Constructing local integrals of motion in the many-body localized phase

Many-body localization provides a generic mechanism of ergodicity breaking in quantum systems. In contrast to conventional ergodic systems, many-body-localized (MBL) systems are characterized by extensively many local integrals of motion (LIOM), which underlie the absence of transport and thermalization in these systems. Here we report a physically motivated construction of local integrals of motion in the MBL phase. We show that any local operator (e.g., a local particle number or a spin-flip operator), evolved with the system's Hamiltonian and averaged over time, becomes a LIOM in the MBL phase. Such operators have a clear physical meaning, describing the response of the MBL system to a local perturbation. In particular, when a local operator represents a density of some globally conserved quantity, the corresponding LIOM describes how this conserved quantity propagates through the MBL phase. Being uniquely defined and experimentally measurable, these LIOMs provide a natural tool for characterizing the properties of the MBL phase, in both experiments and numerical simulations. We demonstrate the latter by numerically constructing an extensive set of LIOMs in the MBL phase of a disordered spin-chain model. We show that the resulting LIOMs are quasilocal and use their decay to extract the localization length and establish the location of the transition between the MBL and ergodic phases.

Figure 1

(a) The disorder-averaged $log\left(M_{ij}\right)$ vs $|i-j|$ for increasing disorder strengths $W=2,3,3.5,5,7$ from top to bottom for LIOMs ${\overline{\sigma }}_i^z,i=1,...,N$ with periodic boundary conditions. (b) The disorder-averaged logarithm of the partial norm $\mathcal{N}\left(r\right)$ [Eq. (20)] vs the range $r$ of LIOMs ${\overline{\sigma }}_i^z$ for $i=1,N$ with open boundary conditions. From top to bottom, $W=2,3,3.5,5,7$.

Figure 2

$\left[M_{ii}\right]$ vs disorder strength $W$ at different system sizes. This quantity can be viewed as an order parameter for the MBL phase. The red dashed line shows the linear in $1/N$ extrapolation to the thermodynamic limit.

Figure 3

The length scale $\xi$ of decay of $M_{ij}\propto exp(-|i-j|/\xi )$ for different system sizes. The localization length in the thermodynamic limit (red dashed curve), obtained by a linear extrapolation of $1/\xi$ for the three largest system sizes to $N\to \infty$. The extrapolated localization length diverges at a critical $W_{*}\approx 3$, indicating a transition into the delocalized phase.