Chain breaking and Kosterlitz-Thouless scaling at the many-body localization transition in the random field Heisenberg spin chain
CChain breaking and Kosterlitz-Thouless scaling at the many-body localizationtransition
Nicolas Laflorencie, ∗ Gabriel Lemari´e, † and Nicolas Mac´e ‡ Laboratoire de Physique Th´eorique, IRSAMC, Universit´e de Toulouse, CNRS, UPS, France
Despite tremendous theoretical efforts to understand subtleties of the many-body localization(MBL) transition, many questions remain open, in particular concerning its critical properties.Here we make the key observation that MBL in one dimension is accompanied by a spin freezingmechanism which causes chain breakings in the thermodynamic limit. Using analytical and numer-ical approaches, we show that such chain breakings directly probe the typical localization length,and that their scaling properties at the MBL transition agree with the Kosterlitz-Thouless scenariopredicted by phenomenological renormalization group approaches.
Introduction—
The field of interacting quantum sys-tems in the presence of disorder has attracted a lot ofattention over the last two decades. Besides tremendoustheoretical efforts [1–11], a growing number of experi-mental realizations have also emerged, either based onultracold atoms or trapped ions [12–16], superconduct-ing qubits [17–19], solid-state setups [20–22]. Building onthis collective movement (for recent reviews, see Refs. [7–11]), it is nowadays well-admitted that upon increas-ing disorder several low-dimensional quantum interactingsystems can undergo a transition towards a many-bodylocalized (MBL) phase. This non-ergodic regime is rea-sonably well-characterized, mostly by exact diagonaliza-tion (ED) techniques [23, 24] showing various propertiesof MBL states: Poisson spectral statistics, area-law en-tanglement, emerging integrability, logarithmic spread-ing of entanglement, eigenstates multifractality. Whiletypically limited to L ∼
20 interacting 2-level systems(due to exponentially small level-spacings ∝ − L at highenergy), ED studies have nevertheless showed a clear er-godicity breaking transition for the random-field spin-1 / H = L (cid:88) i =1 (cid:16) (cid:126)S i · (cid:126)S i +1 − h i S zi (cid:17) , (1)where h i are independently drawn form a uniform distri-bution [ − h, h ]. Despite recent debates [25–31], h c ∼ ξ ∼ | h − h c | − ν with an ex-ponent ν ≈ ∗ nicolas.lafl[email protected] † [email protected] ‡ [email protected] random graphs [37–50] where such unusual scaling prop-erties were first found [45, 49]. Nevertheless, most numer-ical studies violate the Harris bound [51, 52], and a fullyconsistent finite-size scaling theory is still missing [53].In such a puzzling context, several progresses havebeen made to build an analytical theory, able to describethe ergodic-MBL transition [54–62]. The most success-ful description, based on the so-called ”avalanche” sce-nario [63, 64], proposes a phenomenological renormaliza-tion group (RG) treatment, working in the MBL regimewhere large insulating blocks compete with small ergodicinclusions. In this framework, recent works [65–67] founda RG flow of the Kosterlitz-Thouless (KT) form. TheMBL phase is described as a line of fixed points with avanishing density of ergodic inclusions and a finite typ-ical localization length ζ , which controls the spatial ex-tension of the l -bits [68–70]. When the delocalizationtransition is approached from the MBL side, the typi-cal localization length ζ grows and reach a finite criticalvalue ζ c = (ln 2) − at the transition point h c , with asingular behavior [71] ζ − − ζ − c ∝ (cid:112) h − h c . (2)As argued in Ref. [66], a diverging length scale ∼ exp( c/ √ h − h c ) should control finite-size effects, thus ex-plaining some limitations in the numerics.In this Letter, we show that one can overcome suchfinite size constraints by measuring spin polarization,a simple local observable of the S = 1 / ζ of MBL physics. At the thermodynamic limit,the MBL regime is driven by a chain breaking mecha-nism, signalled by a complete freezing of some sites with |(cid:104) S zi (cid:105)| − / ∼ L − γ . The freezing exponent γ >
0, whichcontrols the corrections, can be analytically related tothe typical localization length by γ ∝ ζ − . Building onRefs. [36, 45, 49], a careful scaling analysis of our ED datareveals that they are best described by a volumic scalingvariable N / Λ in the delocalized regime, where N is theHilbert-space size and Λ a non-ergodicty volume, while alogarithmic scaling variable (ln L ) /λ dictates the behav-ior in the MBL regime, with λ − ∝ ζ − − ζ − c followingthe KT singularity Eq. (2). a r X i v : . [ c ond - m a t . d i s - nn ] A p r Distribution of local polarizations and extremevalue statistics—
Let us start the discussion by look-ing at the local polarizations m z = (cid:104) S z (cid:105) , computed byED simulations performed at infinite temperature for theHamiltonian Eq. (1). As already noticed in Refs. [72–75],the histograms P ( m z ) display distinct features across thetwo regimes, as illustrated in Fig. 1 (a) for 3 representa-tive values of the disorder strength ( h = 1 , , m z = 0 and shrinking with increas-ing system size, as clearly visible for h = 1. At h = 3, amore complex form emerges with deviations from gaus-sianity [79, 80]: strongly polarized ( m z (cid:39) ± /
2) sites ap-pear, but their density shrinks down with system size (asevidenced by the decrease of magnetization variance, seeinset). At strong disorder h = 7, the density of stronglypolarized sites no longer shrinks down: ETH is violatedand the distribution is U-shaped, almost free of finite-sizeeffects.To quantify this effect, we introduce the deviation from -12 -8 -400.10.20.30.40.5 L=12L=14L=16L=18L=20L=22 -12 -8 -400.10.20.30.40.5
L=12L=14L=16L=18L=20L=22 -0.5 -0.25 0 0.25 0.5110
L=12L=14L=16L=18 ln δ min −ln 2 P ( l n δ m i n ) P ( m z ) m z -12 -8 -400.10.20.30.40.5 L=12L=14L=16L=18L=20L=22 -12 -8 -400.10.20.30.40.5
L=12L=14L=16L=18L=20L=22 ln δ min −ln 2 P ( l n δ m i n ) h = = P ( m z ) m z -0.5 -0.25 0 0.25 0.5110 L=12L=14L=16L=18 h = -12 -8 -400.10.20.30.40.5 L=12L=14L=16L=18L=20L=22 -12 -8 -400.10.20.30.40.5
L=12L=14L=16L=18L=20L=22 ln δ min −ln 2 P ( l n δ m i n ) h = = P ( m z ) m z -0.5 -0.25 0 0.25 0.5110 L=12L=14L=16L=18 h = h σ m z ( a )( b ) -12 -8 -400.10.20.30.40.5 L=12L=14L=16L=18L=20L=22 -12 -8 -400.10.20.30.40.5
L=12L=14L=16L=18L=20L=22 ln δ min −ln 2 P ( l n δ m i n ) h = = P ( m z ) m z -0.5 -0.25 0 0.25 0.5110 L=12L=14L=16L=18 h = -12 -8 -400.10.20.30.40.5 L=12L=14L=16L=18L=20L=22 -12 -8 -400.10.20.30.40.5
L=12L=14L=16L=18L=20L=22 ln δ min −ln 2 P ( l n δ m i n ) h = = P ( m z ) m z -0.5 -0.25 0 0.25 0.5110 L=12L=14L=16L=18 h = -12 -8 -400.10.20.30.40.5 L=12L=14L=16L=18L=20L=22 -12 -8 -400.10.20.30.40.5
L=12L=14L=16L=18L=20L=22 ln δ min −ln 2 P ( l n δ m i n ) h = = P ( m z ) m z -0.5 -0.25 0 0.25 0.5110 L=12L=14L=16L=18 h = L=8L=10L=12L=14L=16L=18L=20L=22
FIG. 1. Histograms of (a) the local magnetizations m z = (cid:104) S zi (cid:105) displayed for h = 1 , ,
