Saying that a bounded number of exceptions is a stronger requirement than a vanishing fraction of exceptions only means something if the two can actually come apart. This short paper proves they can. It exhibits a real physical system that passes the fraction test but fails the count test, which shows that the sharper counting-based notion carries real content rather than being a technicality. The example is a well-known constrained quantum chain, the Rydberg-blockade or PXP model, famous for holding an exact ladder of special scar states that refuse to thermalize. The number of these scar states grows steadily as the system gets larger, so the count of exceptions is unbounded. Yet they make up only a tiny sliver of an exponentially large space of states, so their fraction still vanishes. In this one concrete, physically realized system the two notions genuinely disagree. A control chain that does thermalize shows no such runaway growth. The paper also checks that the flagged states are real low-entanglement outliers and not an artifact of the data processing: every flag lands on the low-entropy side, and the finding holds up as the tolerance and the method of estimating the thermal trend are varied. Counting exceptions is what tells ordinary chaotic systems apart from the special ones that break thermalization. By finding a real model where the two ways of tallying exceptions come apart, the paper shows that the harder, count-based question is the one worth answering. It is a deliberate step in the program, meant to secure the foundation the stronger thermalization claims will stand on, and it does not present itself as a new physical discovery.
Saying that a bounded number of exceptions is a stronger requirement than a vanishing fraction of exceptions only means something if the two can actually come apart. This short paper proves they can. It exhibits a real physical system that passes the fraction test but fails the count test, which shows that the sharper counting-based notion carries real content rather than being a technicality. The example is a well-known constrained quantum chain, the Rydberg-blockade or PXP model, famous for holding an exact ladder of special scar states that refuse to thermalize. The number of these scar states grows steadily as the system gets larger, so the count of exceptions is unbounded. Yet they make up only a tiny sliver of an exponentially large space of states, so their fraction still vanishes. In this one concrete, physically realized system the two notions genuinely disagree. A control chain that does thermalize shows no such runaway growth. The paper also checks that the flagged states are real low-entanglement outliers and not an artifact of the data processing: every flag lands on the low-entropy side, and the finding holds up as the tolerance and the method of estimating the thermal trend are varied. Counting exceptions is what tells ordinary chaotic systems apart from the special ones that break thermalization. By finding a real model where the two ways of tallying exceptions come apart, the paper shows that the harder, count-based question is the one worth answering. It is a deliberate step in the program, meant to secure the foundation the stronger thermalization claims will stand on, and it does not present itself as a new physical discovery.
The thermalization claim comes in two strengths. The weak version says almost every energy state looks thermal, apart from a vanishing fraction. The strong version forbids exceptions entirely. What sits between them is the rare, misbehaving states, and controlling them is a central open problem. This paper takes one honest step across that gap, sharpening a statement about the vanishing fraction of exceptions into a statement about their vanishing number. The upgrade turns on a single contest between two rates. One is how fast the fluctuations that mark a state as athermal die away as the system grows. The other is how fast the number of states in the energy window multiplies. If the first outpaces the second, the actual count of exceptions cannot keep growing, and the stronger statement follows. Working inside one clean symmetry sector of the benchmark chain, and simulating systems up to eighteen particles, the paper measures the decay rate directly and finds the contest is won with room to spare. That margin holds up even at the pessimistic edge of the statistical error bars, and even after the growth rate is pushed to its largest possible value. A short chain of finite algebraic lemmas reduces the controlling inequality to one standard statistical assumption about the size of fourth-order fluctuations. Each lemma is discharged as a machine-checked certificate showing there is no way for it to fail. The hard goal of controlling every state moves one concrete rung closer once you learn to count the exceptions rather than only weigh them. The paper is careful about what remains. The result holds at a fixed tolerance rather than the sharp thermodynamic one, and it still leans on a plausible but unproven assumption about the tails of the fluctuations, whose proof from first principles is left open. This is a waypoint, not a finish line.
