Abstract
Phase transitions are driven by collective fluctuations of a system’s constituents that emerge at a critical point1. This mechanism has been extensively explored for classical and quantum systems in equilibrium, whose critical behaviour is described by the general theory of phase transitions. Recently, however, fundamentally distinct phase transitions have been discovered for out-of-equilibrium quantum systems, which can exhibit critical behaviour that defies this description and is not well understood1. A paradigmatic example is the many-body localization (MBL) transition, which marks the breakdown of thermalization in an isolated quantum many-body system as its disorder increases beyond a critical value2,3,4,5,6,7,8,9,10,11. Characterizing quantum critical behaviour in an MBL system requires probing its entanglement over space and time4,5,7, which has proved experimentally challenging owing to stringent requirements on quantum state preparation and system isolation. Here we observe quantum critical behaviour at the MBL transition in a disordered Bose–Hubbard system and characterize its entanglement via its multi-point quantum correlations. We observe the emergence of strong correlations, accompanied by the onset of anomalous diffusive transport throughout the system, and verify their critical nature by measuring their dependence on the system size. The correlations extend to high orders in the quantum critical regime and appear to form via a sparse network of many-body resonances that spans the entire system12,13. Our results connect the macroscopic phenomenology of the transition to the system’s microscopic structure of quantum correlations, and they provide an essential step towards understanding criticality and universality in non-equilibrium systems1,7,13.
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Main
The MBL transition8,9,10,11 represents a type of quantum phase transition that fundamentally differs from both its classical and quantum ground-state counterparts2,3,7. Instead of being characterized by an instantaneous thermodynamic signature, it is identified by the system’s inherent dynamic behaviour (Fig. 1a). In particular, the MBL transition manifests itself through a change in entanglement dynamics7,11. Recent years have seen tremendous progress in our understanding of both the thermal and the MBL phases within the frameworks of quantum thermalization6,14,15 and emergent integrability4,5,8,9,10,11, respectively.
The quantum critical behaviour at this transition, however, has remained largely unresolved7. In particular, it is unclear whether the traditional association of collective fluctuations with static and dynamic critical behaviour can be applied to this transition. The high amount of entanglement found at the MBL transition limits numerical studies owing to the required computational power16,17. Several theoretical approaches, despite using disparate microscopic structures, suggest that anomalous transport is the macroscopic behaviour at the quantum critical point12,18,19,20. Experimental studies do indeed indicate a slowdown of the dynamics at intermediate disorder21,22. However, identifying anomalous transport as quantum critical dynamics is experimentally challenging, since similar behaviour can also originate from stochastic effects such as inhomogeneities in the initial state23 or coupling to a classical bath24,25. Additionally, in the case of random disorder, the presence of rare regions, where potential offsets accidentally coincide for consecutive lattice sites, permits several microscopic mechanisms that may govern this critical behaviour and therefore makes it challenging to distinguish between them26,27,28. Our experimental protocol overcomes these challenges by using a quasi-periodic potential, which is free of rare regions, as well as by evolving a pure, homogeneous initial state under unitary dynamics. Using this protocol, we observe quantum critical dynamics via anomalous transport, enhanced quantum fluctuations, and system-size-dependent thermalization. In addition, we microscopically resolve and characterize the structure of the entanglement in the many-body states through their multi-particle quantum correlations.
Our experiments start with a pure state of up to twelve unentangled lattice sites at unity filling (Fig. 1b). We study its out-of-equilibrium evolution after a rapid increase of the tunnelling in the bosonic, interacting Aubry–André Hamiltonian:
where \({{ {\hat{a}} }}_{i}^{\dagger }\) \(\left({{ {\hat{a}} }}_{i}\right)\) is the creation (annihilation) operator for a boson on site i, and \({\hat{n}}_{i}\) is the corresponding particle number operator (H.c. is the Hermitian conjugate). The tunnelling time τ = ħ/J = 4.3(1) ms (with the reduced Planck constant ħ) between neighbouring sites and the pairwise interaction energy U = 2.87(3)J remain constant for all experiments. The potential energy offset hi = cos(2πβi + ϕ) on site i follows a quasi-periodic distribution of amplitude W, period 1/β ≈ 1.618 lattice sites and phase ϕ. After a variable evolution time, we obtain full counting statistics of the quantum state through a fluorescence imaging technique. The applied unitary evolution preserves the initial purity of 99.1(2)% per site, such that all correlations are expected to stem from entanglement in the system11,15.
