Elementary excitations and avalanches in the Coulomb ... - PhysioNet

Elementary excitations andavalanchesin the Coulomb glassMatteo PalassiniwithMartin GoetheDepartament de Física FonamentalUniversitat de Barcelona, SpainTIDS 14 - Transport in Interacting Disordered Systems. Acre (Israel), September 5 - 8, 2011

Classical electron glass model(Efros, 1976)H = e22κ- Localized charges- Long-range Coulomb interaction- Random site energies∑(n i − K) 1 (n j − K)+ ∑ r ijii≠jN∑n i ∈ {0, 1} , n i = KN , N = L di=1[ϕ i ] av =0n i ϕ i[ϕ 2 i ] av = We 2 /lThis talk:LGaussiane 2ϕ iκl =1l =1K = 1 2lPeriodic geometrywith Ewald sum- lightly doped crystalline semiconductors- amorphous semiconductors- granular/discontinuous metals

Classical electron glass model (Efros, 1976)H = e22κ- Localized charges- Long-range Coulomb interaction- Random site energies∑(n i − K) 1 (n j − K)+ ∑ r ijii≠jN∑n i ∈ {0, 1} , n i = KN , N = L di=1[ϕ i ] av =0n i ϕ i[ϕ 2 i ] av = We 2 /lThis talk:LGaussiane 2ϕ iκl =1l =1K = 1 2lOutline:Periodic geometrywith Ewald sum1. Scale-free charge avalanches2. Shape of the Coulomb gap in three dimensions

One-particle energy:ɛbare 1-DOSg 0 (ε)ɛ i = ∑ j≠in j − Kr ij+ ϕ i1-DOS:g(ɛ) = 1 VN∑[〈δ(ɛ − ɛ i )〉] avi=1µWijrg(ɛ)g 0 (µ) ∝ 1 WPollak, Srinivasan (1970): Coulomb interaction depletes the 1-DOS nearµ

One-particle energy:ɛɛ i = ∑ j≠in j − Kr ij+ ϕ i1-DOS:g(ɛ) = 1 VN∑[〈δ(ɛ − ɛ i )〉] avi=1µ∆Wijrg 0 (µ) ∝ 1 Wg(ɛ)r ijPollak, Srinivasan (1970): Coulomb interaction depletes the 1-DOS nearµEfros-Shklovskii (1975) stability argument: Coulomb gap∆E i→j = ɛ j − ɛ i − 1 > 0 =⇒rr ij > |ɛ j − ɛ i | −1ijg(ɛ)≤ c d |ɛ − µ| d−1for|ɛ − µ| ≪ ∆ ∼ W −1/(d−1)

Charged excitations: variable-range hoppingɛµ∆Wijrg(ɛ)g 0 (µ) ∝ 1 WMottIf(TMσ = σ 0 exp −Tg(ɛ) =c d |ɛ − µ| d−1Efros-Shklovskii) 1/4σ = σ 0 exp −(TEST) 1/2Massey and Lee, PRL 1995

Dipole excitations:ω ij = ɛ j − ɛ i − 1/r ijɛjµ∆Wirg(ɛ)r 0g 0 (µ) ∝ 1 WCompact: typical sizer 0 ∼ ∆ −1 , typical separation ∼ ω −1/3

Dipole excitations:ω ij = ɛ j − ɛ i − 1/r ijɛ0.2µ∆g 0 ∝ 1 WWg(ɛ)Compact: typical sizeklr, typical separationr 0 ∼ ∆ −1Baranovskii, Shklovskii, Efros (1980): ∆E i→j,k→l = ω ij + ω kl − 1 r 3 > 0α =1/2jr 0ig L (!) and f L (!, R max =1)0.150.10.052-DOS∼ ω −1/3r3./pi ! 200 0.1 0.2 0.3 0.4 0.5 0.6!1-DOSSingle-particle DOSPair DOS, R max =12-DOS:f(ω,R)= 1 V∑(ij),r ij ≤R[ 〈δ(ω − ω ij )) 〉] av ∝-> Contribute to ac (but not dc) conductivity-> dominate thermodynamics at low Tg 0ln(∆/ω) α

