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Non-ergodic metallic and insulating phases of 5 Josephson junction chains 1 0 2 M. Pinoa, B.L. Altshulerb and L.B. Ioffea,c n a a DepartmentofPhysicsandAstronomy,RutgersTheStateUniversityofNewJersey, J 5 136Frelinghuysenrd,Piscataway,08854NewJersey,USA 1 b PhysicsDepartment,ColumbiaUniversity,NewYork10027,USA ] c n LPTHE,CNRSUMR7589,Boite126,4Jussieu,75252ParisCedex05,France n - s i Abstract the entropy never reaches its thermodynamic value. d Variation of macroscopic parameters, e.g. tempera- . Strictly speaking the laws of the conventional t ture, can delocalize the many body state. However, a Statistical Physics, based on the Equipartition m Postulate[1] and Ergodicity Hypothesis[2], apply the delocalization does not imply the recovery of the equipartition. Such a non-ergodic behavior in iso- - onlyinthepresenceofaheatbath. Untilrecently d this restriction was not important for real phys- lated physical systems is the subject of this Letter. n ical systems: a weak coupling with the bath was We argue that regular Josephson junction arrays o believed to be sufficient. However, the progress (JJA)undertheconditionsthatarefeasibletoimple- c [ in both quantum gases and solid state coherent ment and control experimentally demonstrate both quantum devices demonstrates that the coupling MBL and non-ergodic behavior. A great advantage 1 to the bath can be reduced dramatically. To de- oftheJosephsoncircuitsisthepossibilitytodisentan- v scribe such systems properly one should revisit gle them from the environment as was demonstrated 3 the very foundations of the Statistical Mechanics. 5 by the quantum information devices.[5] At low tem- We examine this general problem for the case of 8 peratures the conductivity σ of JJA is finite (below theJosephsonjunctionchainandshowthatitdis- 3 we call such behavior metallic), while as T →0 JJA plays a novel high temperature non-ergodic phase 0 withfiniteresistance. Withfurtherincreaseofthe becomeseitherasuperconductor(σ →∞)oraninsu- . 1 temperaturethesystem undergoesa transitionto lator(σ →0)[6,7]. Wepredictthatatacriticaltem- 0 the fully localized state characterized by infinite perature T = T JJA undergoes a true phase transi- c 5 resistance and exponentially long relaxation. tion into a MBL insulator (σ = 0 for T > T ). Re- 1 c markably already in the metallic state JJA becomes : v Theremarkablefeatureoftheclosedquantumsys- nonergodic and can not be properly described by the i X tems is the appearance of Many-Body Localization conventional Statistical Mechanics. r (MBL)[3]: under certain conditions the states of a JJA is characterized by the set of phases {φi} and a many-body system are localized in the Hilbert space charges{q }ofthesuperconductingislands, φ andq i i i resembling the celebrated Anderson Localization [4] for each i are canonically conjugated. The Hamilto- of single particle states in a random potential. MBL nian H is the combination of the charging energies impliesthatmacroscopicstatesofanisolatedsystem of the islands with the Josephson coupling energies. depend on the initial conditions i.e. the time averag- Assuming that the ground capacitance of the islands ing does not result in equipartition distribution and dominates their mutual capacitances (this assump- 1 2 tion is not crucial for the qualitative conclusions) we can write H as T/E J (cid:20) (cid:21) (cid:88) 1 H = E q2+E (1−cosφ ) (1) 2 C i J i i Insulator The ground state of the model (1) is determined by the ratio of the Josephson and charging energies, mit Bad metal EJ/EC thatcontrolsthestrengthofquantumfluctu- li ations: JJA is an insulator at E /E < η and a su- al J C c perconductor at EJ/EC >η [6, 7] with η ≈0.63 (see 1.0 Good metal ssi a Supplemental materials). The quantum transition E/E Cl at E /E = η belongs to the Berezinsky-Kosterlitz- J C J C 1.0 2.0 