A constructive theory of the numerically accessible many-body localized to thermal crossover
AA constructive theory of the numerically accessible many-body localized to thermalcrossover
P. J. D. Crowley ∗ and A. Chandran Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA Department of Physics, Boston University, Boston, MA 02215, USA (Dated: February 9, 2021)The many-body localised (MBL) to thermal crossover observed in exact diagonalisation studiesremains poorly understood as the accessible system sizes are too small to be in an asymptoticscaling regime. We develop a model of the crossover in short 1D chains in which the MBL phase isdestabilised by the formation of many-body resonances. The model reproduces several propertiesof the numerically observed crossover, including an apparent correlation length exponent ν = 1 ,exponential growth of the Thouless time with disorder strength, linear drift of the critical disorderstrength with system size, scale-free resonances, apparent /ω dependence of disorder-averagedspectral functions, and sub-thermal entanglement entropy of small subsystems. In the crossover,resonances induced by a local perturbation are rare at numerically accessible system sizes L whichare smaller than a resonance length λ . For L (cid:29) √ λ , resonances typically overlap, and this modeldoes not describe the asymptotic transition. The model further reproduces controversial numericalobservations which Refs. [1, 2] claimed to be inconsistent with MBL. We thus argue that the numericsto date is consistent with a MBL phase in the thermodynamic limit. I. INTRODUCTION
Interacting one-dimensional quantum systems generi-cally many-body localise (MBL) in the presence of strongdisorder. Local subsystems of a MBL system do not ther-malise; they instead retain memory of their initial condi-tions indefinitely. MBL thus provides a remarkable coun-terexample to the ergodic hypothesis, the cornerstone ofquantum statistical mechanics [3–8], and allows for exoticquantum orders at finite energy densities [9–28].Statistical descriptions of both the thermal and MBLphases have been corroborated by numerical studies.Specifically, the thermal phase is found to obey the eigen-state thermalisation hypothesis (ETH) [29–37], whereasthe MBL phase violates the ETH and is instead charac-terised by a complete set of quasi-local conserved quan-tities (or l-bits) [38–44].However, theoretical descriptions and numerical obser-vations of the MBL-thermal transition remain at oddswith one another. Phenomenological models suggestthat the transition has Kosterlitz-Thouless-type scal-ing [45–47], and occurs when the localised phase is desta-bilised by rare thermal regions which seed “thermalisationavalanches” [48–54]. Numerical studies, which are lim-ited to small systems, do not find any evidence of rarethermal regions [55, 56], but are known to be plaguedby unexplained finite-size effects [57–60]. The absence ofa theory of the finite-size crossover leaves unclear whichfeatures of the numerical data may survive in the ther-modynamic limit, and has led Refs. [1, 2] to claim thatthe numerical data precludes the possibility of an MBLphase altogether.We develop a microscopically motivated resonance ∗ [email protected] model for the one-dimensional MBL-thermal crossover atfinite sizes. In this model the MBL phase is not desta-bilised by rare thermal regions, but by many-body res-onances involving macroscopically distinct l-bit states.Although this mode of instability was previously identi-fied [61] and observed in finite size numerics [62], it hasreceived little attention in the literature.Specifically, we consider a presumptively many-bodylocalised chain, analyse the statistics of resonances in-duced by local perturbations, and establish when theseresonances destabilise MBL. The detailed analysis is dif-ferent in the Floquet (Sec. II) and Hamiltonian (Sec. III)settings. However, in both cases, the same set of non-trivial length scales emerge which control the physics.The first of these is the bare localisation length ζ , whichgoverns the exponential decay of off-diagonal matrix el-ements of local operators in the l-bit basis. A site-localperturbation introduces many-body resonances betweeneigenstates. The probability that a given eigenstate findsa first-order resonance involving l-bits within a range r (in the Floquet case) is given by q ( r ) = e − r/ξ λ (1)Here, two additional lengths emerge: the correlationlength ξ sets the typical range of resonances, while the resonance length λ determines their density. The RMpredicts that ξ diverges as the localisation length ap-proaches the critical value ζ c . This marks the transitionbetween a localised phase in which the number of reso-nances is finite and a delocalised phase (dubbed thermalin Fig. 1a) in which the number of resonances grows ex-ponentially with range. The finite-size behaviour nearthe transition depends crucially on the resonance length λ which is much larger than the lattice scale. For systemsize L (cid:28) λ (region I, Fig. 1a), typical eigenstates haveno resonances and non-thermal expectation values. For a r X i v : . [ c ond - m a t . d i s - nn ] F e b I nv e r s e s y s t e m s i ze / L I.II. MBLThermal | ξ - ξ - λ - a ) Inverse localisation length 1 / ζ I nv e r s e s y s t e m s i ze / L MBLThermal
Overlappingresonances ≈ limit of exact diagonalisation | ξ - ξ - λ - / b ) Asymptotic transition
Figure 1. a) The resonance model (RM) predicts a continu-ous transition (orange point) between a localised (blue) anda thermal (red) phase, and an inverse correlation length | ξ | − (orange lines) that vanishes with exponent ν = 1 at the tran-sition. In region I at system sizes smaller than a resonancelength λ (purple), typical eigenstates have no resonances andspectrally averaged properties resemble those of the localisedphase. b) The MBL-thermal finite-size crossover : At large L in the vicinity of the RM transition (hatched region), lo-calisation is inconsistent due to overlapping resonances. TheRM is however self-consistent in the blue regions. The RMthus describes the MBL-thermal crossover in small system nu-merics (horizontal line), even though it does not describe theasymptotic transition (black point). system sizes L (cid:29) λ (region II), typical states participatein L/λ (cid:29) resonances even at first-order [63].The first-order analysis is clearly incomplete in regimeswhere the number of resonances induced by a single lo-cal perturbation grows with L (region II and thermal).In fact, the region of instability is somewhat larger if weconsider locally perturbing the system at every site. Inthis case, a typical eigenstate develops a density ∼ ξ/λ of resonances each of which rearranges a region of size ξ (here and henceforth we measure lengths in units of thelattice constant). For ξ (cid:38) √ λ , the resonances typicallyspatially overlap and we expect them to lead to l-bit re-arrangements on the scale of the system. The hatchedregion in Fig. 1b indicates the parameter regime and fi-nite sizes where localisation in the RM is inconsistent dueto this instability.Nevertheless, we present analytical arguments inSec. IV that the RM is self-consistent outside of thehatched region – i.e. at small enough L in region I andat any L for large enough disorder (i.e. /ζ ). Rough estimates of the resonance length in Floquet and Hamil-tonian disordered chains suggest (cid:46) λ (cid:46) for modelsnumerically studied to date (see Sec. IV) . Thus, we be-lieve that numerically accessible system sizes correspondto the horizontal dashed line in Fig. 1b, so that the ob-served crossovers in spectral quantities, spectral func-tions, finite-size drifts, etc. can all be predicted withinthe region of validity of the RM. Summarising the moredetailed results in Sec. V, the RM reproduces many fea-tures of numerically exact data:• Localised region I:
As typical eigenstates do not finda resonance for L (cid:28) √ λ , the RM predicts that re-gion I displays the phenomenology of the localisedphase: long-time local memory, a logarithmicallygrowing light cone, sub-thermal eigenstate entan-glement entropy of small sub-systems etc.. Spec-trally averaged quantities are thus insensitive tothe boundary between the MBL phase and regionI ( ξ = L , Fig. 1b), in agreement with Ref. [58].• Correlation length exponent ν : The correlationlength exponent in the RM is given by ν = 1 ,consistent with the values extracted from finite-sizescaling in ED [8, 11, 59, 64]. Note that ν = 1 vio-lates the Harris criterion [57, 65, 66].• Drift of the critical disorder strength W c with L: The RM predicts the controversial observation ofRefs. [1, 2] that W c ∝ L at small L .• Apparent /ω low-frequency dependence of spectralfunctions: In region I, disorder-averaged spectralfunctions [ S ( ω )] exhibit a low-frequency power-lawdivergence with a continuously varying exponent.The divergence is strongest in the middle of re-gion I, with [ S ( ω )] ∼ /ω − θ c (Floquet, Fig 2a–b)or [ S ( ω )] ∼ /ω | log ω | / (Hamiltonian, Fig 2d–e). As the corrections are small ( θ c (cid:28) ), the RMexplains the apparent /ω behaviour reported inRefs. [2, 67].• Scale-free resonances:
Within regions I and II, q ( r ) is scale-invariant and resonances form at all ranges,in agreement with a numerically exact calculationof q ( r ) [62].• Apparent sub-diffusion:
On the thermal side of thetransition ( < − ξ < L ), the dynamics at shorttimes t < ω − ξ is critical. The RM describes thisdynamics, and predicts a continuously varying ex-ponent z in spectral functions ∼ /ω − /z (seeFigs 2a–b for Floquet, and Figs 2d–e for the Hamil-tonian case). The RM thus explains the apparentsub-diffusion (as measured by z ) reported in severalstudies [2, 67–71], without invoking rare region ef-fects, which Ref. [55] finds are absent in numericallyaccessible systems.• Exponential increase of Thouless time at weak dis-order W (cid:28) W c : This numerical observation ofRefs. [1, 2] follows from the logarithmic growth ofthe light cone until time t ≈ ω − ξ in the thermalphase of the RM.As the resonance model of the finite-size crossover as-sumes the existence of MBL, and reproduces the numer-ical observations of Refs [1, 2], we conclude to their con-trary, that the numerics to date appears consistent witha stable MBL in the thermodynamic limit.We additionally predict three interesting features ofthe dynamical phase diagram that could be tested nu-merically in the near future.• The exponents controlling the strongest low-frequency divergence of [ S ( ω )] ∼ /ω − θ c in re-gion I: We predict that the exponent θ c is a non-zero non-universal value in the Floquet setting,while θ c → + (corresponding to log corrections)in the Hamiltonian setting with energy conserva-tion. That is, the existence and number of conser-vation laws affects the scaling theory of the finite-size MBL-thermal crossover.• An empirical criterion for MBL: In localised sys-tems, the distribution (cid:37) ( v ) of matrix elementsof a local operator V that couple eigenstates intwo small non-overlapping mid-spectrum energy(or quasi-energy) windows takes the form, (cid:37) ( v ) ∼ v − θ , (2)with < θ ≤ (see Fig. 2c). A simple numericalcriterion follows: ρv ∼ L/ (thermal) , ρv ∼ cons . (MBL) (3)with ρ denoting the mid-spectrum many-bodydensity of states. This criterion generalises theavalanche stability criterion of Ref. [48] to a set-ting without l-bits or rare thermalising regions.• Detecting the crossover between MBL and regionI: In region I, scale free resonances form, but re-main rare. Thus eigenstate averaged observablesare largely insensitive to the formation of reso-nances. However, by analysing the distribution ofan observable over eigenstates, or conditioning onthe formation of resonances, it is possible to nu-merically detect the crossover between MBL andregion I. Such an analysis is performed in Ref. [62].We proceed as follows. In Section II, we describe theFloquet resonance model, couple the RM to a probe spin,compute the statistics of many-body resonances that areference l-bit state is involved in, and thus derive thedisorder-averaged spectral function of a local operator.In Sec. III we repeat the analysis for a Hamiltonian sys-tem. In Sec. IV we establish the regime in which theRM is self-consistent, showing it to apply to small andstrongly disordered systems (small L in region I in Fig. 1).In Sec. V we discuss the implications of this analysis forinterpreting finite-size numerical data, before concludingin Sec. VI. II. FLOQUET RESONANCE MODEL
After a brief overview of the set-up of the Floquet RM(Fig. 3) and the definition of the localisation length ζ ,we detail a careful counting of resonances induced by aprobe spin in Sec. II B. Panels (a), (b) and (f) in Fig. 2summarise the results for the spectral function of theprobe spin in the Floquet RM.Resonances do not span the system for /ζ > /ζ c :=log 2 ; this is the MBL phase of the Floquet RM. The RMMBL phase has infinite time memory of initial conditions,and a power-law divergence of the spectral function atsmall frequency (53).The point /ζ = 1 /ζ c marks the transition out of theRM MBL phase, at which resonances occur on all lengthscales. The statistics of the strongest resonances deter-mine the low-frequency scaling of [ S ( ω )] in regions I andII within Fig. 1a. The exponent θ characterising the low-frequency divergence of [ S ( ω )] in region II jumps at thetransition (57).Although typical states find increasingly many reso-nances at long ranges for /ζ < /ζ c , they remain rareon the scale of the correlation length ξ . Consequently,the RM predicts the behaviour of [ S ( ω )] at intermediatefrequencies (59) in the thermal phase. A. Set-up
1. Chain Hamiltonian
Consider a generic strongly disordered and interactingquantum spin chain with periodic boundary conditions,and subject to a periodic (Floquet) drive. For example,the Heisenberg model with random O (3) fields: H ( t ) = H W = W (cid:88) n v n · σ n ≤ Ω t < πH J = J (cid:88) n σ n · σ n +1 π ≤ Ω t < π (4)where W , J and Ω set the disorder strength, interactionstrength and fundamental frequency of the drive respec-tively, σ n = ( σ xn , σ yn , σ zn ) is the usual vector of Pauli ma-trices acting on the n th site, and σ L +1 = σ enforcesperiodic boundary conditions. The v n are independentand identically distributed (iid) random vectors with zeromean [ v n ] = and unit variance [ v n · v n ] = 1 , with, forexample, iid Gaussian distributed entries. Here [ · ] de-notes disorder averaging.We assume two key properties of H ( t ) : (i) it hasno global conservation laws, and (ii) for some finite Ω , W (cid:29) J , the model is Floquet many-body localised,as per Ref. [72]. The specific form of H ( t ) is otherwiseunimportant.The dynamics of the chain is characterised by the Flo- / ζ c
10 Inverse loc. length 1 / ζ E xpon e n t θ FloquetThermal FloquetMBL [ C zz ( t )] ~ t - θ , [ S ( ω )] ~ ω - + θ θ c = ζ c / λ θ ~ / ζ - / ζ c θ → ) Floquet ThermalFloquet MBL ω H ω c ω ξ J log ω l og [ S ( ω ) ] ~ ω - + θ c + O ( θ ) ~ ω - + θ b ) ω c J log v l og ( v ) ( v ) ~ v - + θ c ) / ζ c
10 Inverse loc. length 1 / ζ E xpon e n t θ Thermal MBL θ c = + θ ~ / ζ - / ζ c θ → ) ThermalMBL ω H ω c ω ξ J log ω l og [ S ( ω ) ] ~ ω - + + + O ( θ ) ~ ω - + θ e ) Floquet - MBL MBL / ζ c Inverse loc. length 1 / ζ [ C zz ] [ C zz ] ~ e - ξ / λ [ C zz ] ~ e - πξ / λ f ) Figure 2.
