Ученые выявили точные признаки перехода к локализации в квантовых цепях через экстремальные значения локальных наблюдаемых

Abstract Recent studies of delocalization-localization transitions in disordered quantum chains have highlighted the role of rare, chain-breaking events that favor localization, in particular for high-energy eigenstates related to many-body localization. In this context, we revisit the random-field spin-1/2 chain at zero temperature with ferromagnetic interactions, equivalent to interacting fermions or hard-core bosons in a random potential with attractive interactions. We argue that localization in this model can be characterized by chain-breaking events with clear signatures in the extreme values of simple local observables, such as the on-site density or the local magnetization, which are readily accessible in both experiments and numerical simulations. Adopting a bosonic language, we study the disorder-induced Berezinskii–Kosterlitz–Thouless (BKT) quantum phase transition from superfluid (SF) to Bose glass (BG), and focus on the strong disorder regime where localization is driven by weak links. Based on high-precision density matrix renormalization group simulations, we numerically show that extreme local densities accurately capture the BKT transition, even for relatively short chains ranging from a few dozen to a hundred sites. We also discuss the SF-BG transition in the weak disorder regime, where finite-size effects pose greater challenges. Overall, our work seeks to establish a solid foundation for using extreme statistics of local observables, such as density, to study delocalization-localization transitions in disordered quantum chains, both in the ground state and at high energy. Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 license. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. 1. Introduction 1.1. Context Many-body quantum interacting systems in the presence of disorder are notoriously difficult to study [1–9], yet they raise fundamental questions in quantum statistical physics and condensed matter. A famously challenging question is to characterize many-body localization (MBL) (for recent reviews, see [10–13]), a possible alternative to thermalization [14–16]. In the infinite-size and infinite-time limit, the behavior of this potential phase of matter has sparked intense debate [17–36], in particular in one dimension where quantum fluctuations play a major role. In this context, the spin-1/2 chain in a random magnetic field is a central example: where is the Ising anisotropy parameter and the are independent and uniformly distributed random variables. The above spin Hamiltonian equation (1.1) is rather generic because it also describes interacting particles in a random potential, either spinless fermions [37] or hard-core bosons [38, 39]. A useful point for later discussions is that the local magnetization in the spin chain Hamiltonian is directly related to the particle density at site as . An ongoing endeavor in the study of MBL is the exploration of rare events. Indeed, extreme value theory (EVT), the study of rare, extreme events which can control the physics of a system, is a fruitful line of research in statistical physics [40, 41]. Due to their geometry, one-dimensional quantum disordered systems form a particularly interesting field of application of EVT, a famous example being the sharp predictions for the distribution of the lowest finite-size energy gap in the random transverse-field Ising model [42–44]. Crucially, delocalization-localization transitions have signatures in the extreme statistics of several observables, such as almost-conserved local operators [45], and entanglement spectra [46, 47], to cite but two. The recent focus has largely been on the role of rare delocalizing events such as avalanches seeded by ergodic regions [26, 34, 48, 49] and long-range many-body resonances notably seen in the presence of large two-body correlations [26, 29–32, 50–52]. However, an opposite but relevant direction is to study the role of rare, highly localized regions that slow down the dynamics and could even prevent transport close to the transition [36, 45, 53–57], or break the chain [47, 58–60]. This has opened the interesting possibility of capturing localization-delocalization transitions through simple local observables [58–65]. Importantly, certain issues related to the MBL transition are connected to aspects of the ground-state phase diagram [58, 63, 66, 67]. In particular, the Berezinskii–Kosterlitz–Thouless (BKT) universality class of the MBL transition is reminiscent of the ground-state superfluid (SF) to Bose-glass (BG) transition [1, 2, 68]. It has been theoretically discussed [19, 69–73], and numerically addressed using extreme magnetization statistics [59], despite the relatively small system sizes accessible to mid-spectrum shift-invert diagonalization techniques [9, 74]. The key idea relies on a drastically different behavior of the distributions of local magnetizations in the ergodic regime—where the variance shrinks, in agreement with the eigenstate thermalization hypothesis (ETH) [14–16]; and in the MBL regime—where these distributions exhibit power-law tails characteristic of localization [58–65, 75]. Both full distributions [64] and extreme statistics of local observables [59] reveal a BKT-type scaling for a critical disorder slightly below the most recent extrapolated estimates [75, 76]. In this context, it is natural to ask whether such ideas—reminiscent of entanglement bottlenecks [77, 78] that are relevant to the ETH–MBL transition at high energy [58–60]—might also apply to ground-state physics. One aim is to validate the use of extreme statistics of local observables to detect and characterize BKT localization-delocalization transitions on small system sizes: at high energy, numerics is limited to short chains (up to ). Does a similar analysis hold up to scrutiny for sites? Specifically, we ask whether the extreme statistics of local densities are also relevant for the zero-temperature SF-BG localization transition. Indeed, this SF-BG transition belongs to the BKT universality class [1, 2, 68], and in its strong-disorder version [68, 79–81] is known to be driven by effective weak links that effectively cut the chain (see below for a definition). Thus, it provides a much more controlled setting in which to test, validate, and highlight the possible limitations of extreme statistics of local observables for studying delocalization-localization transitions. We therefore focus on the role of such rare localized events in the ground state of equation (1.1), with two objectives in mind: (1) exploring the relationship between extreme magnetization and weak links, and (2) showing that extreme magnetization correctly captures the strong disorder transition. 1.2. Model, ground state, and weak links The random-field chain in equation (1.1) has served as a test bed for studying disorder-induced ground-state localization transition in the presence of interactions [1–3], thus going beyond the single-particle Anderson localization framework [83, 84]. This model is directly relevant for the physics of one-dimensional arrays of Josephson junctions [85–89] and low-dimensional quantum magnets with impurities [90–92], and can be experimentally realized in ultracold atoms [93, 94]. More generally, it can be relevant to study the effects of disorder on superconductivity [95, 96]. To set the problem, figure 1 provides a schematic view of the ground-state phase diagram of equation (1.1) as a function of disorder strength , and interactions , focusing on the attractive regime () [97]. In the absence of disorder, the phase is a Luttinger liquid (SF in the bosonic language), characterized notably by the Luttinger parameter which controls the power-law decay of the correlation functions [98]. In their seminal work, Giamarchi and Schulz [1, 2] showed that weak disorder destabilizes the Luttinger liquid SF phase, first reducing the Luttinger parameter, and eventually leading to a BKT transition into a BG insulator at the universal critical value . This result was further confirmed more recently by two-loop calculations [99, 100]. As discussed in these works, this approach, being perturbative in the disorder, is well-controlled near the weak-disorder limit, but does not conclude on the universality of the transition at stronger disorder. A distinct strong-disorder regime, where different physical mechanisms dominate, has been proposed. Strong-disorder RG analyses [68, 79–81] highlighted the role of rare weak links, weak connections between superfluid puddles in the effective description of the chain. They proliferate and effectively cut the chain, yielding a non-universal critical value . Building on this, Prokof’ev, Svistunov and collaborators [101–104] incorporated the Kane–Fisher mechanism [105, 106], whereby quantum fluctuations renormalize these weak links, rendering them either transparent or severed depending on the value of . This leads again to a BKT transition, but governed by a non-universal critical value . The existence of these two regimes has been confirmed by large-scale numerical simulations [82, 107]. In particular, Doggen et al [82] investigated the random-field chain using DMRG and QMC techniques, and reported a transition consistent with the Giamarchi–Schulz scenario with at weak disorder, while finding a distinct strong-disorder regime characterized by . In this latter regime, power-law distributions were observed for several physical observables, supporting the relevance of the weak-link mechanism. The qualitative picture distinguishing the weak-disorder (Giamarchi-Schulz) regime from the strong-disorder (weak-link) behavior [82] is related to the half-bandwidth , which corresponds to the saturation field in the clean chain. Indeed, in the clean case, the polarized ferromagnet obtained for corresponds to an insulator in the bosonic language. Thus, in the presence of disorder, a weak link can take the form of a locally insulating region of length where all . If , this occurs with probability . In contrast, prevents such an origin for weak links. 