Probing Many-Body Localization by Spin Noise Spectroscopy
PProbing Many-Body Localization by Spin Noise Spectroscopy
Dibyendu Roy, Rajeev Singh, Roderich Moessner
Max Planck Institute for the Physics of Complex Systems, N¨othnitzer Str. 38, 01187 Dresden, Germany
We propose to apply spin noise spectroscopy (SNS) to detect many-body localization (MBL) indisordered spin systems. The SNS methods are relatively non-invasive technique to probe sponta-neous spin fluctuations. We here show that the spin noise signals obtained by cross-correlation SNSwith two probe beams can be used to separate the MBL phase from a noninteracting Andersonlocalized phase and a delocalized (diffusive) phase in the studied models for which we numericallycalculate real time spin noise signals and their power spectra. For the archetypical case of thedisordered XXZ spin chain we also develop a simple phenomenological model.
The fate of Anderson localization in the presenceof inter-particle interactions in a disordered quantummedium is an exciting frontier in condensed matterphysics [1–19]. It is well known that all states of a one-dimensional (1D) disordered chain of noninteracting par-ticles are Anderson localized (AL) for any amount of dis-order [20, 21]. Thus, the AL state is a perfect insulator aslong as the particles are not coupled to other degrees offreedom. In the presence of inter-particle interactions, adynamical transition from a delocalized (diffusive) phaseto a many-body localized (MBL) phase has been pre-dicted in disordered quantum media when the strengthof (quenched) randomness is increased [4–9].The MBL state is also a perfect insulator, but it hasdifferent dynamical properties compared to the AL stateof noninteracting particles [10–12, 16]. For example, en-tanglement entropy in the MBL phase shows a slow log-arithmic growth following a global quench in an isolatedsystem [10–12]. However, the entanglement entropy isvery difficult to measure experimentally, and electrons orspins in conventional solid-state systems are coupled toan environment such as a phonon bath. Recent studiesshow that signatures of the MBL phase can survive in thepresence of weak coupling to a thermalizing environment[22, 23]. In particular, the spectral functions of local op-erators can be used to identify the MBL phase in thepresence of weak dissipation [22, 23].A new experimental approach based on a modified non-local spin-echo protocol along with a double electron-electron resonance technique in electron spin resonancehas been proposed to distinguish the MBL phase froma noninteracting AL phase and a delocalized phase atinfinite temperature [24]. The proposed approach canprobe interaction effects, thus is able to separate theMBL phase from the AL phase. The method does in-volve optical pumping or polarization of local spins byexternal pulse fields, which can in principle lead to un-wanted local heating and excitations. Here we propose arelatively non-invasive method based on spin noise spec-troscopy (SNS) to distinguish the MBL phase from theAL phase and the delocalized phase in disordered spinsystems.The optical SNS method has been developed recentlyas an alternative to conventional perturbation-based(pump-probe) techniques for measuring dynamical spin (a) (b)
FIG. 1. (a) Schematic of the two-beam spin noise spec-troscopy set-up with two probe lasers detecting spin fluctua-tions of different spins. (b) Disorder ( η ) vs. energy density( (cid:15) ) phase-diagram of the disordered Heisenberg chain [44]. properties [25, 26]. Intrinsic spin fluctuations of electronsand holes are passively detected in SNS by measuring op-tical Faraday rotation fluctuations of a linearly polarizedprobe laser beam passing through the sample [27–38].SNS with a single probe laser has been useful in char-acterizing various properties (e.g., g-factors, relaxationrates and decoherence times) of different spin ensembles,such as specific alkali atoms [30], itinerant electron spinsin semiconductors [33] and localized hole spins in quan-tum dot ensembles [34]. Last year, an extension of thetraditional SNS has been proposed and demonstrated