Multifractality and self-averaging at the many-body localization transition
MMultifractality and self-averaging at the many-body localization transition
Andrei Sol´orzano, Lea F. Santos, and E. Jonathan Torres-Herrera Tecnol´ogico de Monterrey, Escuela de Ingenier´ıa y Ciencias,Ave. Eugenio Garza Sada 2501, Monterrey, N.L., Mexico, 64849. Department of Physics, Yeshiva University, New York, New York 10016, USA Instituto de F´ısica, Benem´erita Universidad Aut´onoma de Puebla, Apt. Postal J-48, Puebla, 72570, Mexico
Finite-size effects have been a major and justifiable source of concern for studies of many-bodylocalization, and several works have been dedicated to the subject. In this paper, however, wediscuss yet another crucial problem that has received much less attention, that of the lack of self-averaging and the consequent danger of reducing the number of random realizations as the systemsize increases. By taking this into account and considering ensembles with a large number ofsamples for all system sizes analyzed, we find that the generalized dimensions of the eigenstates ofthe disordered Heisenberg spin-1/2 chain close to the transition point to localization are describedremarkably well by an exact analytical expression derived for the non-interacting Fibonacci lattice,thus providing an additional tool for studies of many-body localization.
The Anderson localization in noninteracting systemshas been studied for more than 60 years and it is by nowmostly understood [1–3]. Its interacting counterpart, dis-cussed in [1, 4] and analyzed in [5–11], still presents openquestions. It has received enormous theoretical [12–15]and experimental [16–26] attention in the last decade andis often referred to as many-body localization (MBL).There are some parallels between the two cases, but thereare also differences, such as the issue of multifractality.An eigenstate is multifractal when it is extended, butcovers only a finite fraction of the available physicalspace. Multifractality is characterized by the so-calledgeneralized dimensions D q , for fully delocalized states D q = 1, for multifractal states 1 < D q <
0, and forlocalized states D q = 0. In the thermodynamic limit,all eigenstates of one-dimensional (1D) noninteractingsystems with uncorrelated random onsite disorder areexponentially localized in configuration space for anydisorder strength. It is at higher dimensions that thedelocalization-localization transition takes place and thishappens at a single critical point, where the eigenstatesare multifractal. In contrast, if interactions are added tothese systems, the delocalization-localization transitionhappens already in 1D and for finite disorder strengths,fractality exists even in the MBL phase. For these inter-acting systems, it is still under debate whether before theMBL phase there is a single critical point or an extendedphase where the eigenstates are multifractal [24, 27–42].In fact, even the very existence of the MBL phase hasnow gone under debate [43–45]. One of the reasons whyit is so hard to settle these disputes is the presence of seri-ous finite-size effects. Recent large-scale numerical stud-ies [39, 41] of the disordered spin-1/2 Heisenberg chain,where Hilbert space dimensions of sizes ∼ × havebeen reached, did not question the transition to a local-ized phase, but were not entirely conclusive with respectto the existence of an extended nonergodic phase or asingle critical point, although the latter is strongly advo-cated in Ref. [39]. In this work, we consider the same Heisenberg modeland emphasize another problem that has not receivedas much attention as finite-size effects, but is also cru-cial for studies of disordered systems, that of lack ofself-averaging. This issue becomes particularly alarm-ing as the system approaches the transition to the MBLphase [38, 46, 47]. If a quantity is non-self-averaging,the number of samples used in statistical analysis can-not be reduced as the system size increases [46, 48–58].This reduction is a very common procedure due to thelimited computational resources when dealing with expo-nentially large Hilbert spaces, but it may lead to wrongresults. We show that when the disorder strength of thespin model gets larger than the interaction strength andit moves away from the strong chaotic (thermal) regime,the fluctuations of the moments of the energy eigenstatesincrease as the system size grows, exhibiting strong lackof self-averaging. Decreasing the