7, and (b) of the maximally polarizedsites Eq. (4). Inset: variance σ m z of the local magnetizationplotted as a function of h for various lengths L . Shift-invertED results for infinite temperature eigenstates of Eq. (1), per-formed over thousands of independent random samples. perfect polarization δ = − | m z | . While δ > δ becomes arbitrarily small for large disorder, asevidenced in Fig. 1 (a) at h = 7 where one observes [81] P ( δ ) ∝ δ − γ ( δ → , (3)with an exponent γ ≥ δ min = 12 − max ≤ i ≤ L |(cid:104) S zi (cid:105)| , (4)whose distributions are shown at h = 3 and h = 7 inFig. 1 (b). There, we explicitly report two distinct trendswith increasing system sizes L . In the ETH regime( h = 3), some weight is transferred with increasing L towards small − ln δ min , with a peak developing at − ln 2.The opposite is observed in the MBL regime ( h = 7),where weight is moved towards large values of − ln δ min .In both cases we qualitatively spot that rare eventsof the competing phase (rare insulating bottenecks forthe delocalized phase vs. thermal bubbles in the MBLregime) become less and less relevant upon increasing L .Let us now turn to a more quantitative analysis of theextreme polarization finite size scaling. At strong disor-der, assuming that the magnetizations along the chainare independently drawn from the distribution P ( m z ) ofFig. 1 (a), the deviation from perfect polarization veri-fies (cid:82) δ min P ( x )d x ∼ /L [82], which for a power-law tailEq. (3), yields δ min ( L ) ∼ L − γ . (5)Such a scaling has dramatic consequences since for any γ (cid:54) = 0, we expect δ min ( L ) → L → ∞ , meaningspin freezing, and therefore chain breaking at the ther-modynamic limit. Extreme polarizations: numerical results—
In or-der to numerically check the power-law scaling Eq. (S2),and probe the chain breaking mechanism at the micro-scopic level, the most polarized site is recorded for eachfinite-length sample. Typical values are computed fromshift-invert ED simulations at infinite temperature, anddisorder averaging is performed over a large number ofrealizations, at least 10 samples. Results as a functionof chain lengths L are shown in Fig. 2 (a) for a widerange of disorder strengths. As expected from the ex-treme value argument Eq. (S2), at strong disorder weobserve a power-law decay with L , indicating a chainbreaking at the thermodynamic limit. In contrast, forweak disorder δ min does not vanish with L , but insteadtends to 1 /
2, showing ”healing” in a way similar to Kane-Fisher physics [83, 84]. Such radically different behaviorscorrespond to the two phases of the model, as we arguebelow.In Fig. 2 (b), we show the h -dependence of the freez-ing exponent γ obtained from power-law fits to the form
10 20 3010 -4 -3 -2 -1 δ typmin γ L h A ln( h / h )[8 − 16][10 − 18][12 − 20][14 − 22]
10 20 3010 -4 -3 -2 -1 ), h ~1)8-1610-1812-2014-22 γ c + a h − h c h = 0.4 h = 1.0 h = 1.6 h = 2.0 h = 2.5 h = 3.0 h = 3.2 h = 3.4 h = 3.6 h = 3.8 h = 3.9 h = 4.0 h = 4.1 h = 4.2 h = 4.3 h = 4.5 h = 5.0 h = 5.5 h = 6.0 h = 7.0 h = 8.0 h = 9.0 h = 10 ( a )
10 20 3010 -4 -3 -2 -1 ), h ~1)8-1610-1812-2014-22 δ typmin γ L h A ln( h / h )[8 − 16][10 − 18][12 − 20][14 − 22]
10 20 3010 -4 -3 -2 -1 ), h ~1)8-1610-1812-2014-22 γ c + a h − h c h = 0.4 h = 1.0 h = 1.6 h = 2.0 h = 2.5 h = 3.0 h = 3.2 h = 3.4 h = 3.6 h = 3.8 h = 3.9 h = 4.0 h = 4.1 h = 4.2 h = 4.3 h = 4.5 h = 5.0 h = 5.5 h = 6.0 h = 7.0 h = 8.0 h = 9.0 h = 10 ( a ) ( b ) ( b ) FIG. 2. (a) Typical deviation δ typmin Eq. (4), plotted againstsystem size L for a wide regime of disorder h . Log-log scalereveals the power-law decay Eq. (S2) at large enough h , while δ typmin → / γ gov-erning the decay. Various symbols stand for 4 different fit-ting windows including 5 points in the range [ L min , L max ].At large disorder, the exponent grows logarithmically as γ = A ln( h/h ) with fitting parameters A = 1 . h ≈ Eq. (S2) over 4 different fitting windows. Finite-sizescaling towards perfect polarization is governed by adisorder-dependent freezing exponent γ , which shows alogarithmic divergence at large h , a behavior explainedbelow by analytical arguments. Conversely, at weak dis-order we observe a finite-size (possibly non-monotonous)crossover, and ultimately γ → Analytical derivation at large disorder—
Using theJordan-Wigner transformation, we can rewrite Eq. (1) asinteracting spinless fermions in a random potential H = (cid:88) i (cid:104) (cid:16) c † i c i +1 + c † i +1 c i (cid:17) − h i n i (cid:105) + (cid:88) i n i n i +1 . (6)The first sum, describing free fermions, can be diagonal-ized as H = (cid:80) Lk =1 E k b † k b k with new fermionic operators b k = (cid:80) Li =1 φ ki c i . The single-particle orbitals φ ki are ex-ponentially localized for any h (cid:54) = 0, and the interactingHamiltonian rewrites in this ”Anderson basis” as 4 terms: H = (cid:88) k (cid:16) E k + V (1) k (cid:17) n k + (cid:88) k (cid:54) = l V (2) k,l n k n l (7)+ (cid:88) k (cid:54) = l (cid:54) = m V (3) k,l,m n k b † l b m + (cid:88) k (cid:54) = l (cid:54) = m (cid:54) = n V (4) k,l,m,n b † k b † l b m b n . In the strong disorder limit h (cid:29)
1, the second line canbe neglected [81]. The number operators n k then form acomplete basis of local operators commuting with H andamong themselves: in other words, the Hamiltonian iswritten in l -bit form [55, 68, 70, 85–87], and the n k aregood approximation of the l -bits in the strong disorder limit [88]. The fermion density and thus the spin polar-ization at real-space position i is then simply the sum ofcontributions coming from occupied orbitals: (cid:104) S zi (cid:105) + 1 / (cid:104) n i (cid:105) = (cid:88) k occupied (cid:12)(cid:12) φ ki (cid:12)(cid:12) . (8)Building on previous ideas [89], we are now ready tounderstand the spin freezing mechanism. If we approxi-mate the Anderson orbitals by simple exponential func-tions | φ ki | ∝ exp (cid:16) − | i − i k | ξ (cid:17) , where i k is the localizationcenter of orbital k , and ξ ∼ (ln h ) − the localizationlength at strong disorder [89], we expect the maximaldensity (cid:104) n i (cid:105) max to occur in the middle of the longest re-gion of (cid:96) max consecutive sites that are occupied by anorbital. At half-filling, a configuration with (cid:96) consecu-tive occupied sites appears with probability 2 − (cid:96) , whichfor a finite chain of length L (cid:29) (cid:96) max ≈ ln L/ ln 2.The most polarized site i corresponds to the site withmaximal n i ≈ n i ≈
0) fermionic den-sity, yielding (cid:104) S zi (cid:105) ≈ +1 / (cid:104) S zi (cid:105) ≈ − / δ min ∼ exp( − (cid:96) max / (2 ξ )), which naturally leads to the power-lawdecay Eq. (S2) with a freezing exponent γ ( ξ ) ≈ ξ ln 2 . (9)At large disorder, numerical data perfectly agree witha logarithmic growth γ ( h ) ∝ ln h (panel (b) of Fig. 2),which validates our analytical description of the freez-ing mechanism. Within this description, 1 /γ is identifiedwith the localization length of l -bits deep in the bulk ofthe largest non-thermal region. Being far away from rarethermal inclusions, the inverse freezing exponent providesan estimate of the typical l -bit extension, i.e. the typical localization length ζ : 1 /γ ∼ ζ . As we further elaboratebelow, this has decisive consequences for our understand-ing of the critical behavior. Already in Fig. 2 (b), whenthe transition is approached from the MBL side, a singu-lar behavior develops for γ ( h ) close to h c , in agreementwith Eq. (2), followed by a jump to zero in the ergodicphase. We now provide an explicit description of thiscritical behavior. Scaling analysis: KT behavior—
