Abstract
The thermalization of isolated quantum manybody systems is deeply related to fundamental questions of quantum information theory. While integrable or manybody localized systems display nonergodic behavior due to extensively many conserved quantities, recent theoretical studies have identified a rich variety of more exotic phenomena in between these two extreme limits. The tilted onedimensional FermiHubbard model, which is readily accessible in experiments with ultracold atoms, emerged as an intriguing playground to study nonergodic behavior in a clean disorderfree system. While nonergodic behavior was established theoretically in certain limiting cases, there is no complete understanding of the complex thermalization properties of this model. In this work, we experimentally study the relaxation of an initial chargedensity wave and find a remarkably longlived initialstate memory over a wide range of parameters. Our observations are well reproduced by numerical simulations of a clean system. Using analytical calculations we further provide a detailed microscopic understanding of this behavior, which can be attributed to emergent kinetic constraints.
Introduction
Understanding the complex outofequilibrium dynamics of quantum manybody systems is central to a number of research areas ranging from statistical physics to quantum information theory^{1,2,3}. Stateoftheart experimental platforms are now able to test novel theoretical concepts and approximate descriptions based on experimental observations. Important experimental results were obtained in particular with integrable^{4} or manybody localized (MBL)^{5,6,7} systems. Both phenomena emerge due to the existence of extensively many conserved quantities and have been of considerable interest, because they break the eigenstate thermalization hypothesis, which assumes that each individual eigenstate behaves locally like a thermal ensemble and is believed to hold for generic ergodic systems^{8,9,10}.
In between the two extreme limits of ergodic and localizing dynamics there exists a rich variety of more complex thermalizing behavior. Models with manybody scar states, e.g., host a vanishing fraction of nonthermal eigenstates embedded within an otherwise thermal spectrum^{11,12,13,14,15,16}. They exhibit a weak form of ergodicitybreaking, that strongly depends on the initial state, as has been observed with Rydberg atoms^{14,17,18}. More recently, a whole new class of models has been suggested, where the presence of only few conserved quantities, in particular dipole conservation, results in nonergodic dynamics due to an emergent fragmentation of the Hilbert space into exponentially many disconnected subspaces^{19,20,21,22}. Fragmented models offer an alternative view on a central open question, namely if manybody localization can occur in translationallyinvariant models without disorder^{23,24,25,26,27,28,29}.
In this work, we study nonergodic behavior in the disorderfree tilted onedimensional (1D) FermiHubbard model (Fig. 1a), which lies at the interface of MBL and Hilbertspace fragmentation. In the presence of additional weak disorder or harmonic confinement, theoretical studies have found characteristic MBL phenomenology, known as Stark MBL^{30,31,32,33,34}. This, however, does not hold for a clean system with pure linear potential^{30,33}. While conventional MBL predicts localization for any typical initial state, we do not expect this to hold for our system, where resonances can occur between interaction and tilt energies (regime ① in Fig. 1b). Intriguingly, it has been predicted, that in the limit of large tilts, Δ ≫ J, ∣U∣, nonergodicity may still occur despite the absence of disorder. In this regime, the large tilt energy imposes kinetic constraints, which result in an emergent dipole conservation^{19,20,22,31,33}. This emergent behavior is in fact governed by a fragmented Hamiltonian resulting in nonergodic dynamics. Starting from an initial chargedensity wave (CDW) of singlons (singlyoccupied site), we study relaxation dynamics in the tilted 1D FermiHubbard model for a large range of interaction strengths and moderate values of the tilt (Δ < 4J), where none of the two mechanisms described above should apply and where naively one may expect the system to thermalize^{35,36}. At short times we observe coherent dynamics due to Bloch oscillations, whose amplitude strongly depends on the Hubbard interactions. Surprisingly we find that after intermediate times and even close to resonance (regime ①), the evolution converges to a steadystate, that persists for long evolution times up to 700 tunneling times, signaling a robust memory of the initial CDW throughout.
Using numerical calculations we show that the observed nonergodicity cannot be explained by the phenomenon of StarkMBL, i.e., the robust memory is not due to experimental imperfections, such as residual harmonic confinement or disorder, and the bipartite entanglement entropy does not exhibit the characteristic behavior of MBL systems^{30,37} (Supplementary Fig. 5). Hence, nonergodicity appears to have a different origin, despite similar experimental signatures. This raises the question about the origin of the observed nonergodicity. We construct effective Hamiltonians in two distinct regimes (① and ②, Fig. 1b) by taking the large tilt limit and find stronglyfragmented Hamiltonians in both cases (Supplementary Note 3). While these models are only expected to describe the dynamics at large tilt values and for intermediate times (on the order of a few tens of tunneling times), they allow us to identify the microscopic processes that initiate dynamics at short times (Fig. 1b). In both regimes these are correlated tunneling processes, which result in the formation of doublons (doublyoccupied sites), either resonantly (regime ①) or detuned by the Hubbard interaction energy U (regime ②). Higherorder terms are expected to eventually drive the system towards thermalization^{19}. However, we are able to show that energy penalties for the second or higherorder tunneling processes, which occur naturally in the model, render these dynamics inefficient. This results in extremely slow relaxation (Supplementary Note 3), which appears stable for > 10^{4} tunneling times in our exact diagonalization studies of small systems, in agreement with our experimental observations (Supplementary Fig. 4).
In order to characterize the dynamics across the whole parameter regime studied experimentally, we compute the finitetime connectivity of our initial CDW state \({{{{\mathcal{C}}}}}_{\epsilon }=\dim ({{{{\mathcal{N}}}}}_{\epsilon })/\dim ({{{\mathcal{H}}}})\), which is defined by the fraction of states that participate in the time evolution up to a finite time \({T}_{{{{\mathcal{N}}}}}\); here \({{{{\mathcal{N}}}}}_{\epsilon }\) denotes the subspace in the complete Hilbertspace \({{{\mathcal{H}}}}\), which is defined, such that the residual overlap of the timeevolved state \(\left\psi (t)\right\rangle\) outside of \({{{{\mathcal{N}}}}}_{\epsilon }\) is at most ϵ at any time \(t\le {T}_{{{{\mathcal{N}}}}}\) (Methods). The value of ϵ is typically chosen between 1 and 10%. The finitetime connectivity can be understood as a measure of nonergodicity, similar to the more conventional return probability or other multifractality measures^{38}. While effective Hamiltonians can only be derived explicitly in certain limits, the numerical construction is applicable in the whole parameter regime probed in this work (Fig. 1c). We find that the finitetime connectivity vanishes in the thermodynamic limit for all parameters, suggesting that only a small fraction of the states participates in the dynamics, signaling nonergodic behavior. Our results suggest that the emergent kinetic constraints result in transient nonergodic behavior across the whole parameter range studied in this work. We further show analytically that the relevant microscopic constraints in the resonant ① regime give rise to Hilbertspace fragmentation in the large tilt limit (Supplementary Note 4).