Thoulessuniversalityclass[8]. Awayfromtheground stateinadditiontoE /E thereappearsdimension- J C Fig.1: Phase diagram of one dimensional Joseph- less parameter U/E where U is the energy per su- J son junction array. The MBL phase transi- perconducting island (U = T in the thermodynamic tionseparatesthenon-ergodicbadmetalwith equilibrium at T (cid:28)E ). J exponentially large but finite resistance from Themainqualitativefindingofthispaperistheap- the insulator with infinite resistance. Cooling pearance of a non-ergodic and highly resistive“bad the non-ergodic bad metal transforms it into metal”phaseathightemperatures,T/E >1,which J a good ergodic metal. The points show ap- at T ≥ T ≈ E2/E undergoes the transition to the c J C proximate positions of the effective T/E for MBL insulator. In contrast to the T = 0 behavior, J the quantum problem with a finite number of these results are robust, e.g. are insensitive to the chargingstates. Theredstarsindicateinsula- presence of static random charges. The full phase tor, blue circles bad metal, and squares good diagram in the variables E /E , T/E is shown in J C J metal. Fig. 1. Weconfirmednumericallythatthebadmetal persists in the classical limit although T → ∞; it is c characterized by the exponential growth of the resis- Kauzmannparadoxintoanapparenttemperaturedi- tancewithT andviolationofthermodynamicidenti- vergence (see below). ties. We support these findings by semi-quantitative Qualitative arguments for MBL transition. In a theoreticalarguments. Finally,wepresenttheresults highly excited state U (cid:29) E the charging energy J ofnumericaldiagonalizationandtDMRG(timeDen- dominates: E q2 ∼U (cid:29)E . Accordingly, the value C J sity Matrix Renormalization Group[9]) of quantum of the charge, |q |, and charge difference on neigh- i systems that demonstrate both the non-ergodic bad boring sites, δq = |q − q | are of the order of i i i+1 (cid:112) metal and the MBL insulator. q ∼ δq ∼ U/E . The energy cost of a unit charge C √ It is natural to compare the non-ergodic state of transfer between two sites δE ∼ UE exceeds the c JJA with a conventional glass characterized by in- matrix element of the charge transfer, E /2, as long J finitely many metastable states. The glass entropy asU (cid:29)U =E2/E . Accordingto[4,3]thesys- MBL J C does not vanish at T = 0, i. e. when heated temisanon-ergodicMBLinsulatorunderthiscondi- from T = 0 to the melting temperature the glass re- tion. Thus,weexpectthetransitiontoMBLphaseat leaseslessentropythanthecrystal(Kauzmannpara- T /E ∝ E /E with the numerical prefactor close c J J C dox [10]). Similarly to glasses JJA demonstrates to unity (see supplemental materials). non-ergodic behavior in both quantum and classical If E /E (cid:29) 1 and U ∼ T (cid:28) E the conductivity J C J regimes. However the ergodicity violation emerges limitedbythermallyactivatedphaseslipsisexponen- as high rather than low temperatures transforming tially large, σ ∼ exp(E /T)[11, 12]. As we show be- J 3 low, at T (cid:29)EJ in the metallic phase conductivity is T T T T exponentially small, σ ∼exp(−T/EJ), even far from 2205 1 2 3 w the transition, E (cid:28) T (cid:28) E2/E . The resistance T 15 J J C 15 * in this state can exceed R = h/(2e)2dramatically Q 10 and still display the“metallic”temperature behavior 5 T 10 (dR/dT >0). Th 1.5 2.0 2.5 3.0 3.5 4.0 U Classical regime isrealizedatE →0forfixedE C J 5 and T. One can express the charge of an island q √ through the dimensionless time τ = E E t as q = (cid:112) (cid:112) J C u EJ/ECdφ/dτ. Since q ∼ EJ/EC (cid:29) 1. one can 0 neglectthechargequantizationandusetheequations 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 of motion Fig.2: Energy fluctuations as a function of average d2φ i =sin(φ −φ )+sin(φ −φ ), (2) energy per island for different evolution times dτ2 i+1 i i−1 i (τ = 1, 2, 4, 8,16 × 104 ) in a small sub- ev