Efficient variational diagonalization of fully many-body localized Hamiltonians

We introduce a variational unitary matrix product operator based variational method that approximately finds all the eigenstates of fully many-body localized one-dimensional Hamiltonians. The computational cost of the variational optimization scales linearly with system size for a fixed depth of the UTN ansatz. We demonstrate the usefulness of our approach by considering the Heisenberg chain in a strongly disordered magnetic field for which we compare the approximation to exact diagonalization results.

Figure 1

(a) Schematic representation of an MPS representation of a state $|\psi \rangle$. (b) Variational ansatz for the unitary $U$ that encodes all eigenstates of a fully many-body localized Hamiltonian. The local unitaries $u_{\left[n\right]}^{\left[m\right]}$ are parametrized as $u_{\left[n\right]}^{\left[m\right]}=e^{i{S}_{\left[n\right]}^{\left[m\right]}}$ with real symmetric matrices $S_{\left[n\right]}^{\left[m\right]},n=1\cdots L-1$ and $m=1\cdots N_{\mathrm{layer}}$. (c) Two dimensional generalization of the unitary network.

Figure 2

Comparison of the exact energy levels (blue lines) with the ones found by the variational optimization (red lines) for $W=8$ and $L=8$ as a function of the number of layers of two-site gates. The right panel shows a zoom of some energy levels at the bottom and in the center of the spectrum.

Figure 3

Mean variance of the energy as a function of system size for different number of layers for $W=8$. Inset: Mean variance as a function of $W$ for a fixed $L=8$.

Figure 4

Comparison of the exact spectral function $A\left(\omega \right)$ (black dots) with those obtained using different approximations (see text for details) for $L=10$ and $W=8$. Spectra are shown using a Lorentzian broadening with an imaginary part of $\epsilon =0.1$. Inset: Same data with $W=16$.