We first characterize the system’s dynamical behaviour by studying its transport properties for different disorder strengths. Since the initial state has exactly one atom per site, the system starts with zero density correlations at all length scales. However, during the Hamiltonian evolution, tunnelling dynamics build up anti-correlated density fluctuations between coupled sites of increasing distance (Fig. 2a). Motivated by this picture, we quantify the particle dynamics by defining the transport distance, \(\Delta x\propto {\sum }_{d}d\times \langle {G}_{{\rm{c}}}^{(2)}{(i,i+d)\rangle }_{i}\), as the first moment of the disorder-averaged two-point density correlations, \({G}_{{\rm{c}}}^{(2)}(i,i+d)=\langle {\hat{n}}_{i}{\hat{n}}_{i+d}\rangle -\langle {\hat{n}}_{i}\rangle \langle {\hat{n}}_{i+d}\rangle \) (Supplementary Information section 7). At low disorder, we observe these anti-correlations to rapidly build up and saturate over a timescale of t/τ ≈ L/2. With increasing disorder, we observe a slowdown of particle transport that is consistent with a power-law growth Δx ∝ tα (Fig. 2b)29. We extract the anomalous diffusion exponent α from a subset of the data points that exclude the initial transient dynamics in the system \(\left(L/2 < t/\tau \le 100\right)\) (inset to Fig. 2b). The exponent α is reduced by successively higher disorder, demonstrating the suppression of transport in the MBL regime.
To identify the anomalous diffusion as a signature of quantum critical dynamics, we measure the system-size dependence of two observables in the long-time limit (t = 100τ): the on-site number fluctuations \({\mathscr{F}}\equiv {G}_{{\rm{c}}}^{(2)}(d=0)\) as a probe of local thermalization, and the transport distance Δx as a localization measure (Fig. 2c). At low disorder, the fluctuations agree with those predicted by a thermal ensemble and particles are completely delocalized for both system sizes. This demonstrates that local quantum thermalization occurs independently of system size at low disorder and establishes that this regime corresponds to the system being in the thermal phase. At strong disorder, the physics is governed by the formation of an intrinsic length scale, namely the localization length \(\xi \approx \Delta \mathop{x}\limits^{ \sim }\) (refs 10,11). We observe system-size-independent, sub-thermal fluctuations and measure an intrinsic length scale \(\Delta \widetilde{x}\). This indicates that the strong disorder regime corresponds to the system being in the localized phase. However, at intermediate disorder, we observe that our data are consistent with a theoretically predicted system-size dependence for both observables. This suggests the absence of an intrinsic length scale and the presence of finite-size-limited fluctuations, which identify the critically thermalizing regime (Supplementary Information section 4). These measurements of system-size-dependent thermalization can be visualized as two horizontal cuts in a finite-size phase diagram. The observed finite-size dependence is consistent with the physics associated with a critically thermalizing intermediate phase and a shrinking quantum critical cone (inset to Fig. 2c)1.
We then investigate the multi-particle correlations in the system to probe the presence of enhanced quantum fluctuations in the quantum critical regime. For this study, we employ the n-point connected density-correlation functions (Fig. 2d)30,31,32:
which act on lattice sites with positions x = (x1,..., xn). The disconnected part of this function, \({G}_{{\rm{dis}}}^{(n)}\), is fully determined by all lower-order correlation functions, and therefore does not contain new information at order n. By removing it from the total measured correlation function, \({G}_{{\rm{tot}}}^{(n)}(x)=\left\langle {\prod }_{k=1}^{n}\hat{n}({x}_{k})\right\rangle \), we isolate all n-order correlations that are independent of lower-order processes (Supplementary Information sections 2, 3). This approach gives a direct handle on the level of complexity of the underlying many-body wavefunction and characterizes its entanglement via its non-separability into subsystems of size smaller than n. We quantify the relevance of order n processes by computing the mean absolute value of all correlations arising from both contiguous and non-contiguous n sites in the system (Fig. 2e). We find that in the thermal and the many-body-localized regimes, the system becomes successively less correlated at higher order. The behaviour in the quantum critical regime is strikingly different: we observe that the system is strongly correlated at all measured orders.
To reveal the microscopic origin for the anomalous transport, we now investigate the site-resolved structure of the many-body state (Fig. 3a). We first study how much each lattice site contributes to the transport by considering the site-resolved two-point correlations in the long-time limit (t = 100τ). In the thermal regime, we find similar correlations between all lattice sites, which correspond to uniformly delocalized atoms. In contrast, density correlations are restricted to nearby sites in the MBL regime owing to localization. At intermediate disorder, we observe strongly inhomogeneous correlations, revealing a sparse structure that involves only specific distances between lattice sites, yet spans the entire system size.
