Complete Many-Body Localization in the $t\text{−}J$ Model Caused by a Random Magnetic Field

The many body localization (MBL) of spin-$\frac{1}{2}$ fermions poses a challenging problem. It is known that the disorder in the charge sector may be insufficient to cause full MBL. Here, we study dynamics of a single hole in one dimensional $t\text{−}J$ model subject to a random magnetic field. We show that strong disorder that couples only to the spin sector localizes both spin and charge degrees of freedom. Charge localization is confirmed also for a finite concentration of holes. While we cannot precisely pinpoint the threshold disorder, we conjecture that there are two distinct transitions. Weaker disorder first causes localization in the spin sector. Carriers become localized for somewhat stronger disorder, when the spin localization length is of the order of a single lattice spacing.

Figure 1

(a) The hole density ${\rho }_i$ at time $t=200$ for different values of $W$ as indicated in the inset. The size of the system was $L=29$; (b) extracted charge localization lengths ${\xi }_c$ for different system sizes vs $W$. Thin lines represent fits of the form ${\xi }_c=A/(W-W_0{)}^{\gamma }$. Inset: fit parameters extrapolated towards $1/L=0$; (c) ${\sigma }^2(t)$ for short times below the localization transition, $W=2$ and 3 showing diffusive behavior. Thin black dashed straight lines are guides to the eye. Dashed, dot-dashed, and full lines represent systems sizes $L=21$, 25, and 29, respectively; (d) $\sigma (t)$ on $\log (t)$ scale for $W=5$, 7, and 10 for maximal system size $L=29$ using LFS.

Figure 2

(a) $F(\alpha )$ for different values of $W$. The largest available LFS Hilbert space with $L=29$ was used in this case; (b) $1/\sigma$ scaling with the system size $L$. Inset: Extrapolated values ${\sigma }_0$ (circles) with a fit (full line) on the functional form ${\sigma }_0\propto 1/(W-W_{\sigma }^c{)}^{\gamma }$ with $W_{\sigma }^c\simeq 5$ and $\gamma \simeq 0.95$; (c) variance of $\sigma$ for different system sizes $L$. Calculations in (b) and (c) were performed from eigenstates from the middle of the energy spectrum using complete Hilbert spaces.

Figure 3

(a) $S/L_a$ for various values of $W$. Results were computed using a complete basis on $L=13$ sites chain. Time evolution started from a Néel state with hole located in the middle of the chain. Thin black line represents Anderson’s localized state for $W=7$ and $J=0$; (b) $S/L_a$ computed from eigenstates from the middle of the energy band. The same holds for (c) and (d). Results are shown for various chain lengths, $L=9$, 11, 13, and 15; (c) the variance of $S$ and $S_h$ (symbols connected with dashed and full lines, respectively) vs $W$; (d) hole entropy $S_h$ for different system sizes $L$.

Figure 4

Time evolution of the charge imbalance $P(t)$ of the $t\text{−}J$ model with $L$ sites, $L/3$ holes, and equal number of spin-up and spin-down Fermions. Results after finite-size analysis are shown in (a); (b) results at fixed $L=15$ and different values of $W$ compared with data for $J=0.4$ and $W=10$; (c) schematics portraying diagonal energies of the basis states on neighboring sites for the case of charge and spin disorder.