Results
The experimental setup consists of a degenerate Fermi gas of 50(5) × 10^{3} ^{40}K atoms that is prepared in an equal mixture of two spin components \(\left\uparrow \right\rangle =\left{m}_{F}=7/2\right\rangle\) and \(\left\downarrow \right\rangle =\left{m}_{F}=9/2\right\rangle\) in the F = 9/2 groundstate hyperfine manifold. The atoms are loaded into a 3D optical lattice with lattice constant d_{s} = 266 nm along the x direction and deep transverse lattices, with constant d_{⊥} = 369 nm, to isolate the 1D chains along x (“Methods”). The central 1D chains have a length of about 290 lattice sites. The residual coupling along the transverse directions is less than 3 × 10^{−4}J. The dynamics along x is described by the tilted 1D FermiHubbard model
where \({\hat{c}}_{i\sigma }^{{\dagger} }\) (\({\hat{c}}_{i\sigma }\)) is the fermionic creation (annihilation) operator and \({\hat{n}}_{i,\sigma }={\hat{c}}_{i\sigma }^{{\dagger} }{\hat{c}}_{i\sigma }\). The onsite interaction strength U is controlled by a Feshbach resonance centered at 202.1 G and a magnetic field gradient is used to create the tilt Δ_{σ}, with Δ_{↑} ≃ 0.9Δ_{↓}. The weak spindependence arises due to the different m_{F} quantum numbers (Supplementary Note 9 and 11). The initial state for all subsequent measurements is a CDW of singlons on even sites, which is prepared using a bichromatic optical superlattice (Supplementary Note 7). The initial state can be described as an incoherent mixture of sitelocalized particles with random spin configuration (“Methods”). The subsequent evolution is monitored by extracting the spinresolved imbalance \({{{{\mathcal{I}}}}}^{\sigma }=({N}_{e}^{\sigma }{N}_{o}^{\sigma })/{N}^{\sigma }\); here \({N}_{e(o)}^{\sigma }\) denotes the total number of spinσ atoms on even (odd) sites and \({N}^{\sigma }={N}_{e}^{\sigma }+{N}_{o}^{\sigma }\). A nonzero steadystate imbalance signals a memory of the initial state, where \({{{{\mathcal{I}}}}}^{\sigma }(t=0)=1\).
In a first set of measurements we study the effect of interactions on the coherent shorttime dynamics. In a tilted lattice an initially localized particle exhibits Bloch oscillations^{39}, with a characteristic period T_{σ} = h/Δ_{σ}, set by the spindependent tilt. In the presence of interactions, Bloch oscillations persist, showing a rich variety of dynamics, such as interactioninduced dephasing and amplitude modulation^{40,41,42,43,44,45}. Here, we use the spinresolved imbalance to probe realspace Bloch oscillations in a parityprojected manner. In the noninteracting limit the timedependence can be computed analytically:
which enables a precise calibration of the model parameters Δ_{σ} and J (Fig. 2a) at short times. Here, \({{{{\mathcal{J}}}}}_{0}\) denotes the 0thorder Bessel function of the first kind. The dephasing of the oscillations is caused by a residual harmonic confinement that results in a weak local variation δT_{σ} of the Bloch oscillation period T_{σ} between adjacent sites. An upper bound for the trap frequency ω_{h}/(2π) = 39 Hz was extracted from independent measurements (Supplementary Note 11) and corresponds to δT_{σ}/T_{σ} ≪ 10^{−3}. Since the imbalance dynamics for both spin components is very similar (see Supplementary Fig. 10), we focus on one component \({{{{\mathcal{I}}}}}^{\downarrow }\).
For weak tilt values, Δ_{↓} = 1.2J, we find that the dynamics of the interacting spinmixture (U = 3J) exhibits the same dominant frequency components as the noninteracting Bloch oscillations, while the dephasing is strongly enhanced. This can be seen more directly by calculating the power spectral density (PSD) of the imbalance \( {\tilde{{{{\mathcal{I}}}}}}^{\sigma }(\nu ){ }^{2}\) (inset of Fig. 2a). We find three distinct peaks in the spectrum, the Bloch frequency Δ_{↓} and an admixture of two higher harmonics with the largest spectral weight in the second harmonic at ν_{1} = 2Δ_{↓}/h. For U = 3J its weight is decreased by 70% compared to the noninteracting case. The higherorder harmonics originate from the realspace evolution within one Bloch cycle and are determined by the Bloch oscillation amplitude A_{σ}/d_{s} = 4J/Δ_{σ}. We anticipate frequency components at integer multiples of Δ_{σ}, with an upper bound determined by A_{σ}/d_{s}, in agreement with our data.
Interaction effects are expected to be less relevant once the Bloch oscillation amplitude is smaller than one site, resulting in negligible overlap between neighboring particles for our CDW initial state. In Fig. 2b we show the PSD of the coherent shorttime dynamics for Δ_{↓} = 3.0J. While the largest spectral weight of the PSD is now contained in the Bloch frequency ν_{2} = Δ_{↓}/h, the reduction is still about 50% compared to the noninteracting case. Indeed, the spectral weight is a sensitive measure of the interactioninduced dephasing. Moreover, the onsite interactions lift the degeneracy of the energy levels in the WannierStark spectrum, which results in additional frequency components in the PSD. For our parameters (Fig. 2b) they occur at ≈ ν_{2} ± 0.5Δ_{↓}/h in the timeevolving block decimation (TEBD) simulations^{46,47,48}, which is consistent with our data.
