Experimental probes of Stark many-body localization

Recent work has focused on exploring many-body localization (MBL) in systems without quenched disorder: one such proposal is Stark MBL in which small perturbations to a strong linear potential yield localization. However, as with conventional MBL, it is challenging to experimentally distinguish between noninteracting localization and true MBL. In this paper, we show that several existing experimental probes, designed specifically to differentiate between these scenarios, work similarly in the Stark MBL setting. In particular, we show that a modified spin-echo response shows clear signs of a power-law decay for Stark MBL while quickly saturating for disorder-free Wannier-Stark localization. Furthermore, we observe the characteristic logarithmic-in-time spreading of quantum mutual information in the Stark MBL regime, and an absence of spreading in a noninteracting Stark-localized system. We also show that there are no significant differences in several existing MBL measures for a system consisting of soft-core bosons with repulsive on-site interactions. Lastly, we discuss why curvature or small disorder are needed for an accurate reproduction of MBL phenomenology and how this may be illustrated in experiment. This also connects with recent progress on Hilbert space fragmentation in “fractonic” models with a conserved dipole moment, and we suggest this as an auspicious platform for experimental investigations of these phenomena.

Figure 1

Top panel: Sketch of the system with a linear field ($\propto \gamma$) and quadratic contribution ($\propto \alpha$). Middle panel: Characteristic behavior of the spin-echo and DEER responses for systems in the thermal (left), interacting Stark MBL (middle), and noninteracting Wannier-Stark (right) regimes. Bottom panel: Characteristic behavior of the eigenstate QMI in the same regimes where the magnitude of the normalized QMI between two sites is indicated by the thickness of the lines between them and their color (red being strong and blue being weak).

Figure 2

The short- to medium-time behavior of the thermally averaged DEER response $[\langle \mathcal{D}(t)\rangle ]$ in the Stark MBL, noninteracting Stark, and thermalizing diffusive regimes. We see a clear difference among the three regimes and observe (see the black Stark localization curve) that the DEER response is able to distinguish interacting and noninteracting localization, which a spin-echo protocol would not be able to do (see Fig. 1). The parameters are $L=12,M=50$ with a potential of $\gamma =3,\alpha =2$. The thermal system is exposed to uniformly distributed on-site disorder with half-width 1 rather than a linear potential with curvature. For all cases, $d=3$ and $N=7$.

Figure 3

Logarithmic growth of $\langle X^2\left(t\right)\rangle$ for a system with $\gamma =2,\alpha =2$, and $\mathrm{\Delta }=0.1$ with a noninteracting example for comparison. The results are averaged over at least 500 initial states, except for $L=10$ which is averaged over every Fock state in the magnetization sector.

Figure 4

Top: the $r$ parameter for a system of interacting bosons. Bottom: decay of charge-density wave order. Both measures show signatures of Stark MBL. All data are for a system of $N_b=6$ bosons on a lattice with $L=12$ sites with interaction strength $U=1$ and curvature $\alpha =2$.

Figure 5

Upper panel: the half-chain entanglement $S_{L/2}$ as a function of energy for the eigenstates of a Stark-localized XXZ system with $L=16,\gamma =4$, and $\alpha =2$. Comparisons with the corresponding $\alpha =0$ and noninteracting systems are shown. Lower panel: the half-chain entanglement in a dipole-conserving model with $L=20$ for different relative strengths of $H_4$ and $H_5$. The dotted line shows the average entanglement entropy of a random state, and the inset shows the size of the fluctuations of $S_{L/2}$ as a function of system size. The color scheme in the inset is the same as in the main panel.

Figure 6

Upper panel: the decay of the normalized QMI ${\mathcal{I}}_x/{\mathcal{I}}_1$, plotted as a function of distance for a thermalizing system, a Stark MBL system, and a system with a purely linear potential. The results correspond to a system with $L=18,\mathrm{\Delta }=0.1$, and have been averaged over 100 eigenstates from the middle of the spectrum. The three sites at both ends of the chain have been excluded to minimize boundary effects. Lower panel: the decay of the DEER response with and without curvature. The parameters used were $L=12,M=50,\mathrm{\Delta }=0.1,\gamma =3,d=7$, and region II being of size $N=3$.

Figure 7

The power-law decay of the DEER response at long times on a double-logarithmic scale. The results correspond to a system with $L=12,M=50,\gamma =3,\alpha =2$ and with region II being of size $N=3$.

Figure 8

The decay of the DEER response when simulating product initial states in the physical spin basis. The decay is qualitatively similar to that seen in Fig. 7. The parameters used were $L=12,M=50,\gamma =3,\alpha =2$ and with region II being of size $N=3$.

Figure 9

The decay exponent of the long-time DEER response $\beta$ as a function of the linear potential strength $\gamma$. The fit was obtained from the curves in the inset, which correspond to $L=12,M=50,\alpha =2,d=7,\mathrm{\Delta }=1$, and $N=3$.

Figure 10

Histograms of the inverse participation ratio in the two-particle system, calculated for several system sizes $L$ for n.n. and exponentially decaying interactions with high numerical precision (indicated in the legend). The same histogram for $L=48$ and nearest-neighbor interactions calculated with standard precision is shown for comparison (dotted histogram).