byusing two linearly polarized probe lasers for detectinginter-species spin interactions in a heterogeneous two-component spin ensemble interacting via binary exchangecoupling in thermal equilibrium [39, 40]. In this cross-correlation SNS method, intrinsic spin fluctuations fromtwo different species are independently detected, and in-teraction effects are determined by the cross-correlationof these two spin noise signals. Interaction effects be-tween spins of a single species in a sample can alsobe determined by two-beam SNS when the two probelaser beams are spatially separated as shown in Fig. 1(a)[39, 41]. Thus, we propose that the two-beam SNS mea-surements can be used to separate the MBL phase froman AL phase. In fact, as we show in our example, theresponses from single-beam and two-beam SNS can beefficiently employed to distinguish the MBL phase fromboth the AL and delocalized phases.In order to demonstrate our idea we study the spin-noise signals of a 1D random-field XXZ spin chain, a r X i v : . [ c ond - m a t . d i s - nn ] J un perhaps the best-studied “canonical” model system forMBL. In addition, results for the disordered transverse-field Ising model are given in the Suppl. Mat. [43]. TheHamiltonian of disordered XXZ spin chain is given by H = N (cid:88) i =1 J ⊥ ( S xi S xi +1 + S yi S yi +1 ) + J z S zi S zi +1 + h i S zi , (1)where the random fields h i are chosen from a uniform dis-tribution within the window [ − η, η ], and S x,y,zN +1 = S x,y,z for periodic boundary conditions. For isotropic exchangeinteractions ( J z = J ⊥ ), this is a disordered Heisenbergchain. The spin Hamiltonian in Eq. 1 can be mapped toa model of interacting spinless Jordon-Wigner fermionson a lattice where J ⊥ represents hopping between neigh-boring sites, J z is inter-particle interaction strength and h i denotes a random local chemical potential [42]. For J z = 0, the Hamiltonian in Eq. 1 represents noninter-acting fermions subject only to a random local poten-tial, and hence can be used to study the AL phase ofnoninteracting particles. This model with a non-zero J z has been extensively studied recently in the context ofMBL. A disorder versus energy density phase-diagramseparating the delocalized and MBL phases of an iso-lated chain is shown in Fig. 1(b) following Ref. [44] where (cid:15) = ( (cid:104) H (cid:105)− E max ) / ( E min − E max ) with E max ( E min ) beingmaximum (minimum) energy.For the Hamiltonian in Eq. 1, we now describe thesignals measured in the SNS set-ups. The measured re-sponse of the spin component tranverse to the randommagnetic field in the single-beam SNS is C xI ( t ) = (cid:104)(cid:104){ S xI ( t ) , S xI (0) }(cid:105)(cid:105) , (2)where { ., . } is the anti-commutator and (cid:104)(cid:104) .. (cid:105)(cid:105) denotesboth (canonical) thermal and disorder averaging. Here,the x -component of total spin polarization at the mea-surement spot at time t is S xI ( t ) = (cid:80) l ∈ I S xl ( t ) where thesum runs over all local spin sites within the sample spotilluminated by the probe laser beam I . The correlationfunction C xI ( t ) describes the relaxation of spontaneousspin fluctuations at the spot of the sample probed bythe single-beam SNS. The single-beam (local) spin-noisepower spectrum is obtained by its Fourier transform, P xI ( ω ) = (cid:90) ∞−∞ dt e iωt C xI ( t ) . (3)Let | E n (cid:105) denote a many-body eigenstate with eigenvalue E n , H | E n (cid:105) = E n | E n (cid:105) . The single-beam spin-noise powerspectrum reads, P xI ( ω ) = (cid:68) Z (cid:88) n,p δ ( ω + E n − E p )( e − βE n + e − βE p ) |(cid:104) E n | S xI | E p (cid:105)| (cid:69) , (4)with partition function Z = (cid:80) n e − βE n , and inverse tem-perature β = 1 /k B T . (cid:104) .. (cid:105) in Eq. 4 describes averaging over different disorder realizations. The cross-correlationsignal of the two-beam SNS is C x cr ( t ) = (cid:104)(cid:104){ S xI ( t ) , S xII (0) } + { S xII ( t ) , S xI (0) }(cid:105)(cid:105) , (5)where S xII ( t ) = (cid:80) m ∈ II S xm ( t ), and the sum over m runsthrough all sites which the probe laser II illuminates.The cross-correlation