Emergent dipole moment conservation and subdiffusion in tilted chains

We study transport in an interacting tilted (Stark) chain. We show that the crossover between diffusive and subdiffusive relaxation is governed by $F\sqrt{L}$, where $F$ is the strength of the field, and $L$ is the wavelength of the excitation. While the subdiffusive behavior persists for large fields, the corresponding transport coefficient is exponentially suppressed with $F$ so that the finite-time dynamics appears almost frozen. We (i) explain the crossover scale between the diffusive and subdiffusive transport by bounding the dynamics of the dipole moment for arbitrary initial state, (ii) prove its emergent conservation at infinite temperature for $F\gg 1/L$, and (iii) argue that the numerical data for the tilted chain are consistent with the hydrodynamics of fractons.

Figure 1

Results for NESS in chains with boundary driving. Panels (a) and (b) show the normalized amplitude of the spatial profile of particle density in chains with $L=50$ sites for $V=2$ and 3, respectively. We used two different tilts: $F=0$ (diffusive) and $F=0.8$ (subdiffusive) shown in the legends. A cubic profile, $\langle {\tilde{n}}_l\rangle =ax+b{x}^3,x=2l/L-1$, consistent with the hydrodynamic equation corresponding to the $z=4$ case, fits the data well for $F=0.8$ for both $V=2.0$ and 3.0. Panels (c) and (d) show the $L$-dependence of the NESS current, $I/\mu$, for $V=2.0$ and 3.0, respectively.

Figure 2

Results for NESS in chains with boundary driving. Panels (a) and (b) show the exponent $z$ as a function of $F\sqrt{L}$ for $V=2.0$ and 3.0, respectively. Results obtained for fixed $F$ (shown in the legend) and various $L$ are connected by lines. Panels (c) and (d) show the corresponding exponential suppression of the NESS current with increasing tilt $F$. Note the logarithmic scale in (c) and (d).

Figure 3

Panels (a) and (b) depict time evolution of normalized amplitudes, ${\delta }_t$, of the spatially modulated profile for $L=24$ with the wave vectors $q=2\pi /L$ and $4\pi /L$, respectively. The values of the tilt are shown in the legends. Panels (c) and (d), respectively, show the $F$-dependence of the exponent $z$ and that of the transport coefficient $D$ along with the decay rate, ${\mathrm{\Gamma }}_{\text{min}}=D{q}^z$, for the smallest $q=2\pi /L$. Results in (c) and (d) are obtained from exponential fits to data in the shaded areas in (a) and (b). Panel (d) shows also ${\mathrm{\Gamma }}_{\text{min}}$ obtained for the Hamiltonian Eq. (1) with OBC; see Sec. 4 for details. Results are obtained for $V=3$ and $V^{\prime }=2$.

Figure 4

Normalized dipole-moment correlations $\mathcal{C}\left(t\right)$, defined in Eq. (9), obtained via full diagonalization for an $L=18$ chain with OBC, and $V=3$, $V^{\prime }=2$.

Figure 5

Ratio of the absolute values of the fitting parameters $a,b$ for the NESS particle profile for $L=50$. The ratio approaches the value 3 for larger values of $F$, where we see fractonic subdiffusion with $z=4$. The inset shows the monotonic behavior of the fitting parameters $a,b$ as a function of $F$.

Figure 6

Selected data from Fig. 3 in the main text shifted vertically by a constant $c$ indicated in the legend.