Counting local integrals of motion in disordered spinless-fermion and Hubbard chains

We develop a procedure which systematically generates all conserved operators in the disordered models of interacting fermions. Among these operators, we identify and count the independent and local integrals of motion (LIOM), which represent the hallmark of the many-body localization (MBL). The method is tested first on the prototype disordered chain of interacting spinless fermions. As expected for full MBL, we find for large enough disorder $N_M=2^M-1$ independent and quasilocal LIOM with support on $M$ consecutive sites. On the other hand, the study of the disordered Hubbard chain reveals that $3^M-1<N_M\lesssim 4^M/2$, which is less than required for full MBL but much more than in the case of spinless fermions.

Figure 1

Spinless fermions. Eigenvalues ${\lambda }_{\alpha }$ corresponding to local components of LIOM, see Eq. (4), averaged over disorder for various system sizes $L$, supports $M$ and different disorders $W$.

Figure 2

Spinless fermions (a)–(c) and the Hubbard chain (d). (a) Sorted eigenvalues ${\lambda }_{\alpha }$ (local components of LIOM) for $V=1$ (points) and for Anderson insulator, $V=0$ (lines), (b) total weight of LIOM $\mathrm{\Lambda }$ for various supports $M$ and different disorders $W$, (c) finite–size scaling of $\mathrm{\Lambda }$ with $1/L$ for $M=4$ and different $W$. (d) The same as in (b) but for the Hubbard chain together with two different estimates.

Figure 3

(a) Total weights vs $1/L$ constructed from local operators with at least one hopping term (${\mathrm{\Lambda }}_{\mathrm{hop}}$–points with lines) or without hopping (${\mathrm{\Lambda }}_n$–points) for $W=6$ and 10. (b) ${\mathrm{\Lambda }}_{\text{hop}}$ for $L=10$ as a function of $W$.

Figure 4

Disorder-averaged projections of ${\overline{Q}}_{\alpha }$ on normalized local operators $o_x$ explicitly listed in the first section. Results are obtained for the chain of spinless fermions with $L=12$, $M=4$, and $W=4$.

Figure 5

Hubbard chain. (a) results for various $L$ and $N$ at fixed $M=2$: $(L=8,N=8)$, $(L=10,N=6)$, and $(L=12,N=4)$. (b) Results for LIOM constructed with and without random magnetic field. (c), (d) Points and lines show results for $W=15$ and $W=30$, respectively.

Figure 6

LIOM constructed for the Hubbard chain from local operators with (lines with points) and without (points) hopping terms, respectively.