spin noise power P x cr ( ω ) = (cid:90) ∞−∞ dt e iωt C x cr ( t ) (6)= (cid:68) Z (cid:88) n,p δ ( ω + E n − E p )( e − βE n + e − βE p )Re (cid:0) (cid:104) E n | S xI | E p (cid:105)(cid:104) E p | S xII | E n (cid:105) (cid:1)(cid:69) . (7)We calculate the eigenvalues and eigenstates of theHamiltonian using exact diagonalization, and evaluatethe spin noise signals in Eqs. 2,5 and the correspondingpower spectra in Eqs. 4,7. In the following we presentresults using periodic boundary conditions at high tem-perature, k B T = 50 J ⊥ , averaging over 3000 disorder real-izations. However, the main results of this paper are alsovalid at moderate temperatures [43]. We quote J z , h i , η in the units of J ⊥ , and fix J ⊥ = 1 throughout the paper.The single-beam spin noise response in real time forthe interacting delocalized phase at low disorder ( η = 1)and for the MBL phase at high disorder ( η = 5, at whichevery many-body eigenstate of the isolated chain is local-ized) of the random XXZ model is shown in Fig. 2(a,b).The single-beam responses are clearly different in the twophases. The transverse spin component relaxes exponen-tially in the delocalized phase, while in the MBL phaseit appears to do so algebraically, and it also oscillates.We show the single-beam spin noise response of the ALphase ( J z = 0) in Fig. 2(c), which again indicates analgebraic relaxation of the transverse spin component.The relaxation in this phase can be approximated assin( t/τ A ) / ( t/τ A ) with some characteristic time scale τ A .We use high disorder ( η = 5) in Fig. 2(c) to ensure thatall states are localized even for a finite-length noninter-acting chain. We provide log scale plots in the Suppl.Mat. [43] for the real-time spin noise responses in Fig. 2to highlight the nature of their respective decays.Within the single-beam SNS measurements, the slowspin relaxation due to interactions in the highly disor-dered XXZ model is masked by other mechanisms of re-laxation (e.g., due to random fields and hopping) alsopresent in the noninteracting case. Thus, it becomes dif-ficult to separate the MBL phase from the AL phase bysingle-beam SNS. However, interaction effects are nicelydiagnosed by two-beam SNS when we cross correlate twodifferent noise signals to exclude their self-correlations.The two-beam cross-correlation spin noise responsesin real time for the three different phases are shownin Fig. 2(d,e,f) when two beams are just next to eachother without overlapping (i.e., the separation betweenthe centers of the two beams is two lattice spacings).The responses here are quite different in all three phases. . . . C x I ( t ) (a) J z =1 ,η =1 . . . (b) J z =1 ,η =5 . . . (c) J z =0 ,η =5 t [ J − ] . . . C x c r ( t ) (d) t [ J − ] . (e) t [ J − ] . (f) FIG. 2. Single-beam and two-beam spin noise responses inreal time in the three different phases. (a-c) C xI ( t ) and (d-f) C x cr ( t ) are obtained by exact diagonalization in the delocalized( J z = 1 , η = 1), the MBL ( J z = 1 , η = 5) and the Andersonlocalized ( J z = 0 , η = 5) phase. Here N = 12 and the beam I and II illuminate respectively spins 5,6 and 7,8. A large cross-correlation between the transverse compo-nents of different spins is developed in the diffusive phase,and it relaxes exponentially fast. The cross-correlationin the MBL phase is relatively smaller and it relaxesslowly with many oscillations. In the AL phase, thetransverse component hardly shows any cross-correlationbetween different spins at high temperature. Noticethat C xI (0) = 1 while C x cr (0) ≈ (cid:104) S xl (0) S xm (0) (cid:105) = δ lm / z -direction, and find that it is similar in the MBL andAL phases even within the two-beam SNS measurements[43, 46]. Thus, z -direction response is not a good candi-date to separate these two phases.Turning to the frequency domain, the disorder aver-aged spin noise power spectra of single-beam and two-beam SNS are shown in Fig. 3. We find a Lorentzian line-shape for the single-beam noise power spectrum P xI ( ω )[Fig. 3(a)] in the diffusive regime reflecting the exponen-tial spin relaxation. In the MBL phase, the shape of P xI ( ω ) exhibits a plateau as shown in Fig. 3(b). Theoverall rectangular shape of P xI ( ω ) in the AL regime[Fig. 3(c)] is somewhat similar to that of the MBL phasebut it shows much higher fluctuations.Crucially, the cross-correlation spin noise power spec-tra P x cr ( ω ) of the two-beam SNS in Fig. 3(d-f) differ sig-nificantly in shape and magnitude in the delocalized (dif-fusive), MBL and AL phases. The power spectrum in thedelocalized phase [Fig. 3(d)] can be perceived as the dif-ference between two equal area Lorentzians of differentwidths. Thus, the total area under the cross-correlationcurve is zero, a signature of high-temperature behaviour.The line-shape of P x cr ( ω ) in the MBL phase is very dif-ferent from that in the delocalized and AL phases. It . . . P x I ( ω ) (a) J z =1 ,η =1 . . . (b) J z =1 ,η =5 . . . (c) J z =0 ,η =5 − . . . . P x c r ( ω ) (d) − . . (e) − . . (f) − ω − . . . . P x c r ( ω ) (g) − ω − . . (h) − ω − . . (i) FIG. 3. Single-beam and two-beam spin noise power spectrain the three different phases. (a-c) P xI ( ω ), (d-f) P x cr ( ω ) areobtained by exact diagonalization and (g-i) P x cr ( ω ) are calcu-lated from Eq. 10 in the delocalized ( J z = 1 , η = 1), the MBL( J z = 1 , η = 5) and the Anderson localized ( J z = 0 , η = 5)phase. We use γ = 0 . , γ = 0 .
18 in (g), γ = 0 . , γ = 0 . γ = 0 in (i). The parameters in (a-f) are the sameas in Fig. 2. is also almost one order of magnitude smaller than inthe delocalized phase but is clearly non-zero [Fig. 3(e)].However, P x cr ( ω ) in the AL phase is very small (one or-der of magnitude smaller than that in the MBL phase athigh temperature) and featureless indicating essentiallyno correlation between spins. Note that P x cr ( ω ) showsboth negative and positive values which signify (anti-)correlations between spins at different frequency/timescales. The anti-correlations in Fig. 3(d,e) at higher fre-quencies (shorter time) are induced by fast co-flips be-tween different spins in the presence of spin-exchangecoupling. This feature is present in the two-beam cross-correlation spectra in the delocalized and MBL phasesbut is absent in the AL phase. P x cr ( ω ) depends on the separation between the twoprobe beams. With an increasing separation betweenthe beams, P x cr ( ω ) exhibits more oscillations in the delo-calized and MBL phases, and its magnitude falls rapidlyin the MBL and AL phases as might be expected in thepresence of spatial localization. Also the shape of P x cr ( ω )in the three phases remains unchanged when we explicitlycouple the disordered XXZ chain to a weakly thermaliz-ing environment. Both of these features are discussed inthe Suppl. Mat. [43].In order to understand the cross-correlation noisepower spectra obtained from numerics, we develop a sim-ple phenomenological model inspired by the experimentalset-up of the two-beam SNS. Let us denote the respec-tive total spin polarizations of the spins illuminated bythe probe laser beams I and II by S I and S II . We as-sume that the spins in beam I ( II ) can relax (a) due tospin-exchange interactions with the spins in beam II ( I ),and (b) due to interactions with other spins in the sam-ple. The spin-exchange interactions between the spinsfrom the two illuminated spots conserve total spin butdo transfer spins between the two spots via the spin co-flips with a rate γ . This leads to cross-correlations oftheir fluctuations. For process (b) we define a net spinrelaxation rate γ α for spins in beams α = I, II . Thisprocess does not conserve the total spin in the beams.Combining these with Larmor precession in the randommagnetic field, we obtain the following stochastic evolu-tion equations [40], d S α dt = S α × h α − γ α S α − γ ( S α − S ¯ α ) + ξ α , (8)where ¯ I = II, II = I . Here we have included stochas-tic fluctuations ξ α . For simplicity, we consider these asGaussian white noise with zero mean, (cid:104) ξ αi ( t ) ξ βj ( t (cid:48) ) (cid:105) = δ ( t − t (cid:48) ) δ ij (cid:0) δ αβ γ α + γ ( δ αβ − δ α ¯ β ) (cid:1) , (9)with i, j = x, y, z . The relation in Eq. 9 ensures station-ary equal-time correlation (cid:104) S