Isolated quantum systems never really relax. Their exact evolution keeps perfect memory, and yet the everyday quantities we measure in them do settle to steady thermal values. The leading explanation locates this puzzle inside the system's individual energy states, but it is still largely a well-supported hypothesis whose rigorous status depends heavily on what exactly one asks. Instead of chasing a new phenomenon, this paper takes one thoroughly studied benchmark chain and gathers what is genuinely known about it into a single transparent account, with every step labelled as proved, computed, or taken from the literature. The spin chain it works with is a standard one, chosen because it behaves like a generic chaotic system, and the case is built in layers. First the paper confirms by computer that the chain really is generic: its energy levels follow the universal statistics of chaotic systems, and it hides no extra conserved quantities or other structure that would spoil thermalization. On that basis it proves a weak thermalization theorem for the chain's average magnetization, showing that all but a vanishing fraction of energy states already carry the thermal value. This is done by taking established general theorems and instantiating them for the concrete model with every constant written out. The paper then looks at the tiny residual fluctuations between states and finds they carry a distinctive statistical fingerprint predicted by a modern branch of mathematics called free probability. It calibrates the whole picture against a specially solvable version of the model in which certain quantities can be computed exactly. Its genuinely new technical core is a precise algebraic identity that governs those fluctuations. The value here is discipline rather than spectacle. By taking a single well-worn model and saying which parts of its thermal behaviour are proved, which are seen only in simulation, and which are imported wholesale, the paper converts a diffuse body of belief into one honestly bookkept account. It is open about the limits. It establishes the weak form of thermalization but does not claim the stronger, exception-free version, and it does not claim a sharp bound on the fluctuation structure. Leaving those openly unfinished is part of what makes the claims it does make trustworthy.
One pillar of modern physics is the idea that a sufficiently chaotic isolated quantum system looks thermal even when you examine it one energy state at a time. A sharper version asks about entanglement: does the entanglement between two halves of the system settle to a single predictable value as the system grows, instead of jumping around from state to state? For a genuinely large piece of the system this has resisted proof, because the obvious approach dissolves into an astronomically large sum of contributions. This paper shows that the whole difficulty funnels down to one clean, standard quantity, which turns an unmanageable calculation into a single sharp question. An earlier companion paper had already reduced the problem to controlling a handful of statistical quantities. Here the paper takes the decisive further step and proves that all of those quantities are governed by the size of a single object: a covariance matrix that records how the readings of local measurements vary from one energy state to the next, and in particular by that matrix's largest mode. If the largest mode shrinks as the system grows, the entanglement provably settles down. If it does not, the settling fails. The same object does double duty, since it also delivers the standard thermalization behaviour for all local measurements and fixes the size of the tiny residual fluctuations. The negative side gets equal care, through a companion obstruction result: whenever the system carries a conserved quantity that is not tied to its energy, that one mode refuses to vanish and the entanglement cannot settle. That is what separates the special non-thermalizing systems from ordinary chaotic ones. A problem that looked hopelessly tangled comes down to a single number. Rather than track an exponential explosion of interacting pieces, one only has to ask whether one mode of one matrix fades away as the system grows, and the answer settles whether a chaotic quantum system smooths itself into thermal predictability or holds on to a memory it can never erase. The paper is careful about the boundary of the claim. The reduction and its mechanism are proved and confirmed, but the last step, for a full half of the system, still rests on this one clean and clearly isolated conjecture, which the paper does not claim to have settled.
Quantum scars are nonthermal eigenstates characterized by low entanglement entropy, initially detected in systems subject to nearest-neighbor Rydberg blockade, the so-called PXP model. While most of these special eigenstates elude an analytical description and seem to hybridize with nearby thermal eigenstates for large systems, some of them can be written as matrix product states with size-independent bond dimension. We study the response of these exact quantum scars to perturbations by analyzing the scaling of the fidelity susceptibility with system size. We find that some of them are anomalously stable at first order in perturbation theory, in sharp contrast to the eigenstate thermalization hypothesis. However, this stability seems to break down when all orders are taken into account. We further investigate models with larger blockade radius and find a set of exact quantum scars that we write down analytically and compare with the PXP exact eigenstates. We show that they exhibit the same robustness against perturbations at first order.
