Accessing Many-Body Localized States through the Generalized Gibbs Ensemble

We show how the thermodynamic properties of large many-body localized systems can be studied using quantum Monte Carlo simulations. We devise a heuristic way of constructing local integrals of motion of high quality, which are added to the Hamiltonian in conjunction with Lagrange multipliers. The ground state simulation of the shifted Hamiltonian corresponds to a high-energy state of the original Hamiltonian in the case of exactly known local integrals of motion. The inevitable mixing between eigenstates as a consequence of nonperfect integrals of motion is weak enough such that the characteristics of many-body localized systems are not averaged out, unlike the standard ensembles of statistical mechanics. Our method paves the way to study higher dimensions and indicates that a fully many-body localized phase in 2D, where (nearly) all eigenstates are localized, is likely to exist.

Figure 1

Expectation value of five different $L$ bits with five independent $\lambda$ for $\mathrm{\Delta }=6$ on a support of three sites in a patch of eleven sites for the 1D Heisenberg model in a random magnetic field. Distinct plateaus are seen for each $⟨{\mathbf{L}}^{(p)}⟩$ showing that the $L$ bits represent well-defined quantities.

Figure 2

Expectation value of five distinct $L$ bits for $\mathrm{\Delta }=1$ on a support of three sites in a patch of eleven sites for the 1D Heisenberg model in a random magnetic field. No plateaus are seen in the range $\lambda \in ]-10,10[$; all attempted $L$ bits are hence discarded.

Figure 3

Comparison between the sorted eigenvalues corresponding to the natural orbitals in the ground state and a MBL state for the 1D Heisenberg model for $\mathrm{\Delta }=6$. We constructed 12 $L$ bits on a system of size 100. The ground state energy is $E/J=-180.67(1)$ and the energy of the excited state is $E/J=-135.98(1)$.

Figure 4

Histogram of the quality of the $L$ bit for different disorder strength using Eq. (3), where an ideal $L$ bit has a quality of one.

Figure 5

Illustration of the plateaus found for a single $\mathbf{L}$ as a function of $\lambda$ for a 4-site $L$ bit embedded in a two dimensional $20\times 20$ lattice with $\mathrm{\Delta }=40$. With large $\mathrm{\Delta }$ effective $L$ bits can be optimized even when the embedding Hamiltonian is small.

Figure 6

Sorted eigenvalues of the natural orbitals for the 2D Heisenberg model in the ground state and for a system with 40 $L$ bits obtained by quantum Monte Carlo simulation of size $20\times 20$. The $L$ bits may change the particle number compared to the ground state, causing the shift in the location of the jump.