Abstract
The theory of quantum scarring -- a remarkable violation of quantum unique ergodicity -- rests on two complementary pillars: the existence of unstable classical periodic orbits and the so-called quasimodes, i.e., the non-ergodic states that strongly overlap with a small number of the system's eigenstates. Recently, interest in quantum scars has been revived in a many-body setting of Rydberg atom chains. While previous theoretical works have identified periodic orbits for such systems using time-dependent variational principle (TDVP), the link between periodic orbits and quasimodes has been missing. Here we provide a conceptually simple analytic construction of quasimodes for the non-integrable Rydberg atom model, and prove that they arise from a "requantisation" of previously established periodic orbits when quantum fluctuations are restored to all orders. Our results shed light on the TDVP classical system simultaneously playing the role of both the mean-field approximation and the system's classical limit, thus allowing us to firm up the analogy between the eigenstate scarring in the Rydberg atom chains and the single-particle quantum systems.
Highlights
Quantum scars provide a surprising connection between single-particle quantum billiards and their classical counterpart [1,2]
We focus on the “PXP” model—an idealized effective model of the Rydberg atom experiment [50,51]—which can be formally expressed as a nonintegrable spin-1=2 lattice model without an “obvious” semiclassical limit
The link between the forward-scattering approximation (FSA) and the time-dependent variational principle (TDVP) is obscure, making it difficult to relate the quasimodes with the classical limit of the model, in a manner that had been achieved for single-particle scars
Summary
Quantum scars provide a surprising connection between single-particle quantum billiards and their classical counterpart [1,2]. Quasimodes for the PXP model have been independently constructed by the so-called forward-scattering approximation (FSA) [10,56] This method makes it possible to extract properties of scarred eigenstates in systems much larger than those accessible by exact diagonalization (ED). The link between the FSA and the TDVP is obscure, making it difficult to relate the quasimodes with the classical limit of the model, in a manner that had been achieved for single-particle scars. Our construction relies on building a subspace symmetrized over permutations within each of the sublattices comprising even and odd sites in a chain This approach can be thought of as a “mean-field” approximation which makes it possible to obtain closed-form expression for the projections of states and operators in this symmetric subspace and enables us to numerically approximate highly excited eigenstates of chains of length N ≲ 800, far beyond other methods. Performing time evolution in the symmetric subspace strongly suggests that the classical periodic orbit found in TDVP is stable to the addition of quantum fluctuations in the thermodynamic limit, at least to the mean-field level as described above
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