The sensitivity of the coherent shorttime dynamics on the interaction strength is further highlighted by the strong interactiondependence of the peak power spectral density (PPSD) \( \tilde{{{{\mathcal{I}}}}}({\nu }_{j}){ }^{2}\) of the respective dominant frequency components ν_{j}, j = {1, 2} (Fig. 2c, d). We find a sharp decrease of the PPSD by about 40% already for small interaction strength U =± 0.5J for Δ_{σ} = 1.2J. After reaching a global minimum at intermediate interaction strength, it slowly recovers to the noninteracting value in the limit of large interactions.
For long enough evolution times, the coherent Bloch oscillations are dephased and a finite steadystate imbalance develops in the noninteracting limit (Fig. 3a). Note that, if the dephasing was solely due to residual harmonic confinement, we would expect a coherent revival of the oscillations, which is suppressed in our experiment by additional dephasing mechanisms and ensemble averaging. The observed finite steadystate imbalance is caused by WannierStark localization and can be computed analytically by time averaging the shorttime dynamics:
Excellent agreement between our data and the analytical result provides strong evidence that the effect of the harmonic confinement is negligible for the latetime steadystate imbalance, in contrast to previous fermionic transport experiments^{35,36}. This is further supported by the data in Fig. 3b, where the steadystate value is probed for a larger range of tilt values, even reproducing the nonmonotonous behavior that is found for small values of the tilt. Note, that the vanishing imbalance, as observed for Δ_{↓} ≈ 1.5J (dashed line in Fig. 3b), does not indicate delocalization. It results from localized WannierStark orbitals with equal weight on even and odd sites.
In the presence of weak interactions localization was predicted to survive in the limit of small additional disorder or harmonic confinement, signaled by a finite steadystate imbalance^{30,31}. Here, we find that after a small decay at intermediate times a plateau of the imbalance develops, which persists for long evolution times up to 700τ (Fig. 3a) in the stronglyinteracting regime. A comparison with ED simulations (inset Fig. 3a) in a clean system without spindependent tilt and without harmonic confinement for a Néelordered initial CDW (as opposed to the randomspin initial state realized in the experiment) further highlights that this nonergodic behavior is not due to experimental imperfections at least for the experimentally relevant observation times (see Supplementary Fig. 5 for a systematic finitesize scaling analysis). Moreover, this robust steadystate value survives over a wide range of parameters (Fig. 3b). As a function of the tilt it qualitatively follows the behavior of the noninteracting system, but shows consistently lower steadystate values.
The persistence of nonergodicity down to very small values of the tilt is surprising at first sight. One may expect that for large Blochoscillation amplitudes the interactions between particles result in a dephasing of the coherent dynamics that give rise to WannierStark localization in the noninteracting limit and hence cause ergodic behavior^{35,36,40,41,42,43}. We study the plateau value for Δ_{↓} = 1.1J and find that it is largely independent of interactions (Fig. 3c). In a numerical analysis of this regime for a Néelordered singlon CDW we indeed find that the imbalance decays to zero for evolution times on the order of 10^{4} τ (Supplementary Fig. 5), which further agrees with the finite imbalance measured at ~ 200 τ. The observed inversion of the spinresolved imbalance \({{{{\mathcal{I}}}}}^{\downarrow }\; < \;{{{{\mathcal{I}}}}}^{\uparrow }\) after long evolution times (although Δ_{↓} > Δ_{↑}) is explained by the nonmonotonic dependence of the stationary imbalance on the tilt for Δ_{σ} < 2J as shown in Fig. 3b.
For intermediate values of the tilt Δ/J ≃ 3 on the other hand, we find a surprisingly robust steadystate imbalance, in agreement with numerical calculations, with a clear interaction dependence (Fig. 3d). The behavior is similar for both spin components and well reproduced by numerical simulations. The deviation between experiment and numerical simulations at larger interaction strengths is most likely due to the finite coupling between 1D chains, which plays a larger role for increased interactions^{49}. The steadystate imbalance is symmetric around U = 0 due to a dynamical symmetry [for (Δ_{↓} − Δ_{↑}) ≪ J] between attractive and repulsive interactions (Supplementary Note 2), similar to the homogeneous FermiHubbard model^{50,51}. The curve displays a global minimum for intermediate interactions, which we identify with resonant processes at ∣U∣ ≃ 2Δ, where two singlons separated by two lattice sites form a doublon. This coincides with regime ① in Fig. 1c, where the largest connectivities were found. The precise value of the resonance is slightly shifted, \({U}_{{{{\rm{res}}}}}\simeq 2{{\Delta }}8{J}^{2}/(3{{\Delta }})\), due to perturbative corrections for finite J/Δ, in agreement with our data (dashed line in Fig. 3e). For large interactions and weak spindependence (Δ_{↓} − Δ_{↑}) ≪ J, we expect the system to recover the noninteracting regime (Supplementary Note 3).
In order to gain additional insights into the observed nonergodic behavior, we study the properties of our model perturbatively in the large tilt limit for the two distinct regimes ① and ② (Fig. 1c). In regime ②, Δ ≫ J, ∣U∣, an effective Hamiltonian can be derived in powers of λ = J/Δ. As predicted^{19,20,22,31,33}, we find an emergent dipoleconserving Hamiltonian \({\hat{H}}_{{{{\rm{eff}}}}}^{{{{\rm{dip}}}}}\) [Supplementary Eq. (9)] up to third order in λ (Supplementary Note 3), where the dipolemoment operator is defined as \({\sum }_{i,\sigma }i{\hat{n}}_{i,\sigma }\). The dominant offdiagonal terms of \({\hat{H}}_{{{{\rm{eff}}}}}^{{{{\rm{dip}}}}}\) are of similar nature as those in the fragmented Hamiltonians studied previously^{19,20}, seemingly consistent with the observed nonergodic behavior. Yet, higherorder processes \({{{\mathcal{O}}}}({\lambda }^{4})\), relevant for Δ ≃ 3J, are expected to melt the CDW within the experimentally studied timescales^{19}. These higherorder processes as well as the dominant offdiagonal contribution, however, require the production of doublons, which is penalized by the onsite interaction U. We numerically show that this leads to a significant slowdown of the dynamics (Supplementary Note 5), which explains the robustness of the steadystate value observed in the experiment. Thus, for large values of the tilt, the doublon number is effectively conserved as well, as suggested in Ref. ^{31}.