The sparse structure is expected to be linked to the applied quasi-periodic potential. The average energy offsets of sites d apart in the system are correlated by this potential. This correlation is then inherited by the system’s fluctuations when the interaction energy U compensates for these correlated offsets. To investigate this structure, we compare the two-point density correlations with the autocorrelation function, A(d) = 〈hihi+d〉i, of the quasi-periodic potential. Indeed, we find that the site-averaged connected density correlations \({G}_{{\rm{c}}}^{(2)}(d)=\langle {G}^{(2)}{(i,i+d)\rangle }_{i}\) inherit their spatial structure from A(d) (Fig. 3b). We find that this contribution is maximal in the critical regime but is strongly reduced in the thermal and MBL regimes (Fig. 3c). These observations contrast with the behaviour of a non-interacting system, where the sign of the structure is opposite because resonant tunnelling is favoured for zero potential energy difference. These results illustrate microscopically how the interplay of strong interactions and disorder can lead to anomalous diffusion. However, this picture of effective single-particle hopping that couples distant sites neglects the many-body nature of these systems.
To investigate the system’s many-body structure, we examine the site-resolved contributions of the three-point correlations. Since all non-zero contributions to the three-point correlations involve correlated hopping of at least two particles, they are a signature for multi-particle entanglement31. In the quantum critical regime, we find that these correlations span the entire system and are highly structured, taking on both positive and negative values (Fig. 4a). In contrast to the pattern in the second-order correlation function, this third-order structure is not directly recognizable as the quasi-periodic-potential correlations. To gain further insight into the structure, we analyse the contributions of all possible particle configurations in Fig. 4b. In particular, for \({G}_{{\rm{c}}}^{{\rm{(3)}}}({d}_{1}=3,{d}_{2}=1)\), which is positive, we see that the dominant contribution comes from a particular process that favours multiple atoms hopping to the same site. In contrast, \({G}_{{\rm{c}}}^{{\rm{(3)}}}({d}_{1}=3,{d}_{2}=2)\), which is negative, has a dominant process that favours all atoms leaving the three sites considered. Although this provides some intuition for the emergent many-body resonances, the three-point correlations are, in fact, the result of a superposition of many correlated processes. These observations further demonstrate how the interactions between multiple atoms can compensate for the disorder via correlated tunneling of several atoms. In this way, we can see the additional part that interactions play in the disordered system: they supply higher-order many-body resonances that preserve transport where lower-order processes are energetically suppressed.
Our results demonstrate how a many-body, sparse resonant structure drives the quantum critical behaviour at the MBL transition. This observed microscopic description is consistent with the theoretically suggested mechanisms of a sparse backbone of resonances that can act as a functional bath for the system12,13,33. However, our results provide a new perspective on this description by mapping out the prevalence of high-order processes in the system that facilitate this critical thermalization.
In future experiments, the tunability of our system will allow us to address further open questions on the MBL transition, such as possible discontinuities of the entanglement entropy13, the potential emergence of new dynamic phases near the critical point, and the influence of rare regions in the disorder potential26,27. Furthermore, the demonstrated techniques pave the way to exploring the role of universality in non-equilibrium systems. From a computational perspective, our system’s Hilbert space dimension is comparable to the dimension of 22 spins with zero total magnetization. A moderate increase of the system’s spatial dimension beyond this experiment results in numerically intractable sizes.
Data availability
The data that support the findings of this study are available in the Dataverse repository at https://doi.org/10.7910/DVN/E2ROXU.
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Acknowledgements
We acknowledge discussions with D. Abanin, E. Altman, H. Bernien, C. Chiu, S. Choi, E. Demler, A. Hébert, W. W. Ho, V. Kasper, V. Khemani, J. Kwan, L. Santos and J. Schmiedmayer. We were supported by grants from the National Science Foundation, the Gordon and Betty Moore Foundations EPiQS Initiative, an Air Force Office of Scientific Research MURI programme, an Army Research Office MURI programme and the NSF Graduate Research Fellowship Program. J.L. acknowledges support from the Swiss National Science Foundation.
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All authors contributed extensively to the construction of the experiment, the collection and analysis of the data, and the writing of the manuscript. M.G. supervised the work.
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Supplementary Sections 1–9, including Supplementary Figs. 1–7 and Supplementary Table 1.
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Rispoli, M., Lukin, A., Schittko, R. et al. Quantum critical behaviour at the many-body localization transition. Nature 573, 385–389 (2019). https://doi.org/10.1038/s41586-019-1527-2
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DOI: https://doi.org/10.1038/s41586-019-1527-2
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