We have performeda very careful analysis of our ED data for δ typmin ( L, h ) inorder to address the transition between ergodic and MBLphases. The best data scalings [81], shown in Fig. 3 (a),are obtained using two distinct scaling functions:ln (cid:34) δ typmin ( L, h ) δ typmin ( L, h c ) (cid:35) = (cid:40) f ( N / Λ) if h < h c ,g (cid:0) ln Lλ (cid:1) if h > h c . (10)In the delocalized phase ( h < h c ), we obtain a volu-mic scaling, in agreement with our recent multifractal-ity analysis [36] (see also [45] for the Anderson transi-tion on random graphs), and with ETH predictions [76],given that the density of state scales with Hilbert space (cid:99) -12 -8 -4 N/ (cid:82) -10-505 l n [ (cid:98) m i n t yp ] - l n [ (cid:98) m i n t yp ( h c )] h=4.3h=4.4h=4.5h=5.0h=5.5h=6.0h=6.5h=7.0h=8.0h=9.0h=10h=20h=30h=40h=50h=60h=70h=80h=90 -12 -8 -4 -10-505 h=0.4h=0.8h=1.0h=1.2h=1.6h=2.0h=2.5h=2.6h=2.7h=2.8h=2.9h=3.0h=3.1h=3.2h=3.3h=3.4h=3.5h=3.6h=3.7h=3.8h=3.9h=4.0h=4.1 h=4.3h=4.4h=4.5h=5.0h=5.5h=6.0h=6.5h=7.0h=8.0h=9.0h=10h=20h=30h=40h=50h=60h=70h=80h=90 -12 -8 -4 -10-505 h=0.4h=0.8h=1.0h=1.2h=1.6h=2.0h=2.5h=2.6h=2.7h=2.8h=2.9h=3.0h=3.1h=3.2h=3.3h=3.4h=3.5h=3.6h=3.7h=3.8h=3.9h=4.0h=4.1 h = 4.3 h = 4.4 h = 4.5 h = 5.0 h = 5.5 h = 6.0 h = 6.5 h = 7.0 h = 8.0 h = 9.0 h = 10 h = 20 h = 30 h = 40 h = 50 h = 60 h = 70 h = 80 h = 90 l n δ t yp m i n ( h ) − l n δ t yp m i n ( h c ) (ln L )/ λ 𝒩/Λ d e l o ca li ze d MBL ( a ) collapsepower collapsepowerlog h=4.3h=4.4h=4.5h=5.0h=5.5h=6.0h=6.5h=7.0h=8.0h=9.0h=10h=20h=30h=40h=50h=60h=70h=80h=90 -12 -8 -4 -10-505 h=0.4h=0.8h=1.0h=1.2h=1.6h=2.0h=2.5h=2.6h=2.7h=2.8h=2.9h=3.0h=3.1h=3.2h=3.3h=3.4h=3.5h=3.6h=3.7h=3.8h=3.9h=4.0h=4.1 locdeloctot h c h c ν l o c χ / N d f localizeddelocalizedtotal h c − h h − h c l n Λ − λ h=4.3h=4.4h=4.5h=5.0h=5.5h=6.0h=6.5h=7.0h=8.0h=9.0h=10h=20h=30h=40h=50h=60h=70h=80h=90 -12 -8 -4 -10-505 h=0.4h=0.8h=1.0h=1.2h=1.6h=2.0h=2.5h=2.6h=2.7h=2.8h=2.9h=3.0h=3.1h=3.2h=3.3h=3.4h=3.5h=3.6h=3.7h=3.8h=3.9h=4.0h=4.1 h=4.3h=4.4h=4.5h=5.0h=5.5h=6.0h=6.5h=7.0h=8.0h=9.0h=10h=20h=30h=40h=50h=60h=70h=80h=90 -12 -8 -4 -10-505 h=0.4h=0.8h=1.0h=1.2h=1.6h=2.0h=2.5h=2.6h=2.7h=2.8h=2.9h=3.0h=3.1h=3.2h=3.3h=3.4h=3.5h=3.6h=3.7h=3.8h=3.9h=4.0h=4.1 h = 4.3 h = 4.4 h = 4.5 h = 5.0 h = 5.5 h = 6.0 h = 6.5 h = 7.0 h = 8.0 h = 9.0 h = 10 h = 20 h = 30 h = 40 h = 50 h = 60 h = 70 h = 80 h = 90 h = 0.4 l n δ t yp m i n ( h ) − l n δ t yp m i n ( h c ) (ln L )/ λ 𝒩/Λ h = 0.8 h = 1.0 h = 1.2 h = 1.6 h = 2.0 h = 2.5 h = 2.6 d e l o ca li ze d MBL h = 2.7 h = 2.8 h = 2.9 h = 3.0 h = 3.1 h = 3.2 h = 3.3 h = 3.4 h = 3.5 h = 3.6 h = 3.7 h = 3.8 h = 3.9 h = 4.0 h = 4.1 ( a ) ( b ) ( c )( e )( d ) from collapse from data collapse∝ [ A ln( h / h ) − γ c ] −1 ∝ ( h − h c ) −0.52 from data collapse∝ ( h c − h ) −0.5 collapsepower collapsepowerlog collapsepower collapsepowerlog γ collapsepower collapsepowerlog h=4.3h=4.4h=4.5h=5.0h=5.5h=6.0h=6.5h=7.0h=8.0h=9.0h=10h=20h=30h=40h=50h=60h=70h=80h=90 -12 -8 -4 -10-505 h=0.4h=0.8h=1.0h=1.2h=1.6h=2.0h=2.5h=2.6h=2.7h=2.8h=2.9h=3.0h=3.1h=3.2h=3.3h=3.4h=3.5h=3.6h=3.7h=3.8h=3.9h=4.0h=4.1 locdeloctot h c h c ν l o c χ / N d f localizeddelocalizedtotal h c − h h − h c l n Λ − λ h=4.3h=4.4h=4.5h=5.0h=5.5h=6.0h=6.5h=7.0h=8.0h=9.0h=10h=20h=30h=40h=50h=60h=70h=80h=90 -12 -8 -4 -10-505 h=0.4h=0.8h=1.0h=1.2h=1.6h=2.0h=2.5h=2.6h=2.7h=2.8h=2.9h=3.0h=3.1h=3.2h=3.3h=3.4h=3.5h=3.6h=3.7h=3.8h=3.9h=4.0h=4.1 h=4.3h=4.4h=4.5h=5.0h=5.5h=6.0h=6.5h=7.0h=8.0h=9.0h=10h=20h=30h=40h=50h=60h=70h=80h=90 -12 -8 -4 -10-505 h=0.4h=0.8h=1.0h=1.2h=1.6h=2.0h=2.5h=2.6h=2.7h=2.8h=2.9h=3.0h=3.1h=3.2h=3.3h=3.4h=3.5h=3.6h=3.7h=3.8h=3.9h=4.0h=4.1 h = 4.3 h = 4.4 h = 4.5 h = 5.0 h = 5.5 h = 6.0 h = 6.5 h = 7.0 h = 8.0 h = 9.0 h = 10 h = 20 h = 30 h = 40 h = 50 h = 60 h = 70 h = 80 h = 90 h = 0.4 l n δ t yp m i n ( h ) − l n δ t yp m i n ( h c ) (ln L )/ λ 𝒩/Λ h = 0.8 h = 1.0 h = 1.2 h = 1.6 h = 2.0 h = 2.5 h = 2.6 d e l o ca li ze d MBL h = 2.7 h = 2.8 h = 2.9 h = 3.0 h = 3.1 h = 3.2 h = 3.3 h = 3.4 h = 3.5 h = 3.6 h = 3.7 h = 3.8 h = 3.9 h = 4.0 h = 4.1 ( a ) ( b ) ( c )( e )( d ) from collapse from data collapse∝ [ A ln( h / h ) − γ c ] −1 ∝ ( h − h c ) −0.52 from data collapse∝ ( h c − h ) −0.5 collapsepower collapsepowerlog collapsepower collapsepowerlog h=4.3h=4.4h=4.5h=5.0h=5.5h=6.0h=6.5h=7.0h=8.0h=9.0h=10h=20h=30h=40h=50h=60h=70h=80h=90 -12 -8 -4 -10-505 h=0.4h=0.8h=1.0h=1.2h=1.6h=2.0h=2.5h=2.6h=2.7h=2.8h=2.9h=3.0h=3.1h=3.2h=3.3h=3.4h=3.5h=3.6h=3.7h=3.8h=3.9h=4.0h=4.1 h=4.3h=4.4h=4.5h=5.0h=5.5h=6.0h=6.5h=7.0h=8.0h=9.0h=10h=20h=30h=40h=50h=60h=70h=80h=90 -12 -8 -4 -10-505 h=0.4h=0.8h=1.0h=1.2h=1.6h=2.0h=2.5h=2.6h=2.7h=2.8h=2.9h=3.0h=3.1h=3.2h=3.3h=3.4h=3.5h=3.6h=3.7h=3.8h=3.9h=4.0h=4.1 h=4.3h=4.4h=4.5h=5.0h=5.5h=6.0h=6.5h=7.0h=8.0h=9.0h=10h=20h=30h=40h=50h=60h=70h=80h=90 -12 -8 -4 -10-505 h=0.4h=0.8h=1.0h=1.2h=1.6h=2.0h=2.5h=2.6h=2.7h=2.8h=2.9h=3.0h=3.1h=3.2h=3.3h=3.4h=3.5h=3.6h=3.7h=3.8h=3.9h=4.0h=4.1 h = 4.3 h = 4.4 h = 4.5 h = 5.0 h = 5.5 h = 6.0 h = 6.5 h = 7.0 h = 8.0 h = 9.0 h = 10 h = 20 h = 30 h = 40 h = 50 h = 60 h = 70 h = 80 h = 90 h = 0.4 l n δ t yp m i n ( h ) − l n δ t yp m i n ( h c ) (ln L )/ λ 𝒩/Λ h = 0.8 h = 1.0 h = 1.2 h = 1.6 h = 2.0 h = 2.5 h = 2.6 d e l o ca li ze d MBL h = 2.7 h = 2.8 h = 2.9 h = 3.0 h = 3.1 h = 3.2 h = 3.3 h = 3.4 h = 3.5 h = 3.6 h = 3.7 h = 3.8 h = 3.9 h = 4.0 h = 4.1 ( a ) collapsepower collapsepowerlog h=4.3h=4.4h=4.5h=5.0h=5.5h=6.0h=6.5h=7.0h=8.0h=9.0h=10h=20h=30h=40h=50h=60h=70h=80h=90 -12 -8 -4 -10-505 h=0.4h=0.8h=1.0h=1.2h=1.6h=2.0h=2.5h=2.6h=2.7h=2.8h=2.9h=3.0h=3.1h=3.2h=3.3h=3.4h=3.5h=3.6h=3.7h=3.8h=3.9h=4.0h=4.1 locdeloctot h c h c ν l o c χ / N d f localizeddelocalizedtotal h c − h h − h c l n Λ − λ h=4.3h=4.4h=4.5h=5.0h=5.5h=6.0h=6.5h=7.0h=8.0h=9.0h=10h=20h=30h=40h=50h=60h=70h=80h=90 -12 -8 -4 -10-505 h=0.4h=0.8h=1.0h=1.2h=1.6h=2.0h=2.5h=2.6h=2.7h=2.8h=2.9h=3.0h=3.1h=3.2h=3.3h=3.4h=3.5h=3.6h=3.7h=3.8h=3.9h=4.0h=4.1 h=4.3h=4.4h=4.5h=5.0h=5.5h=6.0h=6.5h=7.0h=8.0h=9.0h=10h=20h=30h=40h=50h=60h=70h=80h=90 -12 -8 -4 -10-505 h=0.4h=0.8h=1.0h=1.2h=1.6h=2.0h=2.5h=2.6h=2.7h=2.8h=2.9h=3.0h=3.1h=3.2h=3.3h=3.4h=3.5h=3.6h=3.7h=3.8h=3.9h=4.0h=4.1 h = 4.3 h = 4.4 h = 4.5 h = 5.0 h = 5.5 h = 6.0 h = 6.5 h = 7.0 h = 8.0 h = 9.0 h = 10 h = 20 h = 30 h = 40 h = 50 h = 60 h = 70 h = 80 h = 90 h = 0.4 l n δ t yp m i n ( h ) − l n δ t yp m i n ( h c ) (ln L )/ λ 𝒩/Λ h = 0.8 h = 1.0 h = 1.2 h = 1.6 h = 2.0 h = 2.5 h = 2.6 d e l o ca li ze d MBL h = 2.7 h = 2.8 h = 2.9 h = 3.0 h = 3.1 h = 3.2 h = 3.3 h = 3.4 h = 3.5 h = 3.6 h = 3.7 h = 3.8 h = 3.9 h = 4.0 h = 4.1 ( a ) ( b ) ( c )( e )( d ) from collapse from data collapse∝ [ A ln( h / h ) − γ c ] −1 ∝ ( h − h c ) −0.52 from data collapse∝ ( h c − h ) −0.5 collapsepower collapsepowerlog collapsepower collapsepowerlog γ collapsepower collapsepowerlog h=4.3h=4.4h=4.5h=5.0h=5.5h=6.0h=6.5h=7.0h=8.0h=9.0h=10h=20h=30h=40h=50h=60h=70h=80h=90 -12 -8 -4 -10-505 h=0.4h=0.8h=1.0h=1.2h=1.6h=2.0h=2.5h=2.6h=2.7h=2.8h=2.9h=3.0h=3.1h=3.2h=3.3h=3.4h=3.5h=3.6h=3.7h=3.8h=3.9h=4.0h=4.1 locdeloctot h c h c ν l o c χ / N d f localizeddelocalizedtotal h c − h h − h c l n Λ − λ h=4.3h=4.4h=4.5h=5.0h=5.5h=6.0h=6.5h=7.0h=8.0h=9.0h=10h=20h=30h=40h=50h=60h=70h=80h=90 -12 -8 -4 -10-505 h=0.4h=0.8h=1.0h=1.2h=1.6h=2.0h=2.5h=2.6h=2.7h=2.8h=2.9h=3.0h=3.1h=3.2h=3.3h=3.4h=3.5h=3.6h=3.7h=3.8h=3.9h=4.0h=4.1 h=4.3h=4.4h=4.5h=5.0h=5.5h=6.0h=6.5h=7.0h=8.0h=9.0h=10h=20h=30h=40h=50h=60h=70h=80h=90 -12 -8 -4 -10-505 h=0.4h=0.8h=1.0h=1.2h=1.6h=2.0h=2.5h=2.6h=2.7h=2.8h=2.9h=3.0h=3.1h=3.2h=3.3h=3.4h=3.5h=3.6h=3.7h=3.8h=3.9h=4.0h=4.1 h = 4.3 h = 4.4 h = 4.5 h = 5.0 h = 5.5 h = 6.0 h = 6.5 h = 7.0 h = 8.0 h = 9.0 h = 10 h = 20 h = 30 h = 40 h = 50 h = 60 h = 70 h = 80 h = 90 h = 0.4 l n δ t yp m i n ( h ) − l n δ t yp m i n ( h c ) (ln L )/ λ 𝒩/Λ h = 0.8 h = 1.0 h = 1.2 h = 1.6 h = 2.0 h = 2.5 h = 2.6 d e l o ca li ze d MBL h = 2.7 h = 2.8 h = 2.9 h = 3.0 h = 3.1 h = 3.2 h = 3.3 h = 3.4 h = 3.5 h = 3.6 h = 3.7 h = 3.8 h = 3.9 h = 4.0 h = 4.1 ( a ) ( b ) ( c )( e )( d ) from collapse from data collapse∝ [ A ln( h / h ) − γ c ] −1 ∝ ( h − h c ) −0.52 from data collapse∝ ( h c − h ) −0.5 collapsepower collapsepowerlog collapsepower collapsepowerlog FIG. 3. (a) Scaling plots for both delocalized (top) and MBL (bottom) regimes following Eq. (10) with h c = 4 .