1.3. Main results In this work, we mostly focus on the strong disorder regime of the SF-BG transition: we argue that the extreme magnetization highlights signatures of a microscopic version of the weak link physics, and show that the local magnetization as a simple probe of localization correctly captures both the location and scaling of the transition, even with modest system sizes in the range [108], compared to sites required in [82]. We also discuss finite-size effects at weaker interactions and characterize the behavior of this observable in the weak disorder (Giamarchi-Schulz) regime of the transition. The importance of these results is threefold. First, they put on a stronger footing an approach previously applied to the ETH–MBL transition with available sizes [59]. Second, they highlight the contrast between the weak-disorder and the strong-disorder regimes of the transition, and might revive the possibility of two different BG phases [1, 82, 100]. Last but not least, they provide a basis for experimentally studying the role of entanglement bottlenecks through much simpler observables such as the local magnetization or density. The key idea is summarized in figure 2. For each disorder realization of the chain, we focus on the local magnetization in the ground state. The distributions of the magnetization show a clear evolution with increasing disorder strength (figure 2(a)), with a higher probability of finding very strongly polarized sites at strong disorder (in the BG phase). Quantum fluctuations, however, forbid a perfectly polarized site for any finite size () [109]. Download figure: Standard image High-resolution image The full distribution, however, provides limited insight into the transition. Qualitatively, its evolution does not suffice to characterize the delocalization-localization transition, since a similar evolution is seen in the chain, which is always localized for strictly non-zero disorder [60] (see section 2 and appendix A). Quantitatively, on the delocalized side, in contrast to the high-energy case controlled by the eigenstate thermalization hypothesis [14–16, 110], the width of the full distribution does not vanish with increasing system sizes. This limits the use of average observables despite their success at high energy [64]. To quantitatively characterize the differences between the distributions of the magnetization in the SF and the BG phase, it is therefore useful to focus on the most polarized sites in the chain (figure 2(b)), i.e. the ones where the deviation to perfect polarization is minimal (figure 2(c)). For reasons that will become clear in section 2, we focus on the typical value of the minimal deviation where the bar indicates disorder average. Studying its scaling for increasing system sizes reveals a fundamental difference between the SF and the BG phases in the strong disorder regime. Indeed, in the SF, we find that the minimal deviation decays to a non-zero value in the thermodynamic limit (figure 2(d), nuanced below), while at strong disorder in the BG, it goes to zero as a power law (figure 2(e)). Both behaviors are captured by the following phenomenological description of our finite length data with going to zero in the BG. Even for modest system sizes, we show that these contrasting behaviors reliably capture the SF-BG transition when it is driven by weak links. The rest of the paper is organized as follows. In section 2 we focus on the non-interacting limit (many-body Anderson case). We recall essential points at high energy and summarize key new results in the ground state, on the basis of exact diagonalization on large chains with periodic boundary conditions (PBC). The remainder of the paper focuses on finite attractive interactions, , for which the results are obtained using DMRG on chains with open boundary conditions. In section 3, we present two ways to use extreme magnetization to spot the transition in the interacting model in the strong disorder (weak-link) regime, specifically at (approximately the top of the SF dome). Section 4 provides the BKT scaling analysis of the transition based on this probe. In section 5, we discuss a possible interpretation of our results, as well as a conjecture for the Giamarchi-Schulz (weak-disorder) regime, where the extreme statistics behave very differently. A summary and outlook are given in section 6. The main appendices are App. A, which provides details related to section 2, and significantly extends the discussion of the new results for the ground state, and App. B which specifies the DMRG parameters and clarifies that we focus on extreme events in the middle part of the chain. Other appendices clarify details on the scaling procedure (App. C), and discuss the main characteristics of the full magnetization distributions (App. D). 