On resonance, ∣U∣ ≃ 2Δ (regime ① in Fig. 1c), doublons can be formed without energy penalties, possibly leading to faster dynamics. Indeed, after an initial faster dynamics, we find a lower steadystate imbalance, which cannot be solely explained by the secondorder resonant tunneling process shown in Fig. 1c, because it leaves the imbalance invariant. In this regime, we derive an effective Hamiltonian \({\hat{H}}_{\,{{\mbox{eff}}}\,}^{{{{\rm{res}}}}}\) [Supplementary Eq. (19)] up to second order in λ (the third order vanishes), conserving the dipole moment, the doublon number or the sum of the two (\({\sum }_{i,\sigma }i{\hat{n}}_{i,\sigma }+2{\sum }_{i}{\hat{n}}_{i,\uparrow }{\hat{n}}_{i,\downarrow }\)). The corresponding symmetry sector exhibits strong fragmentation and results in a finite steadystate imbalance (Supplementary Note 4). In Fig. 4c we show the dominant secondorder tunneling terms for our initial state, illustrating the importance of doublonassisted tunneling processes for the reduction of the steadystate imbalance. For finite λ or longer evolution times, higherorder hopping processes \({{{\mathcal{O}}}}({\lambda }^{4})\) enable additional dynamics. These processes are expected to eventually melt the CDW completely, although the required timescales may be very large. In the experiment, we find robust steadystate values even for rather low values of the tilt (Δ ≃ 3J) up to evolution times of about 700τ (Fig. 3a).
In order to connect the largetilt limit described by \({\hat{H}}_{\,{{\mbox{eff}}}\,}^{{{{\rm{res}}}}}\) to the experimental parameter regime, we investigate the states within the explored subspace \({{{{\mathcal{N}}}}}_{\epsilon }\), which we denote numerical fragment in analogy to the phenomenon of Hilbertspace fragmentation. For simplicity, we study a clean system (ω_{h} = 0, Δ_{↑} = Δ_{↓} ≡ Δ) and a Néelordered CDW initial state. In Fig. 4a, we show the density of states in the Hilbert space \({{{\mathcal{H}}}}\) and compare it to the density of states in the numerical fragment \({{{{\mathcal{N}}}}}_{\epsilon }\) for different values of the cutoff ϵ. Centered around the energy of the initial state, the density of states acquires a finite width within the numerical fragments, that is approximately set by the manybody bandwidth ± 2JN (dashed line in Fig. 4a), where N = N^{↑} + N^{↓} denotes the total number of atoms. In stark contrast to thermal systems, the low finitetime connectivity indicates that only a small number of states is relevant for the dynamics. Moreover, it vanishes exponentially in the thermodynamic limit for finite evolution times up to 1000 τ (Fig. 4b). Since the perturbative Hamiltonian \({\hat{H}}_{\,{{\mbox{eff}}}\,}^{{{{\rm{res}}}}}\) is only valid in the limit of large tilts, the intersection between the numerically constructed fragment and the analytical one \({{{{\mathcal{K}}}}}^{{{{\rm{res}}}}}\) (“Methods”), which was derived using the perturbative Hamiltonian \({\hat{H}}_{\,{{\mbox{eff}}}\,}^{{{{\rm{res}}}}}\) up to third order in λ = J/Δ, is small for our experimental parameters Δ = 3J and U = 5J (Fig. 4c). We expect, however, that the two subsectors coincide for λ → 0. Indeed the normalized intersection saturates to one, although only for Δ/J ≫ 20. For this comparison the cutoff value \(\epsilon ({{{{\mathcal{K}}}}}^{{{{\rm{res}}}}})\) is chosen such that \(\,{{\mbox{dim}}}\,({{{{\mathcal{N}}}}}_{\epsilon ({{{{\mathcal{K}}}}}^{{{{\rm{res}}}}})})=\,{{\mbox{dim}}}\,({{{{\mathcal{K}}}}}^{{{{\rm{res}}}}})\), since generally, \({{{{\mathcal{N}}}}}_{\epsilon }\) contains a much larger number of states. Despite the large value of λ realized in the experiment, we find strong evidence that the slow dynamics is due to kinetic constraints and that the energetically allowed microscopic processes give rise to the phenomenon of Hilbertspace fragmentation in the large tilt limit, as demonstrated for the two regimes (① and ②). This is further supported by the resonance feature that is shown in the inset of Fig. 4c for the resonant regime ①.
Discussion
In conclusion, we have demonstrated both experimentally and numerically nonergodic behavior in the tilted 1D FermiHubbard model over a wide range of parameters and have provided a microscopic understanding based on perturbative analytical calculations. For future studies it would be interesting to study the limit of large tilts, where stronglyfragmented effective Hamiltonians were identified and to investigate the initialstate dependence of the transient dynamics. This is a characteristic feature of Hilbertspace fragmentation, where distinct thermalization properties are expected for different fragments^{19,20,22}. Although experimentally challenging due to finite evolution times, it would be interesting to reconcile the phenomenon of Stark MBL and Hilbertspace fragmentation, by studying the impact of weak disorder or residual harmonic confinement on the longtime dynamics^{33}. Adding periodic modulation as an additional ingredient, other stronglyfragmented models, scarred models and time crystals could be engineered^{52,53,54} or driveinduced localization could be investigated^{55,56}. By tuning the direction of the tilt in a 2D lattice, dipole and highermoment conserving models could be realized^{20,57} enabling studies beyond the hydrodynamic regime^{58}. Moreover, it will be interesting to explore the connection between lattice gauge theories and the phenomenon of Hilbertspace fragmentation^{21,27,29,59,60,61}, which could be addressed experimentally in a similar model^{62}.