Properties of the Resonance Model transition : Panels (a) and (d):
In the MBL phase and at the RM transition( /ζ ≥ /ζ c ), the spectral function diverges at low frequencies [ S ( ω )] ∼ ω − θ . Panels (a) and (d) summarise the behaviourof the exponent θ in the Floquet and Hamiltonian cases respectively. Both panels show θ = θ → deep in the MBL phase( /ζ → ∞ ), and θ → as the transition is approached. At the Floquet RM transition, θ jumps to a finite value θ = θ c (orangepoint, panel (a)), while θ c = 0 + (indicating the presence of log corrections) at the Hamiltonian RM transition. Panels (b) and(e):
In the vicinity of the RM transition, the correlation length | ξ | sets the cross-over frequency scale ω ξ ∼ exp( − / | θ | ) . Thelow-frequency behaviour ( ω (cid:28) ω ξ ) is determined by the phase, while the intermediate frequency behaviour ω (cid:29) ω ξ (cid:29) J − is determined by the transition. The two other frequency scales are set by the system size: the Heisenberg scale ω H is theinverse level spacing, while ω c is the scale of the smallest off-diagonal matrix elements. The thermal-region I crossover occurswhen ω ξ ∼ ω H ∼ ω c . In region I, only the exponent controlling the ω > ω ξ decay is visible. This exponent is continuouslyvarying and is significantly corrected from its value at the transition in region I (as quantified by the O ( θ ) term). The smallestvalue of the exponent is however set by θ c . Panel (c):
The exponent θ may be directly extracted from (cid:37) ( v ) , the distributionof off-diagonal matrix elements of a local operator. In the localised phase, there are exponentially many off-diagonal matrixelements which are exponentially small in range, so (cid:37) ( v ) diverges as a power-law at small v . The exponent defines θ . Panel (f):
The time averaged correlator [ C zz ] serves as an order parameter for the MBL phase. [ C zz ] goes to zero smoothly as /ζ → /ζ c is approached from the MBL side, faster than any power law in both the Hamiltonian and Floquet cases. quet operator U F := T exp (cid:32) − i (cid:90) T H ( t )d t (cid:33) = exp ( − i H J T /
2) exp ( − i H W T / (5)where T = 2 π/ Ω and T is the usual time ordering op-erator. The associated Floquet states | (cid:15) a (cid:105) , and quasi-energies (cid:15) a are defined by U F | (cid:15) a (cid:105) = e − i (cid:15) a T | (cid:15) a (cid:105) . (6)
2. Localisation in the l-bit basis
At sufficiently strong disorder in the MBL phase, weassume that the Floquet states | (cid:15) a (cid:105) may be identifiedwith configurations of quasi-local integrals of motion, or l-bits [7, 38, 39] (in Sec. IV A, we discuss how this as-sumption may be relaxed). Each l-bit τ zn is traceless tr ( τ zn ) = 0 , squares to the identity ( τ zn ) = , is exponen-tially localised around the physical site n , and commuteswith the Floquet operator [ U F , τ zn ] = 0 . (7)Each Floquet state | (cid:15) a (cid:105) can be identified with an l-bitconfiguration τ a ∈ {− , } L . The scalar element τ an = ± of τ a specifies the state of the n th l-bit: τ zn | (cid:15) a (cid:105) = τ an | (cid:15) a (cid:105) . (8)A quasi-local operator U diagonalises the Floquet uni-tary, and maps the physical spin operators to l-bits, U τ αn U † = σ αn . (9)Thus the σ αn are similarly exponentially localised opera-tors in the l-bit basis. ↑ ↓ ↑ ↑ ↓ ↑ ↓ ↑ ↑↑ a ) Disordered chain ( physical basis ) ⋯ σ - α σ - α σ α σ α σ α ⋯ Probe spin Probe - chain coupling ↑ ↓ ↑ ↑ ↓ ↑ ↓ ↑ ↑↑ b ) Disordered chain ( l - bit basis ) ⋯ τ - α τ - α τ α τ α τ α ⋯ Probe spin Probe - chain coupling Figure 3.
Set-up in the physical and l-bit bases respectively :a) A “probe” spin- (orange) couples to a strongly disorderedchain (blue) at the site n = 0 (magenta). b) Transforming tothe l-bit basis renders the Floquet unitary of the chain diag-onal and the probe-chain coupling quasi-local. The couplingstrength decays exponentially with distance from n = 0 . Consider two eigenstates | (cid:15) a (cid:105) , | (cid:15) b (cid:105) . We say two statesdiffer at range r ab if the furthest flipped l-bit is at distance r ab from the site n = 0 . r ab := max {| n | : τ an (cid:54) = τ bn } (10)The range is depicted in Fig. 4b. If the matrix element V ab := (cid:104) (cid:15) a | V | (cid:15) b (cid:105) of an operator V is non-zero, then V ab isalso said to have range r ab .The length scale on which a physical spin operator islocalised in the l-bit basis defines the localisation length ζ . Consider a local operator V acting on the physical siteof index n = 0 . The operator V can be decomposed intoa sum of terms of increasing range V = L/ (cid:88) r =0 V r (11)where all the non-zero matrix elements of V r have range r . The asymptotic decay of the norm of V r defines ζ : log | V r | ∼ − rζ . (12)We use the re-scaled Frobenius norm | V r | := (cid:114) L tr ( V r ) , (13)as it is simple to calculate analytically, and capturesthe typical expectation value of an arbitrary vector |(cid:104) ψ | V r | ψ (cid:105)| ≈ | V r | .
3. Coupling a probe spin to the disordered chain
To probe the dynamical phase of the disordered chain,we introduce a probe spin- σ P subject to a z -field of strength W . The combined Hamiltonian of the probespin and disordered chain, H ( t ) = H ( t ) + H ( t ) , (14)is periodic with fundamental frequency Ω . Here H en-codes the part of the Hamiltonian in which the probespin and disordered chain are decoupled H ( t ) = H ( t ) ⊗ + h ⊗ σ z P , (15)and H ( t ) encodes their coupling. Throughout we usecursive letters to denote properties of the combinedHilbert space of the disordered chain and the probe spin,and roman letters to denote properties of the reducedHilbert spaces. The spin and chain are coupled an inter-action H , we choose H ( t ) = (cid:88) n ∈ Z δ ( n − t/T ) V ⊗ σ x P . (16)Here V is some local operator which acts only on the n = 0 site of the chain, and which is assumed to havenorm | V | = J , e.g. V = Jσ x .The Floquet operator of the combined system is givenby U F = U U (17)where U is the Floquet unitary for H = 0 , and U encodes the interaction U = U F ⊗ exp (cid:0) − i2 W T σ z P (cid:1) (18) U = exp ( − i T V ⊗ σ x P ) . (19)Each eigenstate of the unperturbed Floquet unitary U | ε α (cid:105) = e − i ε α T | ε α (cid:105) is a tensor product of a quasi-energystate of the disordered chain | (cid:15) a (cid:105) and a z -polarised stateof the probe spin | σ (cid:105) , | ε α (cid:105) := | (cid:15) a σ (cid:105) := | (cid:15) a (cid:105) ⊗ | σ (cid:105) ,ε α := (cid:15) a + σW, (20)where α = ( a, σ ) is a composite label. B. Spectral function of σ z P in the RM MBL phase ζ < ζ c Our aim is to calculate the disorder averaged infinitetemperature zz spin correlator, [ C zz ( t )] = 1 D [tr ( σ z P ( t ) σ z P (0))] , (21)in the RM. Here the normalization by D , the Hilbertspace dimension, ensures that [ C zz (0)] = 1 . For simplic-ity, we restrict to stroboscopic observations at the drive σ = ↑ σ = ↓ Q u a s i e n e r gy Range r | ϵ a ↑〉 Reference state | ϵ a ↑〉 r = r = r = ) b ) v ( ) v ( ) | ⋯↑↑↓↑↓↓↑⋯〉 | ⋯↑↑↓↓↓↓↑⋯〉 | ⋯↑↑↑↓↓↓↑⋯〉 | ⋯↑↑↓↑↑↓↑⋯〉 | ⋯↑↑↑↓↓↓↑⋯〉 | ⋯↑↓↑↓↓↑↑⋯〉 | ⋯↑↑↓↓↑↑↑⋯〉 | ⋯↑↓↑↓↑↓↑⋯〉 | ⋯↑↑↑↑↓↑↑⋯〉 | ⋯↑↓↑↑↓↓↑⋯〉 | ⋯↑↓↑↓↑↑↑⋯〉 Figure 4.
Organising resonances by range : a) The many-body spectrum of H in a small quasi-energy window is divided intotwo sectors labelled by the state of the probe spin σ = ↑ , ↓ . | (cid:15) a ↑(cid:105) labels a specific reference state. b) The l-bit configurationcorresponding to reference state (red spectral line) is shown. The states | (cid:15) d ↓(cid:105) in the opposite sector (green lines) can begrouped according to their range r from the reference state (ranges r = 0 , , shown); states at range r differ only on the l-bitswith index | n | ≤ r (highlighted in orange). A state | (cid:15) d ↓(cid:105) at range r is resonant with | (cid:15) a ↑(cid:105) if its quasi-energy separation is lessthan the matrix element size v ( r ) (i.e. if it lies within the magenta region). In the plot, the first resonance occurs at range r = 2 . period t ∈ T N . The Heisenberg operator σ z P ( t ) at integerperiods is given by σ z P ( nT ) = ( U † F ) n σ z P U n F . (22)The spectral function [ S ( ω )] is obtained by Fourier trans-formation of (21), [ C zz ( t )] = (cid:90) ∞−∞ d ω e − i ωt [ S ( ω )] . (23)The basic steps in the calculation are as follows. We re-solve the trace in the correlator (21) over the eigenstates | (cid:15) a σ (cid:105) of H , and argue in Sec. II B 1 that each term iswell approximated by either unity or a pure tone: (cid:104) (cid:15) a σ | σ z P ( t ) σ z P (0) | (cid:15) a σ (cid:105) = (cid:40) (no resonance) cos (cid:0) | V ab | t (cid:1) (resonance) (24)Above, | V ab | is the largest matrix element that couples | (cid:15) a σ (cid:105) to a resonant state | (cid:15) b ¯ σ (cid:105) where ¯ σ is the opposite z -spin projection as compared to σ . Taking the matrixelements at range r to have a characteristic scale v ( r ) ,we obtain [ C zz ( t )] = [ C zz ] + (cid:90) L/ d r p ( r ) cos( v ( r ) t ) (25)where p ( r ) is the probability (upon varying the initialstate, and disorder realisation) that the resonant processwith the largest matrix element is at range r , and [ C zz ] := lim T →∞ T (cid:90) T d t [ C zz ( t )] = 1 − (cid:90) L/ d r p ( r ) (26)is the probability of no resonances. As p ( r ) and v ( r ) are exponentially decaying in r , we find that the spectral ↓ ↑ ↑ ↓ ↑ ↓ ↑↑↓ ↓ ↑ ↑ ↓ ↑ ↑↓ ± Figure 5.
Cartoon of approximate eigenstates : For thepurposes of calculating the spectral function, the resonanteigenstates may replaced with cat states. Here the reso-nance is of range r = 2 , so that only l-bits with indices n ∈ {− , − , , , } (red box) are reconfigured. function is a power law at low frequencies, [ S ( ω )] ∝ ω − θ . (27)The exponent θ approaches zero as ζ → ζ − c from thelocalised side, but jumps to a non-zero θ c precisely at thecritical point ζ = ζ c . Ref. [61] gave a similar resonancecounting argument for the low frequency properties ofthe spectral function in the localised phase.We now detail how these results are obtained. Thefinal expressions for the spin-spin correlator are given inSecs. II B 4, II B 5.
1. Contribution of a resonance to the spectral function
Let us define a resonance. Consider a Floquet state | ε α (cid:105) of combined system, U F | ε α (cid:105) = e − i ε α T | ε α (cid:105) . (28)Expanding these Floquet states to leading order in V , weobtain | ε α (cid:105) = | (cid:15) a ↑(cid:105) + (cid:88) b i V ba T e i( (cid:15) a − (cid:15) b + h ) T − | (cid:15) b ↓(cid:105) + . . . (29)where α = ( a, ↑ ) [73]. We define the two states | (cid:15) a ↑(cid:105) and | (cid:15) b ↓(cid:105) to be resonant if the first-order correction is large,that is, if g ba := max n ∈ Z (cid:12)(cid:12)(cid:12)(cid:12) V ba (cid:15) a − (cid:15) b + h + n Ω (cid:12)(cid:12)(cid:12)(cid:12) > (30)If g ba < for all b , then we approximate | ε α (cid:105) by theunperturbed eigenstate | (cid:15) a ↑(cid:105) .If g ba > for a single b , then degenerate perturbationtheory yields ‘cat’ Floquet states | ε α,β (cid:105) = 1 √ (cid:16) | (cid:15) a ↑(cid:105) ± | (cid:15) b ↓(cid:105) (cid:17) + O( g − ba ) , (31)to good approximation (Fig. 5). These states are de-picted in Fig. 5. The two cat states (31) are split inquasi-energy by the matrix element | V ba | , | ε α − ε β | = | V ba | + O ( | V ba | g − ba ) (32)Ignoring the sub-leading corrections, we thus obtain (cid:104) (cid:15) a ↑ | σ z P ( t ) σ z P (0) | (cid:15) a ↑(cid:105) = cos (cid:0) | V ba | t (cid:1) , t ∈ T N . (33)The corresponding contribution to the spectral functionis two delta function peaks at ω = ±| V ba | . The absenceof weight at zero frequency is a consequence of the equalamplitudes in the RHS of (31). We argue in Appendix Athat extending this calculation to include a small non-zero weight at ω = 0 does not alter the low frequencybehaviour of the disorder-averaged spectral function.If g ba > for multiple indices b , the eigenstates donot have the simple form in (31). Nevertheless, we arguein Appendix A that the strongest resonance , correspond-ing to the largest matrix element, sets the frequency ofoscillation if ζ < ζ c . That is, (cid:104) (cid:15) a ↑ | σ z P ( t ) σ z P (0) | (cid:15) a ↑(cid:105) = cos( ω a ↑ t ) for t ∈ T N (34)with ω a ↑ = max (cid:26) | V ba | : g ba > (cid:27) . (35)In other words, for an initial state | (cid:15) a ↑(cid:105) , the probe spinoscillates at a frequency ω a ↑ for a window of time t (cid:29) ω − a ↑ , and thus the Fourier transform of (34) is sharplypeaked at ± ω a ↑ . Analogous expressions for an initialstate in the down sector are easily obtained.