2. Green and redcurves show the scaling functions f and g of Eq. (10) obtained from Taylor expansions close to h c fitted to the data [81, 90, 91]with two distinct (disorder-dependent) scaling parameters: Λ and λ for the two phases. (d) The non-ergodicity volume Λdiverges exponentially at criticality with an exponent compatible with ν d = 0 . λ diverges at criticality as a power-law (red line) with an exponent ν loc = 0 . A = 1 . h = 1, and γ c = 1 .
7. Panels (b) and(c) display the outcome of such scaling procedures obtained for different choices of h c . In (b) the MBL exponent ν loc smoothlyincreases with h c , while in (c) the total chi-squared statistic for the best fit χ divided by the number of degrees of freedom N df (see text and [81]) displays an O (1) minimum at h c = 4 .
2, where ν = 0 .
52. The freezing exponent γ (red symbols) from Fig. 2(b) is also shown in panel (b), and we get γ c = 1 . size. The scaling variable is the ratio between the Hilbertspace size N ≈ L / √ L and a disorder-dependent non-ergodicity volume Λ, which diverges exponentially at crit-icality: ln Λ ∼ ( h c − h ) − ν d with ν d ≈ .
5, see Fig. 3 (d).Here the fit of the scaling function to the data is equallygood for h c = 3 . h c = 4 .
2, as visible in Fig. 3 (c)where the chi-squared statistic divided by the number ofdegrees of freedom of the fit χ /N df is minimum [81].In contrast, the MBL regime ( h > h c ) is best describedby the function g (cid:0) ln Lλ (cid:1) displayed in Fig. 3 (a). This isa direct consequence (see [36, 49, 81]) of the power-lawdecay Eq. (S2), observed at criticality and in the MBLphase, which yields an MBL scaling function g ∝ (ln L ) /λ for large ln L (cid:29) λ . As an outcome, the scale λ is directlyrelated to the freezing exponent, and therefore with thetypical localization length, such that1 /λ ∝ γ c − γ ∝ ζ − c − ζ − . (11)The larger h c ≥ .
5, the better the goodness of fit for theMBL scaling, see Fig. 3 (c). Indeed, by definition the log-arithmic scaling perfectly describes data with algebraicbehavior such as δ typmin in the MBL phase. To correctlyestimate h c , we therefore need to sum the goodness offit from both ergodic and MBL regimes, see Fig. 3 (c).Then a bootstrap analysis of the data [81] gives a criticaldisorder strength h c = 4 . ≈
10% reflects the difficulties inherent to thisergodic-MBL transition. However, this very number is not decisive for our theo-retical understanding of the critical point [81]. As shownfor h c = 4 . γ c − γ ∝ ( h − h c ) ν loc , (12)with ν loc = 0 . γ c = 1 . L ) /λ implies that finite size effects in real-space areformally controlled by an exponentially diverging lengthscale exp( λ ) ∼ exp( h − h c ) − ν loc , thus confirming a KTscenario, compatible with the Harris bound. Consequences and discussion—
Our key-result isthat the extreme value statistics in the MBL regime givesa direct access to the typical localization length ζ , andtherefore the typical l -bits extension of the MBL states.Maximally polarized sites correspond to entanglementbottlenecks: nearly frozen spins are almost disentangledfrom the rest of the system, with an entanglement en-tropy cutting such sites given by S = − δ ln δ to leadingorder [92]. Consequently, δ controls the leading eigen-values of the entanglement spectrum, whose observedpower-law distribution [93, 94] is also governed by thefreezing exponent γ .Our analysis further indicates that the entire MBLregime, including the critical point, witnesses a chainbreaking mechanism at the thermodynamic limit, withfinite size effects controlled by a power-law behaviorEq. (S2). The power-law exponent, related to the typi-cal localization length γ ∝ ζ − , diverges logarithmicalyat strong disorder, and displays a jump at the transi-tion, with a singular critical behavior Eq. (12) in perfectagreement with a KT mechanism Eq. (2).Besides probing the transition universality class, spinfreezing has deep consequences for MBL physics. Firstly,it provides a simple picture accounting for the absenceof thermalization at the thermodynamic limit. Anotherimportant aspect concerns the recently discussed Hilbert-space fragmentation [95–98] of the MBL regime, whichhere is expected to naturally emerge from spin freezingupon increasing system sizes. In addition, our results arevery encouraging for the development of a perturbativedecimation method [99] discarding the strongly polarized sites. Coupled to exact methods this could provide quan-titative results at strong disorder for system sizes muchlarger than currently accessible, and thus potentially use-ful beyond one dimension. ACKNOWLEDGMENTS
This work is supported by the French NationalResearch Agency (ANR) under projects THER-MOLOC ANR-16-CE30-0023-02, MANYLOK ANR-18-CE30-0017 and GLADYS ANR-19-CE30-0013. Wegratefully acknowledge F. Alet and M. Dupont for col-laborations on related works. Our numerical calculationsstrongly benefited from the HPC resources provided byCALMIP (Grants No. 2018- P0677 and No. 2019-P0677)and GENCI (Grant No. 2018- A0030500225). [1] P. Jacquod and D. L. Shepelyansky, Phys. Rev. Lett. , 1837 (1997).[2] I. V. Gornyi, A. D. Mirlin, and D. G. Polyakov, Phys.Rev. Lett. , 206603 (2005).[3] D. M. Basko, I. L. Aleiner, and B. L. Altshuler, Annalsof Physics , 1126 (2006).[4] M. ˇZnidariˇc, T. Prosen, and P. Prelovˇsek, Phys. Rev.B , 064426 (2008).[5] A. Pal and D. A. Huse, Phys. Rev. B , 174411 (2010).[6] J. H. Bardarson, F. Pollmann, and J. E. Moore, Phys.Rev. Lett. , 017202 (2012).[7] R. Nandkishore and D. A. Huse, Ann. Rev. Cond. Matt. , 15 (2015).[8] J. Z. Imbrie, Journal of Statistical Physics , 998(2016).[9] D. A. Abanin and Z. Papi´c, Annalen der Physik ,1700169 (2017).[10] F. Alet and N. Laflorencie, Comptes Rendus Physique , 498 (2018).[11] D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn,Rev. Mod. Phys. , 021001 (2019).[12] M. Schreiber, S. S. Hodgman, P. Bordia, H. P. Lschen,M. H. Fischer, R. Vosk, E. Altman, U. Schneider, andI. Bloch, Science , 842 (2015).[13] J.-y. Choi, S. Hild, J. Zeiher, P. Schau, A. Rubio-Abadal, T. Yefsah, V. Khemani, D. A. Huse, I. Bloch,and C. Gross, Science , 1547 (2016).[14] J. Smith, A. Lee, P. Richerme, B. Neyenhuis, P. W.Hess, P. Hauke, M. Heyl, D. A. Huse, and C. Monroe,Nat. Phys. , 907 (2016).[15] A. M. Kaufman, M. E. Tai, A. Lukin, M. Rispoli,R. Schittko, P. M. Preiss, and M. Greiner, Science ,794 (2016).[16] A. Lukin, M. Rispoli, R. Schittko, M. E. Tai, A. M.Kaufman, S. Choi, V. Khemani, J. L´eonard, andM. Greiner, Science , 256 (2019).[17] P. Roushan, C. Neill, J. Tangpanitanon, V. M. Bastidas,A. Megrant, R. Barends, Y. Chen, Z. Chen, B. Chiaro,A. Dunsworth, A. Fowler, B. Foxen, M. Giustina, E. Jef-frey, J. Kelly, E. Lucero, J. Mutus, M. Neeley, C. Quin-tana, D. Sank, A. Vainsencher, J. Wenner, T. White, H. Neven, D. G. Angelakis, and J. Martinis, Science (2017).[18] K. Xu, J.-J. Chen, Y. Zeng, Y.-R. Zhang, C. Song,W. Liu, Q. Guo, P. Zhang, D. Xu, H. Deng, K. Huang,H. Wang, X. Zhu, D. Zheng, and H. Fan, Phys. Rev.Lett. , 050507 (2018).