Methods
Experimental sequence
Our sequence begins with loading a degenerate Fermi gas with temperature T/T_{F} = 0.15(1), where T_{F} is the Fermi temperature, into a threedimensional (3D) optical lattice. The wavelength is λ_{l} = 1064 nm along the x direction and λ_{⊥} = 738 nm in the transverse directions. Repulsive interactions during loading in combination with a short, offresonant light pulse after loading ensure an initial state free of double occupancies (Supplementary Note 7). By adding a short lattice with wavelength λ_{s} = λ_{l}/2 along the x direction, we generate a CDW initial state consisting of singlons (Supplementary Note 7). Holding the gas in this deep 3D lattice with a tilted, bichromatic superlattice along the x direction, dephases remaining correlations between neighboring sites and suppresses any residual dynamics, while ramping up a magnetic field gradient and adjusting the interaction strength. The lattice depths are 18 E_{rs} for the short lattice, 20 E_{rl} for the long lattice and 55 E_{r⊥} for the transverse lattices. The depths are given in the respective recoil energies, \({E}_{rj}={\hslash }^{2}{k}_{j}^{2}/(2m)\), with j ∈ {l, s, ⊥}, k_{j} = 2π/λ_{j} the corresponding wave vector, m the mass of ^{40}K and and ℏ = h/(2π) the reduced Planck constant. The deep transverse lattices decouple the 1D chains aligned along x and generate a 2D array of nearly independent 1D systems. The residual coupling along the transverse directions is typically less then 0.03 % of the coupling J along x. The dynamics to probe the tilted 1D FermiHubbard model described by the Hamiltonian in Eq. (1) is initiated by suddenly switching off the long lattice and quenching the short lattice to depths between 6 E_{rs} and 8 E_{rs}. Simultaneously, the strength of the dipole trap is adjusted in order to compensate the anticonfining harmonic potential introduced by the lattice (Supplementary Note 8). After a variable evolution time t the onsite population is frozen by suddenly ramping up the longitudinal lattices to 18 E_{rs} and 20 E_{rl} respectively. Subsequently, we extract the spinresolved imbalance \({{{{\mathcal{I}}}}}^{\sigma }\), by using a bandmapping technique^{63,64} in conjunction with SternGerlach resolved absorption imaging. Note, that the imbalance is defined as a charge imbalance between even and odd lattice sites in our system and does not probe spin imbalances. The magnetization of the systems is conserved during the evolution and it is equal to zero at all times.
Initial state
The initial state in all experiments consists of a CDW of singlons, where \(\left\uparrow \right\rangle\) and \(\left\downarrow \right\rangle\) states are randomly distributed on even lattice sites and odd lattice sites are empty. We work with an equal mixture of both states (N_{↑} = N_{↓}) such that the total magnetization is zero. The fraction of residual holes on even lattice sites, due to imperfections in the loading sequence and due to removed doublons is expected to be about 10%^{65}. Excellent agreement between the data and numerical simulations, which do not consider residual holes on even sites, indicates, that the hole fraction has a negligible effect on our dynamics. The initial state can be modelled as incoherent mixture within the zero magnetization sector with density matrix \(\hat{\rho }=\frac{1}{{{{\mathcal{N}}}}}{\sum }_{\{\sigma \} {\sum }_{i}{\sigma }_{i} = 0}\left{\psi }_{0}(\{\sigma \})\right\rangle \left\langle {\psi }_{0}(\{\sigma \})\right\), where each product state \(\left{\psi }_{0}(\{\sigma \})\right\rangle\), is given by a CDW of singlons and where the sum runs over all \({{{\mathcal{N}}}}\) possible permutations of spin configurations {σ}. The product state \(\left{\psi }_{0}(\{\sigma \})\right\rangle\) is defined as \(\left{\psi }_{0}(\{\sigma \})\right\rangle ={\prod }_{i = {{\mbox{even}}}\in {{{\rm{trap}}}}}{\left({\hat{c}}_{i\uparrow }^{{\dagger} }\right)}^{{n}_{i\uparrow }}{\left({\hat{c}}_{i\downarrow }^{{\dagger} }\right)}^{{n}_{i\downarrow }}\left0\right\rangle\), where \({\hat{c}}_{i\sigma }^{{\dagger} }\) is the fermionic creation operator, n_{iσ} ∈ {0, 1}, σ ∈ {↑, ↓}, n_{i} = n_{i↑} + n_{i↓} ≤ 1 and i is the latticesite index along x.
Note, that in the clean translationallyinvariant FermiHubbard model the initial CDW corresponds to an infinite temperature state and a finite imbalance value is a hallmark signature of localization. In the tilted model, the spectrum is superextensive complicating a meaningful definition of temperature. This is overcome by transforming into the interaction picture with respect to the tilt potential, which leaves all density observables invariant and allows us to establish the imbalance as a good probe for ergodicity breaking (Supplementary Note 1)
Details of numerical calculations
The numerical computations that are compared with the experiment in Figs. 2 and 3 of the main text were performed using ED or TEBD. The parameters J, Δ_{↑} and Δ_{↓} used in the computations were obtained as fit parameters from the corresponding noninteracting data. Additionally, the effect of harmonic confinement present in the experiment was simulated by scaling the trap frequency by a factor \(\sqrt{\frac{{L}_{\exp }}{L}}\) where \({L}_{\exp }=290\) is the system size in the experiment and L is the system size used in the numerical calculation. This is done to appropriately simulate the collapse and revival dynamics in the Bloch oscillations induced by the harmonic confinement (Supplementary Note 11).
We use TEBD for shorttime dynamics (Fig. 2 of the main text) and ED for longtime dynamics (Fig. 3 of the main text). In ED, we consider the Hilbert space as a tensor product \({{{{\mathcal{H}}}}}_{\uparrow }\otimes {{{{\mathcal{H}}}}}_{\downarrow }\) where \({{{{\mathcal{H}}}}}_{\sigma }\) is the Hilbert space of spinσ atoms. In order to efficiently compute the time dynamics, we decompose each time step in the dynamics into three unitary propagators. One each corresponding to the hopping of the two spin components and the third one corresponding to the onsite potential and interactions. We use a TrotterSuzuki approximation in this decomposition (see Supplementary Note 13 for details and error analysis). In Fig. 3a, d, we use L = 16, N_{↑} = N_{↓} = 4. In order to effectively model a mixed CDW initial state, in Fig. 3a, this computation is averaged over 20 randomly chosen pure CDW states. In Fig. 3d we use a superposition of pure CDW product states as we are concerned only with timeaveraged steadystate value. The parameters J, Δ_{σ} and the harmonic confinement are fixed by fitting to the corresponding noninteracting data.
In Fig. 2, we use TEBD calculations with L = 100 and bonddimension χ = 120. The truncation error was less than 10^{−2}. In Fig. 2b, c, we compare the experimental and numerical data in Fourier space. If the two data sets have different number of samplings in the time domain, we scale the numerical data appropriately after the fast Fourier transform.
