Explicit Local Integrals of Motion for the Many-Body Localized State

Recently, it has been suggested that the many-body localized phase can be characterized by local integrals of motion. Here we introduce a Hilbert-space-preserving renormalization scheme that iteratively finds such integrals of motion exactly. Our method is based on the consecutive action of a similarity transformation using displacement operators. We show, as a proof of principle, localization and the delocalization transition in interacting fermion chains with random on-site potentials. Our scheme of consecutive displacement transformations can be used to study many-body localization in any dimension, as well as disorder-free Hamiltonians.

Figure 1

The average of the absolute value of the coefficients $⟨|V_{i_1,\dots ,i_n}|⟩$ present in the classical Hamiltonian. Here we show results deep in the many-body localized phase with $W=8$, for a system size $L=24$ averaged over 20 disorder realizations. The different curves represent the two-body $V_{ij}$, three-body $V_{ijk}$, and four-body $V_{ijkl}$ coefficients, respectively. The strength of the coupling constants is almost independent of the order of the interaction. Inset: The spread of the local integrals of motion at disorder strength $W=8$ and $L=20$, compared to the coefficients of the Hamiltonian at the same distance. Note the difference between the localization length of the integrals of motion and the exponential decay of the interactions.

Figure 2

The average of the absolute value of the coefficients $⟨|V_{i_1,\dots ,i_n}|⟩$ as a function of distance for various disorder strengths $W=3,\dots ,9$ for $L=20$. In the localized phase one can see exponential decay, whereas in the delocalized phase at $W<W_c\approx 5$ the coefficients are independent of distance. Inset: The relative error in the ground-state energy obtained using displacement transformations, compared to exact diagonalization results for $L=14$. The different curves represent the cutoff at a given order $n=4,6,8$. In the localized phase even few-body interactions are sufficient to reproduce the ED results. In the delocalized regime one needs to go up to order 8, whereas around the MBL transition interactions of even higher order need to be included.