2. The probability q ( r ) of resonance at range r We take all the matrix elements at range r to havea single characteristic value v ( r ) which is a monotoni-cally decreasing function of the range r . This recasts theproblem of finding the resonance with the largest matrixelement as the problem of finding the resonance withthe smallest range r . We now calculate v ( r ) , and sub-sequently the probability q ( r ) of finding a resonance atrange r .As described in Sec. II A, V = (cid:80) L/ r =0 V r may be de-composed into terms of increasing range r in the MBLphase. V r couples a given state | (cid:15) a (cid:105) to N r other states | (cid:15) b (cid:105) at range r , where N = 1 , N r> = · r . (36)The characteristic scale v ( r ) of each matrix element isdetermined by, | V r | = 1 D (cid:88) a (cid:104) (cid:15) a | V r | (cid:15) a (cid:105) = N r · v ( r ) . (37)Using | V r | ∼ J e − r/ζ we obtain v ( r ) = | V r |√ N r ≈ J e − r/ξ − r/ζ c , (38)where the correlation length ξ is defined by ξ = 1 ζ − ζ c , ζ c = log 2 . (39)The omission of the unimportant pre-factor of (cid:112) / makes (38) approximate.Two properties of ξ are noteworthy. First, ξ has theinterpretation of a length only in the MBL phase of theRM, in which it is positive. Second, ξ diverges as ζ → ζ − c .When we use results of the RM to discuss the short-time dynamics as ζ → ζ +c , we will be careful to use theabsolute value of ξ .Let ρ ( r )d r denote the density of states per unit quasi-energy with range in the interval [ r, r + d r ] ; from here onwe will coarse grain and treat the range r as a continuousvariable. As the states are uniformly distributed in quasi-energy (cid:15) b ∈ [0 , Ω] , and the total number of states withinrange r is given by r +1 we have (cid:90) Ω0 d ε (cid:90) r d r (cid:48) ρ ( r (cid:48) ) = 2 r +1 = ⇒ ρ ( r ) = 4e r/ζ c ζ c Ω . (40)Consider the d n = Ω ρ ( r )d r states with ranges in theinterval [ r, r +d r ] . As they are uniformly distributed overthe quasi-energy interval [0 , Ω] , the probability that anarbitrarily selected one of them has a quasi-energy in theinterval ε β ∈ ε α + [ − v ( r ) , v ( r )] , and is thus resonant, isgiven by v ( r ) / Ω . It follows that the probability that atleast one of these states is resonant with | (cid:15) a ↑(cid:105) is givenby q ( r )d r = 1 − (cid:18) − v ( r )Ω (cid:19) d n = 2 v ( r ) ρ ( r )d r + . . . (41)where higher-order corrections in v ( r ) / Ω can be droppedfor q ( r ) (cid:28) . Combining (38), (40) and (41) q ( r ) = e − r/ξ λ (42)with ξ as in (39) and the resonance length λ defined as λ := ζ c Ω8 J ≈ Ω J (cid:29) (43)We expect that λ (cid:29) as MBL in the RM requires Ω (cid:29) J .Put another way, deep in the MBL phase where ξ (cid:28) ,the probe spin will typically induce resonances of range r = 0 (i.e involving only the l-bit n = 0 to which itis directly coupled). For stable MBL, the probabilityof such resonances q (0) = 1 /λ should be small so thatnearest-neighbour resonances are atypical.
3. The probability p ( r ) that the strongest resonance is atrange r The fraction F ( r ) of states that have not resonated upto range r satisfies the differential equation ∂F∂r = − q ( r ) F ( r ) (44)with solution F ( r ) = exp (cid:18) − ξλ (cid:16) − e − r/ξ (cid:17)(cid:19) . (45)The probability p ( r )d r that the strongest resonance withthe largest matrix element has range in the interval [ r, r +d r ] is then determined by p ( r ) = − ∂F∂r = 1 λ exp (cid:18) − rξ − ξλ (cid:16) − e − r/ξ (cid:17)(cid:19) . (46)
4. The time domain correlator [ C zz ( t )] and thelogarithmically growing light cone front We now have all the pieces in place to write down thespin-spin correlation function. The strongest resonancefor each state is mediated by a matrix element of size v ( r ) with probability p ( r ) . Plugging this into the pure toneansatz (34), and treating the disorder average as simply sampling the distribution p ( r ) , we obtain the FloquetRM spectral function [ C zz ( t )] = [ C zz ] + (cid:90) L/ d r p ( r ) cos( v ( r ) t ) (47)where the integral runs over all possible ranges ≤ r ≤ L/ , and the infinite time average [ C zz ] := lim T →∞ T (cid:90) T d t [ C zz ( t )] = F (cid:0) L/ (cid:1) (48)is simply the probability that a state of the uncoupledsystem is not resonant with any other state.We were unable to exactly perform the integral (47).However a crude approximation allows us to extract theasymptotic behaviour in the time domain. Specificallywe replace cos( v ( r ) t ) → [[ v ( r ) t < where the Iversonbracket takes values [[ P ]] = 1 , (0) when the proposition P = true , ( false ) . Within this approximation we obtain [ C zz ( t )] = F ( r ( t )) (49)where r ( t ) is obtained by solving v ( r ) t = 1 , r ( t ) = min (cid:16) ζ c (1 − θ ) log( Jt ) , L (cid:17) . (50)where we have defined θ = ζ c ξ + ζ c . (51)The position r ( t ) has a simple interpretation as the frontof a logarithmically growing light cone. Only the catstates formed from l-bits states with r < r ( t ) contributeto the correlation function at time t .
5. The spectral function [ S ( ω )] From (47) it is straightforward to obtain the spec-tral function. For brevity we first recast the matrix el-ement (38) as v ( r ) = J e − r/ ( θ ξ ) using (51). Then byinverse Fourier transform of (47) [ S ( ω )] = 12 (cid:90) L/ d r δ ( | ω | − v ( r )) p ( r )= ξθ | ω | p (cid:18) ξθ log (cid:12)(cid:12)(cid:12)(cid:12) Jω (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) . (52)Inserting the calculated form of p ( r ) (46) into (52) yields [ S ( ω )] = ζ c (1 − θ )4 Jλ · (cid:12)(cid:12)(cid:12) ωJ (cid:12)(cid:12)(cid:12) − θ exp (cid:18) − ξλ (cid:18) − (cid:12)(cid:12)(cid:12) ωJ (cid:12)(cid:12)(cid:12) θ (cid:19)(cid:19) for ω c < | ω | < J [ C zz ] δ ( ω ) for ω c > | ω | (53)in the MBL phase of the Floquet RM. The cutoff scale ω c is set by the smallest matrix elements at distance L/ , ω c = v L/ = J exp (cid:18) − L ξ − Lζ c (cid:19) (54) = ω H λ exp( − L/ ξ ) , (55)where the Heisenberg frequency ω H := Ω2 − L is set bythe typical many-body level spacing.The high-frequency ( ω ≈ J ) behaviour of [ S ( ω )] de-pends on the microscopic Hamiltonian in the immediatevicinity of the spin, and is thus non-universal. In con-trast, the exponent θ characterising the power-law at lowfrequency: [ S ( ω )] ∼ ω − θ (56)is a consequence of distant resonances which reconfigurelarge regions of the chain. Thus, as ζ → ζ − c , we expect θ to have a universal functional dependence on | ζ − ζ c | .For L (cid:29) λ (region II in Fig. 1a), it follows from (53)that θ = θ = ζ c ζ c + 2 ξ ζ < ζ c θ c = ζ c λ ζ = ζ c (57)That is, θ vanishes linearly with | ζ − ζ c | as ζ → ζ − c , butjumps to a non-universal non-zero value at the transition.For L (cid:46) λ (region I in Fig. 1a), θ = θ c + O ( θ ) , so thatthe exponent is continuously varying. The low-frequencydivergence in [ S ( ω )] is strongest when θ = θ c , we returnto this in Sec. V B [74].Eq. (56) implies that that disorder-averaged correlatorsexhibit a power-law decay at long times t (cid:29) J − in theRM MBL phase: [ S ( ω )] ∼ ω − θ ⇐⇒ [ C zz ( t )] ∼ ( Jt ) − θ (58)The decay persists until time ∼ ω − , which is expo-nentially larger than the Heisenberg time ∼ ω − . Thedynamics at these long time scales are due to the ex-ponentially small (in L ) fraction of cat states involvingre-configurations of l-bits on the scale of the system size L .A fraction of the eigenstates | (cid:15) a σ (cid:105) do not hybridise withany other states despite the coupling with the probe spinto the chain, even as L → ∞ . As the probe spin hasa well-defined orientation in these states (even upon in-cluding perturbative corrections), these states contributeto the infinite-time memory [ C zz ] of the MBL phase.We defer more detailed discussion of the finite-size be-haviour of [ S ( ω )] to Sec. V. C. Spectral function of σ zP in the RM thermalphase ζ < ζ c In the thermal phase, we expect that the off-diagonalmatrix elements obey the eigenstate thermalization hy-pothesis. In particular, the off diagonal matrix elements they do not decay exponentially with range r at large r ,as assumed by the RM in (12). Consequently, the RMdoes not apply in this regime.Despite being generally inapplicable, the early timepredictions of the RM are found to hold even in the ther-mal regime. Specifically, as the probability of resonance q ( r ) is small for r (cid:28) | ξ | , [ S ( ω )] exhibits power-law decay(as in (56)) for J (cid:29) ω > ω ξ where, ω ξ := v ( | ξ | ) = J e − / | θ | . (59)That is, the correlator’s dynamics are critical until atime-scale ∼ ω − ξ . This result is obtained exactly as inthe MBL case, with the refinement that, instead of work-ing in a basis of l-bits (which do not exist in the thermalregime), it is necessary to work in a basis of “almost-l-bits” ˜ τ zn . These operators have the same properties asl-bits (mutually commuting exponentially localised etc.),but only “almost commute” with the Hamiltonian | [ H, ˜ τ zn ] | (cid:46) ω ξ . (60) III. HAMILTONIAN RESONANCE MODEL
We describe the computation of the spectral function ofthe RM with Hamiltonian dynamics. Despite the Hamil-tonian case appearing superficially simpler than the Flo-quet case (as it lacks the additional “ingredient” of a drivefrequency) the analysis is more complicated due to theconservation of energy. The associated hydrodynamicmode constrains the late time dynamics, and hence thelow frequency behaviour of the spectral function.For simplicity, we assume that the chain has a singlehydrodynamic mode. The analysis is easily generalised toaccommodate further conservation laws, such as the spinconservation present in the “standard model of MBL” theHeisenberg model with random z -fields. A. Set-up
1. Chain Hamiltonian
Consider a strongly disordered static chain with disor-der strength W and interaction strength J . For speci-ficity, consider the Ω → ∞ limit of the Floquet modelin (4), that is, the Heisenberg model with O (3) randomfields H = J (cid:88) n σ n · σ n +1 + W (cid:88) n v n · σ n . (61)As before, the details of this model will be unimportantexcept for two key properties: (i) energy is the only con-served extensive quantity at any W, J , and (ii) the modelis many-body localised for some finite W (cid:29) J .0
2. The local energy (cid:15) a In addition to its energy eigenvalue E a , each eigenstate | E a (cid:105) of H can be assigned a local energy (cid:15) a ( r ) which canloosely be understood as the expectation value of theHamiltonian restricted to the sites n ∈ [ − r, r ] : (cid:15) a ( r ) ≈ (cid:104) E a | H [ − r,r ] | E a (cid:105) , (62)Here H [ − r,r ] is the Hamiltonian (61) with the summationrestricted to terms acting on the sites n ∈ [ − r, r ] .We make this notion sharp with the following definition (cid:15) a ( r ) = E a − E ( a, r ) (63)where the energy shift E ( a, r ) is obtained by averagingthe energies of the r +1 states within range r of | E a (cid:105) E ( a, r ) = 12 r +1 (cid:88) b : r ab ≤ r E b (64)The local energy has two useful properties. First, for twostates | E a (cid:105) , | E b (cid:105) within range r , energy differences arepreserved exactly E a − E b = (cid:15) a ( r ) − (cid:15) b ( r ) ⇐⇒ r ab ≤ r. (65)Second, given a state | E a (cid:105) , the distribution of the localenergies (cid:15) b ( r ) of the states within range r is Gaussianand centred at (cid:15) = 0 . Specifically, (cid:88) b : r ab ≤ r δ ( (cid:15) − (cid:15) b ( r )) ∼ r +1 s (cid:15) ( r ) √ π exp (cid:18) − (cid:15) s (cid:15) ( r ) (cid:19) (66)where ∼ denotes convergence in distribution at large r .Neglecting sub-leading corrections in J/W , the width ofthe Gaussian is given by s (cid:15) ( r ) = W √ r + 1 . (67)
3. Coupling a probe spin to the disordered chain
The Hamiltonian of the chain coupled to a probe spinis given by H = H + H with H = H ⊗ + h ⊗ σ z P , H = V ⊗ σ x P . (68)The eigenvectors of H , H and H are denoted | E α (cid:105) , | E α (cid:105) and | E a (cid:105) respectively. These vectors play roles in directanalogy with | ε α (cid:105) , | ε α (cid:105) and | (cid:15) a (cid:105) from the Floquet case inSec. II. The eigenvectors and corresponding eigenvaluesof H and H are related by | E α (cid:105) := | E a σ (cid:105) := | E a (cid:105) ⊗ | σ (cid:105) (69) E α := E a + σh (70)Each eigenstate | E a , σ (cid:105) of H is assigned a local energy e ( a,σ ) ( r ) = (cid:15) a ( r ) + σh/ . (71) B. Spectral function of σ zP in the RM MBL phase ζ < ζ c Our aim is to calculate the disorder averaged infinitetemperature spin-spin correlator [ C zz ( t )] = 1 D tr ( σ z P ( t ) σ z P (0)) = (cid:90) d t e − i ωt [ S ( ω )] , (72)for time evolution generated by the Hamiltonian σ z P ( t ) = e i H t σ z P e − i H t . (73)As in Sec. II B, states with resonant partners contributea pure tone, while states with no resonant partners con-tribute unity (see (24)), and hence [ S ( ω )] follows.The key difference between the Floquet and Hamil-tonian cases stems from the energy dependence of thedensity of states at range r . In the Floquet case, at suf-ficiently large range r , the density of states at range r isindependent of quasi-energy, thus all states states havean equal probability of finding a resonance at range r .In contrast, in the energy conserving case, states withunusually high/low local energy e α ( r ) couple to an atyp-ically small density of states at range r . As such theseatypical states find resonances at a significantly lowerrate (see Fig. 6). We thus adapt the calculation to keeptrack of the local energy e α ( r ) of the states. This leadsto a slower decay of F ( r ) , and hence a slower than powerlaw decay of correlations.