Here i = 1,...,L and the boundary conditions are system; the dashed line is the extrapolation φ0 =φL+1 =0. Wesolvetheseequationsforthevar- to infinite times as explained in the text. A ious initial conditions corresponding to a given total single point (red) obtained by direct compu- energy and compute the energy US contained in a tations up to τev =2106 at which time scales part of the whole chain of the length 1(cid:28)l(cid:28)L as a the time dependence practically disappears function of τ. for L (cid:46) 100. The upper (red) curve cor- The ergodicity implies familiar thermodynamic responds to the thermodynamic identity (3). identities. e.g. Theinsertshowstheeffectivetemperaturede- (cid:16)(cid:10)U2(cid:11)−(cid:104)U (cid:105)2(cid:17)/T2 =d(cid:104)U (cid:105)/dT (3) fined by the energy fluctuations determined S S S at different time scales: τ = 2, 4, 8×104 ev (T −T respectively) and by their extrapo- This relation between the average energy of the sub- 1 3 system, (cid:104)U (cid:105) and its second moment (cid:10)U2(cid:11) turns out lation to infinite times (T∗), their comparison S S with the temperature expected in thermody- tobeinvalidforabadmetal. Todemonstratethiswe namic equilibrium, T . evaluated the average energy per site in this subsys- Th tem, u=(cid:104)U (cid:105) /(E l) and the temporal fluctuations S τ J (cid:16) (cid:17) of this energy, wτ = (cid:104)US2(cid:105)τ −(cid:104)US(cid:105)2τ /(EJ2l). Here time,τev,thecomputedwτ(u)-dependencesaturates (cid:104)...(cid:105)τ and the bar denote correspondingly averaging instead of increasing as u2 (4). At a fixed u, wτ(u) over the time 1 and over the ensemble of the initial increases with time extremely slowly. Below we ar- conditions. gue that wτ(u) does not reach its thermodynamic From (1) it follows that u = T/(2E ) at T (cid:29) E value even at τ → ∞. Violation of the thermody- J J (u(T)-function is evaluated for arbitrary T/E in namicidentityimpliesthattemperatureisilldefined, J supplemental materials). One can thus rewrite (3) so the average energy u rather than T is the proper as controlparametér. Theeffectivetemperaturedefined T2 du as T (u)=E / udu/w(u), isshownin the insertto w = ≈2u2 (4) ∗ J 0 E dT Fig. 2. J Results of the numerical solution of (2) are com- Qualitative interpretation. Large dispersion of pared with (3) in Fig. 2. For any given evolution charges on adjacent islands, i,i + 1 at u (cid:29) 1 im- pliesquickchangeofthephasedifferences, φ −φ , 1Giventheevolutiontimeτav (cid:104)...(cid:105)τ averagingmeansaver- with time. Typical current between the twio neiig+h1- agingoverthetimeintervalτev afterinitialevolutionfortime τev. bors EJsin(φi−φi+1) time-averages almost to zero. 4 However, accidentally the frequencies, ω = dφ /dt, 104 i i getclose. Intheclassicallimitthedifferenceω −ω ρ(u) i i+1 can be arbitrary small. Such pair of islands is char- 100 acterized by one periodic in time phase difference. Contrarily, three consecutive islands with close fre- 100 σ(u) quencies |ωi − ωi+1| ∼ |ωi − ωi−1| (cid:46) 1 experience 1.0 10 chaotic dynamics that contains arbitrary small fre- 1 0.1 quencies, similarly to work [13]. For uncorrelated 0.01 frequencies with variances u (cid:29) 1, a triad of islands 0.01 0.001 1/u (i−1, i, i+1)ischaoticwiththeprobability1/u(cid:28)1, 0 1 2 3 4 i.e. such triads are separated by large quiet regions 0.0 1.0 2.0 3.0 u of a typicalsize r ∼u(cid:29)1. The low frequency noise t Fig.3: Resistivity as a function of the internal en- generated by a chaotic triad decreases exponentially ergy. At high energies resistivity is expo- deep inside a quiet region. Provided that ω (cid:28)ω i+m nentially large due to large regions of almost , m = 0´...r the superconducting order parameter frozen charges (see text). Insert: at low z (ω) = dτexp(iφ +iωτ) at site i satisfies the re- i i energies (temperatures) exponentially large