Operator spreading in quantum maps

Abstract

Operators in ergodic spin-chains are found to grow according to hydrodynamical equations of motion. The study of such operator spreading has aided our understanding of many-body quantum chaos in spin-chains. Here we initiate the study of "operator spreading" in quantum maps on a torus, systems which do not have a tensor-product Hilbert space or a notion of spatial locality. Using the perturbed Arnold cat map as an example, we analytically compare and contrast the evolutions of functions on classical phase space and quantum operator evolutions, and identify distinct timescales that characterize the dynamics of operators in quantum chaotic maps. Until an Ehrenfest time, the quantum system exhibits classical chaos, i.e. it mimics the behavior of the corresponding classical system. After an operator scrambling time, the operator looks "random" in the initial basis, a characteristic feature of quantum chaos. These timescales can be related to the quasi-energy spectrum of the unitary via the spectral form factor. Furthermore, we show examples of "emergent classicality" in quantum problems far away from the classical limit. Finally, we study operator evolution in non-chaotic and mixed quantum maps using the Chirikov standard map as an example.arXiv:1808.04889v2 [cond-mat.stat-mech]

Signature of quantum chaos in operator entanglement in 2d CFTs

“…• While we have studied the operator entanglement of unitary time-evolution operators, it would be also interesting to study the operator entanglement of local operators [30,72,73]. In particular, the expectation values of local operators play an order-parameterlike role in eigenstate thermalizaion hypothesis (ETH).…”

Out-of-time-ordered correlator in the quantum bakers map and truncated unitary matrices

“…We now turn to the central quantity f (t) which measures the noncommutativity of P (0) and P (t) as in Eq. (13). Figure (4) shows the growth of f (t) for two different values of N and three different position space projectors P (0), one is the L partition that includes the origin which is a fixed point in the classical limit j min = 0 and j max = N/2 − 1, one that excludes the origin but is still in the L partition with j min = [N/10], j max = [4N/10], and a third one that is in L but does not include either the origin which is a fixed point or the period-2 orbit at (1/3, 2/3).…”

“…These pose significant challenges and have given us various insights, including semiclassical periodic orbit theory and the relevance of random matrix ensembles for even one-particle systems whose classical limit is chaotic and surveys collected in [5] form an excellent introduction. A resurgence of interest in quantum chaotic or nonintegrable systems has occurred around the related themes of scrambling and out-oftime-ordered correlators or OTOC [6][7][8][9][10][11][12][13]. The OTOC, as commonly defined, is connected to the development of non-commutativity of initially commuting operators [14] and therefore this forms a convenient starting point.…”

“…These include quantum maps [18,19], which are Floquet systems or quantizations of finite canonical transformations which have also been invoked in recent studies of OTOC. For example, the standard map or kicked rotor has been studied in [6] while the quantum cat map and its perturbed versions have been studied in the context of operator spreading and OTOC [9,13]. The classical cat map, introduced by Arnold and Avez [20], is a smooth linear area-preserving map of the two-torus into itself.…”

Exact eigenstates in the Lesanovsky model, proximity to integrability and the PXP model, and approximate scar states

“…Many-body localized systems [14][15][16] strongly violate the ETH. However, the PXP model belongs to a group of systems [17][18][19][20][21][22][23][24][25][26][27] where the weak ETH holds; that is, the ETH holds for almost every eigenstate.…”