1. Identifying resonances
Recall the resonance condition: two states | E a ↑(cid:105) and | E b ↓(cid:105) that differ at range r are said to be resonant if | E a − E b + h | < | V ba | . (74)Using (65), this condition is recast as | e ( a, ↑ ) ( r ) − e ( b, ↓ ) ( r ) | < | V ba | . (75)
2. The probability q ( e , r ) of finding a resonance at range r ,and local energy e Define q ↑ ( e , r ) | e = e ( a, ↑ ) ( r ) , the probability that a state | E a ↑(cid:105) with finds a resonant partner state | E b ↓(cid:105) at range r . Analogous to the Floquet case, q ↑ ( e , r ) is given by q ↑ ( e , r ) = 2 ρ ↓ ( e , r ) v ( e , r ) . (76)where ρ ↓ ( e , r ) is the density of states in the down sector(i.e. the opposite spin sector) at local energy e = e ( b, ↓ ) ( r ) and range r , and v ( e , r ) , the characteristic size of matrixelements, coupling states from the two spin sectors atlocal energies e , and range r .Consider the characteristic matrix element v ( e , r ) . Tobegin with, we neglect the energy dependence of v and1 e ↑ e ↓ Local energy e L o ca l d e n s it yo f s t a t e s ρ σ ( e , r ) ρ ↓ ( e , r ) ρ ↑ ( e , r ) s e ( r ) Figure 6.
Local density of states: the local density of statesat range r and energy e , ρ σ ( e , r ) , is plotted versus the localenergy e for the σ = ↑ (red) and σ = ↓ sectors of the probespin. These distributions have the same width s e ( r ) but areoffset from each other due to the probe spin energy ± h/ .The probability of a state in the ↑ sector finding a resonantpartner is proportional to the density of states in the ↓ sector(see (76)). We illustrate this with an arbitrary cut-off: the ↑ states at energies e (cid:54)∈ e ↓ + [ − s e ( r ) , s e ( r )] (red shaded area)have a much reduced probability of resonating versus those inthe bulk of the distribution. assume that the matrix element have the same form asin (38), v ( e , r ) = J e − r/ξ − r/ζ c . (77)We later discuss refinements to this approximation.Next, the density of states ρ σ ( e , r ) follows from (66), (cid:90) r d r (cid:48) ρ σ ( e , r (cid:48) ) = 2 r +1 s e ( r ) √ π exp (cid:32) − (cid:18) e − e σ s e ( r ) (cid:19) (cid:33) . (78)The mean is biased away from zero due to the orientationof the probe spin e σ := σh (79)and the variance s e ( r ) is set by (67). Differentiating (78)and taking the asymptotically dominant behaviour weobtain ρ σ ( e , r ) ∼ ζ c s e ( r ) √ π exp (cid:32) rζ c − (cid:18) e − e σ s e ( r ) (cid:19) (cid:33) . (80)Equivalently stated, the asymptotic behaviour of ρ σ ( e , r ) is dictated by the growth-diffusion equation ∂ρ σ ∂r = W ∂ ρ σ ∂ e + 2 ζ c ρ σ ρ σ (cid:0) e , − (cid:1) = 2 √ ζ c δ ( e − e σ ) (81) where the boundary condition is obtained by matchingthe solutions with (80).Substituting Eqs. (80) and (77) in (76) q ↑ ( e , r ) ∼ √ λr exp (cid:18) − rξ − ( e − e ↓ ) W r (cid:19) . (82)and similarly for q ↓ ( e , r ) . As before /ξ = 1 /ζ − /ζ c ,and the resonance length is defined as, λ = π (cid:18) ζ c W J (cid:19) ≈ W J . (83)The approximation indicates the dropping of an unim-portant numerical factor π/ (4 log 2) ≈ . . As expected, q σ ( e , r ) is decaying in r on the localised side ( ξ > ), andgrowing on the thermal side ( ξ < ).
3. The probability p ( r ) that the strongest resonance is atrange r The growth diffusion equation (81), which describesthe total density of states at local energy e and range r ,is easily modified to describe the density of states whichhave not found a resonant partner by range r . At eachrange r , the hybridisation probability is set by q σ ( e , r ) .We thus obtain: ∂ρ u σ ∂r = W ∂ ρ u σ ∂ e + 2 ζ c ρ u σ − ρ u σ q σ (84)Here the superscript ‘u’ (for unhybridised) distinguishes ρ u σ from the total density of states ρ σ .We now extract the probability p ( r ) that a state | E a σ (cid:105) finds its strongest resonance at a range r . Observe thatthe second term in (84) leads to exponential growth with r . Define a distribution that scales out this exponentialgrowth: f σ ( e , r ) = ζ c √ − r/ζ c ρ u σ ( e , r ) . (85)Substituting in (84), we obtain ∂f σ ∂r = W ∂ f σ ∂ e − f σ q σ f σ (cid:0) e , − (cid:1) = δ (cid:0) e − σh (cid:1) . (86)The substitution (85) has a simple interpretation: F ( r ) = (cid:90) d e f σ ( e , r ) = (cid:82) d e ρ u σ ( e , r ) (cid:82) d e ρ σ ( e , r ) (87)is the fraction of states which have not hybridised byrange r . Eq. (86) is invariant under the replacements ( e , σ ) → ( − e , − σ ) , by this symmetry F ( r ) is indepen-dent of σ . It follows that the probability p ( r )d r that2the strongest resonance of a given state is in the interval [ r, r + d r ] is given by p ( r ) = − ∂F∂r = (cid:90) d e f ↑ ( e , r ) q ↑ ( e , r ) . (88)Eq. (88) is the generalisation of the Floquet result (46)to the energy conserving case. Here it is necessaryto solve the two-dimensional partial differential equa-tion (86) rather than the simpler one-dimensional ordi-nary differential equation (44).What do the solutions of (86) and (88) look like? Wediscuss two regimes. The first regime in Sec. III B 4is most relevant for the numerically accessible MBL-thermal crossover in Fig. 1b. The second regime of L, | ξ | (cid:29) λ determines properties of the Hamiltonian RMin the vicinity of ζ = ζ c as L → ∞ and is discussed inAppendix B.
4. Far from criticality | ξ | < λ , or small critical systems L < λ < | ξ | Neglecting the energy dependence of q σ ( e , r ) , q σ ( e , r ) ≈ e − r/ξ √ λr . (89)Substituting (89) into (88), we obtain an approximateequation for F ( r ) , ∂F ∂r = − e − r/ξ √ λr F (90)which we denote as F ( r ) to distinguish it from a truesolution to the growth diffusion equations (86) and (87).Let us justify the approximation above a posteriori .For ξ (cid:29) L , the solution F ( r ) of (90) decays exponen-tially on the length scale set by λ . Thus for r < λ , the bulk of the weight of the distribution of unhybridisedstates f σ ( e , r ) is at typical energies | e | < s e ( r ) , where theenergy dependence of q σ ( e , r ) can be neglected by makingthe replacement q σ ( e , r ) → q σ (0 , r ) in (88) to obtain (90).The approximation is thus valid for small critical systems L < λ < | ξ | (region I of Fig 1). Far from the crossoveron the MBL side | ξ | < λ , few resonances form after thelength scale ξ and f σ ( e , r ) does not becomes small attypical energies | e | < s e ( r ) . The bulk of the weight ofthe distribution of unhybridised states f σ ( e , r ) is thus attypical energies and the approximation is justified.On longer length scales r (cid:29) λ at /ξ = 0 , the weight of f σ ( e , r ) at typical energies is depleted by the exponentialdecay. The weight of the distribution is instead concen-trated at atypical energies | e | > s e ( r ) where the reso-nance probability q σ ( e , r ) is much smaller. Appendix Bdiscusses the behaviour at r (cid:29) λ in detail.The solution to the approximated equation (90) is F ( r ) = exp (cid:32) − (cid:114) πξ λ Erf (cid:18)(cid:114) rξ (cid:19)(cid:33) ξ > (cid:32) − (cid:114) − πξ λ Erfi (cid:18)(cid:114) − rξ (cid:19)(cid:33) ξ < (91)where Erf( · ) and Erfi( · ) are the usual error function andimaginary error function respectively. The correlatorthen immediately follows [ C zz ( t )] = F ( L/
2) + (cid:90) L/ d rp ( r ) cos( v ( r ) t ) . (92)Using (88) and (52), we obtain the desired result: [ S ( ω )] = J (cid:115) ζ c (1 − θ )2 λ log | J/ω | (cid:12)(cid:12)(cid:12) ωJ (cid:12)(cid:12)(cid:12) − θ exp (cid:32) − (cid:114) πξ λ Erf (cid:32)(cid:115) θ log (cid:12)(cid:12)(cid:12)(cid:12) Jω (cid:12)(cid:12)(cid:12)(cid:12) (cid:33)(cid:33) for ξ > , ω c < | ω | < J J (cid:115) ζ c (1 − θ )2 λ log | J/ω | (cid:12)(cid:12)(cid:12) ωJ (cid:12)(cid:12)(cid:12) − | θ | exp (cid:32) − (cid:114) − πξ λ Erfi (cid:32)(cid:115) − θ log (cid:12)(cid:12)(cid:12)(cid:12) Jω (cid:12)(cid:12)(cid:12)(cid:12) (cid:33)(cid:33) for ξ < , ω ξ , ω c < | ω | [ C zz ] δ ( ω ) for ξ > , ω c > | ω | (93)The spectral function exhibits the same ω − θ low fre-quency behaviour as (53) in the Hamiltonian RM MBLphase and at intermediate frequencies in the thermalphase. However, as the localisation length approachesthe critical value ζ → ζ c , the correlation length diverges /ξ → , the correlation decay exponent θ → + , and the correction to the low-frequency ω − behaviour ofthe spectral function is logarithmic rather than powerlaw. We further discuss the logarithmic corrections inSec. V B 2.3 IV. REGIME OF SELF CONSISTENCY OF THERESONANCE MODEL
The RM assumes a characteristic range-dependence forthe matrix elements v ( r ) of a local operator V acting atsite n = 0 (see (38)). The coupling to the probe spin in-duces hybridisation between the eigenstates of H . Thereader might thus worry that the off-diagonal matrix el-ements of a local operator between the hybridised eigen-states is not consistent with the RM assumption in (38).In other words, the distribution of matrix elements afterhaving introduced the probe spin is inconsistent with thedistribution we assumed at the beginning.We address this question in two parts. First, we showthat [ S ( ω )] ∼ ω − θ at low frequencies even if the ma-trix elements at range r have a generic distribution p ( v | r ) ,as opposed to a single value v ( r ) , so long as the aggregatedistribution of off-diagonal matrix elements (cid:37) ( v ) = L/ (cid:88) r =0 p ( v | r ) ρ ( r ) (94)is distributed as a power-law in v at small v . Thus, we canrelax the assumption in (38) to allow for a pre-existingpopulation of resonant cat pairs states, as the matrixelements between such cat pairs and the reference statecan differ from v ( r ) .Next, we imagine perturbing a MBL RM chain, witha given p ( v | r ) , weakly at every site. The local pertur-bations induce local resonances. When these resonancesdo not overlap, we argue that the distribution p ( v | r ) isunaffected at large r , and thus that the perturbed chainpresents the same statistics of off-diagonal matrix ele-ments v as the unperturbed chain at small v . Conse-quently, the exponent θ that sets the low-frequency di-vergence of [ S ( ω )] is stable to local perturbations.Specifically, we argue that the resonance model isperturbatively stable, and consequently our conclusionshold, in the regime min (cid:18) L , | ξ | (cid:19) (cid:28) √ λ (95)in which resonances do not typically overlap. Eq. (95)holds deep in the RM MBL phase as L → ∞ and inregion I (see Fig. 1) for sufficiently small systems. Threeimportant conclusions follow:1. As the RM is self-consistent deep in the MBLphase, the RM predicts and describes a stable MBLphase in the thermodynamic limit.2. The RM describes the MBL-thermal crossover inshort chains, despite being an inapplicable at large L .3. The RM describes dynamics in the MBL-thermalcrossover at short times as L → ∞ , or equivalentlyon frequency scales: ω > ω th . := max( v ( √ λ ) , v ( | ξ | )) . (96) A. Generalised RM with p ( v | r ) Define the aggregated distribution of off diagonal ma-trix elements (cid:37) ( v ) as the distribution of matrix ele-ments | V ba | that couple two narrow energy windows E a ∈ [ E, E + ∆] and E b ∈ [ E (cid:48) , E (cid:48) + ∆] at maximumentropy: (cid:37) ( v ) := (cid:88) ab δ ( v − | V ba | ) (97)where (cid:37) ( v ) and the distribution of matrix elements p ( v | r ) are related by (94). In Secs. II and III, we took the ma-trix elements at range r to be single valued p ( v (cid:48) | r ) = δ ( v (cid:48) − v ( r )) . In the Floquet case the corresponding ag-gregated distribution of off diagonal matrix elements atsmall v is (cid:37) ( v ) = 1Ω (cid:88) r N r δ ( v − v ( r )) ∝ ( v/J ) − θ v < J v > J (98)where N r = · r as in (36), < θ ≤ is defined in (51),and the power law is obtained by coarse-graining over thescale separating the delta functions.Eq. (53) follows from (98), independent of the precisemodel p ( v | r ) for the matrix elements at range r . Considerthe Floquet RM. A change of variables in (44) yields d F ( v )d v = F ( v ) v (cid:37) ( v ) . (99)The solution F ( v ) = exp (cid:18)(cid:90) ∞ v d v (cid:48) v (cid:48) (cid:37) ( v (cid:48) ) (cid:19) (100)is the fraction of states which do not have a resonanceinduced by a matrix element of size v or larger. Notethat F ( v = ∞ ) = 1 . Similarly we may define p ( v ) := ∂F∂v = v(cid:37) ( v ) exp (cid:18)(cid:90) ∞ v d v (cid:48) v (cid:48) (cid:37) ( v (cid:48) ) (cid:19) , (101)so that p ( v )d v is the fraction of eigenstates of H whosestrongest resonance is due to a matrix element in therange [ v, v + d v ] . The spectral function is then given by, [ S ( ω )] = p ( | ω | ) + δ ( ω ) F ( v = 0) . (102)Substituting (98), we recover the previously calculatedspectral function (53). The calculation presented inSec. III for the Hamiltonian RM can be similarly gen-eralised.Note that a general model for the matrix elementsalters the simple relationship between the localisationlength ζ and the exponent θ , and thus leads to an alteredcritical value of the localisation length ζ c := ζ | θ =0 .4 ↑ ↓ ↑ ↑ ↓ ↑ ↓ ↑ ↑ a ) ⋯ τ - α τ - α τ α τ α τ α ⋯ ξλ / ξξ ≪ λ ↑ ↓ ↑ ↑ ↓ ↑ ↓ ↑ ↑ b ) ⋯ τ - α τ - α τ α τ α τ α ⋯ ξ ≫ λ Figure 7.