cursion relation conductivity is limited by exponentially rare r−1 (cid:89) 1 phase slips. High u points require computa- z (ω)=z (ω) (5) i+r i 2ω2 tion times τ ∼108. m=0 i+m ev which implies the log-normal distribution for the resistances R of quiet regions (see Supplemental dlnρ/du≈3.0 at u=3.5 is consistent with (8) that j,j+r materials): gives dlnρ/du≈2.5+1.5lnu−1/u. The qualitative picture of triads separated by log- (cid:10)ln2(R/Rt)(cid:11) = lnRt (6) normally distributed resistances of silent regions al- lnR =(cid:104)lnR(cid:105) = uln(u) (7) lows one to understand the long time relaxation of t w (u)inthesubsystemoflengthl (seeFig. 2). Each τ where R is the typical resistance of a quiet region. t resistance can be viewed as a barrier with a tunnel- The resistance of the whole array is the sum of the ing rate ∼1/R. For a given time τ the barriers with resistancesofthequietregions. Themeannumberof τ (cid:28) R can be considered impenetrable, whilst the these regions in the chain equals to N = L/r (cid:29) 1, t barriers with τ (cid:29) R can be neglected. As a result, its fluctuations being negligible. For the log normal the barriers with R≥τ break the system into essen- distribution the average resistance of a quiet region, tiallyindependentquasiequilibriumregions(QER)of (cid:104)R(cid:105), is given by(cid:104)R(cid:105)=Rt3/2. For the resistivity, ρ, we the typical size thus have ρ=N(cid:104)R(cid:105)/L=R3/2/r . According to (7) t t r (cid:20)ln2(τ/R )(cid:21) 3 l ∼ √ t exp t . (9) lnρr ≈ (γ+lnu)r ≈u(lnu+γ) (8) τ 2πlnR 2lnR t 2 t t t where γ =0.577 is Euler constant. If l (cid:29) r and τ (cid:46) exp(cid:104)(cid:112)ln(l/r )lnR (cid:105) the subsys- t t τ Inordertodeterminethecurrentcausedbyvoltage tem contains l /l (cid:29) 1 QER, so that w ∝ l /l. At V across the chain we solved the equations (2) with τ τ longer times, l (cid:29)l, the subsystem is in equilibrium modified boundary conditions, φ = 0 , φ = Vt. τ 0 L+1 withaparticularQERandw ∝l/l . Thefulldepen- The results confirm the prediction (8), see Fig. 3. τ dence on time can be interpolated as The range of the resistances set by realistic computation time is too small to detect the logarith- w = w∞ (10) mic factor in (8,7), however, a relatively large slope, τ 1+βexp(cid:0)−αln2(τ/τ )(cid:1) 0 5 14 w 0.0200 T /T eff Th 12 0.0100 10 0.0050 8 u 6 0.0020 U=2.5 S(ω) 4 0.0010 U=3.0 2 U=3.55 ω 0 lnτ 10-6 10-5 10-4 0.001 0.01 6 8 10 12 14 16 Fig.5: Spectrum of current noise fluctuations, Fig.4: Time dependence of energy fluctuations S(ω) for different internal energies, U = for the subsystems of different sizes l = 2.5, 3.0, 3.55. The dashed line shows S(ω) ∼ 10, 20, 40, 80 (left to right) and their fits to ω1/2 dependence. Insert shows the effective log-normal law (10). The length of the full temperature determined from FDT relation. system was L = 5l. The best fit shown here corresponds to the value α = 0.05, β/l = 1.0−1.5 and lnτ ≈3.0−4.5. The values for pect at T = T a MBL phase transition between 0 c larger sizes (β/l ≈ 1.0 and lnτ ≈ 4.5 agree two non-ergodic states: the insulator (ρ = ∞) and 0 very well with the ones expected for R ob- a bad metal (ρ < ∞). For E (cid:29) E the bad metal t J C tainedinthecomputationoftheresistanceat can be described classically at T (cid:28) T ∼ E2/E . c J C u = 3.5 :lnR ≈ 5.0. This yields r ≈ 5.0 , Our previous discussion suggests that the bad metal t t β/l≈1.1 and lnτ ≈5.0. is non-ergodic in a broad range of the parameters, 0 T/E and E /E . To verify this conjecture numer- J J C ically we reduced the Hilbert space of the model (1) (cid:112) whereα=(2lnRt)−1, β = π/α(l/rt)andτ0 =Rt. to a finite number of charging states at each site, Thenumericalsimulationsconfirmthattheenergy q = 0,±1,±2 (RHS model). We analyzed the time i variance w relaxes in agreement with (10) as shown evolution of entanglement entropy, S{Ψ} of the left