number of random real-izations in this case may affect the analysis of the struc-tures of the eigenstates, including the results for the gen-eralized dimensions.The various challenges faced by the numerical studiesof the MBL is a great motivator for theoretical works,which, however, have difficulties of their own. The cur-rent trend is to focus on phenomenological renormaliza-tion group approaches [59–66] that aim at improving ourunderstanding of the MBL transition in 1D systems withquenched randomness, without providing microscopic de-tails. Some of these studies suggest that the transitionis characterized by a finite jump of the inverse localiza-tion length. Similarly, numerical studies indicate thatthe generalized dimensions jump at the critical point [39],and a connection between these two jumps was proposedin [47].Our contribution to those theoretical efforts is to showthat an exact analytical expression for the generalizeddimensions derived for the 1D non-interacting Fibonaccilattice [67, 68] matches surprisingly well our numericalresults for the disordered spin-1/2 Heisenberg chain in a r X i v : . [ c ond - m a t . d i s - nn ] F e b the vicinity of the MBL critical point. This expressionprovides an additional tool in the construction of effectivemodels for the MBL transition. Its derivation is basedon a renormalization group map of the transfer matricesused to investigate the wave functions of the Fibonaccimodel [67, 68].Our 1D lattice system has L interacting spin-1/2 par-ticles subjected to on-site magnetic fields. It is describedby the Hamiltonian H = L (cid:88) k =1 (cid:2) S xk S xk +1 + S yk S yk +1 + S zk S zk +1 (cid:3) + L (cid:88) k =1 h k S zk , (1)where S x,y,zk are spin-1/2 operators, the couplingstrength was set equal to 1, h k are random numbersfrom a flat distribution in [ − h, h ], h being the disor-der strength, and periodic boundary conditions, S x,y,zL +1 = S x,y,z , are imposed. Since H (1) conserves the totalspin in the z -direction, S z = (cid:80) S zk , we work in thelargest subspace corresponding to S z = 0, which hasdimension N = L ! / ( L/ . The model is integrablewhen h = 0 and chaotic, that is, it shows level statis-tics similar to those from full random matrices [69], when h chaos ≤ h < h c . The value of h chaos for the transitionfrom integrability to chaos and of the critical point h c for the transition from delocalization to the MBL phaseare not yet known exactly. Our focus here is on the sec-ond transition, and for that, some works estimate that3 < h c < h c > Multifractality and ensemble size.–
To obtain the gen-eralized dimensions D q , we perform scaling analysis ofthe generalized inverse participation ratios, which are de-fined as IPR αq := (cid:80) N k =1 | (cid:104) φ k | ψ α (cid:105) | q , where q can take,in principle, any real value, | ψ α (cid:105) is an eigenstate of theHamiltonian (1), and | φ k (cid:105) represents a physically rele-vant basis. Since we study localization in the configura-tion space, | φ k (cid:105) is a state where the spins point up ordown in the z -direction, such as | ↑↓↑↓ . . . (cid:105) . We aver-age the generalized inverse participation ratios, (cid:104) IPR q (cid:105) ,over ensembles with n samples that include 0 . N eigen-states with energy close to the middle of the spectrumand n/ (0 . N ) random realizations, and then extract thegeneralized dimensions using (cid:104) IPR q (cid:105) ∝ N − ( q − D q . (2)Multifractality holds when D q is a nonlinear function of q . In practice, D q is obtained from the slope of the linearfit of ln (cid:104) IPR q (cid:105) versus ln N . In Fig. 1, we show some rep-resentative examples of the scaling analysis for differentvalues of h and q , and also for ensembles of different sizes n , varying from n = 10 to n = 3 × . The symbols arenumerical data and the solid lines are the correspondingfitting curves.In the chaotic region, for example when h = 1, thescaling of ln (cid:104) IPR q (cid:105) with ln N is independent of the size of the ensemble, with all points and lines for a given L coinciding and leading to D q ∼
1. This is shown inFig. 1 (a) for q = 1 . q that we studied, 0 . ≤ q ≤ -2-1.5-1-0.5 l n 〈 I P R q 〉 -8-6-4-20-8-6-4-2 l n 〈 I P R q 〉 -6-4-24 6 8 10 12 ln N -4-3-2-1 l n 〈 I P R q 〉 ln N -2-1.5-1-0.5 (a) (b)(d)(c)(e) (f) h = 1.0 h = 1.8h = 2.4h = 3.6 h = 6.0h = 2.2q = 1.2q = 2.0q = 2.4 q = 2.4q = 2.0q = 1.2 FIG. 1. Scaling of ln (cid:104)
