Unusual quantum many-body scars

Graphical abstract

In undergraduate statistical mechanics, we have learned that thermalization is the process of physical bodies reaching thermal equilibrium through mutual interaction. In general, the natural tendency of a system is towards a state of equipartition of energy and uniform temperature that maximizes the system’s entropy. The thermalization process erases the local memory of the initial conditions. One can then naturally ask: How does an isolated quantum many-body system under unitary Hamiltonian evolution reach equilibrium? This question has attracted tremendous attention in recent years [1] not only because of its fundamental importance, but also because of the fact that it can be experimentally investigated with synthetic quantum systems based on atoms, ions, or superconducting circuits [2, 3]. A powerful conjecture explaining the process of quantum thermalization is eigenstate thermalization hypothesis (ETH) [4, 5] which states that individual eigenstates of quantum systems act as thermal ensembles and thus the system’s relaxation does not depend strongly on the initial condition. Systems exhibiting ETH are called quantum ergotic where parts of the system act as heat reservoirs for the other parts.

It is well understood that ETH are strongly violated in quantum integrable systems [6, 7] and in disordered interacting systems that exhibit many-body localization [1, 8], both of which possess an extensive number of conservation laws which prevent thermalization. It came as a surprise when a 2017 experiment [9] on a Rydberg-atom quantum simulator, a system that is disorder-free and not quantum integrable, demonstrated that whether the system thermalizes or not depends on the choice of initial states: some initial states show thermalization expected in an ergotic system, while others do not. The puzzle posed by this experiment has been addressed by a series of subsequent theoretical works. The key to understanding this behavior is the existence of a set of non-thermal eigenstates in the highly excited energy spectrum [10, 11]. Initial states with significant weight on these non-thermal eigenstates, which only comprise a small fraction of the Hilbert space, will exhibit non-thermal behavior. By analogy with chaotic stadium billiards, which also host non-thermal eigenstates visualized as “scars” of classical unstable periodic orbits [12], the non-thermal eigenstates in the Rydberg system have been called “quantum many-body scars” (QMBS), and they represent a weak violation of the ETH.

Over the past few years, great efforts have been made to understand the phenomenology of quantum scarring and to construct Hamiltonians that support QMBS [13]. A common theme of such weak ergodicity-breaking systems is that the Hilbert space is (approximately) fragmented in the sense that it breaks into a small scarred part and a large thermal part, and the two parts are either uncoupled or only weakly coupled, as schematically shown in Fig.1. In the work by Zhang et al. [14], such a Hamiltonian is constructed, but with an unusual twist.

Zhang et al. [14] considered an ensemble of interacting spinless fermions in a one-dimensional tilted lattice under periodic driving which renders the hopping amplitude between near-neighbor sites a periodic function of time with period T. The total number of fermions N=L/2 where L is the number of lattice sites, and open boundary condition is imposed. Using a standard Floquet perturbation technique [15, 16], they derived the time-independent first-order effective Floquet Hamiltonian. The effective Hamiltonian is manifestly disorder-free, and its non-integrability is proven by analyzing the energy spectrum. Furthermore, it possesses an emerging chiral symmetry which breaks the Hilbert space into a chiral symmetric and a chiral antisymmetric subspaces. This has two important effects on the system: (i) Tunneling is only allowed between states with opposite chirality which makes the lattice system bipartite. (ii) Such a bipartite system in general possesses a zero-energy manifold, and the number of zero-energy eigenstates is bound from below by the difference between the dimensions of the two chiral subspaces [17]. Interestingly, they proved that this difference is zero if N is odd, and exponentially large (more specifically, =N!/[(N/2) !(N/2)!]) if N is even, which leads to an even-odd effect.

Upon examination of various types of tunneling process, they identified the scarred subspace which is spanned by a tower of states that are weakly coupled to states outside of this tower, and the domain wall state

| \mathrm{t}\mathrm{p}⟩=|( 1...1 {)}^N(\mathrm{0...0}{)}^N⟩,

where all the fermions are located in the left half of the lattice, belongs in the scarred subspace. Evolution dynamics starting from |\mathrm{t}\mathrm{p}⟩ clearly shows the scarred behavior, with some important difference depending on the odevity of N which stems from the even-odd effect mentioned above. In particular, for even N, some unusual scarred dynamics is revealed. For example, the fidelity F(k)=|⟨\psi (0)|\psi (kT )⟩{|}^2, where |\psi (0) ⟩=|\mathrm{t}\mathrm{p}⟩ is the initial state and |\psi (kT)⟩ the state at time kT (where k is an integer) evolved under the effective Floquet Hamiltonian, oscillates around a finite value without decay. In contrast, F(k) for odd N exhibits decaying collapse and revival which is typical for quantum scarred systems. This can be traced to the fact that a large number of zero-energy states exist for even N, and | \mathrm{t}\mathrm{p}⟩ has a significant overlap with these states. After projecting | \mathrm{t}\mathrm{p}⟩ onto the zero-energy subspace, it is shown that this projected zero-energy state represents a non-thermal QMBS state that is strongly localized in the Hilbert space. By contrast, for odd N, the scarred eigenstates themselves are thermal but can give rise to non-thermal dynamics for certain initial states such as | \mathrm{t}\mathrm{p}⟩. The unsusal scarred dynamical behaviour is also displayed by the local density in the original driven system, which has potential interest for experiments.

The construction of the Hamiltonian in Ref. [14] is quite general: it relies on the bipartite and chiral nature of the system. In the future, it will be interesting to see whether this can be applied to other lattice models, including models in high dimensions.

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