the randomfield Ising model).
Our results bring into question the validity of the meanfield picture of the Coulomb glass Pastor and Dobrosavljević (1999); Pastor et al. (2002); Dobrosavljević et al. (2003); Müller and Ioffe (2004); Pankov and Dobrosavljević (2005); Müller and Pankov (2007), predicting extreme fragility of the ground state to external perturbations. However, the generic absence of SOC for avalanches driven by a uniform electric field may be related to the fact that such large avalanches locally violate charge neutrality. Other dynamical perturbations may couple differently to the elementary excitations and may perhaps serve as a more sensitive probe to the proposed SOC nature of the CG ground state. This could be achieved by applying external fields that do not directly couple to the uniform charge density, such as varying the amplitude of the disorder potential. Such or similar studies represent an opportunity to further elucidate the longstanding mystery of the Coulomb glass, however, exploring this exciting research direction remains a challenge for future work.
Acknowledgements.
We would like to thank R. S. Andrist for many discussions, as well as Mauricio Andresen for providing the necessary motivation to complete this project. V.D. and Y.P. were supported by the NSF (Grant No. DMR1005751). G.T.Z. was supported by the NSF (Grant No. DMR1035468). H.G.K. acknowledges support from the NSF (Grant No. DMR1151387) and would like to thank ETH Zurich for CPU time on the Brutus cluster, as well as Aspall (Suffolk) for inspiration. Part of H.G.K.’s research is based upon work supported in part by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), via MIT Lincoln Laboratory Air Force Contract No. FA872105C0002. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of ODNI, IARPA, or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purpose notwithstanding any copyright annotation thereon.References
 Dobrosavljević et al. (2012) V. Dobrosavljević, N. Trivedi, and J. M. Valles, ConductorInsulator Quantum Phase Transitions (Oxford University Press, Oxford, England, 2012).
 Pollak (1970) M. Pollak, Disc. Faraday Soc. 50, 13 (1970).
 Efros and Shklovskii (1975) A. L. Efros and B. I. Shklovskii, Coulomb gap and low temperature conductivity of disordered systems , J. Phys. C 8, L49 (1975).
 Shklovskii and Efros (1988) B. I. Shklovskii and A. L. Efros, Electronic Properties of Doped Semiconductors (Springer Series in SolidState Sciences, Vol. 45, New York, 1988).
 Lee and Ramakrishnan (1985) P. A. Lee and T. V. Ramakrishnan, Disordered electronic systems, Rev. Mod. Phys. 57, 287 (1985).
 Davies et al. (1982) J. H. Davies, P. A. Lee, and T. M. Rice, Electron glass, Phys. Rev. Lett. 49, 758 (1982).
 Davies et al. (1984) J. H. Davies, P. A. Lee, and T. M. Rice, Properties of the electron glass, Phys. Rev. B 29, 4260 (1984).
 Xue and Lee (1988) W. Xue and P. A. Lee, Monte Carlo simulations of the electron glass, Phys. Rev. B 38, 9093 (1988).
 Grannan and Yu (1993) E. R. Grannan and C. C. Yu, Critical behavior of the Coulomb glass, Phys. Rev. Lett. 71, 3335 (1993).
 Pastor and Dobrosavljević (1999) A. A. Pastor and V. Dobrosavljević , Melting of the Electron Glass, Phys. Rev. Lett. 83, 4642 (1999).
 Pastor et al. (2002) A. A. Pastor, V. Dobrosavljević, and M. L. Horbach, Meanfield glassy phase of the randomfield Ising model, Phys. Rev. B 66, 014413 (2002).
 Dobrosavljević et al. (2003) V. Dobrosavljević, D. Tanasković, and A. A. Pastor, Glassy Behavior of Electrons Near MetalInsulator Transitions, Phys. Rev. Lett. 90, 016402 (2003).
 Müller and Ioffe (2004) M. Müller and L. B. Ioffe, Glass Transition and the Coulomb Gap in Electron Glasses, Phys. Rev. Lett. 93, 256403 (2004).
 Grempel (2004) D. Grempel, Offequilibrium dynamics of the twodimensional Coulomb Glass, Europhys. Lett. 66, 854 (2004).
 Pankov and Dobrosavljević (2005) S. Pankov and V. Dobrosavljević, Nonlinear Screening Theory of the Coulomb Glass, Phys. Rev. Lett. 94, 046402 (2005).
 Kolton et al. (2005) A. B. Kolton, D. R. Grempel, and D. Domínguez, Heterogeneous dynamics of the threedimensional Coulomb glass out of equilibrium, Phys. Rev. B 71, 024206 (2005).
 Müller and Pankov (2007) M. Müller and S. Pankov, Meanfield theory for the threedimensional Coulomb glass, Phys. Rev. B 75, 144201 (2007).