in Fig. 4. The best fit to the equation (10) yields half of the system. The entropy was averaged over parameters close to the expected, lnR = lnρ(u)r , the initial states from the ensemble of product states t t r ≈u. Extrapolation gives w (∞)≈10.0 of w (l) inthechargebasis,S (L)≡<S{Ψ}> thatcorre- t ∞ ∞ t Ψin tolargel, whichissignificantlysmallerthanthermo- spond to zero total charge. As a result, we obtained dynamic value w =14.0 at u=3.5. theGibbsentropyatT =∞(allstateshavethesame Th Another test of the ergodicity follows from the weight exp(−H/T)≡1) fluctuation-dissipation theorem (FDT) that relates The insert to Fig. 6 shows the time dependence conductivity and current fluctuations. In the low of the entropy at E /E = 0.3 which corresponds J C frequency limit the noise power spectrum is S(ω → to the bad metal regime (see below). A slow satura- 0)=2T /RwhereT istheeffectivetemperature tion of the entropy follows its quick initial increase. eff eff (see Fig. 5), which we extracted from the numerical It is crucial that the saturation constant, S (L) is ∞ data. We found that T > T for u (cid:38) 1, where significantly less than its maximal value, S (L) = eff Th Th T is thermodynamic temperature. In particular, Lln5 expected at T =∞ equilibrium. Furthermore, Th T ≈ 1.6T for u = 3.5, which is close to T (u) dS (L)/dL < ln5, indicating that S −S is ex- eff Th ∗ ∞ ∞ Th shown on the inset to Fig. 2. tensive and the system is essentially non-ergodic. Quantum behavior. In contrast to a classical limit Fig. 6 presents S as a function of E /E . Note ∞ J C E → 0 in the quantum regime E > 0 we ex- that S is measurably less than S for E /E < C C ∞ Th J C 6 2 sition happens in the interval 0.05<E /E <0.3. ln(5) J C 1.8 The variances of the charge in the RHS model at 1.6 T = ∞ and the problem (1) at finite T coincide dS/dL at T = 2E . Thus, we expect that the results of 1.4 C S(L=8)/4 the quantum simulations describe the behavior of 1.2 6 the model (1) at T/E ∼ E /E yielding the hy- S(t) J C J 1 5 perbola shown in Fig. 1. The MBL transition at 0.8 4 E /E ∼ 0.2 in T = ∞ RHS model corresponds to J C 0.6 3 the transition temperature T ∼ 10E in model (1). c J 2 The transition line shown in Fig. 1 is a natural con- 0.4 1 nectionofthispointwithT ≈E2/E asymptoticat 0.2 E/E 0 ECt c J C 0 J C 100 102 104 106 108 1010 EJ/EC (cid:29) 1 discussed above. The maximum of the 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 transition temperature is natural because the quantum fluctuations are largest at E ∼E . J C Fig.6: Enropy per spin in thermodynamic limit ex- Possible experimental realization. MBL and the trapolated to t = ∞. The insert shows violation of the ergodicity can be observed only at S(t) dependence in the bad metal regime at sufficiently low temperatures when one can neglect E /E = 0.3 that shows slow relaxation of the effects of thermally excited quasiparticles which J C the entropy to its saturation value. form the environment to model (1). This limits temperatures to T (cid:46) 0.1∆ where ∆ is superconducting gap. In order to explore the phase diagram 0.6−1. For EJ/EC (cid:29) 1, the entropy saturation is one has to vary both T/EJ and EJ/EC in the in- quick and the accuracy of the simulations does not tervals 1 (cid:46) T/E (cid:46) 5 and 0.1 (cid:46) E /E (cid:46) 5. J J C allow us to distinguish S∞ from STh (see Supple- The former condition can be satisfied if each junc- menting material). We thus are unable to conclude tion is implemented as a SQUID loop with individ- if the system is truly ergodic or weakly non-ergodic ual Josephson energy E(0) ∼ T = 0.1∆ so that J max at EJ/EC (cid:29) 1. The former behavior would imply a E = 2cos(eΦ/π(cid:126))E where Φ is flux through the J J genuine phase transition between bad and good met- loop. Thelatterconditionrequiresenhancingground als, while the latter corresponds to a crossover. capacitance