Construction of the Krylov subspace
The Krylov subspace (corresponding to the fragment \({{{{\mathcal{K}}}}}^{{{{\rm{res}}}}}\)) is constructed by using the effective Hamiltonian on resonance \({\hat{H}}_{{{{\rm{eff}}}}}^{{{{\rm{res}}}}}\) in Supplementary Eq. (19). This Hamiltonian is then interpreted as an adjacency matrix in the Wannier basis and the Krylov subspace consists of all states, which are connected to the Néelordered CDW initial state. The Krylov subspace \({{{{\mathcal{K}}}}}^{{{{\rm{res}}}}}\) is closed under timeevolution generated by the effective Hamiltonian \({\hat{H}}_{{{{\rm{eff}}}}}^{{{{\rm{res}}}}}\). Starting from initial states within the Krylov subspace \({{{{\mathcal{K}}}}}^{{{{\rm{res}}}}}\) and including higherorder terms \({{{\mathcal{O}}}}({\lambda }^{4})\), the dynamics is captured only approximately (Supplementary Note 4). An improvement is obtained by further rotating the diagonal basis in which the effective Hamiltonian becomes fragmented with the unitary transformation obtained in powers of λ (as given by the SchriefferWolff perturbative expansion (Supplementary Note 3)). This results in a rotated Krylov subspace.
Construction of the numerical fragment
We define the numerical fragment \({{{{\mathcal{N}}}}}_{\epsilon }\) as the span of a subset \({{{{\mathcal{B}}}}}_{\epsilon }\) of the number basis \({{{\mathcal{B}}}}\) of \({{{\mathcal{H}}}}\), where \({{{\mathcal{H}}}}\) is restricted to quarter filling and zero magnetization. We define the set \({{{{\mathcal{B}}}}}_{\epsilon }\) via its complement, \({{{{\mathcal{B}}}}}_{\epsilon }={{{\mathcal{B}}}}\backslash {{{{\mathcal{B}}}}}_{\epsilon }^{c}\), where \({{{{\mathcal{B}}}}}_{\epsilon }^{c}\) would be ideally defined as the largest subset of \({{{\mathcal{B}}}}\) satisfying \({\max }_{t\,{ < }\,{T}_{{{{\mathcal{N}}}}}}{\sum }_{{n}^{c}\in {{{{\mathcal{B}}}}}_{\epsilon }^{c}}{\left\left\langle {n}^{c} \psi (t)\right\rangle \right}^{2}\, < \,\epsilon\). Here \({T}_{{{{\mathcal{N}}}}}\) defines a time window for the evolution of the initial state \(\left\psi (t=0)\right\rangle\). Equivalently, one could define the subset \({{{{\mathcal{B}}}}}_{\epsilon }\) as the smallest one, satisfying \(\mathop{\min }\limits_{t < {T}_{{{{\mathcal{N}}}}}}{\sum }_{n\in {{{{\mathcal{B}}}}}_{\epsilon }}{\left\left\langle n \psi (t)\right\rangle \right}^{2}\ge 1\epsilon\). We work with the complement, because it is easier to implement numerically. This inequality condition for the complement would ensure that the residual overlap of \(\left\psi (t)\right\rangle\) outside of \({{{{\mathcal{N}}}}}_{\epsilon }\) at any time \(t\le {T}_{{{{\mathcal{N}}}}}\) is bounded by ϵ. Constructing this \({{{{\mathcal{B}}}}}_{\epsilon }^{c}\), however, involves a search in the powerset of \({{{\mathcal{B}}}}\), which is exponential in the dimension of \({{{\mathcal{H}}}}\). This is intractable even for relatively small system sizes such as L = 7. It follows from the inequality \({\max }_{t\,{ < }\,{T}_{{{{\mathcal{N}}}}}}{\sum }_{{n}^{c}}{\left\left\langle {n}^{c} \psi (t)\right\rangle \right}^{2}\le {\sum }_{{n}^{c}}{\max }_{t\,{ < }\,{T}_{{{{\mathcal{N}}}}}}{\left\left\langle {n}^{c} \psi (t)\right\rangle \right}^{2}\) that keeping the latter sum smaller than ϵ will ensure that the former sum is also bounded by ϵ. Moreover, the latter sum is computationally easier to handle and therefore, we use it to define the fragment. We construct the numerical fragment \({{{{\mathcal{N}}}}}_{\epsilon }\) using a \({{{{\mathcal{B}}}}}_{\epsilon }^{c}\), defined such that \({\sum }_{{n}^{c}}{\max }_{t\,{ < }\,{T}_{{{{\mathcal{N}}}}}}{\left\left\langle {n}^{c} \psi (t)\right\rangle \right}^{2}\, < \,\epsilon\). The gap in the inequality \({\max }_{t\,{ < }\,{T}_{{{{\mathcal{N}}}}}}{\sum }_{{n}^{c}}{\left\left\langle {n}^{c} \psi (t)\right\rangle \right}^{2}\le {\sum }_{{n}^{c}}{\max }_{t\,{ < }\,{T}_{{{{\mathcal{N}}}}}}{\left\left\langle {n}^{c} \psi (t)\right\rangle \right}^{2}\) loosely depends on the sum \({\sum }_{n\in {{{\mathcal{B}}}}}{\max }_{t\,{ < }\,{T}_{{{{\mathcal{N}}}}}}{\left\left\langle n \psi (t)\right\rangle \right}^{2}\), which is in general, not normalized. Although this sum can be as large as the dimension of \({{{\mathcal{H}}}}\), in the examples that we study, it remains small, i.e., < 10 for L < 20, and grows logarithmically in the dimension of \({{{\mathcal{H}}}}\).
Data availability
All data files are available from the corresponding author upon reasonable request. Source data are provided with this paper.
Code availability
The code that supports the plots within this paper are available from the corresponding author upon reasonable request.
References
Gogolin, C. & Eisert, J. Equilibration, thermalisation, and the emergence of statistical mechanics in closed quantum systems. Rep. Prog. Phys. 79, 056001 (2016).
D’Alessio, L., Kafri, Y., Polkovnikov, A. & Rigol, M. From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics. Adv. Phys. 65, 239 (2016).