 Surer et al. (2009) B. Surer, H. G. Katzgraber, G. T. Zimanyi, B. A. Allgood, and G. Blatter, Density of States and Critical Behavior of the Coulomb Glass, Phys. Rev. Lett. 102, 067205 (2009).
 BenChorin et al. (1993) M. BenChorin, Z. Ovadyahu, and M. Pollak, Nonequilibrium transport and slow relaxation in hopping conductivity, Phys. Rev. B 48, 15025 (1993).
 Ovadyahu and Pollak (1997) Z. Ovadyahu and M. Pollak, Disorder and Magnetic Field Dependence of Slow Electronic Relaxation, Phys. Rev. Lett. 79, 459 (1997).
 Vaknin et al. (2000) A. Vaknin et al., Aging Effects in an Anderson Insulator, Phys. Rev. Lett. 84, 3402 (2000).
 Bogdanovich and Popovic (2002) S. Bogdanovich and D. Popovic, Onset of Glassy Dynamics in a TwoDimensional Electron System in Silicon, Phys. Rev. Lett. 88, 236401 (2002).
 Vaknin et al. (2002) A. Vaknin, Z. Ovadyahu, and M. Pollak, Nonequilibrium field effect and memory in the electron glass, Phys. Rev. B 65, 134208 (2002).
 Orlyanchik and Ovadyahu (2004) V. Orlyanchik and Z. Ovadyahu, Stress Aging in the Electron Glass, Phys. Rev. Lett. 92, 066801 (2004).
 Jaroszyński et al. (2004) J. Jaroszyński, D. Popović, and T. M. Klapwijk, MagneticField Dependence of the Anomalous Noise Behavior in a TwoDimensional Electron System in Silicon, Phys. Rev. Lett. 92, 226403 (2004).
 Ovadyahu (2006) Z. Ovadyahu, Quenchcooling procedure compared with the gate protocol for aging experiments in electron glasses, Phys. Rev. B 73, 214204 (2006).
 Jaroszyński and Popović (2006) J. Jaroszyński and D. Popović, Nonexponential Relaxations in a TwoDimensional Electron System in Silicon, Phys. Rev. Lett. 96, 037403 (2006).
 Jaroszyński and Popović (2007) J. Jaroszyński and D. Popović, Nonequilibrium Relaxations and Aging Effects in a TwoDimensional Coulomb Glass, Phys. Rev. Lett. 99, 046405 (2007).
 Raičević et al. (2008) I. Raičević, J. Jaroszyński, D. Popović, C. Panagopoulos, and T. Sasagawa, Evidence for Charge Glasslike Behavior in Lightly Doped LaSrCuO at Low Temperatures, Phys. Rev. Lett. 101, 177004 (2008).
 Raičević et al. (2011) I. Raičević, D. Popović, C. Panagopoulos, and T. Sasagawa, NonGaussian noise in the inplane transport of lightly doped LaSrCuO: Evidence for a collective state of charge clusters, Phys. Rev. B 83, 195133 (2011).
 Lin et al. (2012) P. V. Lin, X. Shi, J. Jaroszynski, and D. Popović, Conductance noise in an outofequilibrium twodimensional electron system, Phys. Rev. B 86, 155135 (2012).
 Parisi (1983) G. Parisi, Order parameter for spinglasses, Phys. Rev. Lett. 50, 1946 (1983).
 Rammal et al. (1986) R. Rammal, G. Toulouse, and M. A. Virasoro, Ultrametricity for physicists, Rev. Mod. Phys. 58, 765 (1986).
 Mézard et al. (1987) M. Mézard, G. Parisi, and M. A. Virasoro, Spin Glass Theory and Beyond (World Scientific, Singapore, 1987).
 Young (1998) A. P. Young, ed., Spin Glasses and Random Fields (World Scientific, Singapore, 1998).
 Pázmándi et al. (1999) F. Pázmándi, G. Zaránd, and G. T. Zimányi, Selforganized criticality in the hysteresis of the sherringtonkirkpatrick model, Phys. Rev. Lett. 83, 1034 (1999).
 Andresen et al. (2013) J. C. Andresen, Z. Zhu, R. S. Andrist, H. G. Katzgraber, V. Dobrosavljević, and G. T. Zimanyi, SelfOrganized Criticality in Glassy Spin Systems Requires a Diverging Number of Neighbors, Phys. Rev. Lett. 111, 097203 (2013).
 Young and Katzgraber (2004) A. P. Young and H. G. Katzgraber, Absence of an AlmeidaThouless line in ThreeDimensional Spin Glasses, Phys. Rev. Lett. 93, 207203 (2004).