Charge avalanches1. Find a local minimum stable against all one-electron hops.2. - Add an extra electron at a soft site (charge insertion),or - Excite a soft dipole (dipole insertion)3. Relax an unstable electron-hole pair chosen at random.4. Repeat until another local minimum is reached.5. Measure the avalanche size S = number of hops performed--> probability distribution of avalanche sizes p(S)charge insertiondipole insertion

Avalanches in the infinite-range spin glass (Sherrington-Kirpatrick model):“self-organized criticality” (Pazmandi, Zarand, Zimanyi 1999)H = ∑ i

Infinite-range spin glass3D Coulomb glassequilibriumglass phaseyesequilibriumglass phase?saturatedlinear gapin 1-DOSTABLE I: Critical exponents for the fluid-COP transitionyesalong BC in Fig. 1, compared with the RFIM values [37].γ/ν ¯γ/ν β/ν νCoulomb glass 1.69(17) 2.89(15) 0.06(3) 1.11(12)RFIM 1.44(12) 2.93(11) 0.011(3) 1.37(9)We thank A.Möbius for helpful correspondence. Thisresearch is supported by the Generalitat de Catalunyaand the Ministerio de Ciencia e Innovación (contractsFIS-2006-13321-C02-01 and AP2007-01005). The computationswere performed on the Spanish SupercomputingNetwork (RES) node at Universidad de Cantabria.saturatedMP thanks the Aspen Center for Physics where part ofthis work was carried out.scale-freequadratic gapavalanches∗ Electronic address: palassini@ub.edu[1] M. Pollak, Discuss. Faraday Soc. 50, 13 (1970); G. Srinivasan,Phys. Rev. B 4, 2581 (1971).in 1-DOSFIG. 4: (Color online) Charge-order correlation length along [2] A. L. Efros and B. I. Shklovskii, J. Phys. C 8, 49 (1975).path BC in Fig. 1. Top inset: order parameter Ms along paths [3] E. I. Levin, V. L. Nguyen, B. I. Shklovskii, and A.AB, BC, and DE. Bottom inset: specific heat along path BC. L. Efros, Sov. Phys. JETP 65, 842 (1987).[4] A. Hunt, Philos. Mag. Lett. 62, 371 (1990).[5] T. Vojta, W. John, and M. Schreiber, J. Phys. Condens.GP does not exist above the dashed line in Fig. 1.Matter 5, 4989 (1993).Critical behavior – Since at W = 0 the fluid-COP transitionhas a positive specific-heat exponent [23], disorder [7] J. G. Massey and M. Lee, Phys. Rev. Lett. 75, 4266[6] See e.g. Ref.[7] and references therein.is relevant [34] and the W ≠ 0 transition will be governedby a random fixed point which, by analogy with the [8] B. Sandow, K. Gloos, R. Rentzsch, A. N. Ionov, and W.(1995).RFIM [12], we expect to be at zero temperature [35]. Assumingthe transition is second order, as we ascertainedSchirmacher, Phys. Rev. Lett. 86, 1845 (2001).[9] Q. Li and P. Phillips, Phys. Rev. B 49, 10269 (1994).