Explicit construction of local conserved operators in disordered many-body systems

The presence and character of local integrals of motion—quasilocal operators that commute with the Hamiltonian—encode valuable information about the dynamics of a quantum system. In particular, strongly disordered many-body systems can generically avoid thermalization when there are extensively many such operators. In this work, we explicitly construct local conserved operators in one-dimensional spin chains by directly minimizing their commutator with the Hamiltonian. We demonstrate the existence of an extensively large set of local integrals of motion in the many-body localized phase of the disordered XXZ spin chain. These operators are shown to have exponentially decaying tails, in contrast to the ergodic phase where the decay is (at best) polynomial in the size of the subsystem. We study the algebraic properties of localized operators and confirm that in the many-body localized phase, they are well described by “dressed” spin operators.

Figure 1

A typical integral of motion ${\tau }_i$ in the many-body localized phase of the disordered XXZ model. ${\tau }_i$ decays exponentially away from the central site $i$. The operator ${\tau }_i$ is determined using the method described in Sec. 4, and its spatial profile is obtained by plotting the disorder-averaged logarithm of the operator density, given by Eq. (14). The system is an XXZ spin chain with interaction $V=0.5t$ and disorder $\mathrm{\Delta }=5t$ in Eq. (5).

Figure 2

Minimum eigenvalue of the commutant matrix for the disordered XXZ spin chain (5) as a function of the subsystem size $L$. Parameters are $V=0.5t$ and $\mathrm{\Delta }=0.5t$ (left) or $\mathrm{\Delta }=5t$ (right). Points are averaged over either 10000 ($L=3,4,5,6,7$) or 200 ($L=8$) disorder realizations. Lines are polynomial (left) and exponential (right) fits to the points. Fit parameters are given in the figures.

Figure 3

The lowest 50 eigenvalues of the commutant matrix for systems in the integrable (red), ergodic (blue), and MBL (black) phases, with a subsystem of seven sites. The results for the integrable system are exact; those for the ergodic and MBL systems are averaged over 5000 disorder realizations (error bars are shown in figure). Inset: Relative difference between the eigenvalues of the commutant matrix in the MBL phase, given by Eq. (15). Points that correspond to changes between the distance $d_I$ in Fig. 4 have been marked, and a local maximum is visible at each point.

Figure 4

Top: Table of some combinations of local integrals of motion $P_S$ in a subsystem of width 7 in a spin chain. On each row, the single $\tau$ operators in the product ${\prod }_{i\in S}{\tau }_i$ are indicated, and the distance to the boundary $d_S$ is written on the right column. An asterisk denotes when the distance $d_S$ is to the boundary of slower decay (see text). Bottom: Example of a $P_S$ operator; in this case, we have $P_S={\tau }_3{\tau }_4$. If $P_S$ were approximated by our method inside the boxed region, the closest distance to the boundary would be $d_S=2$ on the left, and we would expect the average $\left|\right|[\left[P_S\right],\widehat{H}]\left|\right|\propto exp(-2/\xi )$, with $\xi$ the localization length.

Figure 5

Operator participation ratio, given by Eq. (17), for various disorder strengths as a function of subsystem size. All points are an average over 10000 disorder realizations (error bars in plot). Lines are a fit to the first few points to emphasize trends of curving above or below a linear increase.

Figure 6

Scatter plot of 10000 individual realizations of the lowest eigenvalue of the commutant matrix against the corresponding operator participation ratio for a range of disorders, colored by the disorder strength $\mathrm{\Delta }$. All points are calculated over a system size of length 7.

Figure 7

2D histogram plot of various correlations between eigenvalues and the spectral and product measures defined in the text. System parameters are $V=0.5t$ and $\mathrm{\Delta }=t$ (left) or $\mathrm{\Delta }=8t$ (right). All data taken at a resolution of $200\times 200$ bins, from a total of 100000 samples.

Figure 8

Plot of the residual from the power-law and ergodic fits as in Fig. 2, for a range of disorder values. Each fit was taken using five data points from $L=3$ to $L=7$, with each data point being in turn averaged over 1000 disorder realizations.