Loschmidt echo in many-body localized phases

The Loschmidt echo, defined as the overlap between quantum wave function evolved with different Hamiltonians, quantifies the sensitivity of quantum dynamics to perturbations and is often used as a probe of quantum chaos. In this work we consider the behavior of the Loschmidt echo in the many-body localized phase, which is characterized by emergent local integrals of motion and provides a generic example of nonergodic dynamics. We demonstrate that the fluctuations of the Loschmidt echo decay as a power law in time in the many-body localized phase, in contrast to the exponential decay in few-body ergodic systems. We consider the spin-echo generalization of the Loschmidt echo and argue that the corresponding correlation function saturates to a finite value in localized systems. Slow, power-law decay of fluctuations of such spin-echo-type overlap is related to the operator spreading and is present only in the many-body localized phase, but not in a noninteracting Anderson insulator. While most of the previously considered probes of dephasing dynamics could be understood by approximating physical spin operators with local integrals of motion, the Loschmidt echo and its generalizations crucially depend on the full expansion of the physical operators via local integrals of motion operators, as well as operators which flip local integrals of motion. Hence these probes allow one to get insights into the relation between physical operators and local integrals of motion and access the operator spreading in the many-body localized phase.

Figure 1

Cartoon of the setup implementing orthogonality catastrophe in the cold atom setting. The spin-1/2 impurity on the left is coupled to the spin chain via the Zeeman interaction and has no internal dynamics. If impurity spin is initialized along the $x$ direction, the expectation value of ${\sigma }_{\text{imp}}^x$ at time $t$ gives the real part of the overlap function.

Figure 2

Fluctuations of the overlap decay as a power law in time, saturating to the value that is exponentially suppressed with the system size. The averaging was performed for at least ${10}^3$ disorder realizations in an XXZ spin chain with disorder $W=6.5$ and interaction $J_z=1$.

Figure 3

Cyan and gray lines show the power-law decay of $\langle {\left|S\left(t\right)\right|}^2\rangle$ in Anderson insulator and MBL phase. Note the much faster overlap decay in the MBL phase. Moreover, the exponent of the decay is more sensitive to the increase in the value of disorder in the MBL phase. System has $L=14$ spins.

Figure 4

Fluctuations of impurity spin-echo signal decay in a similar way to the overlap (solid lines). Value of disorder is $W=5$, interaction $J_z=1$, and $g=2$.

Figure 5

Fluctuations of the spin-echo overlap do not relax in the noninteracting system at long times (solid curves), while presence of even small interactions ($J_z=0.01$, dotted lines) leads to a slow power-law-like decay and residual fluctuations that decrease exponentially with the system size. Increasing interaction strength to $J_z=1$ (solid lines) gives even faster decay of spin-echo overlap. Data is obtained for disorder $W=4$ and perturbation strength is $g=4$.

Figure 6

Absolute value of the averaged coherence does not depend on the system size and interaction strength, and has a weak dependence on the disorder strength (solid lines correspond to $W=6.5$ and dashed lines to $W=7.5$). The numerical data agrees reasonably well with the theory suggesting the log-normal distribution of the localization length.

Figure 7

Entanglement generated by $U_{\text{echo}}\left(t\right)$ across the middle link of the spin chain depends on the presence of interactions. When $J_z>0$ entanglement grows logarithmically and has extensive saturation value (solid lines). In contrast, in the noninteracting system saturation value of entanglement decreases with the system size (dashed lines). Disorder strength is $W=3$ and $g=4$.