[10] M.H. Overlin, L.A. Wong, and C. C. Yu, Phys. Rev. Bby inspecting the distribution of ms for individual samples,we obtain the critical exponents in Table I. We es-[11] M. Sarvestani, M. Schreiber, and T. Vojta, Phys. Rev. B70, 214203 (2004).timated β/ν and ¯γ/ν with the quotient method [36] for 52, R3820 (1995).the observables Ms and ¯χ L = N[〈ms 2 〉]av respectively [12] A. Möbius, M. Richter, and B. Drittler, Phys. Rev. B 45,[the quotient estimates from (L, L ′ ) = (6, 8), (6, 10) and 11568 (1992).(8, 10) agree within the errors], while γ/ν was obtained [13] J. H. Davies, P. A. Lee, and T. M. Rice, Phys. Rev. Bby fitting aL γ/ν29, 4260 (1984).to the height of the peak of the susceptibilityN[〈ms 2 〉−〈|ms|〉 2 [14] See e.g. Ref.[21] and references therein.]av (data not shown). Thespecific heat cL =1/(NT 2 )[〈H 2 〉 − 〈H〉 2 [15] A. Díaz-Sánchez, M. Ortuño, A. Pérez-Garrido, and]av shows a peak E. Cuevas, Phys. Stat. Sol. (b) 218, 11 (2000).that increases slowly with L (Fig. 4, bottom inset), which [16] E. R. Grannan and C. C. Yu, Phys. Rev. Lett. 71, 3335suggests either α < 0 or a logarithmic divergence (α = 0). (1993).We could not estimate ν directly in a reliable way, but [17] B. Surer, H. G. Katzgraber, G. T. Zimanyi,we obtain ν =1.11(12) from the modified hyperscalingrelation [35] (d − θ)ν =2− α, assuming α = 0 and usingθ =¯γ/ν −γ/ν =1.20(20). As shown in Table I, the criticalexponents agree fairly well with the known values formean-fieldB. A. Allgood,and G. Blatter, cond-mat/08054640 (2008).theory[18] A. A. Pastor and V. Dobrosavljević, Phys. Rev. Lett. 83,4642 (1999).[19] S. Pankov and V. Dobrosavljević, Phys. Rev. Lett. 94,046402 (2005).the RFIM [37], which suggests that the system remains [20] M. Müller and L. B. Ioffe, Phys. Rev. Lett. 93, 256403effectively short-range near the transition.(2004).To conclude, our results show that, although meanfieldtheory seems to capture correctly the DOS near the[21] M. Müller and S. Pankov, Phys. Rev. B 75, 144201(2007).[22] R. Dickman and G. Stell, cond-mat/9906364 (1999).Coulomb gap, the interaction remains well screened and[23] A. Möbius and U. K. Rößler, cond-mat/0309001 (1999).the correlations remain short-range down to rather low [24] V. Malik and D. Kumar, Phys. Rev. B 76, 125207 (2007).temperatures. If an equilibrium glass phase exists, its [25] A. Möbius, talk given at TIDS11 (2005).onset must be at exceptionally low temperatures.[26] M. Goethe and M. Palassini, in preparation.4beyond mean fieldscale-free?avalanches?Pankov, Dobrosavljevic 2005Müller, Ioffe 2005Müller, Pankov 2007