[19] B. Chiaro, C. Neill, A. Bohrdt, M. Filippone, F. Arute,K. Arya, R. Babbush, D. Bacon, J. Bardin, R. Barends,S. Boixo, D. Buell, B. Burkett, Y. Chen, Z. Chen,R. Collins, A. Dunsworth, E. Farhi, A. Fowler, B. Foxen,C. Gidney, M. Giustina, M. Harrigan, T. Huang,S. Isakov, E. Jeffrey, Z. Jiang, D. Kafri, K. Kechedzhi,J. Kelly, P. Klimov, A. Korotkov, F. Kostritsa, D. Land-huis, E. Lucero, J. McClean, X. Mi, A. Megrant,M. Mohseni, J. Mutus, M. McEwen, O. Naaman,M. Neeley, M. Niu, A. Petukhov, C. Quintana, N. Ru-bin, D. Sank, K. Satzinger, A. Vainsencher, T. White,Z. Yao, P. Yeh, A. Zalcman, V. Smelyanskiy, H. Neven,S. Gopalakrishnan, D. Abanin, M. Knap, J. Martinis,and P. Roushan, arXiv:1910.06024 (2019).[20] M. Ovadia, D. Kalok, I. Tamir, S. Mitra, B. Sac´ep´e,and D. Shahar, Scientific Reports , 13503 (2015).[21] A. De Luca and A. Rosso, Phys. Rev. Lett. , 080401(2015).[22] K. X. Wei, C. Ramanathan, and P. Cappellaro, Phys.Rev. Lett. , 070501 (2018).[23] D. J. Luitz, N. Laflorencie, and F. Alet, Phys. Rev. B , 081103 (2015).[24] F. Pietracaprina, N. Mac´e, D. J. Luitz, and F. Alet,SciPost Physics , 045 (2018).[25] T. Devakul and R. R. Singh, Phys. Rev. Lett. ,187201 (2015).[26] E. V. H. Doggen, F. Schindler, K. S. Tikhonov, A. D.Mirlin, T. Neupert, D. G. Polyakov, and I. V. Gornyi,Phys. Rev. B , 174202 (2018).[27] R. K. Panda, A. Scardicchio, M. Schulz, S. R. Taylor,and M. ˇZnidariˇc, EPL , 67003 (2020).[28] J. Suntajs, J. Bonca, T. Prosen, and L. Vidmar,arXiv:1905.06345 (2019).[29] T. Chanda, P. Sierant, and J. Zakrzewski, Phys. Rev.B , 035148 (2020). [30] D. A. Abanin, J. H. Bardarson, G. De Tomasi,S. Gopalakrishnan, V. Khemani, S. A. Parameswaran,F. Pollmann, A. C. Potter, M. Serbyn, and R. Vasseur,arXiv:1911.04501 (2019).[31] P. Sierant, D. Delande, and J. Zakrzewski,arXiv:1911.06221 (2019).[32] J. A. Kjall, J. H. Bardarson, and F. Pollmann, Phys.Rev. Lett. , 107204 (2014).[33] F. Pietracaprina, V. Ros, and A. Scardicchio, Phys.Rev. B , 054201 (2016).[34] S. Schiffer, J. Wang, X.-J. Liu, and H. Hu, Phys. Rev.A , 063619 (2019).[35] H.-Z. Xu, S.-Y. Zhang, Z.-Y. Rao, Z. Zhou, G.-C. Guo,and M. Gong, arXiv:1910.06601 (2019).[36] N. Mac´e, F. Alet, and N. Laflorencie, Phys. Rev. Lett. , 180601 (2019).[37] R. Abou-Chacra, D. J. Thouless, and P. W. Anderson,J. Phys. C: Solid State Phys. , 1734 (1973).[38] A. D. Mirlin and Y. V. Fyodorov, Phys. Rev. Lett. ,526 (1994).[39] C. Monthus and T. Garel, J. Phys. A , 075002 (2008).[40] G. Biroli, A. Ribeiro-Teixeira, and M. Tarzia,arXiv:1211.7334 (2012).[41] A. De Luca, B. L. Altshuler, V. E. Kravtsov, andA. Scardicchio, Phys. Rev. Lett. , 046806 (2014).[42] K. S. Tikhonov, A. D. Mirlin, and M. A. Skvortsov,Phys. Rev. B , 220203 (2016).[43] K. S. Tikhonov and A. D. Mirlin, Phys. Rev. B ,184203 (2016).[44] G. Biroli and M. Tarzia, Phys. Rev. B , 201114(R)(2017).[45] I. Garc´ıa-Mata, O. Giraud, B. Georgeot, J. Martin,R. Dubertrand, and G. Lemari´e, Phys. Rev. Lett. ,166801 (2017).[46] G. Biroli and M. Tarzia, arXiv:1810.07545 (2018).[47] V. E. Kravtsov, B. L. Altshuler, and L. B. Ioffe, Ann.Phys. (Amsterdam) , 148 (2018).[48] K. S. Tikhonov and A. D. Mirlin, Phys. Rev. B ,024202 (2019).[49] I. Garc´ıa-Mata, J. Martin, R. Dubertrand, O. Giraud,B. Georgeot, and G. Lemari´e, Phys. Rev. Research ,012020 (2020).[50] M. Tarzia, arXiv:2003.11847 (2020).[51] A. B. Harris, J. Phys. C: Solid State Phys. , 1671(1974).[52] A. Chandran, C. R. Laumann, and V. Oganesyan,arXiv:1509.04285 (2015).[53] Note that the very applicability of this bound has beencriticized by C. Monthus in Ref. [107].[54] R. Vosk and E. Altman, Phys. Rev. Lett. , 067204(2013).[55] D. A. Huse, R. Nandkishore, and V. Oganesyan, Phys.Rev. B , 174202 (2014).[56] D. Pekker, G. Refael, E. Altman, E. Demler, andV. Oganesyan, Phys. Rev. X , 011052 (2014).[57] R. Vasseur, A. C. Potter, and S. A. Parameswaran,Phys. Rev. Lett. , 217201 (2015).[58] E. Altman and R. Vosk, Ann. Rev. Cond. Matt. , 383(2015).[59] R. Vosk, D. A. Huse, and E. Altman, Phys. Rev. X ,031032 (2015).[60] A. C. Potter, R. Vasseur, and S. Parameswaran, Phys.Rev. X , 031033 (2015).[61] P. T. Dumitrescu, R. Vasseur, and A. C. Potter, Phys. Rev. Lett. , 110604 (2017).[62] M. Schiro and M. Tarzia, Phys. Rev. B , 014203(2020).[63] T. Thiery, F. Huveneers, M. Mller, and W. De Roeck,Phys. Rev. Lett. , 140601 (2018).[64] T. Thiery, M. M¨uller, and W. De Roeck,arXiv:1711.09880 (2017).[65] A. Goremykina, R. Vasseur, and M. Serbyn, Phys. Rev.Lett. , 040601 (2019).[66] P. T. Dumitrescu, A. Goremykina, S. A. Parameswaran,M. Serbyn, and R. Vasseur, Phys. Rev. B , 094205(2019).[67] A. Morningstar and D. A. Huse, Phys. Rev. B ,224205 (2019).[68] M. Serbyn, Z. Papi´c, and D. A. Abanin, Phys. Rev.Lett. , 127201 (2013).[69] L. Rademaker, M. Ortu˜no, and A. M. Somoza, Annalender Physik , 1600322 (2017).[70] J. Z. Imbrie, V. Ros, and A. Scardicchio, Annalen derPhysik , 1600278 (2017).[71] See [49] for a similar behavior at the Anderson transitionon random graphs.[72] V. Khemani, F. Pollmann, and S. L. Sondhi, Phys. Rev.Lett. , 247204 (2016).[73] S. P. Lim and D. N. Sheng, Phys. Rev. B , 045111(2016).[74] M. Dupont and N. Laflorencie, Phys. Rev. B , 020202(2019).[75] M. Hopjan and F. Heidrich-Meisner, arXiv:1912.09443(2019).[76] J. M. Deutsch, Phys. Rev. A , 2046 (1991).[77] M. Srednicki, Phys. Rev. E , 888 (1994).[78] L. D’Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol,Advances in Physics , 239 (2016).[79] D. J. Luitz and Y. B. Lev, Phys. Rev. Lett. , 170404(2016).[80] D. J. Luitz and Y. Bar Lev, Annalen der Physik ,1600350 (2017).[81] For details see supplemental material.[82] S. N. Majumdar, A. Pal, and G. Schehr, Physics Re-ports , 1 (2020).[83] C. L. Kane and M. P. A. Fisher, Phys. Rev. Lett. ,1220 (1992).[84] G. Lemari´e, I. Maccari, and C. Castellani, Phys. Rev.B , 054519 (2019).[85] A. Chandran, I. H. Kim, G. Vidal, and D. A. Abanin,Phys. Rev. B , 085425 (2015), 00112.[86] C. Monthus, J. Stat. Mech. , 033101 (2016).[87] L. Rademaker and M. Ortu˜no, Phys. Rev. Lett. ,010404 (2016).[88] G. De Tomasi, F. Pollmann, and M. Heyl, Phys. Rev.B , 241114 (2019).[89] M. Dupont, N. Mac´e, and N. Laflorencie, Phys. Rev. B , 134201 (2019).[90] K. Slevin and T. Ohtsuki, Phys. Rev. Lett. , 382(1999).[91] A. Rodriguez, L. J. Vasquez, K. Slevin, and R. A.R¨omer, Phys. Rev. B , 134209 (2011).[92] E. S. Sørensen, M.-S. Chang, N. Laflorencie, and I. Af-fleck, Journal of Statistical Mechanics: Theory and Ex-periment , P08003 (2007).[93] A. Samanta, K. Damle, and R. Sensarma,arXiv:2001.10198 (2020).[94] R. Pal and A. Lakshminarayan, arXiv:2002.00682 (2020).[95] S. Roy, D. E. Logan, and J. T. Chalker, Phys. Rev. B , 220201 (2019).[96] S. Roy, J. T. Chalker, and D. E. Logan, Phys. Rev. B , 104206 (2019).[97] G. De Tomasi, D. Hetterich, P. Sala, and F. Pollmann,Phys. Rev. B , 214313 (2019).[98] F. Pietracaprina and N. Laflorencie, arXiv:1906.05709(2019).[99] F. Igloi and C. Monthus, Physics Reports , 277(2005).[100] H. Tasaki, Phys. Rev. Lett. , 1373 (1998).[101] W. Beugeling, R. Moessner, and M. Haque, Phys. Rev.E , 042112 (2014).[102] T. N. Ikeda, Y. Watanabe, and M. Ueda, Phys. Rev. E , 021130 (2011).[103] M. Rigol and M. Srednicki, Phys. Rev. Lett. , 110601(2012).[104] T. N. Ikeda and M. Ueda, Phys. Rev. E , 020102(2015).[105] D. J. Luitz, Phys. Rev. B , 134201 (2016).[106] T.-C. Lu and T. Grover, Phys. Rev. E , 032111(2019).[107] C. Monthus, Entropy , 122 (2016). Supplemental Material for ”Chain breaking and Kosterlitz-Thouless scaling at themany-body localization transition”
In this supplemental material we provide addi-tional informations regarding: 1. The power-lawdivergence of the distributions P ( − | m z | ) , in par-ticular finite-size effects. 2. The analytical deriva-tion in connection with free-fermions. 3. Thefinite-size scaling analysis of the MBL transition. S1. POWER-LAW DIVERGENCE OF THEDISTRIBUTIONS: FINITE-SIZE EFFECTS