IPR q (cid:105) with respect to the natural log-arithm of the Hilbert space dimension N for various valuesof h and q , as indicated in the panels. Different numbers ofsamples are considered for the average of IPR q : 10 (blacksquares), 5 × (turquoise diamonds), 1 × (blue up tri-angles), 5 × (green down triangles), 1 × (maroon lefttriangles), 2 × (magenta right triangles), and 3 × (redcircles). The solid lines are the linear fittings for the numer-ical points and have the same colors as their correspondingpoints. In contrast, when h >
1, the numerical points stronglydepend on the number of samples used, as seen fromFig. 1 (b) to Fig. 1 (f). Notice that this dependencebecomes more evident for the larger system sizes. In theparticular cases of Figs. 1 (b)-(e), where 1 < h < ∼ h c , thefittings lead to larger slopes when the ensemble sizes aresmaller. For these smaller n ’s, the values of D q wouldget even larger if we would neglect the smallest systemsizes when doing the fittings. These results illustrate thedanger of reducing the number of samples as the systemsize increases.We verify in Figs. 1 (b)-(f) that the convergence ofour numerical points happens for ensembles with n > ∼ × . Indeed the points for n = 2 × and n = 3 × are nearly indistinguishable, so in all of our subsequentstudies, we use n = 3 × for all L ’s. It may be, however,that for system sizes larger than the ones considered here,convergence would require even larger ensembles. Self-averaging.–
The fluctuations of the values of IPR q bring us to the discussion of self-averaging. A givenquantity O is self-averaging when its relative variance R O = ( (cid:10) O (cid:11) − (cid:104)O(cid:105) ) / (cid:104)O(cid:105) decreases as the system sizeincreases [48–56]. This implies that in the thermody-namic limit, the result for a single sample agrees withthe average over the whole ensemble of samples.In quantum many-body systems, the eigenstates canspread over the many-body Hilbert space, which is ex-ponentially large in L , so we study the scaling of R IPR q with N [46, 57], R IPR q ∝ N ν . (3)If ν <
0, then IPR q is self-averaging and one can reducethe number of samples for the average as the system sizeincreases. This cannot be done when ν ∼
0, and it iseven worse in the extreme scenario where ν > -6 -4 -2 R I P R q -4 -2 h -4 -2 R I P R q h -6 -4 -2 ν ν ν ν (b)(a)(c) (d) q = 1.8q = 2.0 q = 2.4q = 1.2 FIG. 2. Relative variance R IPR q versus disorder strength h for different q ’s, as specified in the panels, 3 × statisticaldata, N = 70 (black), N = 256 ,(maroon), N = 924 (blue), N = 3 ,
432 (dark green), N = 12 ,
870 (light green), and N =48 ,
620 (red). Insets: Exponent ν [Eq. (3)] versus disorderstrength h . Dashed line marks ν = 0. Error bars are standarderrors from a linear fitting. In the main panels of Fig. 2, we show the dependenceof R IPR q on the disorder strength h for different values of q and each line represents one system size. It is clear thatdeep in the chaotic region, the relative variance decreasesas the system size grows, implying self-averaging of IPR q .This is also illustrated in the insets, where ν < h chaos ≤ h < ∼
1, which is consistent with Fig. 1 (a), wherethe scaling of IPR q does not depend on the number ofsamples.There is, however, a turning point at h > ∼
1, where ν suddenly jumps above zero and R IPR q grows signifi-cantly with system size. As seen in Figs. 2 (a)-(d), this isparticularly bad in the region preceding the MBL phase, 1 < ∼ h < ∼
4. For this range of disorder strength, as theinsets indicate, ν > q > ∼ h > ∼
4, where the system should already be inthe MBL phase, the relative variance R IPR q continuesto grow with system size, but ν is close to zero and thecurves for L = 16 and L = 18 are not far from each other.The results in Fig. 1 and Fig. 2 show that extra careneeds to be taken when performing scaling analysis awayfrom the chaotic region, not only due to finite-size effects,but also due to the lack of self-averaging. No matter howlarge the system size is, large numbers of samples arerequired and may even need to be increased as L grows.One can reduce the fluctuations of the generalized in-verse participation ratios by using their logarithm, knownas participation R´enyi entropies. In fact, using a toymodel, it was shown in [46] that in the MBL phase, R IPR grows with system size, while R − ln IPR decreases with L . However, for 1 < ∼ h < ∼