2. Spin freezing and chain breaking in the non-interacting case We first focus on the Hamiltonian equation (1.1) in the limit. There, the model is an chain in random field [111], which maps to an Anderson localized chain of non-interacting spinless fermions in a random potential through the Jordan-Wigner transformation [37, 83, 112]. The local magnetization is related to the local fermion density, the total magnetization conservation is equivalent to the particle number conservation, and the sector corresponds to the Anderson localized chain with half the levels filled. In the discussion below, we use the term ‘many-body’ Anderson localization despite the non-interacting spinless fermion model to highlight the role of the filling. Section 2.1 briefly recalls key elements linking extreme magnetization, chain breaking and localization length at high energy, previously discussed in [58–60, 63]. We then provide in section 2.2 central results in the ground state of this non-interacting case. 2.1. Known results at high energy For high-energy eigenstates in the non-interacting, half-filled Anderson localized case, the typical minimal deviation defined in equation (1.3) decays as a power-law of the system size for any finite disorder [60]: This implies that in the thermodynamic limit, there is at least one spin in the chain that is perfectly polarized. This behavior is well captured by a simple toy model [58–60], which we briefly recall here. Building on the spin-particle correspondence, we model the random-field chain (which corresponds to a many-body Anderson localized chain) by a collection of half-filled localized orbitals (labeled by ), each centered on a different site and all with the same localization length : . This is a minimal description of the many-body Anderson insulator, which nevertheless captures the essential features of extreme statistics of local densities. In this simplified framework, the most polarized site is located in the middle of the longest series of neighboring orbitals which are all either empty or all occupied. It is easy to show that the power-law exponent is directly related to the localization length (see appendix A and [58, 60] for details): Upon lowering the disorder, the toy model breaks down for (), but the power-law decay remains valid with an exponent for large enough system sizes. 2.2. Ground state properties We now summarize a few main messages on the minimal deviation from perfect polarization in the ground state of the chain. These key results support the discussion of the Giamarchi-Schulz regime in section 5. We refer the reader to appendix A for a detailed presentation of our results at , in particular for both the full and the extreme values distributions. The main difference between the high-energy eigenstates of the chain in the sector and its ground state is which single-particle Anderson-localized orbitals are occupied. In the latter, the lower half of the spectrum is full, and the full range of the localization lengths in the single-particle spectrum appear. Indeed, both states with very short length at the band edge, and states with much larger localization length in the middle of the spectrum are occupied. For strong disorders, when all localization lengths are short, we do not expect fundamental differences in the decay of the typical minimal deviation, and it is indeed captured by equation (2.1). The main difference is that the minimal deviation is controlled mostly by much shorter localization lengths than the average (see appendix A for a detailed argument). As a result, we observe that the exponent controlling the decay in the ground state is larger than controlling that at high energy, for the same disorder. This result is valid for . For weaker disorder, strong finite-size effects appear. From the toy model, it is naturally expected that the power-law decay of the minimal deviation cannot be captured correctly when the localization length far exceeds , and this is indeed what we observe. In practice, this suggests a lower-bound on the exponent that we can capture correctly for a given system size : for the large ( sites) PBC chains we study with exact diagonalization in appendix A, and for the more modest sizes () we reach in the interacting systems in the rest of the paper. 3. Extreme statistics and strong disorder transition We now focus on the strong disorder regime of the SF-BG transition. We present two procedures to spot the SF-BG transition by taking advantage of the two contrasting behaviors seen in figure 2. The first one (section 3.1) is inspired by the approach used for MBL in [59], which relies on the crossing of size-dependent effective exponents. The second one comes from the observation that the numerical results are well described by equation (1.4) with in the SF, see section 3.2. 3.1. Crossing of the effective exponents To locate the transition in the strong-disorder (weak-link) regime, we use the effective exponent extracted from successive power-law fits using sliding fitting windows involving larger and larger system sizes. The idea is that for a pure power-law decay, is independent of the fitting window. In contrast, for a slower-than-power-law decay, with increasing sizes, and for a faster-than-power-law decay, grows. As already seen in figure 2(e), we can reasonably expect the minimal deviation in the BG to behave similarly as in the Anderson localized case (). Namely, we expect it to decay as a power-law with increasing system size ( in equation (1.4)), with a well-defined exponent . However, similar analysis at high energy (both in the non-interacting model [60] and in the random-field Heisenberg chain [59]), and in the non-interacting ground state (appendix A) show that is not fully free of finite-size effects even on the localized side. Rather, the effective exponent grows to from below [113]. In contrast, in the SF, we expect the minimal deviation to go to a non-zero value and as . The point where