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Escape the thermal fate

Nature Physicsvolume 14pages637638 (2018) | Download Citation

Many-body quantum systems fail to reach thermalization only under specific circumstances. An analysis now reveals a new, different kind of non-equilibrating dynamics based on the many-body analogue of quantum scars in single-particle quantum chaos.

The state of thermal equilibrium is the destiny of any sufficiently large isolated system after a sufficiently long time. Until now we knew only one way to avoid this fate: the presence of a set of independent conserved quantities, where the size of the set grows proportionally to the size of the system. This is the case for integrable systems, or if quenched disorder gives rise to many-body localization1. Writing in Nature Physics, Christopher J. Turner and co-workers2 have now introduced a substantially different way to avoid thermalization: owing to the presence of a limited number of special, atypical energy eigenstates, a system can exhibit persistent oscillations of observable quantities.

To understand what happens, let us review how systems thermalize. The key is the eigenstate thermalization hypothesis (ETH)3,4. Despite its name, related to historical reasons, the ETH is now believed to underlie thermalization in essentially every isolated quantum system that thermalizes1.

In any isolated quantum system, the time-dependent expectation value of the operator representing a certain observable can be written down as a sum of two terms. The first one is time-independent, and is given by the sum over all the diagonal elements of the operator in the basis formed by the energy eigenstates of the system, weighted by the occupation numbers — the modulus squared of the overlap between the system’s initial state and each energy eigenstate. All the off-diagonal elements of the operator are included in a second term, whose individual sub-terms oscillate in time. Now, if an equilibrium state — thermal or not1,5 — exists, the equilibrium value of the observable must be equal to the infinite time average of its time-dependent expectation value. This corresponds to the diagonal part alone, because the infinite-time average of each oscillating sub-term in the off-diagonal part is zero. This outcome is sometimes called the prediction of the diagonal ensemble.

On the other hand, according to statistical mechanics, the value of an observable at thermal equilibrium should follow the prediction of the microcanonical ensemble. This is given by an average over the same diagonal matrix elements as above, but without any weights and with the average running only over eigenstates that are very close in energy to the mean energy of the initial state of the system. And now, a conundrum: how can the predictions of the microcanonical and diagonal ensembles agree, when the former does not depend on the occupation numbers but the latter does? The only way they can agree for all initial states that are narrow in energy — a condition that is generically valid1 — is if the diagonal matrix elements have the same value and factor out of the sum in any energy window whose width is comparable to the energy width of the initial state. This is the core idea of the ETH.

It is at this point where systems with many additional conserved quantities behave differently: to obtain diagonal elements that are the same, one must consider not only an energy-narrow window, but additional windows that are narrow in all the conserved quantities. This is the microcanonical version of the generalized Gibbs ensemble1.

Furthermore, to have equilibration, the off-diagonal matrix elements should be exponentially small in the size of the system, which ensures that the expectation value will be close to the diagonal ensemble’s prediction for an overwhelming majority of the time.

When the ETH first became widely known, some researchers cautioned that even if most eigenstates of a generic system satisfy the ETH, embedded among them could be rare states whose diagonal matrix elements might be outliers6,7. The consensus nowadays is that this does not generically happen. Turner and colleagues now offer a vindication of this scenario: a system where rare eigenstates prevent thermalization of particular initial states. Additionally, this system does not even equilibrate, which means that some of the off-diagonal matrix elements are anomalously large, as well as some occupation numbers — leading to a violation of Riemann’s equilibration conditions5. Remarkably, though, this is an experimentally relevant scenario: indeed, explaining the recently observed persistent oscillations in a 51-Rydberg-atom quantum simulator8 was a motivation for the authors’ work.

Intriguingly, the authors suggested that these rare and special states, whose number grows only linearly with the system size, are many-body analogues of quantum scars in single-particle quantum chaos (Fig. 1). The argument goes as follows. The system considered by the authors is mostly chaotic, because the energy-level spacing approaches the Wigner–Dyson distribution, and most eigenstatesare are ergodically spread out over the Hilbert space, as measured by the participation ratio. However, the participation ratio of the special states is larger, meaning that they are more concentrated in certain parts of the Hilbert space, similarly to how quantum scars are more concentrated in certain parts of the real space. Just as quantum scars concentrate around classical periodic orbits, special states are found concentrated around certain product states. This gives rise, in an effective tight-binding model, to periodic motion connecting two product states.

Fig. 1: Quantum scars.
Fig. 1

A typical scarred state of the Bunimovich stadium quantum billiard. The scar is the region of high amplitude, which arises due to a closed classical orbit.

The effect on the full system is persistent oscillations at the experimentally observed frequency in the time evolution of a particular initial state, labelled \(\left|{{\mathbb{Z}}}_{2}\right\rangle\), which has high overlaps with the special states. Many-body localization can be ruled out, since the system is translationally invariant, is not disordered, and shows ballistic propagation of entanglement. Turner et al. suggested that what is behind the existence of special states is a dynamical constraint, as their system —a spin-1/2 chain — does not allow relaxation of several adjacent up-spins. Indeed, their \(\left|{{\mathbb{Z}}}_{2}\right\rangle\) state is the alternated up–down spins forming a charge-density wave.

Numerous further questions suggest themselves. The relevant concept of many-body periodic trajectory — namely the analogy with the usual quantum scars — must be generalized and made more rigorous to enable us to understand and classify all initial states that result in persistent oscillations. We should also try to better understand what class of system exhibits this new kind of dynamics. On the basis of these results, one can eventually envision engineering systems that have states with very long coherence times, to be used in quantum information processing.


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  1. Department of Physics, University of Massachusetts Boston, Boston, MA, USA

    • Vanja Dunjko
    •  & Maxim Olshanii


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Correspondence to Vanja Dunjko or Maxim Olshanii.

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