Absence of exponential sensitivity to small perturbations in nonintegrable systems of spins 1/2

Abstract

We show that macroscopic nonintegrable lattices of spins 1/2, which are often considered to be chaotic, do not exhibit the basic property of classical chaotic systems, namely, exponential sensitivity to small perturbations. We compare chaotic lattices of classical spins and nonintegrable lattices of spins 1/2 in terms of their magnetization responses to an imperfect reversal of spin dynamics known as Loschmidt echo. In the classical case, magnetization is exponentially sensitive to small perturbations with a characteristic exponent equal to twice the value of the largest Lyapunov exponent of the system. In the case of spins 1/2, magnetization is only power-law sensitive to small perturbations.

Thermodynamics of quantum information scrambling

“…This reflects the irreversible information scrambling occurring in the non-integrable case. Note also that, despite non-integrability, no exponential behavior is observed in the time dependence of the OTOC, this is in agreement with previous numerical [38] and experimental findings [20]. The plots reveal a small mean work as compared to the second moment w 2 w 2 and a proportionality between them in accordance with linear response theory w β( w 2 − w 2 )/2 β w 2 /2 [39,40].…”

Estimating ergodization time of a chaotic many-particle system from a time reversal of equilibrium noise

“…However, such an approach is impractical experimentally, because it requires tracking all phase-space coordinates of the system. An alternative, more practical approach was proposed in [16,31]. That approach is based on monitoring the effect of Loschmidt echo on equilibrium noise of almost any observable (see appendix A).…”

“…Various aspects of this work are relevant to the previous investigations of lattice gauge models [9][10][11][12][13] and spin lattice models [14][15][16][17]. We also note that our method involves the classical counterpart of out-of-timeorder quantum correlators (OTOCs) [18] that have been actively investigated in recent years in the context of quantum thermalization [19][20][21][22][23] and many-body localization problems [24][25][26][27][28][29].…”

Reversibility of dynamics and multiple-quantum coherences

“…A direct diagonalization of the Hamiltonian can be applied to N ~ 15 spins ½ [34]. In the simulation [9] for N = 25 spins ½, an estimation of traces [35] has been used. Even though the number of quantum energy levels is huge in such clusters, the systems are still too small to adequately reproduce irreversibility of either the Loschmidt or the partial echo.…”

“…In a recent simulation [9] a comparison has been made between the systems of "classical spins" and spins ½ in the Loschmidt echo experiment. While the classical system demonstrated an exponential growth of δM, consistent with the estimated value of the Lyapunov exponent, the quantum system (5 x 5 lattice of spins ½ with periodic boundary conditions) showed strongly non-exponential behavior.…”