the behavior of changes is therefore expected to correspond to the transition. To validate this approach, we study this behavior at [114] in figure 3. Panel (a) shows the typical minimal deviation as a function of system sizes in the SF. The dashed lines correspond to fits on the smaller system sizes and the solid lines to fits on the larger system sizes (cf caption), highlighting the finite-size crossover of the effective exponent. Panel (b) presents the decay of the minimal deviation in the BG. The log–log scale used in both panels highlights the power-law behavior in the localized phase, contrasting with the SF. For , a slightly slower decay of for smaller system sizes is visible, in agreement with the expected finite-size effect from below. Download figure: Standard image High-resolution image As a result of these contrasting behaviors in the SF and the BG, figure 3(c) shows a crossing of in the grey region, which nicely coincides with the SF-BG transition point previously extracted using a standard BKT analysis of the superfluid stiffness obtained from quantum Monte Carlo simulations [82]. At very strong disorder, we also find the expected logarithmic dependence (see appendix D). Indeed, at high energy, and if the disorder is strong enough, one can argue using simple perturbative ideas [58] that the freezing exponent , which matches the behavior of the localization length at strong disorder. Finally, the fact that is finite at the transition tells us that this freezing exponent cannot be simply interpreted as an inverse localization length in the interacting case—or, at least, not at criticality. Indeed, both the weak-link and the Giamarchi-Schulz critical points are expected to show a finite SF density at the transition [82] and a diverging correlation length [2, 99, 100]. The result that at thus seems surprising at first glance. We clarify in section 5 that this result is compatible with algebraically decaying long-range correlations provided that there is a healing at long distance across the highly polarized site exactly at the transition. 3.2. Large system limit of the minimal deviation As sketched in the introduction, an alternative way to spot the transition is to compare pure power-law fits to fits of the form equation (1.4), i.e. with an additional constant. As shown in figure 2, the latter seem to capture well the results in the SF phase. We now make this more quantitative. In this type of fit, the exponent of the power-law part is not expected to go to zero. Instead, figure 4(a) suggests that is finite and depends only weakly on in the SF phase for . This motivates the following analysis: - (i)at weak disorder (i.e. deep in the SF phase), we find the exponent providing the best fit for the power-law part of the decay, see inset figure 4(a). - (j) Download figure: Standard image High-resolution image The result for with this fixed exponent is shown in figure 4(b). For , we expect to be close to (see figure 4(a)) and the resulting to be a good estimate of the infinite-size limit of the minimal deviation. We first remarkably observe that passes exactly through zero at the transition point . For , is no longer a good choice for the exponent, and accordingly, the negative indicates that the fit is poor. An alternative fit (not shown) in which is not fixed results in above the transition (but much larger errors on for ). We confirm this by comparing the fixed power fit to a pure power-law equation (2.1) with flexible exponent. The former provides a consistently better description than the pure power-law all the way up to , as shown by the larger values in the inset [115]. In contrast, for , the pure power-law with flexible exponent offers a better fit, consistent with the conclusion that in this regime [116]. Thus, the behavior of in equation (1.4) sharply locates the strong-disorder transition. Note that the result that below the transition also implies that , while for is compatible with at and above the transition (see inset of figure 3(c)), suggesting a jump of the (pure) power-law exponent of equation (2.1) at the transition in the infinite size limit. These results raise three main questions. First, beyond locating the transition, can the extreme magnetization also capture the nature of the transition, in particular, the BKT scaling expected in the present case? Second, can one have a consistent interpretation of the value of at the transition (and thus of the behavior of the power-law exponent )? And third, would such an approach be as successful in the weak-disorder Giamarchi-Schulz regime? We address the first question in section 4, and discuss the two others in section 5. 