“Equilibrium glass phase”Infinite-range spin glass3D Coulomb glassTT goverlapP (q)q = 1 N∑Si a Sibparamagnetqi3D mean field theory :(locator approximation)Pankov, Dobrosavljevic 2005Müller, Ioffe 2005Müller, Pankov 2007T < T g ∼ 1/ √ WCHARGE ORDERP (q)Almeida-Thouless lineglassqGoethe, Palassini 2009∼ e αN- Many (# ) equilibrium states- “Marginal criticality”:power law correlations belowT g(〈S i S j 〉−〈S i 〉〈S j 〉) 2 ∼ 1/r θ ijhT0.150.10.050ACFluidBEChargeOrderD0 0.1 0.2 0.3 0.4 0.5WEquilibrium MC:no glass phasedown to verylow T(see alsoSurer et al. 2009)

Avalanches and nonlinear screeningBaranovskii, Shklovskii, Efros 1984screening of a point charge: large spatial fluctuations of the potentialCapacitance experimentsD. Monroe et al., PRL 1987(Markus Müller)Screening in conductance experimentsDelahaye, J. Honoré, T. Grenet 2011rev] [Markus Mueller (ICTP, Trieste) 06] [NEXT> [last>

obust and does not require fine tuning. In fact, we observe a divergent cutoff for other valuesof of K near 1/2, when performing hopsinin orderof of increasing hopping length insteadof of uniformlyat at random, and even for huge disorder (W = 32).Avalanche distribution for dipole insertionp(S) p(S)10 10 -2 -210 10 -3 -310 10 -4 -410 10 -5 -5W =2L L = = 8816 1630 3060 60! S -3/2A(a,L) L 3/2 3/2 p(S)10 10 0010 10 -1 -110 10 -2 -210 10 -3 -3L L = 8816 1630 3060 60(2 (2") ") -1/2 xx -3/2 exp(-x/a)a = 0.4810 10 -1 -1 10 10 00 10 10 11 10 10 2210 10 11 10 10 -1 -1 10 10 00SS // LFigure cutoff 3. 3. diverging Avalanche linearly size distribution with L scale free Figure p(S) 4. ∼ Scaling S −τ extrapolating plotof of the avalanche to L = ∞for dipole-triggered avalanches.Thesize distribution assuming Eq.(5) withdata are averaged over 20000, 48000,τ = 3/2 and S c c = aL. A(a, L) is isp(S) ∼ S −τ exp(−S/S 29000, 2135 samples for L =8, 16, c ) τ ≃ 1.530, 60a normalization Consistent factor with slowly mean-fieldvaryingrespectively. S with L, exponent computed analytically τ =3/2from thec = aL a ≃ 0.5branching process model.normalization:also Although well fitted these by results p(S) may ∼ Ssuggest −τ exp[−(S/S an analogy c ) 2 ]with the self-organizedA(a, L) critical=λ(L)/(1 avalanches− e −λ(L) ) of ofthe SK model, we argue that they τ ≃ 1.7 can be understood more simplyλ(L) in in terms − 1 −of of ln λ(L) the elementary = (aL) −1

Avalanche size distribution for dipole insertionwith restricted hopping length≤ R maxP (Y ) ∼ Y −τ exp(−Y/S c )Y= avalanche sizeR max independent of L R max proportional toL10 1 5 10 15 20p(Y | R max = 2,4)10 010 -110 -210 -3R max = 2R max = 4(shifted by factor 10)L=163060p(Y | R max = L/10)10 -110 -210 -310 -4L=16306010 0 5 10 15 20 25 3010 -4Y10 -5Y1S c ∼ LS c ∼ const- Long O(L) hops are necessary to sustain scale-free avalanches- In a glass phase we expect scale-free avalanches also for finiteR max

Average number of dipoles destabilizedcharge insertion2-DOSf(ω) =g 0ln(∆/ω) α ∼ g 0dipole insertionrdrrdrN u =∫ Lr 0N u (r) ∝ g 04πr 2 dr∫ 1/r20f(ω)dω ∝ g 0 L N u =∫ Lr 0N u (r) ∝ g 0 /r4πr 2 dr∫ 1/r30f(ω)dω ∝ g 0 log L

# of unstable long hops created (3D):M u =∫ Lr 04πr 2 dr∫ 1/r20g(ɛ)dɛ ∝ O(1)O(L)rdrGalton-Watson branching processS = tree size = λλ =1 p(S) ∼ S −3/2λ < 1 p(S) ∼ S −3/2 e −S/S c

Long hops explain why cutoff S c ∝ LS c ∼ number of successive events created bya hop that reaches the boundaryHence in 2D we expect S c ∼ log LAvalanche size distribution in 2D:p(S) in 2D10 0 0.60.4p(0) ∼ exp(−g0.20 log L)~ 1 / [ C log(L) ]030 60 100240LL= 30601202400 5 10 15 20 2510 -110 -210 -310 -410 -5Scharge insertionsqrt(2 π) λ (1-exp(-λ)) Y c3/2 p(Y)10 1 ( Y/Y c ) -3/2 exp(- Y/Y c )10 0Dipole-triggered avalanches in 2DL= 3060120240110 -110 -210 -310 -410 -5Y / Y cdipole insertionwith Y c = log(L/3)