4, even though we observea reduction of the fluctuations, ln IPR q remains non-self-averaging and we still have ν ∼ (cid:104) IPR q (cid:105) , we use (cid:104) ln IPR q (cid:105) . Multifractality and analytical expression for D q .– Af-ter taken the necessary precautions for performing thescaling analysis of the generalized inverse participationratios, as discussed in Fig. 1, we now proceed with thestudy of how D q depends on q and h .In Refs. [67, 68], an exact analytical expression was de-rived for the structure of the eigenstate at the center ofthe spectrum of the off-diagonal version of the Fibonaccimodel in the thermodynamic limit, leading to the gener-alized dimensions D Fibonacci q = 13( q − σ (cid:8) q ln[ λ ( h )] − ln[ λ ( h q )] (cid:9) , (4)where σ = ( √ / λ ( h ) = (cid:110) ( h + 1) + (cid:2) ( h + 1) + 4 h (cid:3) / (cid:111) / (2 h ) is the maximumeigenvalue of the transfer matrix [67, 68]. For the Fi-bonacci model, h denotes the ratio between its two hop-ping constants, which are arranged in a Fibonacci se-quence.In the case of our interacting spin model, we use Eq. (4)as an ansatz. Since in this case, the eigenstates are ex-tended for h chaos ≤ h < ∼
1, while Eq. (4) predicts a mono-tonic decrease of D Fibonacci q for h <
1, we compare our re-sults with the expression for Θ( h − D Fibonacci q +Θ(1 − h ),where Θ is the Heaviside step function. We find that thisexpression matches the numerical values of D q for thespin chain extremely well for disorder strengths in thevicinity of the critical value, 3 < h c < , is the most com-monly used quantity in studies of localization, so we startby analyzing D . and D . as a function of the disorderstrength. They are shown in Fig. 3 (b) and Fig. 3 (c), h h h D q q=0.2q=0.4q=0.6 h q=3.0q=3.4 q D q q q q h=2.2 h=3.6 h=4.2 h=6.0q=2.0 q=2.4 (c)(a) (b) (d)(e) (f) (g) (h) FIG. 3. Top: D q versus disorder strength h for different values of q , as indicated in the panels. Bottom: D q versus q for differentdisorder strengths h , as indicated in the panels. Points are numerical results obtained through scaling analysis, averages overan ensemble of 3 × samples. Solid curve gives Θ( h − D Fibonacci q + Θ(1 − h ). Error bars are standard errors from linearfittings. The numerical results in Fig. 3 (a) and Fig. 3 (d) cross the analytical curve within the shaded vertical area where thecritical point should lie. respectively. The numerical results agree very well withthe analytical expression for D Fibonacci q for all h ’s. How-ever, for smaller q ’s, the agreement is not as good, asillustrated in Fig. 3 (a), the same happening for larger q ’s, as seen in Fig. 3 (d).We cannot say whether the agreement of the curves for D q vs h with D Fibonacci q for q away from 2 would improveor get even worse if larger system sizes were considered.If it would improve, that would point to the existence ofan extended phase of multifractal eigenstates before theMBL phase and described by the analytical expression ofthe Fibonacci lattice. Large-scale numerical studies [39,41] indicate that if such a phase exists, it should appearfor h > q ’s cross the curve of the analytical expressionof D Fibonacci q at h ∼ h c [shaded area in Fig. 3 (a) andFig. 3 (d)]. This indicates that at least in the vicinityof (or right at) the critical point, the generalized dimen-sions of the disordered spin model is indeed extremelywell described by Eq. (4).The bottom panels of Fig. 3 give further support forthis observation. There, we plot D q as a function of q fordifferent values of the disorder strength. For 1 < h < h c ,as illustrated by Fig. 3 (e), there is no good agreementbetween the numerical points and D Fibonacci q . The samehappens for h > h c , as seen in Fig. 3 (h), although the mismatch in this case is not as large. However, for h ∼ h c , as shown in Fig. 3 (f) and Fig. 3 (g), the agreementis extremely good. Conclusions.–
Our analysis of the disordered spin-1/2Heisenberg chain calls attention to the strong lack of self-averaging of the generalized inverse participation ratiosfor a range of disorder strengths that precedes the crit-ical point of the MBL transition. This implies that intheoretical and experimental studies of this region, oneshould not decrease the number of samples as the sys-tem size increases. We notice also that the logarithm ofthe generalized inverse participation ratios can be usedto reduce fluctuations, but it still does not lead to self-averaging in that region.Our studies indicate a strong relationship between mul-tifractality, 0 < D q <