Mori, T., Ikeda, T. N., Kaminishi, E. & Ueda, M. Thermalization and prethermalization in isolated quantum systems: a theoretical overview. J. Phys. B: Atom. Molec. Opt. Phys. 51, 112001 (2018).
Calabrese, P., Essler, F. H. L. & Fagotti, M. Quantum quench in the transversefield ising chain. Phys. Rev. Lett. 106, 227203 (2011).
Nandkishore, R. & Huse, D. A. Manybody localization and thermalization in quantum statistical mechanics. Annu. Rev. Condens. Matter Phys. 6, 15 (2015).
Altman, E. & Vosk, R. Universal dynamics and renormalization in manybodylocalized systems. Annu. Rev. Condens. Matter Phys. 6, 383 (2015).
Abanin, D. A., Altman, E., Bloch, I. & Serbyn, M. Colloquium : manybody localization, thermalization, and entanglement. Rev. Mod. Phys. 91, 021001 (2019).
Deutsch, J. M. Quantum statistical mechanics in a closed system. Phys. Rev. A 43, 2046 (1991).
Srednicki, M. Chaos and quantum thermalization. Phys. Rev. E 50, 888 (1994).
Rigol, M., Dunjko, V. & Olshanii, M. Thermalization and its mechanism for generic isolated quantum systems. Nature 452, 854 (2008).
Shiraishi, N. & Mori, T. Systematic construction of counterexamples to the eigenstate thermalization hypothesis. Phys. Rev. Lett. 119, 030601 (2017).
Mondaini, R., Mallayya, K., Santos, L. F. & Rigol, M. Comment on "systematic construction of counterexamples to the eigenstate thermalization hypothesis”. Phys. Rev. Lett. 121, 038901 (2018).
Moudgalya, S., Rachel, S., Bernevig, B. A. & Regnault, N. Exact excited states of nonintegrable models. Phys. Rev. B 98, 235155 (2018).
Turner, C. J., Michailidis, A. A., Abanin, D. A., Serbyn, M. & Papić, Z. Weak ergodicity breaking from quantum manybody scars. Nat. Phys. 14, 745 (2018a).
Iadecola, T. & Schecter, M. Quantum manybody scar states with emergent kinetic constraints and finiteentanglement revivals. Phys. Rev. B 101, 024306 (2020).
Chattopadhyay, S., Pichler, H., Lukin, M. D. & Ho, W. W. Quantum manybody scars from virtual entangled pairs. Phys. Rev. B 101, 174308 (2020).
Bernien, H. et al. Probing manybody dynamics on a 51atom quantum simulator. Nature 551, 579 (2017).
Turner, C. J., Michailidis, A. A., Abanin, D. A., Serbyn, M. & Papić, Z. Quantum scarred eigenstates in a Rydberg atom chain: entanglement, breakdown of thermalization, and stability to perturbations. Phys. Rev. B 98, 155134 (2018).
Sala, P., Rakovszky, T., Verresen, R., Knap, M. & Pollmann, F. Ergodicity breaking arising from hilbert space fragmentation in dipoleconserving hamiltonians. Phys. Rev. X 10, 011047 (2020).
Khemani, V., Hermele, M. & Nandkishore, R. Localization from Hilbert space shattering: from theory to physical realizations. Phys. Rev. B 101, 174204 (2020).
Rakovszky, T., Sala, P., Verresen, R., Knap, M. & Pollmann, F. Statistical localization: From strong fragmentation to strong edge modes. Phys. Rev. B 101, 125126 (2020).
Moudgalya, S., Prem, A., Nandkishore, R., Regnault, N. & Bernevig, B.A. Thermalization and its absence within Krylov subspaces of a constrained Hamiltonian. arXiv:1910.14048. Preprint at https://arxiv.org/abs/1910.14048 (2019).
van Horssen, M., Levi, E. & Garrahan, J. P. Dynamics of manybody localization in a translationinvariant quantum glass model. Phys. Rev. B 92, 100305 (2015).
Schiulaz, M., Silva, A. & Müller, M. Dynamics in manybody localized quantum systems without disorder. Phys. Rev. B 91, 184202 (2015).
Yao, N. Y., Laumann, C. R., Cirac, J. I., Lukin, M. D. & Moore, J. E. Quasimanybody localization in translationinvariant systems. Phys. Rev. Lett. 117, 240601 (2016).
Papić, Z., Stoudenmire, E. M. & Abanin, D. A. Manybody localization in disorderfree systems: the importance of finitesize constraints. Annals of Physics 362, 714 (2015).
Smith, A., Knolle, J., Kovrizhin, D. L. & Moessner, R. Disorderfree localization. Phys. Rev. Lett. 118, 266601 (2017).
Smith, A., Knolle, J., Moessner, R. & Kovrizhin, D. L. Absence of ergodicity without quenched disorder: from quantum disentangled liquids to manybody localization. Phys. Rev. Lett. 119, 176601 (2017).
Brenes, M., Dalmonte, M., Heyl, M. & Scardicchio, A. ManyBody localization dynamics from gauge invariance.
Schulz, M., Hooley, C. A., Moessner, R. & Pollmann, F. Stark manyBody localization. Phys. Rev. Lett. 122, 040606 (2019).
Nieuwenburg, E. V., Baum, Y. & Refael, G. From Bloch oscillations to manybody localization in clean interacting systems. PNAS 116, 9269 (2019).
Wu, L.N. & Eckardt, A. Bathinduced decay of Stark manybody localization. Phys. Rev. Lett. 123, 030602 (2019).
Taylor, S. R., Schulz, M., Pollmann, F. & Moessner, R. Experimental probes of Stark manybody localization. Phys. Rev. B 102, 054206 (2020).
Yao, R. & Zakrzewski, J. Manybody localization of bosons in an optical lattice: dynamics in disorderfree potentials. Phys. Rev. B 102, 104203 (2020).
Ott, H. Collisionally induced transport in periodic potentials. Phys. Rev. Lett. 92, 160601 (2004).
Gustavsson N. et al. Interactioncontrolled transport of an ultracold fermi gas. Phys. Rev. Lett. 99, 220601 (2007).
Bardarson, J. H., Pollmann, F. & Moore, J. E. Unbounded growth of entanglement in models of manybody localization. Phys. Rev. Lett. 109, 017202 (2012).