αi ( t ) S βj ( t ) (cid:105) ∝ δ αβ δ ij / z -direction following theHamiltonian in Eq. 1, i.e., h α = h α ˆ z where h α are twoi.i.d. random numbers chosen from a uniform distribu-tion over [ − η, η ]. Averaging over noise, we find the cross-correlator of spin polarizations in Eq. 7 in compact form, P x cr ( ω ) = γ (cid:88) q = ± (cid:68) χ q χ q + κ q (cid:69) , (10)where χ ± = γ γ − ( ω ± h I )( ω ± h II ) , κ ± = γ (2 ω ± ( h I + h II )). Here γ I = γ II = γ , γ = γ + 2 γ and γ = γ + γ are characteristic rates that influence broad-ening of the peaks. (cid:104) .. (cid:105) in Eq. 10 denotes averaging over the random fields h α . Due to the presence of the ran-dom field throughout the sample, the phenomenologicalrelaxation rates γ, γ I and γ II vary with disorder strength η , and we use them as fitting parameters. We plot thecross-correlator P x cr ( ω ) after averaging over the randomfields for η = 1 , P x cr ( ω ) = 0 for γ = 0, in theAL phase ( J z = 0). We note, however, that our modelassumes instantaneous transfer of polarization densitythrough direct co-flips of spins, appropriate for directlyadjacent spots. If this condition is not fulfilled, a delayfor the propagation of polarization density over the dis-tance between the illuminated spots will have to be addedto our simplified model. Secondly, our model works onlyat relatively high temperatures and the results obtainedfrom the phenomenological model would deviate from thenumerical results at low temperature [43, 47].To summarize, we have calculated spin noise signalsin the diffusive, AL and MBL regimes of the disorderedXXZ and transverse-field Ising spin chains, for differ-ent temperatures as well as beam separations. We haveshown how these signals can be used to identify the pres-ence of MBL. 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Dibyendu Roy, Rajeev Singh, Roderich Moessner − − − | C x I ( t ) | (a) J z =1 ,η =1 (b) J z =1 ,η =5 (c) J z =0 ,η =5 t [ J − ] − − − | C x c r ( t ) | (d) − t [ J − ] (e) − t [ J − ] (f) FIG. S1. Log scale plots of single-beam and two-beam spinnoise responses of transverse spin component in real time inthree different phases of disordered XXZ spin chain. (a,d) inlog-linear scale in the delocalized phase ( J z = 1 , η = 1) and(b,c,e,f) in log-log scale in the MBL ( J z = 1 , η = 5) and theAnderson localized ( J z = 0 , η = 5) phase. Here N = 12 andthe beam I and II illuminate respectively spins 5,6 and 7,8. We show plots in log-linear and log-log scales of real-time spin noise responses of the transverse spin com-ponent of disordered XXZ spin chain to elucidate thenature of their relaxation. We plot absolute values of C xI ( t ) and C x cr ( t ) in log scale and time t in linear scalein Fig. S1(a,d). Both the real-time responses in the de-localized (diffusive) phase appear to exhibit exponentialdecay in time. In Fig. S1(b,c,e,f) we plot both absolutevalues of C xI ( t ) and C x cr ( t ) and time t in log scale. Whilethe single-beam spin noise response of the transverse spincomponent in real time shows algebraic (power-law) de-cay both in the MBL and AL phases, cross-correlationspin noise response shows nearly algebraic decay only inthe MBL phase. The transverse spin component showssignificantly less cross-correlation in the AL phase, as isevident in Fig. S1(f).In Fig. S2 we show single-beam and two-beam spinnoise responses of the longitudinal spin component ( z -direction) of disordered XXZ spin chain in real time in thethree different phases. Both spin noise responses showdifferent relaxation behaviour in the delocalized and thelocalized phases. However, the relaxation of the longi-tudinal spin component is quite similar in the MBL andthe AL phases within both single-beam and two-beamcross-correlation measurements. . . . C z I ( t ) (a) J z =1 ,η =1 (b) J z =1 ,η =5 (c) J z =0 ,η =5 t [ J − ] . . C z c r ( t ) (d) t [ J − ] (e) t [ J − ] (f) FIG. S2. Single-beam and two-beam spin noise responses oflongitudinal spin component in real time in the delocalized,MBL and Anderson localized phases of disordered XXZ spinchain. The parameters are the same as in Fig. S1.