4. Scaling in the strong disorder regime We focus on the BG side of the transition in the strong disorder regime. The power-law behavior of the typical minimal deviation demonstrated in the previous section (figure 3) suggests a logarithmic scaling of the form . Instances of such scaling forms (but with either the volume of the Hilbert space or the volume of the random graph playing the role of ) can be found in [117–121]. An analogous form in time was recently discussed in small-world graphs [122]. More directly relevant here, a logarithmic scaling for the extreme magnetization on the localized side of the ergodic-MBL transition was discussed in [59]. In that work as in the present case, it is justified by the expected behavior Thus, ignoring the disorder-dependency of the amplitude for the moment, we can write a scaling hypothesis as with for large . This, together with the interpretation of as an inverse localization length in the non-interacting case, suggests that here plays a role similar to that of a length scale in usual collapse analysis. To validate the above hypothesis, we use a controlled finite-size scaling analysis [59, 123, 124]. The idea is to seek a global fit of the following form for all the data for for all sizes: where close to the critical point. This is done through a systematic fit in the form of a Taylor expansion (see appendix C for details). For fixed critical disorder , a collapse with such a scaling gives a divergence We systematically perform this analysis for various tentative critical disorders , as shown in figure 5. In panel (a), the result for is shown with the associated scaling function . The reduced values as a function of are shown in panel (b), illustrating that the fits are of good quality for . For each of these fits, we show the resulting exponent controlling the power-law divergence of the logarithmic length scale near the critical point (panel c). However, because of the scaling form equation (4.2), this power-law divergence with corresponds to an effectively exponentially diverging length scale controlling the finite-size effects. This is in agreement with the expected BKT scenario, even though the obtained exponent is larger than the usually expected . It is often challenging to obtain such exponents precisely, and we do not a priori believe this difference to be very meaningful. This result also shines light on the similar analysis for the high-energy MBL transition in the random-field Heisenberg chain model mentioned in the introduction [59]. Indeed, while this analysis gives for a critical disorder strength , the results are also perfectly compatible with for the stronger critical disorder strength recently discussed in the MBL literature on this model and extracted from similar numerics [17, 75, 76]. We note that the critical value (also shown in panel (c)) depends on and matches the obtained in figure 3 at , as should be expected. Download figure: Standard image High-resolution image Note that actually depends on disorder, with playing a role in the scaling. In fact, in the crude approximation where the full distribution of deviations is modeled as a pure power-law, is related to through the normalization of the distribution. As we discuss further in appendix C, this type of term, also present in the non-interacting case [60], can be tentatively characterized by irrelevant logarithmic corrections of the form [59]. This does not change the results we obtain for the BKT-like scaling exponent . 5. Discussion 5.1. Spin freezing: a local signature of weak-link physics? Given the successful analysis of the strong-disorder (weak-link-driven) transition using minimal deviations, it is tempting to ask whether one can interpret spin freezing and chain breaking as a local signature of weak-link physics. To this end, we first recall a few key points about the weak links in the existing literature. In the strong disorder scenario à la Altman et al [68, 79, 80, 125], the weak links are weak Josephson couplings in the effective description of the model reached after several strong-disorder-RG steps. The exponent characterizing the power-law tails of weak-links is therefore a size-dependent quantity, and the critical point is characterized by a universal value of (but a non-universal value of ). Taking into account a Kane-Fisher healing of the weak-links, the scratched-XY universality scenario [101–104, 126] also suggests that is non-universal, but depends on as [104] [127]. Crucially, in the numerical literature, the weak links are not observed directly. Rather, their exponent is accessed indirectly through the effect of the weakest link on a macroscopic observable. For instance, QMC is used to probe the weak-link exponent by studying the power-law tail of the superfluid stiffness distributions either directly [81, 82] or by looking at rare events [103]. Doggen et al [82] also focus on the tail of the distribution of the one-body density matrix (transverse correlators in spin language). In these analyses, an underlying hypothesis is that the weakest link controls the tail of the distributions at long distances. While this makes sense in the SF, the proliferation of weak-links in the BG naturally gives rise to a breakdown of the power-law tails, both in the superfluid stiffness and in the long-range correlator [82]. Instead, as we have seen, the extreme magnetization exhibits a pure power-law decay on the BG side of the transition (at least at strong enough disorder, see below), converging to a well-defined exponent for sufficiently large sizes. Furthermore, the extreme magnetization events also seem to exhibit healing in the SF phase. It may therefore be tempting to interpret the extreme magnetization events, related to entanglement bottlenecks, as a local signature of the weak link