Resonances : The spectral function calculation inthe RM is self-consistent if the eigenstates in the RM-MBLare well characterised as l-bit configurations dressed with lo-cal resonances. a) A l-bit state dressed with two resonancesof range r = 1 centred at sites n = − and n = 2 . Eachresonance is represented by an arc encompassing the patch ofrearranged spins. Resonances typically rearrange a patch ofsize ξ and have density ξ/λ , and thus are well separated for ξ (cid:28) λ . b) For ξ (cid:38) λ , these resonances typically overlapforming large resonant patches that destabilise MBL. B. Self-consistent and stable localisation
To be self-consistent, the RM must have the same sta-tistical distribution of resonances before and after a localperturbation.Consider a perturbation V of strength | V | ≈ J appliedat a single site n = 0 (as in Sec. (II B)). The effect ofthis perturbation is straightforward: first the eigenstateenergies are corrected by the diagonal elements of V (i.e. E a → E a + V aa ) and second, each state | E a (cid:105) finds aresonance at range r (i.e. | V ab | > | E a − E b | , where r ab = r ) with probability q ( r ) = e − r/ξ /λ . This leads to a pairof resonant ‘cat’ states (cid:18) | E (cid:48) a (cid:105)| E (cid:48) b (cid:105) (cid:19) ≈ √ (cid:18) − (cid:19) (cid:18) | E a (cid:105)| E b (cid:105) (cid:19) (103)with corresponding energies E (cid:48) a , E (cid:48) b and splitting | E (cid:48) a − E (cid:48) b | ≈ | V ab | .We now apply a second perturbation U , also ofstrength | U | ≈ J , at a site m a finite distance from n = 0 .Naively, the arguments of Sec. (II B) imply each such sub-sequent perturbation causes more long range resonancesto develop. However, this is not the case. The matrixelement (cid:104) E (cid:48) a | U | E (cid:48) b (cid:105) ≈ J e − s/ξ where s = max(0 , m − r ab ) acts to disentangle cat state pairs (103) whose splittingis small | V ab | (cid:28) J e − s/ξ . This removes all resonances dueto V which are of long range r ab (cid:29) m/ . This disentan-gling of resonances is counterbalanced by the formationof new long range resonances due to the combined actionof U and V . Their distribution is statistically identical to that induced by a single local perturbation. Specifically,the range of typical resonances remains O( ξ ) .Short range ( r ab (cid:46) m/ ) resonances induced by V survive the second perturbation. When the survivingresonances overlap with those induced separately by U ,the eigenstate entanglement further increases. Specif-ically, two cat pairs | E (cid:48) a (cid:105) , | E (cid:48) b (cid:105) (103) and | E (cid:48) c (cid:105) , | E (cid:48) d (cid:105) with respective level splittings | V ab | and | V cd | survive if (cid:104) E (cid:48) a | U | E (cid:48) b (cid:105) (cid:46) | V ab | and (cid:104) E (cid:48) c | U | E (cid:48) d (cid:105) (cid:46) | V cd | hold. Thestates | E (cid:48) a (cid:105) , | E (cid:48) c (cid:105) may hybridise if (cid:104) E (cid:48) a | U | E (cid:48) c (cid:105) (cid:38) | E (cid:48) a − E (cid:48) c | yielding | E (cid:48)(cid:48) a (cid:105) ≈ ( | E (cid:48) a (cid:105) + | E (cid:48) c (cid:105) ) / √ . In the state | E (cid:48)(cid:48) a (cid:105) , asmall subsystem in the vicinity of n = 0 has entangle-ment entropy S ≈ . Similarly two “cats of cats” | E (cid:48)(cid:48) a (cid:105) , and | E (cid:48)(cid:48) e (cid:105) may be hybridised by a third perturba-tion W to form | E (cid:48)(cid:48)(cid:48) a (cid:105) ≈ ( | E (cid:48)(cid:48) a (cid:105) + | E (cid:48)(cid:48) e (cid:105) ) / √ , with entropy S ≈ . Here we have illustrated the increase of en-tanglement entropy due to overlapping resonances for thecase (cid:104) E (cid:48)(cid:48) a | W | E (cid:48)(cid:48) e (cid:105) < (cid:104) E (cid:48) a | U | E (cid:48) c (cid:105) < (cid:104) E a | V | E b (cid:105) . (104)The general case is more complex. However we suspectsimilar increases of the entanglement entropy when reso-nances overlap.The merging of local resonances into larger resonantclusters with larger entanglement entropies represents aninstability of the “l-bits + local resonances” picture as-sumed by the RM unless the localisation length is suffi-ciently short ξ (cid:28) √ λ . Consider perturbing the RM atevery site. At each site, the probability of inducing atleast one resonance between the reference state | E a (cid:105) anda second state | E b (cid:105) is − F ( r = ∞ ) ≈ ξ/λ . If the typ-ical spacing between these resonances λ/ξ exceeds theirtypical size ξ , then they remain spatially separated. Weconclude that for ξ /λ (cid:28) resonances do not merge,and do not alter the asymptotic distribution of matrixelements at low frequencies. The RM is thus self con-sistent and stable to local perturbations in this regime.This case is depicted in Fig. 7a where the extent of eachresonance is indicated by the red arcs. We note thatrare states participate in long range resonances r (cid:29) ξ ;however these do not destabilise the localisation.Repeating the above arguments for systems of finite-size L , we find that resonances occur with density − F ( r = L/ ≈ min( ξ, L/ /λ and involve min( ξ, L/ sites. This yields the condition (95).Finally, we note that the RM describes dynamics inthe thermodynamically large thermalising phase at shorttimes, or equivalently at frequencies satisfying (96). Atthese short times, resonances are rare and thus the RM iscontrolled. As noted in Sec. II C, the derivation of [ S ( ω )] proceeds through “almost-l-bits” that almost commutewith the Hamiltonian.5 V. RM PREDICTIONS FOR FINITE-SIZENUMERICS
The RM is self-consistent in short chains L (cid:46) √ λ (105)in region I and provides a simple model for the MBL-thermal crossover. Could the RM describe the numer-ically accessible MBL-thermal finite size crossover? Anaive estimate of the resonance length λ comes fromEqs. (83) and (43) using numerical and experimentallyreported values for the critical frequency or critical dis-order strength [6, 64, 75]. This gives (cid:46) λ (cid:46) . Phys-ically, λ has to far exceed the lattice scale, as q (0) = 1 /λ is the probability of a nearest neighbour resonance in theMBL phase. We thus reason that numerically accessiblechain lengths L are smaller than or comparable to √ λ ,and that the RM is an analytically tractable model forthe numerics.In what follows, we describe several properties of theRM in short chains that explain numerical observa-tions about the finite-size MBL-thermal crossover. Thecrossover occurs around the line | ξ | = L/ separating thethermal phase from region I in Fig. 1a). We also explainthe numerical observations of Refs. [1] and [2] within theRM. As the RM has a stable MBL phase, we weigh inon the controversy of the existence of MBL in favour ofMBL. A. Correlation length exponent ν = 1 The thermal-MBL crossover in the resonance model ischaracterised by a correlation length | ξ | : | ξ | ∝ | ζ − ζ c | − ν (106)which diverges with exponent ν = 1 . This value isclose to the numerically reported values of . ≤ ν ≤ . reported for data collapses of different quantities inRef [64]. Note that the RM exponent, as well as thenumerically reported ones, violate the Harris bound forrandomly disordered systems ν ≥ [57, 65, 66], as theyonly capture the pre-asymptotic in L scaling. B. Apparent /ω divergence of the spectralfunction The RM predicts a power-law divergence in [ S ( ω )] atlow frequencies in the MBL phase and in region I: [ S ( ω )] ∼ ω − θ . (107)Above ∼ indicates asymptotic equality up to a constantfactor, and θ > .Deep in MBL phase, the following hierarchy of fre-quency scales hold: ω c (cid:28) ω H (cid:28) ω ξ , < ξ (cid:46) L/ (108) and [ S ( ω )] takes the form in (107) for ω > ω c with theexponent θ given by θ > in (57).In region I in Fig. (1), | ξ | (cid:38) L/ , and the frequencyscales are arranged as: ω ξ (cid:46) ω c ∼ ω H , | ξ | (cid:38) L/ . (109)Below, we show that the low-frequency divergence of [ S ( ω )] is strongest in the middle of region I and is givenby [ S ( ω )] ∝ ω − up to logarithmic corrections.Ref. [2] interpreted the apparent ω − behaviour as in-consistent with MBL. The RM however predicts this be-haviour near the finite-size MBL-thermal crossover in re-gion I and allows for a stable MBL phase.
1. Floquet systems
The exponent θ in (57) vanishes as | ξ | → ∞ in theRM. The strongest low-frequency divergence [ S ( ω )] ishowever not ∼ /ω (indeed, as noted in [2] such a strongdivergence would violate an elementary sum rule) be-cause the exponential term in (53) modifies the exponent.The RM instead predicts the following spectral functionin the middle of region I: [ S ( ω )] ∼ ω − θ c , ω (cid:29) ω c , ω H and | ξ | (cid:29) λ, (110)with θ c = ζ c / λ , as given by (57).As λ (cid:29) and ζ c is on the lattice scale, we conclude θ c = ζ c / λ (cid:28) . The strongest low-frequency divergencein (110) is thus close to /ω .Note that (110) implies a power law decay of correla-tions at late times. Such decay can only be consistentwith a logarithmically spreading light cone (50) in theabsence of any conserved quantities, such as in a Floquetsystem.
2. Hamiltonian systems
Hamiltonian systems conserve energy, which results ina logarithmic, rather than power law, correction to /ω scaling of [ S ( ω )] . Specifically, for | ξ | (cid:29) λ (cid:29) L , we sim-plify (93) to obtain: [ S ( ω )] ∼ ω (cid:112) λ log | J/ω | , ω (cid:29) ω c , ω H . (111)Here ∼ indicates equivalence up to an ω independent pre-factor.Observe that this decay is not asymptotically consis-tent with hydrodynamics. The light-cone only grows log-arithmically in time in the RM (see Fig. 8), but (111) im-plies critical correlations that decay faster than / log( Jt ) as t → ∞ , lim ξ →∞ [ C zz ( t )] ∼ exp (cid:32) − (cid:114) ζ c λ log Jt (cid:33) . (112)6 ↑ ↓ ↑ ↑ ↓ ↑ ↓ ↑ ↑↑ Disordered chain σ α σ α σ α ⋯⋯ Probe spin σ P α Probe - chain coupling t Support of σ P z ( t ) r ∝ log ( Jt ) Figure 8.
Logarithmically growing light cone : the Heisenbergoperator σ z P ( t ) (in (22)) is localised to the probe spin sitetime t = 0 . Under time evolution, the support spreads anddefines a light cone. After a time t , this light-cone has width r ( t ) ∝ log Jt (green). More careful analysis of the Hamiltonian resonancemodel finds that below a frequency timescale ω λ := v ( λ ) ,the decay of [ C zz ( t )] is dictated by a form [ C zz ( t )] ∼ √ log Jt , t > /ω λ (113)consistent with hydrodynamics. We note this corre-sponds to a time averaged value which goes to zero as [ C zz ] ∼ ξ − / . However, as (113) applies outside of theregime of self-consistency of the resonance model, we rel-egate further discussion to Appendix B. C. Localised finite-size crossover
As the resonance probability is small for L (cid:28) √ λ ,the RM predicts a localised finite-size crossover (i.e. alocalised region I).First, the time-averaged correlator [ C zz ] is close tounity in both the Floquet and energy conserving cases,and thus retains long-time memory: lim ξ →∞ [ C zz ] = (cid:40) e − L/ λ (Floquet) e − √ L/ λ (Energy conserving)(114)Next, the late-time memory implies that small sub-systems of the chain have sub-thermal entanglement en-tropy. This prediction is in agreement with numericalobservations in Ref. [58].Finally, dynamics in the finite size crossover is char-acterised by a dynamical exponent z = ∞ as per thelogarithmically growing light cone (see (50) and Fig. 8).The length-energy relationship set by the matrix ele-ments t ∼ v ( r ) − determines the light cone; any l-bitsoutside the light cone are not entangled with the probespin. In the thermal phase, we expect that the logarith-mic expansion of the light cone crosses over to ballisticor diffusive expansion for t > ω − ξ in Floquet and Hamil-tonian systems respectively. - ( W - W c ) / W c - C zz = S P / l og2 - L ( W δ - W c ) / W c δ = Figure 9.