Next, we discuss how cross-correlation spin noise spec-tra P x c r ( ω ) of transverse spin component of the disorderedXXZ spin chain depend on temperature and separationbetween the probe beams in the three different phases.For this we change temperature and separation betweenthe probe beams from those in Fig. 3 of the main text.For comparison in Fig. S3(d-f) we use the same temper-ature and separation between the probe beams as in themain text. We reduce temperature to 10 J ⊥ in Fig. S3(a-c), and we find that the line-shapes of P x c r ( ω ) in all threephases remains similar to that in Fig. S3(d-f) at high tem-perature 50 J ⊥ . Thus, the two-beam SNS is still a goodprobe to separate the three different phases. However,fluctuations in P x c r ( ω ) in the AL phase at low tempera-ture are increased by one order of magnitude from that athigh temperature. In fact, P x c r ( ω ) is similar in magnitudeboth in the MBL and AL phases but still has differentshapes. In Fig. S3(g-i) we move the beam II by onelattice spacing, and we find extra oscillations in P x c r ( ω )compared to Fig. S3(d,e) in the delocalized and MBLphases. We also notice that the magnitude of P x c r ( ω )is reduced by one order of magnitude in the MBL phaseand by two orders in the AL phase compared to Fig. S3(e-f). We use periodic boundary conditions and averagingover 3000 disorder realizations in Figs. S1,S2,S3, exceptin Fig. S3(h) where 10000 disorder realizations are used. − . . . . P x c r ( ω ) (a) J z =1 ,η =1 − . . (b) J z =1 ,η =5 − . . (c) J z =0 ,η =5 − . . . . P x c r ( ω ) (d) − . . (e) − . . (f) − ω − . . . P x c r ( ω ) (g) − ω − . . (h) − ω − . . (i) FIG. S3. Dependence of cross-correlation spin noise powerspectra P x cr ( ω ) on temperature and separation between theprobe beams in the delocalized, MBL and Anderson localizedphases of the disordered XXZ spin chain. Here N = 12 andtemperature is 10 J ⊥ in (a-c) and 50 J ⊥ in (d-i). The beam I and II illuminate respectively spins 5,6 and 7,8 in (a-f) andspins 5,6 and 8,9 in (g-i). We check the nature of cross-correlation spin noisepower spectra of disordered XXZ spin chain by explic-itly coupling the chain (system) to a thermalizing bath.We model the bath here by a weakly disordered XXZspin chain in the diffusive phase (nonintegrable). TheHamiltonians of the bath and system-bath coupling arerespectively, H bath = N (cid:88) i =1 J b (cid:126)σ i .(cid:126)σ i +1 + h bi σ zi , (S1) H c = N (cid:88) i =1 J c (cid:126)S i .(cid:126)σ i , (S2)where the random fields h bi are chosen from a uniformdistribution within the window [ − η b , η b ], and (cid:126)σ N +1 = (cid:126)σ for periodic boundary conditions. Here J c deter-mines strength of the system-bath coupling, and the bathshould be effective in thermalizing the system when thecoupling matrix element is of the order of the many-bodylevel spacing in the bath. The full Hamiltonian of the sys-tem plus bath is H f = H + H bath + H c where H is givenin Eq. 1 of the main text. We calculate spin noise signalsin an eigenstate | n (cid:105) of H f at an energy E n correspondingto that of the thermal ensemble at inverse temperature β , E n = (cid:104) n | H f | n (cid:105) = Tr[ e − βH f H f ] / Tr e − βH f . In Fig. S4 weshow two-beam cross-correlation spin noise spectra of thetransverse spin component in the three different phasesfor two different values of J c . Noise spectra are shownafter averaging over 30000 (except 10000 in Fig. S4(a,d))disorder realizations. We find that the shapes of two-beam spin noise spectra in Fig. S4 are similar to thoseobtained earlier without explicitly coupling the system − . . . . P x c r ( ω ) (a) J z =1 ,η =1 − . . (b) J z =1 ,η =4 (c) J z =0 ,η =4 − ω − . . . . P x c r ( ω ) (d) − ω − . . (e) − ω (f) FIG. S4. Cross-correlation spin noise power spectra P x cr ( ω )in the delocalized ( J z = 1 , η = 1), the MBL ( J z = 1 , η = 4)and the Anderson localized phases ( J z = 0 , η = 4) of thedisordered XXZ spin chain coupled to a thermalizing bath.The bath coupling J c = 0 .