physics. However, a proper identification between microscopic entanglement bottlenecks and weak-links would require an analysis of the impact of such bottlenecks for the emergence of effective weak couplings between superfluid puddles in the RG procedure. A key point is what happens at the strong-disorder transition. As we observed at , and at the transition. Yet, it is known that the critical point is superfluid, so the correlations at long distances must be (and are) healed: at this point, the correlations across the weakest link are affected only at short distances by its presence. This suggests a crucial difference between the microscopic observable used here, which goes to zero at the transition, and the renormalized weak links, which have to be healed at the transition for the critical point to remain SF. 5.2. Giamarchi-Schulz regime We now turn to the Giamarchi-Schulz regime of the SF-BG transition, see figure 1. We argue that the nature of the transition, together with the slow decay of the minimal deviation, makes it difficult to pinpoint the transition using extreme-magnetization statistics. Nevertheless, we make a conjecture about the behavior of the minimal deviation at the transition based on (1) numerical results for , and (2) a continuity argument near (where is the half-bandwidth of the clean problem, see figure 1). First, recall that, in the weak disorder regime, the SF is destroyed not by the effect of weak links systematically getting weaker, but rather by the proliferation of topological defects in the phase (quantum phase slips) [1, 2, 98]. Thus, we do not expect the local density to play a key role at the transition. Second, we saw that for , finite-size effects are expected to play an important role and possibly lead to the false conclusion that . In practice, these two effects limit the use of extreme statistics to locate the Giamarchi-Schulz transition. This is shown in figures 6 and 7. The first figure shows that seems to decrease with increasing system sizes for . However, this occurs for already small exponents, where we expect the finite-size effects to underestimate the power-law exponent. Relying on the observation that in the SF, (see appendix D), we attempt a different analysis, by performing a power-law fit with , where is free. Comparing the fit qualities in figure 7 (for and for ), we see that the results are rather inconclusive. They suggest that the fit with a constant is marginally better in the SF regime, and in part of the BG phase for , but the large oscillations reflect large errors. Download figure: Standard image High-resolution image Download figure: Standard image High-resolution image Can we make a conjecture about the thermodynamic limit? Let us first focus on the transition. Recall that the power-law exponent controlling the distribution of the weak links diverges at the Giamarchi-Schulz transition [82], i.e. the probability of very weak links vanishes, consistent with the transition being controlled by quantum phase slips. This implies that the weakest link scales with , with diverging. For the same reason, we expect to be zero along the Giamarchi-Schulz transition, and . A further argument supporting this conjecture is the observation that in the SF, the distributions seem systematically lower-bounded by the deviation in the clean case (see appendix D). If this holds in the limit of large system sizes, then to have a continuity of at the transition point for , should become non-zero at the Giamarchi-Schulz transition. The question then becomes whether or in the BG above the Giamarchi-Schulz transition. One possible scenario would be that in the BG below . This would suggest that, in this regime, the local density does not see a trace of the localization transition [128], re-opening the possibility of two distinct BG phases [1, 82, 100], here below and above the half-bandwidth. However, we observe from the analysis at and a verification at very weak interactions that the minimal deviation can go below on the localized side below the half-bandwidth, and that is compatible with the results at . This rather suggests a scenario where in the BG, but where strong finite-size effects hide this result. This is summarized in figure 8: at the weak-link transition, the minimal deviation captures a proxy of the weak-links driving the transition and our numerics predicts that goes to zero at , whereas in the Giamarchi-Schulz regime, we conjecture that jumps at the transition. Download figure: Standard image High-resolution image 6. Summary and outlook In this work, we have investigated the use of a simple observable which is the extreme value of the magnetization (density) to capture a delocalization-localization transition in the ground state of the random-field chain. This basic metric highlights the sharp contrast between the two regimes of the SF-BG transition in this paradigmatic model. Indeed, we have demonstrated that in the strong disorder regime driven by weak-links, it successfully captures the location of the transition and the existence of a BKT-like scaling, with relatively modest system sizes. In the weak-disorder (Giamarchi-Schulz) regime, the extreme magnetization is essentially unaffected by the transition for