Drift of the critical disorder strength W c ( L ) with L at small sizes: The main plot shows the RM probe spinentanglement entropy [ S P ] averaged over all eigenstates vsthe scaled disorder strength for L = 5 , , , . . . (colouredsolid lines). The dashed black dashed line indicates the L →∞ limit. Inset: Numerically extracted W δ with δ = 0 . (green solid line) vs L , and the corresponding analytic curvefrom (117) (black dotted line). The red dotted line is a linearfit at small L . We see that W c ( L ) ∝ L at small L . Parameters: /ξ = log( W/W c ) , W c = 10 , λ as given by (83), and J = 1 . Ref. [62] numerically observed stretched exponentialdecay of typical spatial correlations in eigenstates in theMBL-thermal crossover region and noted the similar-ity of their numerical results to that near an infinite-randomness fixed point. Although we do not flesh out theconnection between the RM transition and the infinite-randomness transition here, we note that both theoriespredict z = ∞ and logarithmically growing light cones. D. Scale-free resonances near the finite-sizecrossover
In region I (and II), the probability of resonance atrange r is scale free lim ξ →∞ q ( r ) = λ (Floquet) √ rλ (Energy conserving) (115)resulting in the formation of resonances on all lengthscales. This feature of the thermal-MBL crossover insmall systems has been observed numerically in Ref. [62]. E. Linear drift of critical disorder strength with L The RM predicts a ubiquitous feature of small sys-tem numerics on disordered chains: that the critical dis-order strength increases approximately linearly with L .Refs. [1] and [2] argued this drift to be inconsistent with7the existence of MBL; the RM however provides an al-ternative explanation.The origin of the drift lies in the localised nature of re-gion I. On increasing /ζ at small sizes, the chain crossesover from thermal to localised behaviour when the cor-relation length first exceeds the system size | ξ | ≈ L (seeFig. 1). The critical /ζ (and equivalently the criticaldisorder strength) thus increase with L .This drift can be quantified: let W δ ( L ) denote thedisorder strength at which the time-averaged correlator [ C zz ] deviates from its value in the infinite temperatureGibbs ensemble by some small amount δ , [ C zz ( W δ )] = δ (cid:28) . (116)For the Hamiltonian RM, algebraic manipulation of (91)with /ξ = log( W/W c ) yields: W δ ( L ) ≈ W c e − (cid:96) δ / ( L +1) . (117)for some δ -dependent constant (cid:96) δ . Over a regime of suf-ficiently small L , this function is approximately linearlyincreasing with L (see Appendix D for derivation).More generally the linear growth of W δ follows fromTaylor expanding ξ near W = W δ . Precisely, if we iden-tify ξ ( W δ ( L )) ∝ L , (for some δ -dependent constant ofproportionality), and consider the taylor expansion ξ ( W ) = ξ ( W δ ( L )) + ( W − W δ ( L )) ξ (cid:48) ( W δ ( L )) + . . . (118)about the point W = W δ ( L + ∆ L ) we obtain ∆ W δ := W δ ( L + ∆ L ) − W δ ( L ) ∝ ∆ Lξ (cid:48) ( W δ ( L )) . (119)Eq. (119) and the linear-in- L drift of the critical pointfollow provided W is sufficiently far from the transitionthat i) the Taylor expansion is valid (i.e. | W − W δ ( L ) | < | W δ ( L ) − W c | ) and ii) that ξ (cid:48) ( W δ ( L )) is slowly varyingin L .Fig. 9 plots the probe spin entanglement entropy [ S P ] averaged over all eigenstates, [ S P ] = log 2 (cid:0) − [ C zz ] (cid:1) , (120)in the Hamiltonian RM vs the re-scaled disorder strength(using /ξ = log( W/W c ) ). The probe spin entropy ismaximal in the cat states, and is zero is the fraction [ C zz ] = F ( L/ of states that do not resonate. The insetconfirms that the deviation ( W δ − W c ) increases linearlywith L at small L , before converging to zero from belowat large L .A similar analysis in the Floquet RM predicts a lineardrift of the critical frequency at which localisation setsin with L for fixed disorder strength. F. Exponential increase of the Thouless time withdisorder strength
Refs. [1] and [2] numerically studied the scaling of theThouless time with disorder strength in the thermalising phase. The Thouless time is defined as the time-scaleabove which random matrices govern quantum dynamicsin chaotic systems, or equivalently as the inverse of theenergy scale below which the random matrices governeigenstate properties. Through a detailed study of thespectral form factor and [ S ( ω )] , Refs. [1] and [2] arguedthat the inverse of the Thouless time ω Th . exponentiallydecreases with disorder strength: ω Th . ∝ e − cW/J . (121)Should this behaviour continue asymptotically as L → ∞ , then the numerically observed MBL-thermalcrossover is simply a finite-size effect caused by ω Th . be-coming smaller than the Heisenberg time ω − . That is,the observed localisation is simply a consequence of thesmall sizes accessible to exact numerics.The RM provides an alternate explanation for (121)while allowing for a MBL phase. In a diffusive system,the Thouless time is set by the time taken by a localisedpacket of energy to spread over the system. For diffusionconstant D , thus ω th . = D/L . As the packet takes time ω − ξ to spread a distance ξ , D = ω ξ ξ . Combining theseestimates ω Th . = DL = ω ξ ξ L ≈ Jξ L e − | ξ | /ζ c (122)where ≈ indicates the dropping of an O (1) factor.Next, consider the correlation length ξ ( W ) . It is asmooth function of the disorder strength W and divergesat the critical disorder W c defined by ζ = ζ c . As dis-cussed in Sec. V E, the crossover from spectrally averagedstatistics being close to their thermal values, to close totheir localised values occurs at disorder strength W δ , amuch weaker disorder strength than W c in small systemssizes. We may thus Taylor expand ξ near W = W δ (asin (118)) from which the exponential dependence of theThouless time on the disorder strength W of (121) fol-lows. G. Apparent sub-diffusion in the RM thermalphase
Eqs. (53) and (91) predict a continuously varying ex-ponent for the spectral function [ S ( ω )] ∼ ω − θ above athreshold frequency scale ω ξ in the thermal phase. TheRM thus explains the apparent sub-diffusion (as mea-sured by the dynamic exponent /θ ) reported in sev-eral studies [67–71] without any reference to rare re-gions, and indeed predicts such apparent sub-diffusivebehaviour even in Floquet systems without any conser-vation laws. This prediction of the RM may resolve amystery about the absence of broad distributions of theconductivity (across disorder realisations) that are ex-pected in a sub-diffusive regime characterised by weaklinks [55, 56].We note that Ref. [76] (in the supplementary material)previously speculated that rare resonances may lead toapparent sub-diffusive behaviour in the thermal phase.8 H. Exponentially enhanced sensitivity toeigenstates or ‘maximal chaos’
The fidelity susceptibility χ a measures the sensitivityof an eigenstate | E a (cid:105) to perturbation by a local operator U . It is defined as χ a = (cid:88) b (cid:54) = a (cid:12)(cid:12)(cid:12)(cid:12) (cid:104) E b | U | E a (cid:105) E b − E a (cid:12)(cid:12)(cid:12)(cid:12) . (123)The mean of the logarithm of χ (defined as the averageof log χ a across infinite temperature eigenstates and dis-order realisations) shows the following scaling with L : [log χ ] ∼ L · log 2 thermal L MBL . (124)Ref. [2] made two observations about the distributionof log χ a at numerically accessible sizes. First, there isa regime of maximal chaos separating the thermalisingand MBL regimes in which [log χ ] ∼ L · , ( “maximal chaos”. ) (125)Second, the tails of the distribution in the putative MBLregime (in which [log χ ] saturates) are fatter than ex-pected from a Poisson distribution. The authors ex-plained both observations through the exponential en-hancement of matrix elements between eigenstates withenergy differences comparable to the many-body levelspacing, and concluded that such enhancement is incon-sistent with MBL.The RM explains both observations in Ref. [2] assum-ing a thermodynamic MBL phase.Consider a pair of resonant cat states | E (cid:48) a,b (cid:105) = ( | E a (cid:105) ±| E b (cid:105) ) / √ involving the re-arrangement of l-bits at range r = L/ and splitting comparable to or less than themany-body level spacing. A generic local perturbation U will couple these states as (cid:104) E (cid:48) a | U | E (cid:48) b (cid:105) = O ( | U | ) [77].Consequently, their fidelity susceptibility is very large,increasing as ∼ L .In the numerically accessible MBL-thermal crossover, afinite fraction q ( L/ L of the eigenstates are involved inresonances with range between L and L + ∆ L and split-ting comparable to the many-body level spacing. TheRM thus predicts maximum chaos (125) at the finite-sizecrossover. More precisely, in regions I and II of the Flo-quet RM [log χ ] = (cid:90) L s ( r ) log (cid:0) | U | ρ ( r ) (cid:1) = L (2 log 2 + O( λ/L )) (126)where ρ ( r ) sets the typical inverse level spacing for a res-onance at range r , and s ( r ) = q ( r ) exp( − (cid:82) L/ r q ( r (cid:48) )d r (cid:48) ) ,is the probability that the longest range resonance for agiven state is at range r . Thu, maximum chaos is ap-proached as L becomes closer to λ . In the RM MBL phase, the fraction of states involvedin system-wide resonances q ( L/ is exponentially smallin L . These states thus do not contribute to [log χ ] , whichis independent of L . Nevertheless, these rare states leadto increased weight in the tail of the distribution of log χ .This explains the second observation of Ref. [2]. I. Absence of a cut-off at the Heisenberg time inthe MBL phase
We find that the dynamics in the MBL phase are notcut-off by the Heisenberg time t H ∼ ω − ∼ J − L . In-stead, the RM is cut-off by an exponentially larger in L time-scale set by ω − : ω c = v ( L/
2) = ω H e − L/ ξ (127)The dynamics on the time-scales t (cid:29) ω − are due tothe rare cat states with energy splittings that are smallerthan the typical level spacing.The existence of a timescale longer than the Heisen-berg time t H contradicts commonly held lore that at t H the system “realises” that it is finite, the discreteness ofthe spectrum is resolved, the dynamics becomes quasi-periodic, and thus there cannot be physically meaningfuldynamics beyond t H . This lore neglects that in the lo-calised phase all local operators have discrete (i.e. pure-point) spectra even before t H , so there is nothing to “re-alise” at t H . J. A simple numerical stability criterion for MBL
Following the discussion in Sec. IV B, MBL requiresthat the expected number of resonances induced by alocal perturbation V in a typical eigenstate of the chainis much smaller than unity: (cid:90) ∞ d r q ( r ) (cid:28) . (128)Using the tools developed in Sec. IV A, we can re-writethe above criterion in-terms of the aggregated distribu-tion (cid:37) ( v ) of off-diagonal matrix elements of V : (cid:90) ∞ d v v(cid:37) ( v ) = ρ ¯ v (cid:28) . (129)Here ρ is the many body density of states in some smallmid spectrum window of width ∆ , and ¯ v = 1∆ ρ (cid:88) b | V ba | (130)is the mean matrix element in the same window for amid-spectrum state a .Eq. (130) provides a simple numerically tractable cri-terion for MBL. As L → ∞ , the quantity ρ ¯ v grows expo-nentially with L in a thermalising phase that satisfies the9 [ S ( ω )] ~ C /( W ω ) , Ref. [ ][ S ( ω )] ~ C ' /( ω | log ( ω ) - / ) , Eq. ( ) ( W ω ) - ( [ S ( ω ) ] ω W ) - × - Figure 10.