15 in (a-c) and J c = 0 .
25 in (d-f).Here N = 6, J ⊥ = J b = η b = 1, temperature is 50 J ⊥ andbeams I and II illuminate respectively spins 2,3 and 4,5. to a thermalizing bath. This shows the robustness oftwo-beam spin noise signals to distinguish the three dif-ferent phases. We note from Fig. S4 that the fluctuationin the noise spectra falls with increasing bath couplingas long as the coupling is smaller than the characteristicenergy scales in the system. It signals better thermal-ization with increasing bath coupling without destroyingthe signatures of the three different phases in the isolateddisordered XXZ chain. We mention that the large fluc-tuations in the noise spectra shown in Fig. S4 are alsodue to relatively shorter chain length which we were ableto simulate for this case.Finally, we present cross-correlation spin noise signalsmeasured by two-beam SNS for a disordered transverse-field Ising model with next-nearest neighbor coupling.The Hamiltonian of this model is [1] H I = − N − (cid:88) i =1 J i S zi S zi +1 + J N − (cid:88) i =1 S zi S zi +2 + h N (cid:88) i =1 S xi , (S3)where the nearest neighbor couplings J i = J + δJ i , withall random δJ i chosen from a uniform distribution withinthe window [ − η, η ]; h is a constant magnetic field. Forfinite J , the above model has a delocalized phase at lowdisorder and an MBL phase at high disorder [1]. The sys-tem is in the AL phase when J = 0. We numerically cal-culate cross-correlation spin noise signals of the spin com-ponent along y -direction. We quote J , h, η in the unitsof J , and fix J = 1. The cross-correlation spin noise sig-nals in real time in the delocalized, MBL and AL phasesare shown in Fig. S5(a-c) at high temperature. A largecross-correlation between the transverse spin componentsof separate spins is developed in the delocalized phase t [ J − ] − . . C y c r ( t ) (a) J =0 . ,η =1 − ω − . . P y c r ( ω ) (d) t [ J − ] − . . (b) J =0 . ,η =5 − ω − . . (e) t [ J − ](c) J =0 ,η =5 − ω (f) FIG. S5. Two-beam spin noise responses of transversespin component in the three different phases of disorderedtransverse-field Ising chain. (a-c) C y cr ( t ) and (d-f) P y cr ( ω ) areobtained numerically in the delocalized ( J = 0 . , η = 1),the MBL ( J = 0 . , η = 5) and the Anderson localized( J = 0 , η = 5) phase. Here N = 12, J = 1, h = 0 .
6, tem-perature is 50 J and beams I and II illuminate respectivelyspins 5,6 and 7,8. and it relaxes relatively fast, while the cross-correlationis much weaker in the MBL phase and it relaxes veryslowly. The transverse spin component hardly shows anycross-correlation between different spins at high tempera-ture in the AL phase. We plot cross-correlation spin noisepower spectra in Fig. S5(d-f) which are different in shapeand magnitude in the three phases. Especially, the cross-correlation power spectra in the MBL and AL phases arevery different in shape. We use periodic boundary con-ditions, averaging over 3000 (in Fig. S5(a-c)), 6000 (inFig. S5(d)) and 20000 (in Fig. S5(e,f)) disorder realiza-tions. [1] J. A. Kj¨all, J. H. Bardarson, and F. Pollmann, Phys. Rev.Lett.113