Solution of a minimal model for many-body quantum chaos

We solve a minimal model for quantum chaos in a spatially extended many-body system. It consists of a chain of sites with nearest-neighbour coupling under Floquet time evolution. Quantum states at each site span a $q$-dimensional Hilbert space and time evolution for a pair of sites is generated by a $q^2\times q^2$ random unitary matrix. The Floquet operator is specified by a quantum circuit of depth two, in which each site is coupled to its neighbour on one side during the first half of the evolution period, and to its neighbour on the other side during the second half of the period. We show how dynamical behaviour averaged over realisations of the random matrices can be evaluated using diagrammatic techniques, and how this approach leads to exact expressions in the large-$q$ limit. We give results for the spectral form factor, relaxation of local observables, bipartite entanglement growth and operator spreading.Comment: Accepted in PR

Classification of symmetry-protected topological many-body localized phases in one dimension.

We provide a classification of symmetry-protected topological (SPT) phases of many-body localized (MBL) spin and fermionic systems in one dimension. For spin systems, using tensor networks we show that all eigenstates of these phases have the same topological index as defined for SPT ground states. For unitary on-site symmetries, the MBL phases are thus labeled by the elements of the second cohomology group of the symmetry group. A similar classification is obtained for anti-unitary on-site symmetries, time-reversal symmetry being a special case with a [Formula: see text] classification (see [Wahl 2018 Phys. Rev. B 98 054204]). For the classification of fermionic MBL phases, we propose a fermionic tensor network diagrammatic formulation. We find that fermionic MBL systems with an (anti-)unitary symmetry are classified by the elements of the (generalized) second cohomology group if parity is included into the symmetry group. However, our approach misses a [Formula: see text] topological index expected from the classification of fermionic SPT ground states. Finally, we show that all found phases are stable to arbitrary symmetry-preserving local perturbations. Conversely, different topological phases must be separated by a transition marked by delocalized eigenstates. Finally, we demonstrate that the classification of spin systems is complete in the sense that there cannot be any additional topological indices pertaining to the properties of individual eigenstates, but there can be additional topological indices that further classify Hamiltonians

Out-of-time-order correlator, many-body quantum chaos, light-like generators, and singular values

We study out-of-time-order correlators (OTOCs) of local operators in spatial-temporal invariant or random quantum circuits using light-like generators (LLG) -- many-body operators that exist in and act along the light-like directions. We demonstrate that the OTOC can be approximated by the leading singular value of the LLG, which, for the case of generic many-body chaotic circuits, is increasingly accurate as the size of the LLG, $w$, increases. We analytically show that the OTOC has a decay with a universal form in the light-like direction near the causal light cone, as dictated by the sub-leading eigenvalues of LLG, $z_2$, and their degeneracies. Further, we analytically derive and numerically verify that the sub-leading eigenvalues of LLG of any size can be accessibly extracted from those of LLG of the smallest size, i.e., $z_2(w)= z_2(w=1)$. Using symmetries and recursive structures of LLG, we propose two conjectures on the universal aspects of generic many-body quantum chaotic circuits, one on the algebraic degeneracy of eigenvalues of LLG, and another on the geometric degeneracy of the sub-leading eigenvalues of LLG. As corollaries of the conjectures, we analytically derive the asymptotic form of the leading singular state, which in turn allows us to postulate and efficiently compute a product-state variational ansatz away from the asymptotic limit. We numerically test the claims with four generic circuit models of many-body quantum chaos, and contrast these statements against the cases of a dual unitary system and an integrable system.Comment: 6 + 15 pages, 3 + 11 figures. Comments are welcome. Updated on 2023-10-1

Many-body quantum chaos and emergence of Ginibre ensemble

We show that non-Hermitian Ginibre random matrix behaviors emerge in spatially-extended many-body quantum chaotic systems in the space direction, just as Hermitian random matrix behaviors emerge in chaotic systems in the time direction. Starting with translational invariant models, which can be associated with dual transfer matrices with complex-valued spectra, we show that the linear ramp of the spectral form factor necessitates that the dual spectra have non-trivial correlations, which in fact fall under the universality class of the Ginibre ensemble, demonstrated by computing the level spacing distribution and the dissipative spectral form factor. As a result of this connection, the exact spectral form factor for the Ginibre ensemble can be used to universally describe the spectral form factor for translational invariant many-body quantum chaotic systems in the scaling limit where $t$ and $L$ are large, while the ratio between $L$ and $L_{\mathrm{Th}}$, the many-body Thouless length is fixed. With appropriate variations of Ginibre models, we analytically demonstrate that our claim generalizes to models without translational invariance as well. The emergence of the Ginibre ensemble is a genuine consequence of the strongly interacting and spatially extended nature of the quantum chaotic systems we consider, unlike the traditional emergence of Hermitian random matrix ensembles.Comment: 7+16 pages, 2+12 figure

Many-body quantum chaos in stroboscopically-driven cold atoms

Seeking signatures of quantum chaos in experimentally realizable many-body systems is of vigorous interest. In such systems, the spectral form factor (SFF), defined as the Fourier transform of two-level spectral correlation function, is known to exhibit random matrix theory (RMT) behaviors, namely a 'ramp' followed by a 'plateau' in sufficiently late time. Recently, a generic early-time deviation from the RMT behavior, which we call the 'bump', has been shown to exist in random quantum circuits and spin chains as toy models for many-body quantum chaotic systems. Here we demonstrate the existence of the 'bump-ramp-plateau' behavior in the SFF for a number of paradigmatic, stroboscopically-driven cold atom models of interacting bosons in optical lattices and spinor condensates. We find that the scaling of the many-body Thouless time $t_{\text{Th}}$ -- the time of the onset of the (RMT) ramp behavior -- and the increase of the bump amplitude in atom number are significantly slower in (effectively 0D) chaotic spinor gases than in 1D optical lattices, demonstrating the role of locality in many-body quantum chaos. Moreover, $t_{\text{Th}}$ scaling and the bump amplitude are more sensitive to variations in atom number than the system size regardless of the hyperfine structure, the symmetry classes, or the choice of the driving protocol. We obtain scaling functions of SFF which suggest power-law behavior for the bump regime in quantum chaotic cold-atom systems. Finally, we propose an interference measurement protocol to probe SFF in the laboratory.Comment: 10 pages, 7 figures, supplementary materia