Spectral function data from Ref. [2] : Disorderaveraged spectral function data for the random XXZ modelfrom Fig. 2a of Ref. [2] (same colour scheme). Different seriescorrespond to different disorder strengths W (legend above).Here we plot ([ S ( ω )] ωW ) − as a function of ( W ω ) − so thatthe pure /ω divergence predicted by Ref. [2] appears as ahorizontal line (black solid, C = 0 . ) whereas the formpredicted in this work, (111), appears as line of constant gra-dient (black dashed). Agreement with (111) is seen for . decades for ( W ω ) − ∈ [1 . , . eigenstate thermalization hypothesis, but saturates in aMBL phase: ρv ∝ L/ (thermal) , ρv = cons . (cid:28) (MBL) . (131)Note that (130) makes no reference to a l-bit basis.colourWhen (cid:37) ( v ) ∝ v − θ at small v , the stability cri-terion implies that ≤ θ < for MBL.Eq. (130) generalises the stability criterion to thermal-ising avalanches introduced in Ref. [48]. Ref. [48] stud-ied the stability of a MBL system composed of l-bits toa thermalising inclusion, and argued that ζ (the lengthscale controlling the localisation of a physical spin opera-tor in the l-bit basis) must be smaller than ζ c = 1 / log 2 .Re-writing the avalanche criterion in terms of propertiesof off-diagonal matrix elements, we obtain (130) with noreference to either rare regions or to l-bits. VI. DISCUSSION
We have presented the RM, a model of the finite-sizeMBL-thermal crossover in which the localised phase isdestabilised by many-body resonances, rather than rarelow-disorder regions. The RM is consistent with a stableMBL phase, and reproduces several numerically observedfeatures of the MBL-thermal finite-size crossover, includ-ing the controversial observations of Refs. [1, 2].Fig. 10 re-plots the [ S ( ω )] data in Fig. 2 of Ref. [2].The plot shows the frequency dependence of [ S ( ω )] at several disorder strengths . ≤ W ≤ . in the putativethermalising phase of the disordered spin- XXZ chain.Ref. [2] argued that the data is consistent with the scalinglaw [ S ( ω )] ∼ C/ ( W ω ) (black horizontal line) over an in-creasing range of frequencies. We instead argue that thedata is consistent with the scaling law predicted by theHamiltonian RM with a logarithmic correction (dashedblack line). Indeed, the curves for W (cid:38) align with theRM prediction over ≈ . decades in frequency, whileevidence of the plateau predicted by Ref. [2] is visibleonly in two of the curves with W ≈ . , . , and overless than a decade in frequency. The behaviour of thecurves with W ≈ . , . is however noteworthy, andnot immediately explained by the RM. To settle the de-bate between the two scaling predictions requires moresystematic numerical investigation of the effects of sys-tem size on the curves in Fig. 10. Specifically, numericsat larger L should reveal which of the two regimes (thelinear growth or the plateau) expands with increasing L .The RM makes several numerically testable predic-tions about Floquet and quasi-periodically modulatedspin chains. First, Sec. V applies without alteration tothe quasi-periodic case. Second, the exponent θ c con-trolling the strongest low-frequency divergence of thespectral function in region I in the Floquet case is non-universal and non-zero, in contrast to the HamiltonianRM with θ c → + . Third, Floquet systems on the ther-malising side of the finite-size crossover would also exhibitapparent sub-diffusive scaling in their spectral functions.The origin of this apparent sub-diffusion is the formationof many-body resonances on length scales shorter than ξ .Fourth, irrespective of the type of disorder or the numberof conservation laws, we predict logarithmically growinglight cones in the thermalising phase for t (cid:46) ω − ξ . Finally,observables conditioned on the formation of resonancescould detect the MBL-region I crossover in Fig. 1a.Eq. (131) offers a new numerical criterion to differen-tiate localised and thermalising systems. Analogous tothe G parameter in Ref. [78] and the typical fidelity sus-ceptibility [2], ρv is exponentially larger in L in the ther-malising phase as compared to the MBL phase. Prelimi-nary work on a disordered Ising model suggests that (131)bounds the transition out of the localised phase to largerdisorder strengths than other standard criteria basedon energy level statistics or eigenstate entanglement en-tropies.Future work could explore the RM along several axes.The first is to establish whether the distribution of sam-ple conductivities (across disorder realisations) predictedby the RM is consistent with the observations of Ref. [55].This would add further evidence to the claim that many-body resonances, and not rare regions, give rise to theapparent sub-diffusion observed numerically.The second is to compare the eigenstate correlationspredicted by the Hamiltonian RM to those from the An-derson model on the random regular graph (RRG) [79,80]. The RRG Anderson transition is believed to modelthe MBL-thermal transition if one identifies each site of0the RRG with a computational basis state of a disor-dered spin chain [81]. Using Mott-type resonance ar-guments similar to those of Sec. III, Ref. [79] recentlyargued that in the RRG localized phase, the correlator [tr (Π n ( t )Π n (0))] (where Π n ( t ) is the time evolved sin-gle site projector onto the site n ) has a Fourier spectrum β ( ω ) which diverges as a power law as ω → . Identifyingeach Π n with | E a σ (cid:105)(cid:104) E a σ | , a product state of the probespin and the disordered chain, the RM predicts that β ( ω ) diverges exactly as [ S ( ω )] (27). The reconcilation of theRM with the RRG is however less apparent in the ther-mal phase, where the latter predicts a correlation lengththat diverges with a different exponent than in the RM.The third is to attempt an extension of the RM tothe asymptotic limit in systems with correlated disor-der. The RM neglects the effects of rare low-disorderregions; these regions dictate the asymptotic transitionin randomly disordered systems [45, 46, 57, 61, 65, 82– 87]. Contrarily, in MBL chains with quasiperiodic [88–90]or sufficiently hyperuniform [91] disorder, as there are nosuch rare regions [57, 92, 93], MBL may be destabilised bymany-body resonances even in the thermodynamic limit. ACKNOWLEDGMENTS
We are grateful to S. Gopalakrishnan, D. Huse, A.Polkovnikov, A. Scardicchio, and D. Sels for insightfulcomments and useful discussions, to P. Krapivsky for in-sight into the treatment and regimes of (84), and to C.R.Laumann and V. Khemani for detailed comments on adraft of the manuscript. We are additionally grateful toD. Sels for providing the data of Fig 2. from Ref. [2],here plotted in Fig. 10. P.C. is supported by the NSFSTC “Center for Integrated Quantum Materials” underCooperative Agreement No. DMR-1231319. This workis supported by NSF DMR-1813499 (A.C.). 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However, we refrain fromthis nomenclature as the region is masked by the collec-tive instability of overlapping resonances discussed next.[64] David J Luitz, Nicolas Laflorencie, and Fabien Alet,“Many-body localization edge in the random-field heisen-berg chain,” Physical Review B , 081103 (2015).[65] A Brooks Harris, “Effect of random defects on the criticalbehaviour of ising models,” Journal of Physics C: SolidState Physics , 1671 (1974).[66] JT Chayes, L Chayes, Daniel S Fisher, and T Spencer,“Finite-size scaling and correlation lengths for disorderedsystems,” Physical review letters , 2999 (1986).[67] Maksym Serbyn, Z. Papić, and Dmitry A. Abanin,“Thouless energy and multifractality across the many-body localization transition,” Phys. Rev. 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(29) recovers the standard first-order term in Hamil-tonian perturbation theory in the high-frequency limit T → .[74] We note that the RM predicts that θ = θ < for ζ > ζ c leading to a stronger divergence than at θ = θ c . However,as this prediction hinges on the exponential growth of q ( r ) on the thermal side for ranges r < ξ , this prediction is unphysical and and may be disregarded.[75] Pranjal Bordia, Henrik Lüschen, Ulrich Schneider,Michael Knap, and Immanuel Bloch, “Periodically driv-ing a many-body localized quantum system,” NaturePhysics , 460–464 (2017).[76] Pranjal Bordia, Henrik Lüschen, Sebastian Scherg,Sarang Gopalakrishnan, Michael Knap, Ulrich Schnei-der, and Immanuel Bloch, “Probing slow relaxation andmany-body localization in two-dimensional quasiperiodicsystems,” Physical Review X , 041047 (2017).[77] To see this note that if U = τ zn on a site n in which τ an (cid:54) = τ bn , then U has an order one matrix element between thetwo cat states (and similarly for any string of τ zn withan odd number of such terms). (cid:104) E (cid:48) a | U | E (cid:48) b (cid:105) = O ( | U | ) thenfollows as a generic local operator U has O ( | U | ) overlaponto such terms.[78] Maksym Serbyn, Z. Papić, and Dmitry A. Abanin, “Cri-terion for many-body localization-delocalization phasetransition,” Phys. Rev. X , 041047 (2015).[79] Konstantin S. Tikhonov and Alexander D. Mirlin,“Eigenstate correlations around many-body localizationtransition,” (2020), arXiv:2009.09685 [cond-mat.dis-nn].[80] K. S. Tikhonov and A. D. Mirlin, “Statistics of eigen-states near the localization transition on random regulargraphs,” Phys. Rev. B , 024202 (2019).[81] Boris L Altshuler, Yuval Gefen, Alex Kamenev, andLeonid S Levitov, “Quasiparticle lifetime in a finite sys-tem: A nonperturbative approach,” Physical review let-ters , 2803 (1997).[82] Ronen Vosk, David A Huse, and Ehud Altman, “The-ory of the many-body localization transition in one-dimensional systems,” Physical Review X , 031032(2015).[83] Andrew C Potter, Romain Vasseur, andSA Parameswaran, “Universal properties of many-body delocalization transitions,” Physical Review X ,031033 (2015).[84] Liangsheng Zhang, Bo Zhao, Trithep Devakul, andDavid A Huse, “Many-body localization phase transition:A simplified strong-randomness approximate renormal-ization group,” Physical Review B , 224201 (2016).[85] Philipp T Dumitrescu, Romain Vasseur, and Andrew CPotter, “Scaling theory of entanglement at the many-body localization transition,” Physical review letters ,110604 (2017).[86] Thimothée Thiery, Markus Müller, and WojciechDe Roeck, “A microscopically motivated renormaliza-tion scheme for the mbl/eth transition,” arXiv preprintarXiv:1711.09880 (2017).[87] Vedika Khemani, DN Sheng, and David A Huse, “Twouniversality classes for the many-body localization tran-sition,” Physical review letters , 075702 (2017).[88] Shankar Iyer, Vadim Oganesyan, Gil Refael, andDavid A Huse, “Many-body localization in a quasiperi-odic system,” Physical Review B , 134202 (2013).[89] Marko Žnidarič and Marko Ljubotina, “Interaction insta-bility of localization in quasiperiodic systems,” Proceed-ings of the National Academy of Sciences , 4595–4600(2018).[90] Nicolas Macé, Nicolas Laflorencie, and Fabien Alet,“Many-body localization in a quasiperiodic fibonaccichain,” SciPost Phys , 050 (2019).[91] Philip JD Crowley, CR Laumann, and Sarang Gopalakr-ishnan, “Quantum criticality in ising chains with random hyperuniform couplings,” Physical Review B , 134206(2019).[92] J.M. Luck, “Critical behavior of the aperiodic quantumising chain in a transverse magnetic field,” Journal ofStatistical Physics , 417–458 (1993).[93] J. M. Luck, “A classification of critical phenomena onquasi-crystals and other aperiodic structures,” EPL (Eu-rophysics Letters) , 359 (1993). Appendix A: Multiple and imperfect resonances inthe Resonance Model1. Imperfect cat states
In Sec. II B 1, we assume that pairs of resonant eigen-states of H form perfect cat states with equal weights, | ε α,β (cid:105) = 1 √ (cid:0) | (cid:15) a ↑(cid:105) ± | (cid:15) b ↓(cid:105) (cid:1) . (A1)Their contribution to [ S ( ω )] is thus pure tone with noweight at zero frequency, (cid:104) (cid:15) a σ | σ z P ( t ) σ z P (0) | (cid:15) a σ (cid:105) = cos( | V ba | t ) . (A2)A more refined ansatz for the hybridised states wouldincorporate the resonance parameter g ba and lead to im-perfect cat states: | ε α,β (cid:105) = √ p | (cid:15) a ↑(cid:105) + (cid:112) − p e i φ | (cid:15) b ↓(cid:105) . (A3)Above, p ≈ / O ( g − ba ) . Imperfect cat states contributedelta function peaks at ω = 0 and ω = ω a ↑ ≈ | V ba | + O ( | V ba | g − ba ) (cid:104) (cid:15) a σ | σ z P ( t ) σ z P (0) | (cid:15) a σ (cid:105) = (1 − p ) + 4 p (1 − p ) cos( ω a ↑ t ) . (A4)Accounting for the distribution of g ba in (25) corrects λ ,the weight at zero frequency and the exact form of [ S ( ω )] .However, it does change universal features, such as thevanishing of the exponent θ with | ζ − ζ c | and the expo-nential decay in r of F ( r ) , the weight at zero frequencyafter all range r (cid:48) ≤ r processes have been accounted for,as per (45).