Luca, A. D. & Scardicchio, A. Ergodicity breaking in a model showing manybody localization. EPL 101, 37003 (2013).
Ben Dahan, M., Peik, E., Reichel, J., Castin, Y. & Salomon, C. Bloch oscillations of atoms in an optical potential. Phys. Rev. Lett. 76, 4508 (1996).
Buchleitner, A. & Kolovsky, A. R. Interactioninduced decoherence of atomic Bloch oscillations. Phys. Rev. Lett. 91, 253002 (2003).
Kolovsky, A. R. & Buchleitner, A. FloquetBloch operator for the BoseHubbard model with static field. Phys. Rev. E 68, 056213 (2003).
Tomadin, A., Mannella, R. & Wimberger, S. Manybody interband tunneling as a witness of complex dynamics in the BoseHubbard model. Phys. Rev. Lett. 98, 130402 (2007).
Tomadin, A., Mannella, R. & Wimberger, S. Manybody LandauZener tunneling in the BoseHubbard model. Phys. Rev. A 77, 013606 (2008).
Gustavsson, M. et al. Control of interactioninduced dephasing of Bloch oscillations. Phys. Rev. Lett. 100, 080404 (2008).
P. M. Preiss, P. M. et al. Strongly correlated quantum walks in optical lattices. Science 347, 1229 (2015).
Schollwöck, U. The densitymatrix renormalization group in the age of matrix product states. Ann. Phys. 326, 96 (2011).
Paeckel, S. et al. Timeevolution methods for matrixproduct states. Ann. Phys. 411, 167998 (2019).
Hauschild, J. & Pollmann, F. Efficient numerical simulations with Tensor Networks: Tensor Network Python (TeNPy). SciPost Phys. Lect. Notes, 5 (2018).
Bordia, P. et al. Coupling Identical onedimensional ManyBody Localized Systems. Phys. Rev. Lett. 116, 140401 (2016).
Schneider, U. et al. Fermionic transport and outofequilibrium dynamics in a homogeneous Hubbard model with ultracold atoms. Nat. Phys. 8, 213–218 (2012).
Schreiber, M. et al. Observation of manybody localization of interacting fermions in a quasirandom optical lattice. Science 349, 842 (2015).
Pai, S. & Pretko, M. Dynamical scar states in driven fracton systems. Phys. Rev. Lett. 123, 136401 (2019).
Zhao, H., Vovrosh, J., Mintert, F. & Knolle, J. Quantum manybody scars in optical lattices. Phys. Rev. Lett. 124, 160604 (2020).
Kshetrimayum, A., Eisert, J. & Kennes, D. M. Stark time crystals: Symmetry breaking in space and time. Phys. Rev. B 102, 195116 (2020).
Bairey, E., Refael, G. & Lindner, N. H. Driving induced manybody localization. Phys. Rev. B 96, 020201 (2017).
Bhakuni, D. S., Nehra, R. & Sharma, A. Driveinduced manybody localization and coherent destruction of Stark manybody localization. Phys. Rev. B 102, 024201 (2020).
Feldmeier, J., Sala, P., de Tomasi, G., Pollmann, F. & Knap, M. Anomalous Diffusion in Dipole and HigherMomentConserving Systems. Phys. Rev. Lett. 125, 245303 (2020).
GuardadoSanchez, E. et al. Subdiffusion and Heat Transport in a Tilted TwoDimensional FermiHubbard System. Phys. Rev. X 10, 011042 (2020).
Pai, S. & Pretko, M. Fractons from confinement in one dimension. Phys. Rev. Res. 2, 013094 (2020).
Verdel, R., Liu, F., Whitsitt, S., Gorshkov, A. V. & Heyl, M. Realtime dynamics of string breaking in quantum spin chains. Phys. Rev. B 102, 014308 (2020).
Yang, Z.C., Liu, F., Gorshkov, A. V. & Iadecola, T. Hilbertspace fragmentation from strict confinement. Phys. Rev. Lett. 124, 207602 (2020).
Yang, B. et al. Observation of gauge invariance in a 71site Bose–Hubbard quantum simulator. Nature 587, 392–396 (2020).
SebbyStrabley, J., Anderlini, M., Jessen, P. S. & Porto, J. V. Lattice of double wells for manipulating pairs of cold atoms. Phys. Rev. A 73, 033605 (2006).
Fölling, S. et al. Direct observation of secondorder atom tunnelling. Nature 448, 1029–1032 (2007).
Scherg, S. et al. Nonequilibrium Mass Transport in the1D FermiHubbard Model. Phys. Rev. Lett. 121, 130402 (2018).
Acknowledgements
We thank D. Abanin, G. De Tomasi, M. Filippone, M. Knap, N. Lindner, R. Moessner, T. Rakovszky, and N. Yao for inspiring discussions. We thank C. Schweizer for very useful discussion about the experimental results and their interpretation. We thank M. Buser for illuminating discussions regarding the ED calculations. This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC2111 – 39081486. The work at LMU was additionally supported by DIP and B.H.M. acknowledges support from the European Union (Marie Curie, Pasquans). The work at TU was additionally supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No. 771537).
Funding
Open Access funding enabled and organized by Projekt DEAL.
Author information
Authors and Affiliations
Contributions
S.S., T.K. and B.H.M. conceived and performed the experiments and analyzed the data. B.H.M. and P.S. carried out the numerical simulations. P.S. carried out the analytic derivations. M.A., F.P. and I.B. supervised the work. All authors contributed critically to the writing of the manuscript and the interpretation of experimental and numerical results.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Peer review information Nature Communications thanks Leo Radzihovsky and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Source data
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Scherg, S., Kohlert, T., Sala, P. et al. Observing nonergodicity due to kinetic constraints in tilted FermiHubbard chains. Nat Commun 12, 4490 (2021). https://doi.org/10.1038/s41467021247260
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s41467021247260
This article is cited by

Absence of localization in interacting spin chains with a discrete symmetry
Nature Communications (2023)

Reviving product states in the disordered Heisenberg chain
Nature Communications (2023)

Observation of manybody Fock space dynamics in two dimensions
Nature Physics (2023)

Observation of Stark manybody localization without disorder
Nature (2021)

Quantum manybody scars and weak breaking of ergodicity
Nature Physics (2021)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.