Encoding the structure of many-body localization with matrix product operators

Anderson insulators are noninteracting disordered systems which have localized single-particle eigenstates. The interacting analog of Anderson insulators are the many-body localized (MBL) phases. The spectrum of the many-body eigenstates of an Anderson insulator is efficiently represented as a set of product states over the single-particle modes. We show that product states over matrix product operators of small bond dimension is the corresponding efficient description of the spectrum of an MBL insulator. In this language all of the many-body eigenstates are encoded by matrix product states (i.e., density matrix renormalization group wave functions) consisting of only two sets of low bond dimension matrices per site: the $G_i$ matrices corresponding to the local ground state on site $i$ and the $E_i$ matrices corresponding to the local excited state. All $2^n$ eigenstates can be generated from all possible combinations of these sets of matrices.

Figure 1

Diagram of (a) a matrix product state $|\psi \rangle$, (b) a matrix product operator $U$, and (c) the operator product $U^{†}{\widehat{\sigma }}_i^{+}U$. The boxes correspond to tensors and lines to tensor indices. Lines connecting two boxes are indices to be contracted while dangling lines are external indices. The matrix product state $|\psi \rangle$ “eats” the external indices $|{\sigma }_1{\sigma }_2{\sigma }_3\cdots \rangle$ (these are the configurations of our spin-1/2 chain) and “spits” out a complex number—the amplitude of that configuration. Similarly, the matrix product operator “eats” two sets of external indices—the “ket” set and the “bra” set—and “spits” out the value of the corresponding matrix element.

Figure 2

Saturation of the bond dimension $D$ of the matrix product operator representing the unitary that diagonalizes the Hamiltonian as a function of the chain length for a number of values of disorder strength. $D$ was measured on the middle bond. Points represent numerical data (from 200 disorder realizations) and the solid lines are fits (see text). The change of regime—saturating $D$ vs growing $D$—corresponds to the transition from the localized to the delocalized phase. Inset: Length $\xi$ as a function of disorder strength, computed using (1) the decay of couplings in the effective Hamiltonian $H_{\text{eff}}$ and (2) the saturation of $D$ [to match (1) and (2) we scaled ${\xi }_{H_{\text{eff}}}\to 8.7*{\xi }_{H_{\text{eff}}}$]. The dashed line indicates maximum chain length ($L=12$).

Figure 3

The probability distribution function (PDF) of the bond dimension across the center bond of the spin chain. (a) PDF for 12-site chains at various disorder strengths. For weak disorder, $\mathrm{\Delta }=1$, the PDF is clustered around the maximal allowed value $D=4^6=4096$. Near the many-body localization-delocalization transition, $\mathrm{\Delta }=3$, the PDF becomes spread over a wide range of bond dimensions. In the localized phase, the PDF becomes clustered around the $D=4^{\xi }$ with a power-law tail extending to larger bond dimensions. (b) PDF for chains of various length and disorder strengths; power laws can be observed for $\mathrm{\Delta }\gtrsim 3$.