Many-body localization beyond eigenstates in all dimensions

Isolated quantum systems with quenched randomness exhibit many-body localization (MBL), wherein they do not reach local thermal equilibrium even when highly excited above their ground states. It is widely believed that individual eigenstates capture this breakdown of thermalization at finite size. We show that this belief is false in general and that a MBL system can exhibit the eigenstate properties of a thermalizing system. We propose that localized approximately conserved operators $\left({\mathrm{l}}^{*}\text{-bits}\right)$ underlie localization in such systems. In dimensions $d>1$, we further argue that the existing MBL phenomenology is unstable to boundary effects and gives way to ${\mathrm{l}}^{*}\text{-bits}$. Physical consequences of ${\mathrm{l}}^{*}\text{-bits}$ include the possibility of an eigenstate phase transition within the MBL phase unrelated to the dynamical transition in $d=1$ and thermal eigenstates at all parameters in $d>1$. Near-term experiments in ultracold atomic systems and numerics can probe the dynamics generated by boundary layers and emergence of ${\mathrm{l}}^{*}\text{-bits}$.

Figure 1

Phase diagram in $d>1$ indicating dynamical phases and eigenstate properties. The fully MBL phase is shaded pink and described by ${\mathrm{l}}^{*}\text{-bits}$ at all disorder strengths. ETH is satisfied everywhere in the phase diagram with the weight in the spectral function vanishing as $L\to \infty$ in the fully MBL phase. The black line marks the delocalization transition.

Figure 2

Phase diagram in $d=1$ indicating dynamical phases and eigenstate properties. At strong disorder, the system is fully MBL with l-bits and violates ETH. At intermediate disorder (shaded pink), the system is fully MBL with ${\mathrm{l}}^{*}\text{-bits}$. The red line marks the eigenstate phase transition between ETH and non-ETH states in this regime. The true delocalization transition is denoted by the black line and can lie entirely to the left of the eigenstate phase transition.

Figure 3

Schematic diagram showing the $f$ function in Eq. (7) for a ${\mathrm{l}}^{*}\text{-bit}$ system, which satisfies ETH in $d>1$. The $f$ function is smooth on the scale of the many-body level spacing $\mathrm{\Delta }E$, while vanishing on the much longer frequency scale ${\tau }_{\min }^{-1}$.

Figure 4

(Left) An MBL system with l-bits ${\tau }_{\mathbf{i}}^z$. The shading indicates that the operators are quasilocal with exponentially decaying weights away from the localization center. (Middle) The l-bit system coupled to a thermal boundary layer at temperature $T$. The boundary induces incoherent l-bit flips in the bulk with a rate that decreases exponentially with the distance from the boundary. (Right) As $t\to \infty$ at finite size, the bulk and boundary are locally thermal at temperature $T$.

Figure 5

(a)–(d) Real space pictures of potential ultracold atomic experiments to probe the effect of thermal layers on localized bulk. Red indicates the cloud of atoms and grey the speckle potential applied to only part of the cloud. White spots indicate holes in the initial density. (e) The mean parity of density in the hole as a function of time in the experiments indicated in (a)–(d).