Infinite-range spin glass3D Coulomb glassequilibriumglass phaseyesequilibriumglass phasenosaturatedlinear gapin 1-DOSTABLE I: Critical exponents for the fluid-COP transitionalong BC in Fig. 1, compared with the RFIM values [37].γ/ν ¯γ/ν β/ν νCoulomb glass 1.69(17) 2.89(15) 0.06(3) 1.11(12)RFIM 1.44(12) 2.93(11) 0.011(3) 1.37(9)We thank A.Möbius for helpful correspondence. Thisresearch is supported by the Generalitat de Catalunyayesand the Ministerio de Ciencia e Innovación (contractsFIS-2006-13321-C02-01 and AP2007-01005). The computationswere performed on the Spanish SupercomputingNetwork (RES) node at Universidad de Cantabria.MP thanks the Aspen Center for Physics where part ofthis work was carried out.saturatedscale-freequadratic gap?∗ Electronic address: palassini@ub.edu[1] M. Pollak, Discuss. Faraday Soc. 50, 13 (1970); G. Srinivasan,Phys. Rev. B 4, 2581 (1971).FIG. 4: (Color online) Charge-order correlation length alongpath BC in Fig. 1. Top inset: order parameter Ms along pathsAB, BC, and DE. Bottom inset: specific heat along path BC.GPavalanches[2] A. L. Efros and B. I. Shklovskii, J. Phys. C 8, 49 (1975).[3] E. I. Levin, V. L. Nguyen, B. I. Shklovskii, and A.L. Efros, Sov. Phys. JETP 65, 842 (1987).does not exist above the dashed line in Fig. 1.Critical behavior – Since at W = 0 the fluid-COP transitionhas a positive specific-heat exponent [23], disorderin 1-DOS[4] A. Hunt, Philos. Mag. Lett. 62, 371 (1990).[5] T. Vojta, W. John, and M. Schreiber, J. Phys. Condens.Matter 5, 4989 (1993).[6] See e.g. Ref.[7] and references therein.[7] J. G. Massey and M. Lee, Phys. Rev. Lett. 75, 4266is relevant [34] and the W ≠ 0 transition will be governedby a random fixed point which, by analogy with the [8] B. Sandow, K. Gloos, R. Rentzsch, A. N. Ionov, and W.(1995).RFIM [12], we expect to be at zero temperature [35]. Assumingthe transition is second order, as we ascertainedSchirmacher, Phys. Rev. Lett. 86, 1845 (2001).[9] Q. Li and P. Phillips, Phys. Rev. B 49, 10269 (1994).[10] M.H. Overlin, L.A. Wong, and C. C. Yu, Phys. Rev. Bby inspecting the distribution of ms for individual samples,we obtain the critical exponents in Table I. We es-[11] M. Sarvestani, M. Schreiber, and T. Vojta, Phys. Rev. B70, 214203 (2004).timated β/ν and ¯γ/ν with the quotient method [36] for 52, R3820 (1995).the observables Ms and ¯χ L = N[〈ms 2 〉]av respectively [12] A. Möbius, M. Richter, and B. Drittler, Phys. Rev. B 45,[the quotient estimates from (L, L ′ ) = (6, 8), (6, 10) and 11568 (1992).(8, 10) agree within the errors], while γ/ν was obtained [13] J. H. Davies, P. A. Lee, and T. M. Rice, Phys. Rev. Bby fitting aL γ/ν29, 4260 (1984).to the height of the peak of the susceptibilityN[〈ms 2 〉−〈|ms|〉 2 [14] See e.g. Ref.[21] and references therein.]av (data not shown). Thespecific heat cL =1/(NT 2 )[〈H 2 〉 − 〈H〉 2 [15] A. Díaz-Sánchez, M. Ortuño, A. Pérez-Garrido, and]av shows a peak E. Cuevas, Phys. Stat. Sol. (b) 218, 11 (2000).that increases slowly with L (Fig. 4, bottom inset), which [16] E. R. Grannan and C. C. Yu, Phys. Rev. Lett. 71, 3335suggests either α < 0 or a logarithmic divergence (α = 0). (1993).We could not estimate ν directly in a reliable way, but [17] B. Surer, H. G. Katzgraber, G. T. Zimanyi, B. A. Allgood,and G. Blatter, cond-mat/08054640 (2008).we obtain ν =1.11(12) from the modified hyperscaling[18] A. A. Pastor and V. Dobrosavljević, Phys. Rev. Lett. 83,relation [35] (d − θ)ν =2− α, assuming α = 0 and using4642 (1999).θ =¯γ/ν −γ/ν =1.20(20). As shown in Table I, the criticalexponents agree fairly well with the known values forthe RFIM [37], which suggests that the system remainseffectively short-range near the transition.To conclude, our results show that, although meanfieldtheory seems to capture correctly the DOS near themean-field theory[19] S. Pankov and V. Dobrosavljević, Phys. Rev. Lett. 94,046402 (2005).[20] M. Müller and L. B. Ioffe, Phys. Rev. Lett. 93, 256403(2004).[21] M. Müller and S. Pankov, Phys. Rev. B 75, 144201(2007).[22] R. Dickman and G. Stell, cond-mat/9906364 (1999).Coulomb gap, the interaction remains well screened and[23] A. Möbius and U. K. Rößler, cond-mat/0309001 (1999).the correlations remain short-range down to rather low [24] V. Malik and D. Kumar, Phys. Rev. B 76, 125207 (2007).temperatures. If an equilibrium glass phase exists, its [25] A. Möbius, talk given at TIDS11 (2005).onset must be at exceptionally low temperatures.[26] M. Goethe and M. Palassini, in preparation.4beyond mean fieldyes!(also in 2D)scale-freeavalanches

