Many-body localization in a quantum Ising model with the long-range interaction: Accurate determination of the transition point

TL;DR

This work investigates the many-body localization transition in a quantum Ising model with infinite-range interactions, where long-range couplings suppress finite-size fluctuations that plague short-range models. By mapping the problem to a Bethe lattice localization scenario and cross-validating with exact diagonalization, the authors derive an analytic threshold $\Gamma_c$ that scales as $\Gamma_c \sim J_0/(\sqrt{N}\ln N)$ (up to order-one factors) and verify it numerically for up to $N \approx 19$ spins, extracting $\beta \approx 1.4$ and $\eta \approx 0.41$. The transition threshold is found to be about a factor of three larger than the Bethe-lattice prediction due to destructive interference of multi-spin pathways, underscoring the role of loop structure in the actual system. The results demonstrate that the Bethe-lattice framework can reliably capture critical behavior near the MBL transition in long-range interacting systems and have implications for a broad class of physical platforms with dipole-dipole, elastic, or indirect exchange interactions, enabling exploration of critical phenomena beyond short-range models.

Abstract

Many-body localization (MBL) transition emerges at strong disorder in interacting systems, separating chaotic and reversible dynamics. Although the existence of MBL transition within the macroscopic limit in spin chains with a short-range interaction was proved rigorously, the transition point is not found yet because of the dramatic sensitivity of the transition point to the chain length at computationally accessible lengths, possible due to local fluctuations destroying localization. Here we investigate MBL transition in the quantum Ising model (Ising model in a transverse field) with the long-range interaction suppressing the fluctuations similarly to that for the second-order phase transitions. We estimate the MBL threshold within the logarithmic accuracy using exact results for a somewhat similar localization problem on a Bethe lattice problem and show that our expectations are fully consistent with the estimate of the transition point using exact diagonalization. In spite of unlimited growing of the critical disorder within the thermodynamic limit, this result offers the opportunity to probe the critical behavior of the system near the transition point. Moreover, the model is relevant for the wide variety of physical systems with the long-range dipole-dipole, elastic or indirect exchange interactions.

Figures (6)

• Figure 1: (a) Hypecube with vertices representing the phase space of three spins. (b) Bethe lattice with each site coupled to three neighbors.
• Figure 2: General behavior of (a) the average gap ratio$\langle r \rangle$ and (b) the fluctuation$\Delta r$ depending on the dynamic field $\Gamma/J_{0}$. The data is shown for $N=13$.
• Figure 3: Dependence of $\Delta r$ on $\Gamma/J_{0}$ for $N=13$ and ${\cal N}_{dis}=31200$ (circles) with the corresponding parabolic fit (solid line). The shaded area corresponds to the estimate of the error in determining position of $\Gamma_c$.
• Figure 4: Dependence of the transition point $\Gamma_{c}/J_{0}$ on the system size. The fitting coefficients are $\beta=1.4$ and $\eta=0.41$.
• Figure 5: Dependence of the value of mean gap ratio at the MBL transition point $r_{c}$ on the system size. The best horizontal fit corresponds to approximately $r_c^{\rm (fit)}\approx0.472$.