of each island which should exceed the Deep in the insulator the time dependence of the capacitance of the junctions in order for the model entropy is extremely slow, roughly linear in lnt in a (1) to be relevant. Realistic measurements of such widetimeinterval(seeSupplementalmaterials). This array include resistance R(T,E ) and current noise. J resembles the results of the works [14, 15] for the Here we predict a fast growth with temperature and conventional disordered insulators. The extremely divergence at T of the resistivity and violation of c long relaxation times can be attributed to rare pairs FDT. of almost degenerate states localized within different In conclusion, we presented strong numerical halvesofthesystem. Theexponentialdecaywithdis- evidences for the MBL transition and its semi- tanceofthetunnelingamplitudethatentanglesthem quantitative description in a regular, disorder free2 leads to the exponentially slow relaxation. Josephson chain. Probably the most exciting find- In order to locate the MBL transition we analyzed ingistheintermediatenon-ergodicconductingphase the time dependence of the charge fluctuations. In a (bad metal) between the MBL insulator and good metalthechargefluctuationsrelaxationratedepends ergodic metal. This phase distinguishes Joseph- weakly(asapowerlaw)onthesamplesizeincontrast totheexponentialdependenceintheinsulator. Com- 2Whenthisworkwasfinishedwelearntaboutthenumerical studies[16,17]thatreportsMBLindisorderfree1Dsystems, paringthedependencesoftheratesonthesystemsize in cotrast the work [18] reports the absence of MBL in such fordifferentEJ/EC (seeFig. 7)weseethatthetran- systems. Alltheseworksstudiedmodelsdifferentfromours. 7 6<n >2 1080 Simulation of the JJA in the classical regime. At L tcr EJ/EC=0.05 large u averaging out temporal fluctuations requires 5 1060 EJ/EC=0.3 EJ/EC=1.75 exponentially long times. Moreover, at u (cid:29) 1 the 1040 resistance increases with u factorially leading to a 4 1020 strong heating in the computation of resistance un- 100 L lessthemeasurementcurrentisfactoriallysmall. Ob- 3 3 4 5 6 7 8 9 101112 servation of a small current against the background of a low frequency noise requires increasingly long 2 L=4 times. Accordingly, for the realistic evolution times L=6 τ (cid:46) 108 the resistance can be computed only for 1 ev L=8 u<4.0. 0 L=10 Ect Simulation of the quantum problem. The time de- 100 102 104 106 108 1010 pendence of the entropy and the charge fluctuations for system of sizes L = 4,6,8 was computed using Fig.7: Charge relaxation in a bad metal (E /E = J C exact diagonalization in a symmetric subspace under 0.3). The characteristic time scale, t , can cr charge conjugation. The tDMRG method was em- be defined by the extrapolated crossing point of (cid:10)n2(cid:11) with t-axis. The crossing point t ployedforlargersizeswhichaccuracylimitstherange cr of times that we could study. 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The work was supported in part by 1Freeenergyofthechaininthermodynamicequilibrium 9 grantsfromtheTempletonFoundation(40381),ARO Supplemental materials (W911NF-13-1-0431) and ANR QuDec. 1 Free energy of the chain in thermodynamic equilibrium Evaluation of the free energy with Hamiltonian (1) gives (cid:18) (cid:19) 1 E F = T ln C −T ln[I (E /T)] 2 2πT 0 J which allows us to express u in terms of T: (cid:18) (cid:19) T I (E /T) u= + 1− 1 J (11) 2E I (E /T) J 0 J Here I (x) are modified Bessel functions. Using (11) α onecancheckthevalidityof(3),whichtakestheform w =T2du/dT . 