Numerical determination of the 1-DOSThermalfluctuations|ɛ − µ| TɛFinite temperature equilibrium simulation:Equilibration time ∼ e Lψ /TCurrent limit: in 3DL ≤ 12Finite sizeeffects|ɛ − µ| L −1µ∆ ∼ W − 1 2Zero temperature energy relaxation:- much larger size: L ≤ 300 in 3D- systematic error (local minima are nottrue ground states)g(ɛ)g 0 (µ) ∝ 1 W

Equilibrium finite temperature 1-DOSM. Goethe, M. Palassini (PRL 2009)W= 2L= 10ScalingL ≤ 10g(ɛ,T)=|ɛ| δ˜g(ɛ/T )δ =2.0 ± 0.1 forɛ ≥ 0.2 ∆g L (!,T)10 -210 -3!10 -1 -5 -4 -3 -2 -1 0 1 2 3 4 5T= 0.1260.0350.010g L (!,T) / ! 210 110 2 10 0 10 1 10 2T= 0.00260.00490.00770.01050.01380.0170.0250.0350.0473/"Raikh10 0W = 4L = 10! / TEarlier results:Li, Phillips (1994): δ =2.38 (3D)Möbius, Richter, Drittler (1992):Sarvestani, Schreiber, Vojta (1995):Overlin, Wong, Yu (2004):δ =2.6 ± 0.2 (3D); 1.2 ± 0.1 (2D)2.1 ≤ δ ≤ 2.6 (3D)δ =2.7 (3D); 1.75 (2D)No exponentialgap!