Quantum many-body scars break ergodicity and evade thermalization, resulting in sub-volume law entanglement entropy even with high energy density. While their quantum correlations and entanglement have been elaborated previously, their capacity in storing extractable energy, quantified by the notion ergotropy, remains an open question. Here we focus on the representative PXP model, and unveil the extensive ergotropy scaling of a family of states interpolating between quantum many-body scars and thermal states, the latter of which are known to be passive with vanishing ergotropy in the thermodynamic limit. A phenomenological relation between ergotropy and entanglement is uncovered, which generalizes the existing free fermion integrable results to an interacting scenario. The ergotropy in a dynamical protocol shows that a reset with a global uniform coherent rotation can inject extractable energy, as a proof of principle way to charge a quantum "battery". Our protocol is tailored for near term Rydberg neutral atoms array, while also being feasible for other quantum processors. Our results establish that quantum many-body scars, despite the tiny fraction of the Hilbert space they occupy, can be efficiently exploited for storing extractable energy, and "scarring" a many-body system as a promising route for engineering quantum many-body battery.
This thesis investigates the anomalous dynamics observed in a Rydberg atom experiment in the broadest sense: a strongly interacting spin chain, without integrability or disorder, exhibits coherent many-body revivals ensuing only from specific initial states. This non-thermal behavior is attributed to the presence of quantum many-body scars (QMBS), a novel class of ergodicity breaking phenomena. Unlike other non-thermalizing systems such as integrable models and many-body localization, the mechanism of ergodicity breaking in QMBS is quite subtle, relying on a (vanishingly) small subset of anomalous eigenstates, known as scar states: one could observe ergodicity breaking only if initial states have appreciable overlap with these scar states. In other words, systems hosting QMBS are indistinguishable from typical chaotic, thermalizing systems, except for scar states. While the theoretical understanding of QMBS has advanced through the investigation of various toy quantum models, ranging from the Affleck-Kennedy-Lieb-Tasaki (AKLT) model to the Hubbard model, these insights do not fully resolve the mysterious oscillations in the Rydberg experiment: crucially the wavefunction corresponding to the experiment, and its relation to many other toy models in the literature remain open questions. Furthermore, beyond mathematical curiocity, the observability of non-thermal dynamics suggests that QMBS may yield distinct physical effects. the nature of which remains to be determined. This dissertation addresses these issues by solving two concrete problems: (i) the mathematical characterization of scar states in the PXP model, the effective model for the experiment, and (ii) the identification of physical consequences of QMBS in both short- and long-time dynamics. In the first part, we clarify the algebraic structure underlying the scar states of the PXP model and derive analytic ansatz for these wavefunctions. The proposed forms make explicit the hidden algebraic structure of the PXP scars and demonstrate how this structure is shared with many other QMBS Hamiltonians, including the AKLT model. The second part investigates dynamical signatures and robustness. Given the fragility of many-body eigenstates to perturbations, identifying which scar characteristics survive realistic perturbations is crucial. In closed systems, we show that out-of-time-ordered correlators (OTOCs) and the associated Lyapunov exponent can distinguish QMBS from generic chaotic states, at least within a semiclassical regime. Specifically, the OTOC growth associated with scar states is parametrically slower and remains slow under small ``scar-breaking'' perturbations. For open systems governed by non-Hermitian dynamics, we introduce a novel non-equilibrium phase, the QMBS phase, where the long-time wavefunction is localized in a small corner of the Hilbert space. We further identify a first-order transition between the QMBS phase and a thermal phase, where the wavefunction becomes delocalized.