Interferometric Probes of Many-Body Localization

We propose a method for detecting many-body localization (MBL) in disordered spin systems. The method involves pulsed coherent spin manipulations that probe the dephasing of a given spin due to its entanglement with a set of distant spins. It allows one to distinguish the MBL phase from a noninteracting localized phase and a delocalized phase. In particular, we show that for a properly chosen pulse sequence the MBL phase exhibits a characteristic power-law decay reflecting its slow growth of entanglement. We find that this power-law decay is robust with respect to thermal and disorder averaging, provide numerical simulations supporting our results, and discuss possible experimental realizations in solid-state and cold-atom systems.

Figure 1

Schematic illustration of the proposed protocols. (a) Spins are manipulated with lasers in two spatially separated regions I and II. (b) A Hahn spin-echo sequence is applied to region I while leaving region II untouched. (c) The DEER protocol differs by $\pi /2$ rotations in region II which are performed after half of the evolution time. Both protocols work for arbitrary initial states (product states, thermal state, etc.). (d) Schematic response of a system in the diffusive (left), noninteracting localized (center), and many-body localized (right) phases, to spin-echo and DEER protocols, respectively. The combined information from both sequences allows one to distinguish the different phases.

Figure 2

(a) Typical behavior of thermally averaged spin-echo $\mathcal{F}(t)$ and DEER response $\mathcal{D}(t)$ for the random-field $XXZ$ model [Eq. (6)] and a single disorder realization ($W=6$). $\mathcal{D}(t)$ slowly decays, while $\mathcal{F}(t)$ quickly saturates. (b) Disorder-averaged $\mathcal{F}(t)$ saturates to a nonzero value in thermodynamic limit, and approaches 1 for strong disorder $W$.

Figure 3

Thermally averaged DEER response at weak interactions ($J_z=0.1J_{\perp }$) and moderate interactions ($J_z=J_{\perp }$) in the $XXZ$ model. In both cases, we show two particular disorder realizations. Despite strong sample-to-sample fluctuations, the response is consistent with Eq. (5).

Figure 4

(a) DEER response, after thermal and disorder averaging, for different values of the interaction $J_z$ and separations $d$ between spin I and region II. Power-law decay of the response, spanning multiple decades, can be seen in all cases. (b) Exponent of the power law, $\alpha$, scales as $\alpha =c_1/\ln (c_{2}W)$ with disorder $W$, consistent with $\alpha =\xi \ln 2$ and the scaling of the localization length, $\xi \sim 1/\ln (W)$, at strong disorder. (c) Saturated DEER response as a function of size of region II (denoted $N$) decreases as $c/1.8^N$, and is insensitive to interaction $J_z$.