2 Noise distribution. At high temperatures the frequencies of individual phases are typically large, ω (cid:29) 1 which allows to i solve the equations (2) in perturbation theory. Fur- thermore, because the effect of noise decreases expo- nentiallywithdistanceonecanusetheforwardprop- agation approximation in which the evolution of the next phase is determined exclusively by the previous one d2φ i =−sin(φ −φ ) (12) dτ2 i i−1 Looking for the solution in the form φ = ω τ + i i δφ (τ) and introducing the notation i ˆ eiφi = dωe−iωτz (ω) i for the noise at site i we get 1 δφ = Imz (ω)ei(ω−ωi)τ (13) i (ω −ω)2 i−1 i We are interested in the propagation of the low frequency harmonics of the noise,ω (cid:28) ω . Neglecting i the high frequency components of the noise one ob- tains from (13) the noise recursion 1 z (ω)= z (ω) (14) i 2(ω −ω)2 i−1 i 3Chargepropagationinquantuminsulatingphase. 10 Iterationsof(14)inthelimitω (cid:28)ω leadtotheequa- -1.0 i tion(5)ofthemaintext. Theexponentialdecreaseof f(r) the noise away from its source in the classical regime -1.5 is in agreement with the numerical computation in -2.0 the quantum regime (see section 3). Therecursion(5)ofthemaintextimpliesthatthe -2.5 noisegeneratedbyasingletriadatdistance,r (cid:29)1is givenbyaproductofalargenumberoffactors. Being -3.0 asumofalargenumberofrandomindependentterms r the logarithm of the noise has Gaussian distribution. -3.5 Thus, at the distance r (cid:29) r (cid:29) 1, from the closest 3 4 5 6 7 8 9 10 t triad the noise distribution has a log normal form Fig.8: Logarithm of the dimensionless amplitude of 1 (cid:34) (ζ−ζ (r))2(cid:35) the charge transfer, f(r) determined in the P (ζ)= exp − ∗ (15) forward propagation approximation. r (cid:112)2πζ∗(r) 2ζ∗(r) where ζ =lnz, ζ∗(r)=(γ+lnu)r, and γ =0.577 is where (cid:104)...(cid:105) means average over charge configu- q Euler constant. The exponential decay of the effect rations normally distributed with variance T/E . C ofasingletriad(5,15)impliesthechargelocalization. Rescaling the charge and factoring out the dimen- The DC charge transport in a macroscopically large sionfull parameters we find that array requires interaction of different triads through the quiet regions. A quiet interval (j,j + r) be- (cid:18) E (cid:19) (cid:104)ln|Ψ (r)|(cid:105)=rln √ J +f(r) (16) tween islands can be characterized by the resistance FWD TE c R ∼ z /z . Convolution of distribution (15) j,j+r j j+r withthePoissondistributionforthesizesofthequiet regions, r, yields the log-normal distribution for the (cid:42) (cid:43) (cid:88) 1 resistances Rj,j+r of quiet regions. f(r)= ln (cid:12)(cid:80) (cid:12) (17) P 2(cid:12) k(qP(k)+1−qP(k))(cid:12) q 3 Charge propagation in quantum where average is performed over charge configura- insulating phase. tions with unit variance. The numerical evaluation of f(r) shows that f(r)≈ηr with η ≈−0.3(see Fig. In the insulating phase the charge transfer by dis- 8). tance r appears in the rth order of the perturba- √ We estimate the transition temperature by de- tion theory in the parameter E / TE . For small √ J c manding (cid:104)ln|Ψ (r)|(cid:105) ∼ const which gives E / TE (cid:28) 1 the main contribution to the ampli- FWD J c TFWD ≈ e2ηE2/E . The forward propagation ap- tude, Ψ(r) of this processes comes from exactly r−1 c J C proximationneglectstheeffectsofthebackscattering virtualhopsbetweenneighboringsitesinthesamedi- and level repulsion. All these apparently small ef- rection (forward propagation approximation). These fects favor the insulating behavior, thus, the actual hopes can occur in any order. Each sequence of the transition is below T (cid:46) TFWD. We conclude that hops corresponds to a particular permutation, P, of c c the numerical prefactor, η˜, in the equation for the 1...r−1. After n<r−1 hops the charging energy changesby∆E =E (cid:80) (q −q ),sothat critical temperature, Tc = η˜EJ2/EC is close to one, n c k≤n P(k)+1 P(k) η˜(cid:46)0.5. (cid:42) (cid:43) The estimate of the charge transfer (16) neglects (cid:104)ln|ΨFWD(r)|(cid:105)= ln(cid:88) (cid:12)(cid:80) EJ (cid:12) the charge discreteness which is expected to become P 2Ec(cid:12) k(qP(k)+1−qP(k))(cid:12) qirrelevant at T/EC → ∞. To estimate the effect of