In the MBL regime, the deviation form perfect polar-ization δ = − | m z | displays power-law tails in the dis-tributions, according to P ( δ ) ∝ δ − γ ( δ → δ : P (ln δ ) ∝ exp (cid:18) ln δγ (cid:19) . (S1)Histograms of ED data for L = 8 , , , . . . ,
22 areshown in Fig. S1 (a) in the MBL regime ( h = 7). Inthe left panel, P ( δ ) is displayed in log-log scale, and apower-law behavior is visible, but does not correspond tothe extreme value tail which we aim at exploring in thiswork. In order to better see such a tail, we show P (ln δ )in the right panel where an exponential tail is clearly visi-ble, thus targeting much smaller values of δ , and thus theextreme values. However, the apparent prefactor 1 /γ ex-tracted from the exponential tail shows important finite-size effects, as reported in the inset of Fig. S1 (a) wherewe have compared this estimate with the one extractedfrom the extreme value scaling δ min ( L ) ∼ L − γ , (S2)which has almost no finite-size effects. Considering thelimited system sizes available to shift-invert ED tech-niques for the interacting problem, we have also exploredthis effect for the non-interacting case of many-body free-fermions, described by the random-field XX model H = L (cid:88) i =1 (cid:0) S xi S xi +1 + S yi S yi +1 − h i S zi (cid:1) (S3)in the S z tot = 0 sector. Using standard free-fermion tech-niques, one can easily access to large systems, typically L ∼ without particular numerical effort. Results areshown in Fig. S1 (b) for the very same system sizes as forthe MBL case, with the following additional lengths L =24 , , , , , , , , , , h = 4 ensures that non-interacting fermionshave short localization lengths. Interestingly, we clearlyobserve a slow convergence of the tail exponent 1 /γ , ascompared to the extreme value Eq. (S2) which, like for the MBL case, shows almost no finite-size effects. There-fore, we clearly see that focusing on the extreme value δ min ( L ) offers the most direct way to get the asymptoticvalue γ , as compared to the distribution tail. -20 -15 -10 -5 010 -4 -3 -2 -1 -20 -15 -10 -5 010 -4 -3 -2 -1 δ ln δ P ( δ ) P ( l n δ ) -20 -15 -10 -5 010 -4 -3 -2 -1 -20 -15 -10 -5 010 -4 -3 -2 -1 -20 -15 -10 -5 010 -4 -3 -2 -1 from distributionsfrom δ typmin ( L ) ∼ L − γ average from L ∈ [12,22] γ L -20 -15 -10 -5 010 -4 -3 -2 -1 -20 -15 -10 -5 010 -4 -3 -2 -1 -20 -15 -10 -5 010 -4 -3 -2 -1 from distributionsfrom δ typmin ( L ) ∼ L − γ average from L ∈ [12,1024] P ( δ ) P ( l n δ ) δ ln δ ( a ) MBL h = ( b ) free fermions h = L γ FIG. S1. Distributions of the deviations from perfect po-larization δ = 1 / − | m z | for (a) MBL at h = 7, and (b)non-interacting XX model Eq. (S3) at h = 4. Log-log ploton the left panels focus on the non-universal power-law be-havior for P ( δ ), diverging with an exponent ≈ P (ln δ ), see Eq. (S1), which fo-cuses on the universal tail. For the MBL case (a), the slope1 /γ slowly decreases towards its asymptotic value (thick blueline), as visible in the inset (black symbols) where 1 /γ es-timates from the extreme value scaling are also shown forcomparison: either from 3-point fits (red symbols) of δ typmin ( L )to the form Eq. (S2), or from a global fit (thick blue line)including all sizes L ≥
12. The slow convergence is better vis-ible for the non-interacting problem (b) with the same systemsizes as for the MBL case, together with the additional sizes L = 24 , , , , , , , , , , S2. ANALYTICAL DERIVATION IN THEANDERSON BASISs1. Fermionic representation of the spin problem
The spin- XXZ Hamiltonian H = L (cid:88) i =1 (cid:104) (cid:0) S + i S − i +1 + S − i S + i +1 (cid:1) − h i S zi + ∆ S zi S zi +1 (cid:105) , (S4)can be decomposed in two terms: H = H + V . H is the random-field XX model, Eq. (S3), whichcorresponds to free-fermions (using the Jordan-Wignertransformation) in a random potential H = (cid:88) i (cid:104) (cid:16) c † i c i +1 + c † i +1 c i (cid:17) − h i n i (cid:105) . (S5)The interaction terms read V = ∆ L (cid:88) i n i n i +1 + C , (S6)where C is an irrelevant constant. The non-interactingpart Eq. (S5) is diagonalized by b k = (cid:80) Li =1 φ ki c i ( φ k beingthe single particle orbitals, Anderson localized for anyfinite disorder in 1D), yielding H = L (cid:88) k =1 (cid:15) k b † k b k . In this Anderson basis, the interaction term Eq. (S6) is V = (cid:88) k,l,m,n V k,l,m,n b † k b l b † m b n , with V k,l,m,n = ∆ L (cid:88) i =1 φ ki φ li φ mi +1 φ ni +1 , (S7)where we have assumed real orbitals. As proposed in themain text, we decompose the interacting part in 4 terms: V = (cid:88) k V (1) k n k + (cid:88) k (cid:54) = l V (2) k,l n k n l + (cid:88) k (cid:54) = l (cid:54) = m V (3) k,l,m n k b † l b m + (cid:88) k (cid:54) = l (cid:54) = m (cid:54) = n V (4) k,l,m,n b † k b l b † m b n . (S8)The first two terms V (1 , are diagonal, and can be in-terpreted as a first approximation for the so-called l -bitHamiltonian, while V (3 , are off-diagonal. s2. Strong disorder limit Below, we show that off-diagonal terms can be ignoredat strong disorder. In such a limit, the Hamiltonian re-mains diagonal in the Anderson basis, i.e. H diag = (cid:88) k (cid:16) (cid:15) k + V (1) k (cid:17) n k + (cid:88) k,l V (2) k,l n k n l . (S9)In this case, the maximum of the fermionic density canbe readily obtained. We first use a simplified expressionfor the non-interacting orbitals, assuming the followingexponential form | φ ki | = tanh (cid:18) ξ (cid:19) exp (cid:18) − | i − i k | ξ (cid:19) . (S10)The particle density at a given site i given by (cid:104) n i (cid:105) = (cid:88) occupied k | φ ki | , (S11)is maximal in the middle of a region of (cid:96) max consecutiveoccupied orbitals. Indeed, for i ∼ (cid:96) max / n max (cid:39) (cid:96) max / (cid:88) r = − (cid:96) max / tanh (cid:18) ξ (cid:19) exp (cid:18) − | r | ξ (cid:19) , (S12)where we have neglected the contribution of orbitals out-side of the occupied region of size (cid:96) max . This yields forthe deviation δ min = 1 − n max the following finite-sizescaling δ min ∝ L − γ with γ ≈ ξ ln 2 ≈ ln h ln 2 . (S13)In the above expression we have used the fact thatat strong disorder ξ ≈ (2 ln h ) − . Indeed, a simpleperturbative expansion of any wavefunction awayfrom its localization center yields amplitudes vanishing ∼ h − r , where r is the distance to the localization center.We now provide simplified expressions for variousterms in Eq. (S8) in the limit h (cid:29)
