Spectral tensor networks for many-body localization

Subsystems of strongly disordered, interacting quantum systems can fail to thermalize because of the phenomenon of many-body localization (MBL). In this paper, we explore a tensor network description of the eigenspectra of such systems. Specifically, we will argue that the presence of a complete set of local integrals of motion in MBL implies an efficient representation of the entire spectrum of energy eigenstates with a single tensor network, a spectral tensor network. Our results are rigorous for a class of idealized systems related to MBL with integrals of motion of finite support. In one spatial dimension, the spectral tensor network allows for the efficient computation of expectation values of a large class of operators (including local operators and string operators) in individual energy eigenstates and in ensembles.

Figure 1

(a) A projector ${\mathcal{P}}_{{\mu }_n}^{\left(n\right)}$ in $D=1$ acting on ${\ell }_{\mathcal{Z}}=3$ sites. (b) ${\mathcal{P}}_{{\mu }_n}^{\left(n\right)}$ expressed as a MPO. (c) All $q$ projectors $\left\{{\mathcal{P}}_{{\mu }_n}^{\left(n\right)}\right\}$ represented as a MPO with additional index ${\mu }_n$. (d) A projector $P_{{\mu }_n}^{\left(n\right)}$ in $D=2$ acting on ${\left({\ell }_{\mathcal{Z}}\right)}^2=9$ sites. (e) The projector in (d) expressed as a MPO. The bond dimension is all cases is independent of $N$.

Figure 2

(a)–(c) Construction of the spectral tensor network ${\mathcal{P}}_{\vec{\mu }}$ in $D=1 \left(N_{\mathcal{Z}}=N\right)$. (a) Product of $N$ projectors. (b) Each projector decomposed as a MPO (Fig. 1). (c) The spectral tensor network ${\mathcal{P}}_{\vec{\mu }}$. The bond dimension is independent of $N$. (d) The number of sIOM overlapping with site $i$ is upper bounded by the number of sites in $A$.

Figure 3

(a) Bond dimension of the MPO representing $e^{iW}{\sigma }_4^z{e}^{-iW}$ versus the bond index. (b) Number of truncated qIOM overlapping with site $j$ in a typical disorder realization of free fermions with $W/t=3,L=1001$. Inset: Averaged value of $N\left(j\right)$ vs $\epsilon$.