Entanglement entropy and out-of-time-order correlator in the long-range Aubry–André–Harper model

Abstract

We investigate the nonequilibrium dynamics of entanglement entropy and out-of-time-order correlator (OTOC) of noninteracting fermions at half-filling starting from a product state to distinguish the delocalized, multifractal (in the limit of nearest neighbor hopping), localized and mixed phases hosted by the quasiperiodic Aubry-André-Harper (AAH) model in the presence of long-range hopping. For sufficiently long-range hopping strength a secondary logarithmic behavior in the entanglement entropy is found in the mixed phases whereas the primary behavior is a power-law the exponent of which is different in different phases. The saturation value of entanglement entropy in the delocalized, multifractal and mixed phases depends linearly on system size whereas in the localized phase (in the short-range regime) it is independent of system size. The early-time growth of OTOC shows very different power-law behaviors in the presence of nearest neighbor hopping and long-range hopping. The late time decay of OTOC leads to noticeably different power-law exponents in different phases. The spatial profile of OTOC and its system-size dependence also provide distinct features to distinguish phases. In the mixed phases the spatial profile of OTOC shows two different dependences on space for small and large distances respectively. Interestingly the spatial profile contains large fluctuations at the special locations related to the quasiperiodicity parameter in the presence of multifractal states.

Characterizing entanglement in non-Hermitian chaotic systems via out-of-time ordered correlators

“…1]. Thus, the above findings again prove that the OTOCs are closely relevant to entanglement, [9][10][11][12][58][59][60][61][62][63][64][65][66] even in the non-Hermitian case.…”

Evolution of circuit complexity in a harmonic chain under multiple quenches

“…Now it can be seen from the formulas for these functions that for these to be zero, the auxiliary functions and their derivatives must satisfy the conditions in equation ( 5), but at a later time, t * > 0. From the expressions for these functions in equation (11), we see that, though the above mentioned conditions are satisfied by b j s for all values j, the numbers t * are different for all the modes. Therefore, the initial conditions are not satisfied at any later times by all b j s, and hence, the complexity does not show exact revivals.…”

“…For our case, since after the quench at t = 0 the frequency λ j takes a constant value, the solution for the auxiliary equation can be written with y = λ j t as b j (t) = A j cos 2 y + 2B j sin y cos y + C j sin 2 y = α j cos(2y) + β j sin(2y) + γ j . (11) The relation between the two sets of constants can be worked out easily. Now, using the initial conditions and the relation of equation (10) between the three constants, we can obtain these constants to be…”

“…The canonical prescription to see the effect of scrambling, namely the out of time ordered correlator (OTOC) has been studied in a variety of systems mainly by using numerical methods [7][8][9][10][11][12][13]. Even though the general consensus is that the saturation of OTOC at late times indicate scrambling of information in chaotic systems, there are also results in the literature those point to the contrary.…”

Late time dynamics in SUSY saddle-dominated scrambling through higher-point OTOC

“…In recent times, the out-of-time-order correlator (OTOC) has been extensively used as a probe for operator scrambling in various branches of physics such as condensed matter [1][2][3][4][5], holographic theories [6][7][8][9][10], random matrix theory [11][12][13], and quantum computation [14][15][16][17] etc. The OTOC acts as an indicator of chaotic dynamics in a system.…”