Finding Matrix Product State Representations of Highly Excited Eigenstates of Many-Body Localized Hamiltonians

A key property of many-body localized Hamiltonians is the area law entanglement of even highly excited eigenstates. Matrix product states (MPS) can be used to efficiently represent low entanglement (area law) wave functions in one dimension. An important application of MPS is the widely used density matrix renormalization group (DMRG) algorithm for finding ground states of one-dimensional Hamiltonians. Here, we develop two algorithms, the shift-and-invert MPS (SIMPS) and excited state DMRG which find highly excited eigenstates of many-body localized Hamiltonians. Excited state DMRG uses a modified sweeping procedure to identify eigenstates, whereas SIMPS applies the inverse of the shifted Hamiltonian to a MPS multiple times to project out the targeted eigenstate. To demonstrate the power of these methods, we verify the breakdown of the eigenstate thermalization hypothesis in the many-body localized phase of the random field Heisenberg model, show the saturation of entanglement in the many-body localized phase, and generate local excitations.

Figure 1

Pictorial representation for optimizing $\parallel O|\phi ⟩-|\psi ⟩{\parallel }^2$. Here, as an example, we consider optimizing the third site (colored orange) for an $L=4$ chain. The MPO $O$ (pink blocks) and the MPS $|\psi ⟩$ (cyan blocks) are given. To update the orange block, we solve a linear equation, treating the orange block as an unknown vector $x$; the other part of the network on the left-hand side, after being contracted, amounts to a symmetric semi-positive-definite matrix $A$, and the matrices on the right-hand side of the equation becomes a known vector $b$. The entire MPS is optimized by sweeping.

Figure 2

Left: Convergence to eigenstates 503–514. Magenta lines indicate the 12 eigenvalues in this range. SIMPS: Blue and yellow bars indicate the basins of convergence, as a function of parameter $\lambda$, found for two random MPS initial conditions. Yellow and blue dots indicate whether sweeps of the two respective initial conditions find a given eigenvalue. ES-DMRG: Green dots indicate whether ES-DMRG found the eigenvalue after being run with 6144 random product state initial conditions. Right: Energy vs $\lambda$ for $L=30$, $W=10$. Twenty-nine MPSs obtained using SIMPS with $M=40$ are displayed on this plot, with eight distinct energies. Inset: Fidelity function $F({\lambda }_i)=|⟨\mathrm{MPS}({\lambda }_i)|\mathrm{MPS}({\lambda }_{i+1})⟩|$. $F(\lambda )$ also shows eight distinct states and matches exactly with the energy curve.

Figure 3

Values of $⟨{\log }_{10}(\sigma )⟩$ for various algorithms and parameters at different disorder strengths $W$. The blue line ($L=30$) and the orange line ($L=40$) are SIMPS runs at $M=60$ with $\lambda$ tuned exactly to the middle of the spectrum for each disorder realization, where the lowest and highest eigenvalues are obtained via DMRG and the disorder configurations are the same scaled pattern at different $W$ for $L=30$ and $L=40$, respectively. Values of $W\le 5$ are not necessarily reliably converged. The green point is ES-DMRG at $M=20$, $L=30$, and the pink point is ES-DMRG at $M=20$, $L=100$. The dark green point is SIMPS at $M=20$, $L=30$, showing that ES-DMRG and SIMPS converge to similar accuracy at the same bond dimension and chain length. The red line is SIMPS $M=200$, $L=30$ computed using the faster sweeping. Insets are distributions of $\sigma$ at the respective parameters.

Figure 4

Left: Site value of $⟨S_z^6⟩$ as a function of energy at $L=30$ for a fixed disorder configuration with disorder strength $W=10$. Eigenstates are obtained using SIMPS, with $M=40$. Note the small energy range. The energy scan was performed with $200\lambda$ evenly spaced in $[-0.1,0.1]$. In the $S_z^T=-3$, $-2$, $-1$, 0 sectors, a total of 74 distinct MPSs were found. Right: Site value of $⟨S_z^6⟩$ for $L=16$, $W=0.8$ generated using ED.

Figure 5

Left: Dependence of midbond entanglement entropy $S$ on the disorder strength for various chain lengths. Fifty disorder realizations were used for $L=14$ and 100 for $L=30$ and $L=40$. Open markers indicate points at low disorder strengths for which SIMPS failed to obtain states with low standard deviation of the energy. For SIMPS, we set $M=60$ and tuned $\lambda$ to the exact middle of the eigenspectrum for each disorder realization. This saturation confirms entanglement in the MBL is independent of system size. Right: Spectrum of 62 excitations (colored bars) produced by local modifications of a reference MPS eigenstate (long magenta line). The horizontal axis marks the site on which the MPS is being modified. Bars of the same color correspond to the same excitation as determined by their energies. ($L=30$, $W=10$, $M=15$, and excited eigenstates all have $\sigma <0.0005$.)