1. In particular wejustify why it is safe to ignore off-diagonal terms whichvanish at large h . s3. Diagonal terms In the large system size limit we have for the one-bodyterm V (1) k = ∆ tanh (cid:16) ξ k (cid:17) sinh (cid:16) ξ k (cid:17) ∝ ∆ exp (cid:18) − ξ k (cid:19) ∝ ∆ /h when h (cid:29) . (S14)0The two-body contributions, assuming constant ξ k (thisis justified at strong disorder where the distribution P ( ξ )is strongly peaked) reads V (2) k,l = ∆ tanh (cid:18) ξ (cid:19) (cid:104) r exp (cid:18) − r − ξ (cid:19) + exp (cid:16) − rξ (cid:17) sinh ξ − (cid:105) ∝ ∆ rh r − when h (cid:29) , (S15)where r = | i k − i l | ≥ k and l . Therefore the two-body interaction,which reads (cid:88) k (cid:54) = l V (2) k,l n k n l ≈ ∆ (cid:88) k (cid:16) n k n k +1 + 2 h n k n k +2 (S16)+ 3 h n k n k +3 + · · · (cid:17) , is clearly dominated by the nearest neighbor repulsion ∆. s4. Off-diagonal terms We first discuss the three-body terms of the form V (3) k,l,m n k b † l b m . From Eq. (S7) we see that the amplitude can be randomlypositive or negative. Nevertheless, one can easily antic-ipate that V (3) k,l,m will be dominated by maximally over-laping orbitals, more precisely by three nearest-neighbororbitals k, l, m , such that | i k − i l | = | i l − i m | = 1. Insuch a case (cid:12)(cid:12)(cid:12) V (3) k,k +1 ,k +2 (cid:12)(cid:12)(cid:12) ∝ ∆ h when h (cid:29) , (S17)where each orbital k is labelled such that φ ki ∼ exp (cid:16) − | i − k | ξ (cid:17) , meaning that its localization center i k = k .In the more generic case, we have (cid:12)(cid:12)(cid:12) V (3) k,k + r,k + r (cid:48) (cid:12)(cid:12)(cid:12) ∝ ∆ h r + r (cid:48) − when h (cid:29) . (S18)A similar reasoning applies to the 4-body terms V (4) k,l,m,n b † k b l b † m b n , with (cid:12)(cid:12)(cid:12) V (4) k,k + r,k + r (cid:48) ,k + r (cid:48)(cid:48) (cid:12)(cid:12)(cid:12) ∝ ∆ h r (cid:48) + r (cid:48)(cid:48) − r − if h (cid:29) . (S19)This contribution is dominated by situations where the4 orbitals are nearest neighbors, i.e. when r = 1 , r (cid:48) =2 , r (cid:48)(cid:48) = 3, thus giving (cid:12)(cid:12)(cid:12) V (4)max (cid:12)(cid:12)(cid:12) ∝ h . (S20)When the relative distances between orbitals increases,the amplitudes V (4) rapidly vanishes. Both 3-body and4-body processes are illustrated in Fig. S2. b † k + r b † k + r ′ b † k b † k + r ′ ′ ⟵ r ⟶ ⟵ − − − − − − r ′ ′ − − − − − − ⟶ b † k + r b † k + r ′ ⟵ − − − − r ′ − − − − ⟶ ⟵ r ⟶ ⟵ − − − − r ′ − − − − ⟶( a ) 3 − body process( b ) 4 − body process | ϕ k | | ϕ k | k k FIG. S2. Schematic picture of the 3- and 4-body processes:3 rd and 4 th lines in Eq. (S8). S3. FINITE-SIZE SCALING ANALYSIS OF THEMBL TRANSITION
In this section, we detail our analysis by finite-size scal-ing of the polarization data. s1. Non-standard scaling approach
The scaling theory of the MBL transition is subtle fora number of reasons:1. The finite-size scaling analysis of the MBL transi-tion was first made as if it were a standard second-order phase transition, with a behavior of the type y = y c F ( L/ξ ) (where y is the observable and y c = y ( L, h c )) with ξ a characteristic length diverg-ing algebraically at the transition ξ ∼ | h − h c | − ν .Many works converged on a critical exponent ν ≈ ξ ∼ e a | h − h c | − / , with an RG flow that resembles that ofthe Kosterlitz-Thouless transition. It is well knownthat finite-size scaling analysis of KT type transi-tions is particularly delicate.3. An analogy exists between the MBL transition andthe Anderson transition on random networks, avery rich problem, but nevertheless simpler, e.g. al-lowing analytical predictions [37–49]. The scalingtheory of the Anderson transition has only beenclarified recently and is particularly subtle [45, 49].1Thus according to the observables considered, thereis a so-called “ linear ” scaling in y = y c F ( L/ξ )where L is the linear size of the random graph (ie itsdiameter), or a “volumic” scaling in y = y c G ( N / Λ)where N denotes the volume of the graph (i.e. thenumber of sites) and Λ is a characteristic volumediverging exponentially at the transition. Indeed,on the random graphs considered, the volume in-creases exponentially with the system size L andthe linear or volumic scalings are not equivalent(contrary to the case of finite dimension). Differ-ent scalings have been found on both sides of thetransition and two critical exponents exist in thelocalized phase: the average localization length di-verges as ξ ∼ ( W − W c ) − ( where W is the disorderstrength) while the typical localization length fol-lows a behavior: ζ − = ζ − c + c √ W − W c (i.e. acritical exponent 1 /
2) identical to the critical be-havior recently predicted by phenomenological RGsfor the MBL transition [65–67].4. Recently, the MBL transition has been analyzed inthe Hilbert space where critical and MBL regionshave been shown to host multifractal many-bodystates occupying a sub-volumic Hilbert space region N D , with D < h -dependent fractal dimen-sion [36]. This type of behavior is associated witha linear scaling. In the ETH phase on the contrary,the scaling is found to be volumic , indicating thatthe many-body states are ergodic in the Hilbertspace at the thermodynamic limit. Supporting therandom network analogy, the linear scaling of theMBL region is associated to a diverging length scale ξ ∼ ( W − W c ) − . , while on the ETH side, the vo-lumic scaling is associated to a diverging volumeΛ ∼ e a | h − h c | − . .These observations invite us to consider non-standardscaling hypotheses: scaling in N / Λ, in
L/ξ or even inln
L/λ , which can be different on either side of the tran-sition. We will explain below the preferred scaling hy-potheses and then describe the method we used to val-idate or not these hypotheses. Finally we will describethe results of these analyzes.The phenomenological RG studies suggest very differ-ent critical and MBL behaviors for “average” observ-ables, dominated by rare events (thermal bubbles), andfor “typical” observables, controlled by the typical local-ization length ζ [63–67]. An idea to access the typicalquantities is to consider observables that are the leastaffected by rare thermal bubbles. Maximally polarizedsites, being found in the bulk of localized regions, wellseparated from thermal inclusions, make up for a goodcandidate to probe typical properties. This is why wefocused on δ typmin . s1. Scalings in the critical and MBL regions The power-law behavior δ typmin ∼ L − γ ∼ exp( − γ ln L )observed at the largest sizes and predicted analyticallyat strong disorder suggests a scaling in ln L/λ , sinceln (cid:34) δ typmin δ typmin ( h c ) (cid:35) ∼ − ( γ − γ c ) ln L ∼ g (cid:18) ln Lλ (cid:19) , (S21)with g ( X ) ≈ A − A X for large ln L (cid:29) λ .Moreover, the observed power-law N D for the eigen-states multifractality on the Hilbert space [36] (see also[45, 49]), which has been shown to be associated with thescaling ln N /ξ indicates, with L playing the role of N ,that the same hypothesis is also valid here.Note that we have observed a smooth dependenceof the prefactor A ( h ) on the algebraic law for δ typmin ≈ A ( h ) L − γ ( h ) . Such dependence could be described byan irrelevant correction to the logarithmic scaling inthe form g (ln L/λ ) + h (ln L/λ ) / ln L with h (0) = 0 and h ( X ) ≈ CX for large ln L (cid:29) λ , C a constant. Due tothe limited range of variation of L , we did not take intoaccount that correction. s2. Scalings in the delocalized region In the ETH regime, canonical typicality [100] guaran-tees for local observables such as maximal polarization δ typmin self-averaging with a variance ∝ N − / [101–106].We can therefore expect that this observable obeys a vo-lumic scaling with N / Λ, similarly to the participationentropies on the Hilbert space [36]. s2. Controlled finite-size scaling approach
To quantitatively test the compatibility of the nu-merical data with these different scaling hypotheses, wehave adapted the controlled finite-size scaling approach[90, 91] as detailed below: • We first assume a value of h c and consider thedata for the observable ln[ δ typmin /δ typmin ( h c )] along withtheir uncertainties given by (cid:112) σ + σ c , where σ ( σ c ,respectively) is the standard deviation of ln[ δ typmin ](ln[ δ typmin ( h c )]). We aim at asserting whether thesedata can be fitted by a function of the form:ln (cid:34) δ typmin δ typmin ( h c ) (cid:35) = G ( ρ S /ν ) , (S22)where S can be either ln L , L or N , ρ ∼ ( h − h c ) close to h c , and ν is a critical exponent to bedetermined by the fitting procedure. Importantly,we will fit the data for h > h c independently fromthe data for h < h c , as they can follow differentscalings.2 h c χ / N d f (ln L )/ λ L / ξ N / Λ Localized h c Delocalized