2. Multiple resonances
Suppose an eigenstate | (cid:15) a , ↑(cid:105) is resonant with multipleother eigenstates of H . Here we argue that the strongestresonance (defined by (35)) sets the frequency of oscilla-tion of (cid:104) (cid:15) a ↑ | σ zP ( t ) σ zP (0) | (cid:15) a ↑(cid:105) .Consider the case of two resonances at different ranges.Let | ε α (cid:105) = √ ( | (cid:15) a ↑(cid:105) + | (cid:15) b ↓(cid:105) ) denote the cat state result-ing from the strongest resonance (at the shorter range).Suppose that | ε α (cid:105) is now resonant with another state | (cid:15) c ↓(cid:105) at larger range with some matrix element (cid:104) ε α | V | (cid:15) c ↓(cid:105) = V αc := √ ( V ac + V bc ) . (A5) This matrix element is much smaller than | V ba | as | V ac | , | V bc | (cid:28) | V ba | . Treating this resonance within de-generate perturbation theory splits the peak at ω = | V ba | into two peaks at ω = | V ba | ± | V αc | . As this further split-ting is small, we neglect it and assume that the spectralweight remains sharply peaked around ω = | V ba | .In the time domain this statement is as follows: aninitial state | (cid:15) a ↑(cid:105) oscillates between | (cid:15) a ↑(cid:105) and | (cid:15) b ↓(cid:105) ona time scale | V ba | − and tunnels into the state | (cid:15) c ↓(cid:105) onthe much longer timescale | V αc | − .We generalise the above argument to many-resonancecase. Suppose | ε α (cid:105) has a resonance meditated by a matrixelement | V αc | , which leads to hybridised states | ε (cid:48) α ± (cid:105) = 1 √ | ε α (cid:105) ± | (cid:15) c ↓(cid:105) ) . (A6)Take one of these states | ε (cid:48) α + (cid:105) . Suppose this state hasa longer-range resonance mediated by a matrix element | V (cid:48) αd | . We obtain two new cat states. Suppose one ofthese two cat states | ε (cid:48)(cid:48) α + (cid:105) has an even longer-range res-onance mediated by | V (cid:48)(cid:48) αe | and so on. The initial peak at ω a ↑ = | V ba | splits into several peaks at ω = | V ba | − | V αc | , | V ba | + | V αc | − | V (cid:48) αd | , | V ba | + | V αc | + | V (cid:48) αd | ± | V (cid:48)(cid:48) αe | . . . (A7)An analogous procedure splits each of the peaks with aminus sign in the RHS above into many sub-peaks.To show that such shift ∆ ω remain unimportant wecalculate the root-mean-square size shift ∆ ω as showthat ∆ ω (cid:28) ω a ↑ . To do this we first note that the matrixelements v ( r ) (cid:48) connecting an already hybridised state toother unhybridised states at range r are a factor √ smaller v (cid:48) ( r ) = 1 √ v ( r ) , (A8)where as the density of states is twice as large ρ (cid:48) ( r ) = 2 ρ ( r ) (A9)yielding a probability of hybridising at range r of q (cid:48) ( r ) = √ q ( r ) . (A10)Thus, supposing that the initial resonance is at a range r (i.e. that ω a ↑ = v ( r ) ) we find ∆ ω = ∞ (cid:88) r (cid:48) = r +1 v (cid:48) ( r (cid:48) ) X ( r (cid:48) ) (A11)where X ( r ) is a random variable which takes values X ( r ) = 1 , − , with probabilities q (cid:48) ( r ) / , q (cid:48) ( r ) / , − q (cid:48) ( r ) respectively. Thus ∆ ω has mean ∆ ω = 0 and, mea-sured in units of the initial resonant frequency ω a ↑ , hasvariance ∆ ω ω a ↑ = (cid:90) ∞ r +1 d sq (cid:48) ( s ) (cid:18) v (cid:48) ( s ) v ( r ) (cid:19) = e − (3+ r ) /ξ √ λ (4 /ζ c + 3 /ξ ) (A12)4On the localised half of the phase diagram ( ξ > ) thisquantity is exponentially decaying in r , indicating thisapproximation scheme is asymptotically improving at lowfrequencies. In the crossover region it is bounded by itscritical value, which is much smaller than unity ∆ ω ω a ↑ ≈ ζ c √ λ (cid:28) , (A13)and so does not alter the asymptotic form of the spectralfunction [ S ( ω )] , whereas on the thermal this approxima-tion breaks down only for r > ξ , outside the regime ofvalidity of our calculation. Appendix B: The spectral function [ S ( ω )] in theHamiltonian RM for large systems in the vicinity ofthe MBL transition: L, | ξ | > λ In this regime hydrodynamic constraints become im-portant. These constraints highlight the limitations ofthe approximation made in (90), as F predicts unphys-ical behaviour. Specifically lim ξ →∞ F ( r ) = e − √ r/λ (B1)which using [ C zz ( t )] = F ( r ( t )) (49), and the logarithmi-cally growing light cone r ( t ) ∝ log t implies that the cor-relations decay as a stretched exponential in log t . Thisdecay is slower than any power law, but much faster thanthe maximum possible decay rate permitted by energyconservation of [ C zz ( t )] ∝ r ( t ) ∝ t . (B2)This maximum rate follows as the z -field on the probespin σ z P has overlap with the Hamiltonian tr (cid:0) σ z P H (cid:1) = W ,and any initial energy on the probe spin cannot havespread further than the light cone front r ( t ) .In order to address this inconsistency we turn to amore careful treatment of Eqs. (86) and (88). By di-rect numerical integration (see Appendix C 1) we findthat the stretched exponential decay is cut-off at r (cid:38) λ by an asymptotic decay F ( r ) ∼ r − , implying a decay [ C zz ( t )] ∼ log − t . This decay is still too fast to be consis-tent with hydrodynamics, however, the weakness of thisviolation means there are many small corrections whichyield a late time dynamical regime consistent with hy-drodynamics. For example, a sub leading power law in r on the matrix elements v ( e , r ) will suffice. However, herewe explore the effect of energy dependency of the matrixelements.Instead of the energy independent form for the matrixelements (77), we now consider v ( e , r ) = J exp (cid:18) − r ˜ ζ ( e /r ) − rζ c (cid:19) . (B3) where we now allow the localisation length to vary asa function of the energy density e /r of the patch of thesystem which must be rearranged to relate the two states | E a ↑(cid:105) and | E b ↓(cid:105) (As we are interested only in behaviourat asymptotically large r , we consider these states to beat the same energy density, despite their energy differ-ence of ± W due to the probe spin). We consider onlythe leading order dependence on energy density of thelocalisation length ζ ( e /r ) = 1 ζ (cid:18) e rη + e r µ + . . . (cid:19) (B4)where ζ is the localisation length at maximum entropy,the constant energy densities µ, η determine scales overwhich ζ varies, and we have suppressed higher powers of e /r . We will assume η = ∞ as the statistical symmetryof the model implies ˜ ζ should be an even function, and µ positive and finite. This corresponds to a localisationlength which is shorter away from maximum entropy.The energy dependence of the matrix elements thenalters the form of q σ ( e , r ) : q σ ( e , r ) ∼ √ λr exp (cid:18) − rξ − e ζrµ − ( e + e σ ) W r (cid:19) . (B5)For µ positive and finite q σ ( e , r ) is asymptotically nar-rower than ρ σ ( e , r ) at large r , we can extract the asymp-totic behaviour of f σ by replacing q σ ( e , r ) with a deltafunction ∂f σ ∂r = W ∂ f σ ∂ e − γδ ( (cid:15) + σW ) f σ f σ (cid:0) e , − (cid:1) = δ (cid:0) e − σW (cid:1) . (B6)where γ = (cid:82) d e q σ ( e , r ) is an r independent constant atthe critical point. Solving (B6) (see Appendix (C 2)) wefind asymptotic decay F ( r ) = (cid:90) d e f σ ( e , r ) ∼ √ r (B7)where here ∼ indicates asymptotic equality up to an over-all constant. This yields [ C zz ( t )] ∼ log − / Jt (B8) [ S ( ω )] ∼ | ω | − log − / | J/ω | (B9)consistent with hydrodynamic restrictions. Appendix C: Solutions to the loss-diffusion (86)
In this appendix we consider the loss-diffusion equa-tion (86) ∂f σ ∂r = W ∂ f σ ∂ e − f σ q σ f σ (cid:0) e , − (cid:1) = δ (cid:0) e − σW (cid:1) . (C1)We study two regimes:5 × × × × × × × - - - - - - - Figure 11.
Decay in F ( r ) for energy independent matrix ele-ments: Values of λ | d F/ d r | are plotted versus r/λ , these areobtained by numerically solving (C2) and (C4). The point r/λ = 1 is marked with a vertical grey line. For r/λ < ,the behaviour is consistent with F ( r ) = exp( − (cid:112) r/λ ) (dot-ted line). For r/λ > , the decay is slower F ( r ) ∝ ( λ/r ) (dashed). Different series correspond to different values of λ (legend inset). • We first study the critical dynamics ( ζ = ζ c ) withenergy independent matrix elements ( v a functionof r only). We show that the asymptotic decayof F ( r ) = (cid:82) d e f σ ( e , r ) is given by F ( r ) ∝ r − asquoted in the main text. This behaviour is notpermitted asymptotically due to hydrodynamic re-strictions.• We then study the asymptotic critical dynamicsfor energy dependent matrix elements (B3) with η = ∞ , and < µ < ∞ . We show that in this case F ( r ) ∼ r − / , behaviour consistent with hydrody-namics.
1. Critical point with energy independent matrixelements
Here we study the equation defined in the main text,specifically ∂f ↑ ∂r = W ∂ f ↑ ∂ e − f ↑ q ↑ ( e , r ) f ↑ (cid:0) e , − (cid:1) = δ (cid:0) e − W (cid:1) . (C2)for the loss function q ↑ ( e , r ) ∼ √ λr exp (cid:18) − ( e + W ) s e ( r ) (cid:19) . (C3)where s e ( r ) = W √ r + 1 .We numerically solve these equations by stochasticsampling of trajectories. In Fig 11 we plot d F/ d r fordifferent values of the parameter λ where as before F ( r ) = (cid:90) d e f ↑ ( e , r ) . (C4) We see that for all trajectories the initial decay at small r (cid:46) λ is consistent with the approximate solution F ( r ) =exp( − (cid:112) r/λ ) (grey vertical line marks r = λ ) at whichthere is a crossover to F ( r ) ∝ r − behaviour. For theseequations this latter behaviour continues asymptotically.In Fig. 12 we show the variation of f ↑ ( e , r ) with e ,specifically we plot f ↑ ( e , r ) for a series of fixed log-spacedvalues of r . For clarity we also re-scale e by the widthof the distribution s e ( r ) = W √ r + 1 (i.e. so thatfor λ = ∞ the plots would collapse for all r ). Fromthese plot it is clear that the centre of the distributionis depleted faster than the mean, that is f ↑ (0 , r ) decaysasymptotically faster than F ( r ) . This behaviour is ex-hibited for r (cid:29) λ and violates the approximation schemeof Sec. III B 4.
2. Critical point with energy dependent matrixelements
We now study the same loss-diffusion equation (C1) fordynamics in the crossover region with energy dependentmatrix elements. Specifically we now set q ↑ ( e , r ) ∼ √ λr exp (cid:18) − e ζ c rµ − ( e + W ) s e ( r ) (cid:19) . (C5)for some finite µ in the range < µ < ∞ .To simplify the problem we make several approxima-tions which do not alter the asymptotic behaviour ofthese equations. First, as the width of q σ is asymptoti-cally smaller (in r ) than s e ( r ) , for r (cid:29) λ we can approx-imate q ↑ ( e , r ) with a delta function placed at the originwith weight γ = (cid:90) d e q ↑ ( e , r ) = W µ (cid:113) λζ c π (4 W + ζ c µ ) + O ( r − ) . (C6)Second, we neglect the sub-leading r -dependent correc-tion to γ , and thirdly we neglect the initial energy offsetof f ↑ . This yields the equation ∂f ↑ ∂r = W ∂ f ↑ ∂ e − γf ↑ δ ( e ) , (C7)with boundary condition f ↑ ( e , r = 0) = δ ( e ) .To solve this equation we decompose f ↑ as f ↑ ( e , r ) = ∞ (cid:88) n =0 f n ( e , r ) (C8)which satisfy the equations ∂f ∂r = W ∂ f ∂ e (C9)with boundary condition f ( e , r = 0) = δ ( e ) for n = 0 and ∂f n ∂r = W ∂ f n ∂ e − γf n − δ ( e ) (C10)6 - - - - - e / s e ( r ) f ( e , r ) · s e ( r ) λ = × - - - - - e / s e ( r ) f ( e , r ) · s e ( r ) λ = × - - - - - e / s e ( r ) f ( e , r ) · s e ( r ) λ = Figure 12.
Decay in F ( r ) for energy independent matrix elements: the distributions f σ ( e , r ) are plotted for log-spaced intervalsof r , using the same numerical solutions to (C2) and (C4) as Fig 11. In each case it is clear that at large ranges the distributionis depleted at energies e (cid:46) s e ( r ) . with boundary condition f n ( e , r = 0) = 0 for n > .With this f is straightforwardly identified f ( e , r ) = e − e / (4 rW ) √ πrW , (C11)and it further follows that for n > f n ( e , r ) = − γ (cid:90) r d sf ( e , r − s ) f n − (0 , s ) (C12)this equation is obtained by simply treating f n − (0 , s ) as a source term for f n , in accordance with (C10), andintegrating with the heat equation Kernel f . To makeprogress we note that it is sufficient to obtain the f n (0 , s ) ,which are related by a recursion relation f n (0 , r ) = − γ (cid:90) r d s (cid:112) W π ( r − s ) f n − (0 , s ) . (C13)and related to our desired result, F ( r ) , by F ( r ) = ∞ (cid:88) n =0 (cid:90) d e f n ( e , r ) = 1 − γ ∞ (cid:88) n =1 (cid:90) r d sf n − (0 , r ) (C14)where we have substituted (C12).Solving this recursion relation (C13) yields f n (0 , r ) = ( − n γ(cid:96) Γ (cid:0) n +12 (cid:1) (cid:16) r(cid:96) (cid:17) n − . (C15)where (cid:96) = 4 W /πγ . The function F ( r ) is then obtainedby substituting (C15) into (C14), performing the integral γ (cid:90) r d sf n − (0 , r ) = ( − n Γ (cid:0) n +32 (cid:1) (cid:16) r(cid:96) (cid:17) n +12 (C16)and recognising the resulting summation as a Taylor se-ries, this yields F ( r ) = e r/(cid:96) Erfc (cid:16)(cid:112) r/(cid:96) (cid:17) (C17) where
Erfc( x ) = 1 − √ π (cid:90) x − x e − t d t (C18)is the usual complementary error function. From (C17)it follows that F ( r ) decays asymptotically as F ( r ) ∼ (cid:114) (cid:96)πr = 2 Wπγ √ r (C19)as quoted in the main text. The constant pre-factor hereis liable to be altered by the simplifications we made ear-lier in the calculation, however the asymptotic behaviour F ( r ) ∝ r − / is robust. Appendix D: Linear drift of the deviation fromthermal behaviour
In this appendix we derive (117) from the main text W δ ( L ) ≈ W c e − (cid:96) δ / ( L +1) . (D1)where (cid:96) δ is some δ dependent constant, and W δ ( L ) isdefined as the disorder strength at which the time aver-aged correlator [ C zz ] deviates from thermal behaviour bysome small amount δ [ C zz ]( W δ ) = δ (cid:28) (D2)Recalling that [ C zz ] = F ( L/ and using the form (91)for F ( r ) on the thermal side δ = exp (cid:32) − (cid:115) π | ξ ( W δ ) | λ ( W δ ) Erfi (cid:32)(cid:115) L | ξ ( W δ ) | (cid:33)(cid:33) (D3)where we have explicitly labelled disorder dependence ofthe correlation length ξ and the resonance length λ . Weuse ξ ( W ) ≈ W/W c ) (D4)7whereas λ is given by (83).Let us extract from (D3) how W δ varies with L . Awayfrom the crossover region the imaginary error functioncan be written in terms of more familiar functions Erfi( √ x ) = e x √ πx (cid:0) O ( x − ) (cid:1) (D5)Substituting both (D5) and λ ( W δ ) = ( W δ /W c ) λ ( W c ) into (D3) and rearranging we obtain L | ξ ( W δ ) | + log W δ W c = 2 log (cid:32) (cid:112) Lλ ( W c ) | ξ ( W δ ) | | log δ | (cid:33) + O (cid:18) | ξ ( W δ ) | L (cid:19) (D6)Consider the RHS of (D6): for sufficiently small δ we arefar from the crossover L (cid:29) | ξ | and the corrections may be neglected. Now consider the leading term on the RHSof (D6): this term exhibits weak logarithmic dependenceof L , and, recalling that ξ ( W δ ) ≈ / log( W δ /W c ) , doublylogarithmic dependence on W δ , thus to first approxima-tion the RHS may be replaced by a (negative) constant − (cid:96) δ : L | ξ ( W δ ) | + log W δ W c = − (cid:96) δ (D7)Then, again using ξ ( W δ ) ≈ / log( W δ /W c ) , by rearrang-ing we obtain the desired result (D1).This function is approximately linear for sufficientlysmall L . To see this, note that the RHS of (117) has aninflection point at L = (cid:96) δ / − , and thus has zero curva-ture at this point. Taylor expanding about the inflectionpoint and demanding that the cubic term is not largerthan the linear term reveals the approximate linearity topersist for L + 1 (cid:46) (cid:96) δ (1 / (cid:112) /4 )