Self-consistent equation (SCE):(Efros 1976)g(ε) =g 0 (ε) exp[− 2π 3∫ ∞0g(ε ′ )dε ′ ](ε ′ + |ε|) 33DAsymptotic solution forε → 0: g(ε) = 3 π ε2Improved asymptotics: oscillations around g(ε) = 3 π ε2(see Martin Goethe’s poster)Modified SCE:Efros, Skinner, Shklovskii,arXiv:1105.1712v1[g(ε) =g 0 (ε)P (ε) exp − 2π 3[ ( ) ( ∆ ∆P (ε) exp −γ ln −7/4ɛ ɛ∫ ∞g(ε ′ )dε ′ ](ε ′ + |ε|) 3) ( )] ∆fɛ0interpolating functionf(x) = (1 − x) η θ(1 − x)Asymptotic solutiong(ε) ∼ g 0 (ε)P (ε)ε ≪ ∆Approximatelyg(ε) = 3 π ε2forε ∼ ∆

Zero temperature 1-DOSL = 100L = 10010 -2W g L (!)10 -310 -410 -510 -1 10 -2 10 -1 10 0W = 2W = 4(3/") ! 2(2 ") -1/2SCE (W=2, #=3)SCE (W=2, #=4)Asympt. (W=2, #=3)Asympt. (W=2, #=4)g L (!) / ! 210 -110 0 10 -2 10 -1 10 0W = 2W = 43/"Slope 0.4SCE (W=2, #=4)SCE (W=4, #=4)! W 1/2! W 1/2Numerical solution of the modified SCEof Efros, Skinner, Shklovskii- Clear deviation from ES quadratic behavior- Consistent with g(ε) =a|ε| δ W 1−δ/2 δ ≃ 2.4 a ≃ 2.0- Qualitatively but not quantitatively consistent with modified SCE- Larger L needed to resolve exponential vs power-law behaviorrelated resultsby A. Möbius

Zero temperature 1-DOSFinite-size effectsSystematic error: repeated runsfor the same sampleW = 2α = 0.499W = 2L = 60" = 0.49910 0 10 -2 10 -1g L (ε) / ε 210 -1L= 163060100PTMC L= 10Shifting range / 23/πg L (!) / ! 2" = 0.510 0 Min of 20 quenchesMax of 20 quenchesOne quench many samples3/#10 -1ε!Shift to reduce chemical potential fluctuations:g(ε) → g(ε − (ε a + ε b )/2)∫ εa−∞dε g(ε) =∫ ∞ε bdε g(ε) =α