FIG. S3. Quantitative estimate of the compatibility betweenthe numerical data and the scaling assumptions. The local-ized ( h > h c ) and delocalized ( h < h c ) data have been fittedindependently by Eq. (S23) with S = ln L , L or N . Thevalue χ of the chi-squared statistic for the best fit divided bythe number of degrees of freedom N df is plotted as a function h c . χ/N df should be of order one for an acceptable fit. Vo-lumic scaling N / Λ prevails in the delocalized regime, whilethe data in the localized regime are compatible with a scalingas ln
L/λ . The best value of h c , corresponding to the min-imum of χ/N df , corresponds to h c = 3 . h c = 4 . h c (see text). If we combine thelocalized and delocalized data, the optimal h c is h c = 4 . • A controlled finite-size scaling analysis [90, 91] con-sists in a Taylor development of the scaling functionaround h = h c : G ( ρ S /ν ) = n (cid:88) j =0 a j ρ j S j/ν , (S23)with ρ ( h ) = ( h − h c ) + m (cid:88) j =2 b j ( h − h c ) j . (S24)The orders of expansion have been set to n = 5and m = 3. The total number of parameters to bedetermined in the fit is N p = n + m +1 (including ν ).We assessed the goodness of the fit by calculatingthe value χ of the chi-squared statistic for the bestfit divided by the number of degrees of freedom N df = N D − N p where N D is the number of data,which should be of order one for an acceptable fit. • We then tested systematically different values of h c and different scaling hypotheses, i.e. differentchoices S = ln L , L or N for the localized h > h c and delocalized h < h c phases. The best value of h c corresponds to the minimum of χ/N df , takinginto account only the localized or delocalized data,or all the data by considering χ tot /N totdf ≡ ( χ loc + χ deloc ) / ( N locdf + N delocdf ). • To determine the uncertainties on h c and ν , we haveused a bootstrap procedure. From the data, wegenerated 100 synthetic data sets by sampling ran-domly from normal distributions centered on thetrue data and with standard deviations given bythe σ s. We then fitted these data sets and calcu-lated the χ/N df . The best value of h c for each syn-thetic data set was determined and the uncertaintycorresponds to the standard deviation of these h c s.The fluctuations of ν within the data sets was alsorecorded. s3. Results We present here additional results to the ones reportedin the letter. The analysis by finite-size scaling has beendone without excluding any size or value of the disorder.The sizes are between L = 8 and L = 22 and the disorderbetween h = 0 . h = 90.The figure S3 represents χ/N df as a function of h c inthe localized ( h > h c ) and delocalized ( h < h c ) regimes.This quantitative test of the different scaling hypothe-ses confirm our expectation that volumic scaling N / Λprevails in the delocalized regime, while the data in thelocalized regime are compatible with a scaling as ln
L/λ .It is not surprising to see this ln
L/λ scaling in thelocalized phase favors large values of h c , since it is byconstruction good for algebraic data, which is the casefor large values of h .On the delocalized regime, the volumic scaling has aoptimum around h c = 3 . h c = 4 .
2, with a value of χ deloc /N delocdf = 1 . h .If we combine the localized and delocalized data, theoptimal h c is h c = 4 . χ tot /N totdf = 1 . h .This scaling procedure assumes a power law divergenceof the characteristic volume Λ in the delocalized phase.This is done for practical reasons. We have not beenable to implement an exponential divergence in this pro-cedure, but we have added non-linear corrections for ρ to allow for a more complex behavior than just a strictpower-law divergence. The best fit gives a very largevalue of the critical exponent ν d ≈ . which indicates thatthe divergence of Λ close to h c is most probably an expo-nential divergence. Indeed, the figure 3(d) shows that thedata for Λ are compatible with Λ ∼ e a ( h c − h ) − . . Such abehavior has been observed for the scaling of the partic-ipation entropy in the Hilbert space [36], and is knownto control the Anderson transition on random graphs in3the delocalized phase [45, 49]. Importantly, the volumicscaling implies that the whole delocalized phase obeysETH at the thermodynamic limit.The bootstrap procedure gives for the delocalized data( h < h c ) an average (cid:104) h c (cid:105) = 3 .
96 and a standard devia-tion of σ h c = 0 .
22. For the localized data, (cid:104) h c (cid:105) = 6 . σ h c = 0 .
64 which confirms that the scaling as ln
L/λ is not able determine the critical value of h c . How-ever, combining the localized and delocalized data gives (cid:104) h c (cid:105) = 4 . σ h c = 0 .
32. We therefore combine thedelocalized and total estimations to give the estimation h c = 4 . λ ∼ ( h − h c ) − ν loc corresponds to values of the critical exponent ν loc ≈ . h c increases (see fig-ure 3(b)). The uncertainty on h c translates into alarger uncertainty on ν loc than that of the fluctuationsof ν loc found from the bootstrap procedure. We find ν loc = 0 . e b ( h − h c ) − . close to the transition. Together with thealgebraic critical behavior δ typmin ( h c ) ∼ L − γ c and the rela-tion found between γ and the typical localization length ζ , Eq.(9) of the letter, this also confirms that the typicallocalization length reaches a finite value at the transi-tion with a square-root singularity, as predicted by phe- nomenological RGs [63–67]. s4. Non-parametric finite-size scaling In parallel with the controlled finite-size scaling ap-proach whose results are presented above and in the maintext, we have performed a non-parametric finite-size scal-ing. This approach does not assume a parametrizationfor the scaling functions f and g (see Eq. (10) of the maintext) and scaling parameters Λ( h ) and λ ( h ), but insteadfinds the best collapse of the data by optimizing an objec-tive function which quantifies how close the data pointsare from being collapsed onto a unique curve. The ob-jective function we chose to optimize is Spearman’s rankcorrelation coefficient, R , which is a measure of how closea cloud of points are from forming the graph of a mono-tonic function. This approach did not allow us to quan-titatively determine the best value of h c nor the uncer-tainties on the critical exponent ν loc . However, assuming h c = 4 .
2, we obtain scaling results consistent with theparametric approach, as Fig. S4 shows. In particular, wefind a power-law divergence of the MBL scaling parame-ter with an exponent ν loc ≈ .
44, close to the previouslyreported ν loc = 0 .
52. In the ETH region, we observea behavior compatible with an exponentially divergentnon-ergodicity volume with exponent ν d = 0 . . . . . . . L ) /λ − − l n δ t y p m i n ( h ) − l n δ t y p m i n ( h c ) − − −
10 0 10ln N L − ln Λ0.1 1 h c − h l n Λ ( h c − h ) − . h − h c λ ( h − h c ) − . h FIG. S4. Top figure: Scaling plots for both delocalized (top)and MBL (bottom) regimes with h c = 4 .
2. Best collapse isobtained via a non-parametric procedure (see text). Bottomfigures: Λ (left) and λ (right) obtained form the collapse.The non-ergodicity volume Λ diverges exponentially with anexponent ν d ≈ . λ diverges at criticality as a power-law (red line) with anexponent ν loc ≈ ..