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Laflorencie, Hilbert spacefragmentation and many-body localization (2019),arXiv:1906.05709. Supplemental Material:Multifractality and self-averaging at the many-body localization transition
Andrei Sol´orzano, Lea F. Santos, and E. Jonathan Torres-Herrera Tecnol´ogico de Monterrey, Escuela de Ingenier´ıa y Ciencias,Ave. Eugenio Garza Sada 2501, Monterrey, N.L., Mexico, 64849. Department of Physics, Yeshiva University, New York, New York 10016, USA Instituto de F´ısica, Benem´erita Universidad Aut´onoma de Puebla, Apt. Postal J-48, Puebla, 72570, Mexico
In this supplemental material (SM), we show that theapplication of a logarithmic transformation to the gener-alized inverse participation ratios of the energy eigen-states reduces the size of the fluctuations, but is notenough to achieve self-averaging, specially around thecritical point. This implies that reducing the numbersof statistical data to compute averages as the system sizeincreases remains a problem also in this case.We also present results for the errors δD q involved inthe computation of the generalized dimensions. We findthat the errors obtained for the linear fit of ln (cid:104) IPR q (cid:105) ver-sus ln N and those for (cid:104) ln IPR q (cid:105) versus ln N are very sim-ilar. They are larger in the vicinity of the critical pointand get significantly larger for all values of h > h ∼
1, we have that δD q ∼ LOGARITHM OF THE GENERALIZEDPARTICIPATION RATIOS
An alternative approach to compute the generalizeddimensions D q consists in using the logarithm of the gen-eralized participation ratios, S R´enyi q = 1 q − q , (S1)which corresponds to the so-called participation R´enyientropies. In this case, instead of doing the linear fit ofln (cid:104) IPR q (cid:105) versus ln N , as in the main text, we study (cid:104) ln IPR q (cid:105) = − D q ( q −
1) ln N . (S2)That is, instead of computing the logarithm of the av-eraged generalized inverse participation ratios, which isan arithmetic mean, we now compute the average of thelogarithm of the generalized participation ratios, whichis a geometric mean. The logarithmic transformation iscommonly applied to reduce the fluctuations of a set ofdata and it has a significant effect on the tails of thedistributions. In Fig. S1, we show ν for R − ln IPR q ∝ N ν . Comparingit with the insets of the Fig. 2 in the main text, we seetwo main differences. One is that in the localized phase, − ln IPR q becomes self-averaging, in agreement to whatwas discussed in Ref. [42]. The other difference is thatthe values of ν around the critical point become muchsmaller, but it is still not negative, so the lack of self-averaging persists. -1.5-1-0.500.5 ν h -1.5-1-0.500.5 ν h (b)(a)(c) (d) q = 1.8q = 2.0 q = 2.4q = 1.2 FIG. S1. Exponent ν extracted from the scaling of R − ln IPR q with the size of the Hilbert space versus the disorder strength h for different q ’s, as specified in the panels. R − ln IPR q wascomputed from 3 × statistical data and the dashed linemarks ν = 0. Error bars are standard errors from linearfittings. ERRORS
As show in Fig. S2, there is no significant differencebetween the error δD q obtained by extracting the gener-alized dimensions D q from the scaling of ln (cid:104) IPR q (cid:105) withln N [ δD ln (cid:104) IPR q (cid:105) q in Figs. S2 (a)-(b)] and that from thescaling of (cid:104) ln IPR q (cid:105) with ln N [ δD (cid:104) ln IPR q (cid:105) q in Figs. S2 (c)-(d)]. The results in all four panels are very similar. In thechaotic region, h chaos ≤ h < ∼ δD q is close to zero for allensemble sizes considered. The errors increase monoton-ically for 1 < ∼ h < ∼
2, but remain almost independent ofthe number of samples. It is in the vicinity of the criticalpoint, 2 < ∼ h < ∼ .
5, that the errors become clearly largeras the number of samples gets decreased. For h > ∼ .
5, theerrors still depend on the number of samples, but theyare smaller that in the preceding region, specially for theensembles with 3 × samples for which a sudden dropis seen at h ∼ . δ D q h δ D q h (b)(a)(c) (d) q = 2.0q = 1.2 FIG. S2. Standard error δD ln (cid:104) IPR q (cid:105) q (a)-(b) and δD (cid:104) ln IPR q (cid:105) q (c)-(d) versus the disorder strength h . Symbols and colorsrepresent results for ensembles with different number of sam-ples: 10 (black squares), 5 × (turquoise diamonds), 1 × (blue up triangles), 5 × (green down triangles), 1 × (maroon left triangles), 2 × (magenta right triangles), and3 ×4