Exploring Localization in Nuclear Spin Chains

TL;DR

A novel correlation metric, capable of distinguishing MBL from AL in high-temperature spin systems, is presented and the use of this metric is demonstrated to detect localization in a natural solid-state spin system using nuclear magnetic resonance (NMR).

Abstract

Characterizing out-of-equilibrium many-body dynamics is a complex but crucial task for quantum applications and the understanding of fundamental phenomena. A central question is the role of localization in quenching quantum thermalization, and whether localization survives in the presence of interactions. The localized phase of interacting systems (many-body localization, MBL) exhibits a long-time logarithmic growth in entanglement entropy that distinguishes it from the noninteracting Anderson localization (AL), but entanglement is difficult to measure experimentally. Here, we present a novel correlation metric, capable of distinguishing MBL from AL in high-temperature spin systems. We demonstrate the use of this metric to detect localization in a natural solidstate spin system using nuclear magnetic resonance (NMR). We engineer the natural Hamiltonian to controllably introduce disorder and interactions and observe the emergence of localization. In particular, while our correlation metric saturates for AL, it keeps increasing logarithmically for MBL, a behavior reminiscent of entanglement entropy, as we confirm by simulations. Our results show that our NMR techniques, akin to measuring out-of-time correlations, are well suited for studying localization in spin systems.

Figures (7)

• Figure 1: Quantum many-body correlations (top) grow from an initial localized state (left) but are restricted to a finite size by disorder (bottom). The average correlation length $L_c$ measure the spread of the correlations.
• Figure 2: Experimental measurements of spin correlations in noninteracting spin chains. Correlation length $L_c$ for uniform (A) and disordered transverse fields (B). In both cases we set $u=0.2$ and $v=0$ and varied the disorder strength $g$ and field magnitude $b$ (see Eq. \ref{['eq:Hamiltonian']}). (C) Comparison of MQC intensities in the $\chi$ sector used as a litmus test for disorder, see also SM. Solid markers are for uniform field with $b=0.826$ krad/s, open markers are for disordered field with $g=0.15$. Errorbars are determined from the noise in the free induction decay, the solid lines are guides to the eye.
• Figure 3: A: Experimental measurements of spin correlations in interacting spin chains. We plot in log-linear scale the measured $L_c$ dynamics in the presence of disorder and for varying interaction strengths $v$. Data are for $u=0.24$, $g=0.12$, and $b=0$. After an initial growth of correlations, $L_c$ saturates for the non-interacting systems, while it shows a slow growth in the presence of interactions, thus indicating many-body localization. B: Simulations of spin correlation and entanglement entropy. We compare the entropy of the reduced half chain (solid lines, left axis) with the correlation length $L_c$ (dotted lines, right axis) and the approximate $L_c$ obtained from measuring the MQC (dashed lines). The similar behavior (including logarithmic growth) confirm that the chosen metric is as good an indicator of MBL as the more commonly used entanglement entropy. Here we renormalized the entanglement entropy to vary between 0 and 1, see SM for details.
• Figure 4: A Fluorapatite crystal structure, showing the Fluorine and Phosphorus spins in the unit cell. B NMR scheme for the generation and detection of MQC. In the inset (C) an exemplary pulse sequence for the generation of the double-quantum Hamiltonian. Note that thanks to the ability of inverting the sign of the Hamiltonian, the scheme amounts to measuring out-of-time order correlations.
• Figure 5: Different correlation metrics used to distinguish MBL from AL. For pure states, the bipartite entanglement entropy is used (A). For $\rho_\text{eq}$, the mutual information (B), correlation distance (C), and correlation length (D) can be used. In D, the dashed lines corresponding to the approximated $L_c$ extracted from MQC intensities. All plots are for $L=8$ with open boundary conditions, the disorder $h_j$ is drawn uniformly from $[-W, W]$ with $W=8$.