 Katzgraber and Young (2005) H. G. Katzgraber and A. P. Young, Probing the AlmeidaThouless line away from the meanfield model, Phys. Rev. B 72, 184416 (2005).
 Boettcher et al. (2008) S. Boettcher, H. G. Katzgraber, and D. Sherrington, Local field distributions in spin glasses, J. Phys. A 41, 324007 (2008).
 Jörg et al. (2008) T. Jörg, H. G. Katzgraber, and F. Krzakala, Behavior of Ising Spin Glasses in a Magnetic Field, Phys. Rev. Lett. 100, 197202 (2008).
 Katzgraber et al. (2009) H. G. Katzgraber, D. Larson, and A. P. Young, Study of the de AlmeidaThouless line using powerlaw diluted onedimensional Ising spin glasses, Phys. Rev. Lett. 102, 177205 (2009).
 Larson et al. (2013) D. Larson, H. G. Katzgraber, M. A. Moore, and A. P. Young, Spin glasses in a field: Three and four dimensions as seen from one space dimension, Phys. Rev. B 87, 024414 (2013).
 Palassini and Goethe (2012) M. Palassini and M. Goethe, Elementary excitations and avalanches in the Coulomb glass, J. Phys.: Conf. Ser. 376, 012009 (2012).
 Newman and Barkema (1999) M. E. J. Newman and G. T. Barkema, Monte Carlo Methods in Statistical Physics (Oxford University Press Inc., New York, USA, 1999).
 Middleton (2004) A. A. Middleton, Improved extremal optimization for the ising spin glass, Phys. Rev. E 69, 055701(R) (2004).
 Sarvestani et al. (1995) M. Sarvestani, M. Schreiber, and T. Vojta, Coulomb Gap at Finite Temperatures, Phys. Rev. B 52, R3820 (1995).
 Wang and Holm (2001) Z. Wang and C. Holm, Estimate of the cutoff errors in the Ewald summation for dipolar systems, J. Chem. Phys. 115, 6351 (2001).
 Press et al. (1995) W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C (Cambridge University Press, Cambridge, England, 1995).
 Dong et al. (1993) M. Dong, M. C. Marchetti, A. A. Middleton, and V. Vinokur, Elastic string in a random potential, Phys. Rev. Lett. 70, 662 (1993).
 Newman and Ziff (2000) M. E. J. Newman and R. M. Ziff, Efficient Monte Carlo Algorithm and HighPrecision Results for Percolation, Phys. Rev. Lett. 85, 4104 (2000).
 Xi et al. (2015) B. Xi, M. B. Luo, V. M. Vinokur, and X. Hu (2015), (arxiv:condmat/1501.04436).
 (53) We also applied a stretched exponential fitting function to the CG model avalanche distributions. The stretched exponent for the CG model are in good agreement with an exponential function for fields . Close to the exponent is . Nevertheless, a pure exponential function still gives satisfactory fitting quality values.
 Bak et al. (1987) P. Bak, C. Tang, and K. Wiesenfeld, SelfOrganized Criticality: An Explanation of Noise, Phys. Rev. Lett. 59, 381 (1987).
 Drossel and Schwabl (1992) B. Drossel and F. Schwabl, SelfOrganized Critical ForestFire Model, Phys. Rev. Lett. 69, 1629 (1992).
 Schenk et al. (2002) K. Schenk, B. Drossel, and F. Schwabl, SelfOrganized Criticality in ForestFire Models, in Computational Statistical Physics, edited by K. H. Hoffmann and M. Schreiber (SpringerVerlag, Berlin, 2002), p. 127.
 Sethna et al. (1993) J. P. Sethna, K. Dahmen, S. Kartha, J. A. Krumhansl, B. W. Roberts, and J. D. Shore, Hysteresis and hierarchies: Dynamics of disorderdriven firstorder phase transformations, Phys. Rev. Lett. 70, 3347 (1993).
 Perkovic et al. (1995) O. Perkovic, K. A. Dahmen, and J. P. Sethna, Avalanches, Barkhausen Noise, and Plain Old Criticality, Phys. Rev. Lett. 75, 4528 (1995).
 Perkovic et al. (1999) O. Perkovic, K. A. Dahmen, and J. P. Sethna, Disorderinduced critical phenomena in hysteresis: Numerical scaling in three and higher dimensions, Phys. Rev. B 59, 6106 (1999).
 Kuntz et al. (1998) M. C. Kuntz, O. Perkovic, K. A. Dahmen, B. W. Roberts, and J. P. Sethna, Hysteresis, Avalanches, and Noise: Numerical Methods (1998), (arXiv:condmat/9809122v2).
 Sethna et al. (2004) J. P. Sethna, K. A. Dahmen, and O. Perkovic, RandomField Ising Models of Hysteresis (2004), (arXiv:condmat/0406320v3).