available system sizes, due to extremely important finite-size effects and the completely different mechanism driving the transition. We nevertheless proposed a conjecture for the behavior of this observable in the infinite system size limit, where the chain breaking should be recovered in the BG. In this sense, this part of the study may serve as a cautionary tale for an apparent absence of localization depending on the observable. Yet, our results are also reminiscent of discussion around the possible existence of two distinct BG phases [1, 82, 100]: indeed, from the perspective of the local density on finite-size chains, the BG phase looks quite different above and below the half-bandwidth. We also note that, if a transition exists between two distinct BG regimes, the proximity of a tricritical point could account for the observed , which departs from the BKT value . The rare events we have investigated are intrinsically linked to entanglement bottlenecks [58], but are experimentally easier to measure, as compared to the entanglement entropy, which is limited to very small systems [129, 130]. For instance, extreme magnetization may be probed in bulk material through NMR spectroscopy, which can resolve inhomogeneous magnetic moment profiles [131–133]. Considering possible use in quantum simulation platforms, besides the precision on the local density, we expect a possible limitation to be the relatively large number of disorder realizations needed to take advantage of the extreme value statistics. In quantum gas microscopes, site-resolved access to local densities (already used in measurements of the Edwards-Anderson parameter) could enable the study of extreme statistics and be considered as a complementary route to the study of coherence through Talbot interferometry [134]. In Rydberg spin chains [135], direct access to the local magnetization may provide an interesting route for the study of the effects of disorder on dipolar Tomonaga–Luttinger liquids [136, 137]. We note (appendix B) that it may not be necessary to use the absolute most polarized site in the chain: the distribution of a few highly polarized sites already contains sufficient information. The role of extreme densities in our work highlights related interesting questions in MBL and out-of-equilibrium dynamics. In particular, their possible connection with weak-links, which are key players in the destruction of the SF phase, suggests a similar question at high energy: in the strongly discussed avalanche phenomenology [26, 48, 49, 138, 139], what is the role played by such bottlenecks? More precisely, it would be relevant to study how extremely polarized sites in an initial state respond to the presence of a thermal bubble in their vicinity. Finally, we note that our approach is conceptually simple in spirit. It may be helpful to better understand glassy aspects of the BG [140–142], and could be easily tested in other models, for instance quasiperiodic potentials [143–145], long-range hopping models [146] or other types of interactions or degrees-of-freedom [147, 148]. Various non-interacting disordered models (for instance, with chiral symmetry) could also provide an interesting testing ground for the robustness of the approach, in particular to shine a clearer light on our results near the Giamarchi-Schulz transition. In general, when considering possible further applications of the approach, we emphasize that the central questions to ask are whether extreme statistics of local observables can capture transport or entanglement bottlenecks in the system under consideration, and whether such events are expected to drive the transition. Acknowledgments It is a pleasure to thank Nicolas Dupuis for numerous insightful discussions on this topic. We are also indebted to Georges Bouzerar, Sylvain Capponi, Weitao Chen, Thierry Giamarchi and Nicolas Roch for enlightening discussions and comments. We are grateful to Maxime Dupont for sharing his C++ DMRG code at the beginning of the project. All calculations are performed with the ITensor library in Julia [149, 150]. Data availability statement The data that support the findings of this study will be openly available following an embargo at the following URL/DOI: https://doi.org/10.5281/zenodo.17913085 [151]. Data will be available from 31 August 2026. Supplementary material available at: https://doi.org/10.1088/1367-2630/ae9397/data1. Funding NL and GL are supported by the ANR research grant ManyBodyNet No. ANR-24-CE30-5851. This work has been partly supported by the EUR Grant NanoX No. ANR-17-EURE0009 in the framework of the ‘Programme des Investissements d’Avenir’, is part of HQI initiative (www.hqi.fr) and is supported by France 2030 under the French National Research Agency award number ANR-22-PNCQ-0002 and ANR-23-PETQ-0002, and also benefited from the support of the Fondation Simone et Cino Del Duca. JC, GL, and SA acknowledge support from the Singapore Ministry of Education AcRF Tier 2 Grant MOE-T2EP50222-0005. NC acknowledges LPT Toulouse (CNRS) for hospitality and the support from Delft Technology Fellowship. We acknowledge the use of HPC resources from CALMIP (Grants 2023-P0677 and 2024-P0677) and GENCI (Projects A0150500225 and A0170500225).