conclusionsyessaturatedquadratic gapin 1-DOSyesequilibriumglass phaseno!noyes!2D & 3Dscale-freeavalanchesTABLE I: Critical exponents for the fluid-COP transitionalong BC in Fig. 1, compared with the RFIM values [37].γ/ν ¯γ/ν β/ν νCoulomb glass 1.69(17) 2.89(15) 0.06(3) 1.11(12)RFIM 1.44(12) 2.93(11) 0.011(3) 1.37(9)We thank A.Möbius for helpful correspondence. Thisresearch is supported by the Generalitat de Catalunyaand the Ministerio de Ciencia e Innovación (contractsmean-field theory beyond mean FIS-2006-13321-C02-01 fieldand AP2007-01005). The computationswere performed on the Spanish SupercomputingNetwork (RES) node at Universidad de Cantabria.MP thanks the Aspen Center for Physics where part ofthis work was carried out.Long jumps explain scale free avalanches∗ Electronic address: palassini@ub.edu[1] M. Pollak, Discuss. Faraday Soc. 50, 13 (1970); G. Srinivasan,Phys. Rev. B 4, 2581 (1971).FIG. 4: (Color online) Charge-order correlation length along [2] A. L. Efros and B. I. Shklovskii, J. Phys. C 8, 49 (1975).path BC in Fig. 1. Top inset: order parameter Ms along paths [3] E. I. Levin, V. L. Nguyen, B. I. Shklovskii, and A.AB, BC, and DE. Bottom inset: specific heat along path BC. L. Efros, Sov. Phys. JETP 65, 842 (1987).[4] A. Hunt, Philos. Mag. Lett. 62, 371 (1990).[5] T. Vojta, W. John, and M. Schreiber, J. Phys. Condens.GP does not exist above the dashed line in Fig. 1.Matter 5, 4989 (1993).Critical behavior – Since at W = 0 the fluid-COP transitionhas a positive specific-heat exponent [23], disorder [7] J. G. Massey and M. Lee, Phys. Rev. Lett. 75, 4266[6] See e.g. Ref.[7] and references therein.is relevant [34] and the W ≠ 0 transition will be governedby a random fixed point which, by analogy with theRFIM [12], we expect to be at zero temperature [35]. Assumingthe transition is second order, as we ascertainedBranching process model by inspecting reproduces(1995).the distribution of ms for individual samples,we obtain the critical exponents in Table I. We es-[11] M. Sarvestani, M. Schreiber, and T. Vojta, Phys. Rev. Bwell[8] B. Sandow, K. Gloos, R. Rentzsch, A. N. Ionov, and W.Schirmacher, Phys. Rev. Lett. 86, 1845 (2001).[9] Q. Li and P. Phillips, Phys. Rev. B 49, 10269 (1994).[10] M.H. Overlin, L.A. Wong, and C. C. Yu, Phys. Rev. B70, 214203 (2004).timated β/ν and ¯γ/ν with the quotient method [36] forthe observables Ms and ¯χ L = N[〈ms 2 〉]av respectively[the quotient estimates from (L, L ′ ) = (6, 8), (6, 10) and(8, 10) agree within the errors], while γ/ν was obtainedby fitting aL γ/ν to the height of the peak of the susceptibilityN[〈ms 2 〉−〈|ms|〉 2 [14] See e.g. Ref.[21] and references therein.the avalanche size distributions52, R3820 (1995).[12] A. Möbius, M. Richter, and B. Drittler, Phys. Rev. B 45,11568 (1992).[13] J. H. Davies, P. A. Lee, and T. M. Rice, Phys. Rev. B29, 4260 (1984).]av (data not shown). Thespecific heat cL =1/(NT 2 )[〈H 2 〉 − 〈H〉 2 [15] A. Díaz-Sánchez, M. Ortuño, A. Pérez-Garrido, and]av shows a peak E. Cuevas, Phys. Stat. Sol. (b) 218, 11 (2000).that increases slowly with L (Fig. 4, bottom inset), which [16] E. R. Grannan and C. C. Yu, Phys. Rev. Lett. 71, 3335suggests either α < 0 or a logarithmic divergence (α = 0). (1993).We could not estimate ν directly in a reliable way, but [17] B. Surer, H. G. Katzgraber, G. T. Zimanyi, B. A. Allgood,and G. Blatter, cond-mat/08054640 (2008).we obtain ν =1.11(12) from the modified hyperscaling[18] A. A. Pastor and V. Dobrosavljević, Phys. Rev. Lett. 83,relation [35] (d − θ)ν =2− α, assuming α = 0 and using4642 (1999).θ =¯γ/ν −γ/ν =1.20(20). As shown in Table I, the criticalexponents agree fairly well with the known values forthe RFIM [37], which suggests that the system remainseffectively short-range near the transition.To conclude, our results show that, although meanfieldtheory seems to capture correctly theshape[19] S. Pankov and V. Dobrosavljević, Phys. Rev. Lett. 94,Avalanches have an elongated046402 (2005).[20] M. Müller and L. B. Ioffe, Phys. Rev. Lett. 93, 256403(2004).[21] M. Müller and S. Pankov, Phys. Rev. B 75, 144201DOS near (2007).[22] R. Dickman and G. Stell, cond-mat/9906364 (1999).Coulomb gap, the interaction remains well screened and[23] A. Möbius and U. K. Rößler, cond-mat/0309001 (1999).the correlations remain short-range down to rather low [24] V. Malik and D. Kumar, Phys. Rev. B 76, 125207 (2007).temperatures. If an equilibrium glass phase exists, its [25] A. Möbius, talk given at TIDS11 (2005).onset must be at exceptionally low temperatures.[26] M. Goethe and M. Palassini, in preparation.4open questionsPhysical dynamics: length-dependenthopping rates -> time introduces acutoff in the avalanche size.Avalanches at finite temperatureRelevance for screeningCan avalanches be measured inexperiments?SCE works only qualitatively