Localization in a random XY model with long-range interactions: Intermediate case between single-particle and many-body problems

Many-body localization in an XY model with a long-range interaction is investigated. We show that in the regime of a high strength of disordering compared to the interaction an off-resonant flip-flop spin-spin interaction (hopping) generates the effective Ising interactions of spins in the third order of perturbation theory in a hopping. The combination of hopping and induced Ising interactions for the power-law distance dependent hopping $V\left(R\right)\propto R^{-\alpha }$ always leads to the localization breakdown in a thermodynamic limit of an infinite system at $\alpha <3d/2$ where $d$ is a system dimension. The delocalization takes place due to the induced Ising interactions $U\left(R\right)\propto R^{-2\alpha }$ of “extended” resonant pairs. This prediction is consistent with the numerical finite size scaling in one-dimensional systems. Many-body localization in an XY model is more stable with respect to the long-range interaction compared to a many-body problem with similar Ising and Heisenberg interactions requiring $\alpha \ge 2d$ which makes the practical implementations of this model more attractive for quantum information applications. The full summary of dimension constraints and localization threshold size dependencies for many-body localization in the case of combined Ising and hopping interactions is obtained using this and previous work and it is the subject for the future experimental verification using cold atomic systems.

Figure 1

Many-body interaction of two resonant pairs of spins $(i,i^{\prime })$ and $(j,j^{\prime })$ assisted by neighboring spin $k$. Wave lines indicate significant spin-spin hopping interactions $V$.

Figure 2

Original (inset) and rescaled dependencies of ergodicity parameter $Q$ on disordering $W$ for $\alpha =1$.

Figure 3

Original (inset) and rescaled dependencies of ergodicity parameter $Q$ on disorder $W$ for $\alpha =1.25$.

Figure 4

Original (inset) and rescaled dependencies of ergodicity parameter $Q$ on disorder $W$ for $\alpha =1.5$.

Figure 5

Dependence of ergodicity parameter $Q$ on disorder $W$ for $\alpha =1.75$.

Figure 6

Dependence of ergodicity parameter $Q$ on disorder $W$ for $\alpha =2$.

Figure 7

Scaling factors $c$ vs a number of spins $N$ for different interactions ($\alpha$) and the fit of $c\left(N\right)$ dependencies using functions $f_a\left(N\right)$ with optimum fitting parameters $a$ estimating scaling exponents for localization transition size dependence. Inset shows the dependence of exponents $a$ on the interaction power-law exponents $\alpha$. The solid (red) line indicates the threshold case $a=0$.