Quantum coherence and the localization transition

A dynamical signature of localization in quantum systems is the absence of transport which is governed by the amount of coherence that configuration space states possess with respect to the Hamiltonian eigenbasis. To make this observation precise, we study the localization transition via quantum coherence measures arising from the resource theory of coherence. We show that the escape probability, which is known to show distinct behavior in the ergodic and localized phases, arises naturally as the average of a coherence measure. Moreover, using the theory of majorization, we argue that broad families of coherence measures can detect the uniformity of the transition matrix (between the Hamiltonian and configuration bases) and hence act as probes to localization. We provide supporting numerical evidence for Anderson and many-body localization (MBL). For infinitesimal perturbations of the Hamiltonian, the differential coherence defines an associated Riemannian metric. We show that the latter is exactly given by the dynamical conductivity, a quantity of experimental relevance which is known to have a distinctively different behavior in the ergodic and in the many-body localized phases.

Figure 1

(a) Log-log plot of the average return probability $1-\langle C_B^{\left(2\right)}\left({\mathcal{V}}_W\right)\rangle$ as a function of the system size $L$ for different values of the disorder strength $W$. The system is in the localized phase for all $W>0$, since the asymptotic escape probability is strictly less that 1 for $L\to \infty$. (b) Log-linear plot of $\langle C_B^{(rel)}\left({\mathcal{V}}_W\right)\rangle$ as a function of the system size $L$ for different values of the disorder strength $W$. The system is in the localized phase for all $W>0$, in which the asymptotic value is finite. In the ergodic phase ($W=0$) $\langle C_B^{(rel)}\left({\mathcal{V}}_W\right)\rangle$ diverges logarithmically. The number of realizations range from 30000 for small sizes to just 8 for the largest size. Error bars represent one standard deviation. Entropy has logarithm with base 2.

Figure 2

Asymptotic behavior for the slope of the quantities: ${log}_2\left(1-\langle C_B^{\left(2\right)}\left({\mathcal{V}}_W\right)\rangle \right)={log}_2\left(P_{\text{return}}\right), \langle C_B^{\left(\text{rel}\right)}\left({\mathcal{V}}_W\right)\rangle , {log}_2\left(1-\langle f_B^{(det)}\left(X_{{\mathcal{V}}_W}\right)\rangle \right), {log}_2\left(1-\langle f_B^{\left(\text{time-avg}\right)}\left(X_{{\mathcal{V}}_W}\right)\rangle \right)$, and ${log}_2\left(1-\langle f_B^{\left(\infty \right)}\left(X_{{\mathcal{V}}_W}\right)\rangle \right)$ for large $L$ as a function of the disorder strength $W$ for the Hamiltonian $H_{\mathrm{XXX}}$ at $h_x=0.3$. The slope was extracted for sizes $L=4,6,...,14$, with sample sizes $20000,20000,20000,8000,2000,800$; except at $W=3.7$, where the sample sizes were doubled. The error bars represent the standard error of the linear fit (see Appendix pp8 for more details). Entropy has logarithm with base 2.

Figure 3

Plot of the escape probability $\langle C_B^{\left(2\right)}\left({\mathcal{V}}_{\mathrm{\Gamma }}\right)\rangle$ as a function of the disorder strength $\mathrm{\Gamma }$ for the Lloyd model Hamiltonian $H_{\mathrm{\Gamma }}$, as predicted analytically by the heuristic Eq. (24a) (solid line) and the numerical simulations (points). For the case of the numerical simulation, $L\to \infty$ is extrapolated by averaging over disorder for sizes up to $L=2^{12}$. Standard deviations are within the point radius.

Figure 4

Plot of the (a) average escape probability $\langle C_B^{\left(2\right)}\left({\mathcal{V}}_W\right)\rangle$ and (b) $\langle C_B^{(rel)}\left({\mathcal{V}}_W\right)\rangle$ as a function of the system size $L$ for different values of the disorder strength $W$. The disorder values displayed are $W=0.4,1.0,1.4,1.8,2.5,3.1,3.7,5.0,7.0,9.0$ (monotonically from the top to bottom in the plots) for $L=4,6,\cdots ,12$ with sample sizes $20000,20000,20000,8000,2000$, except at $W=3.7$, where the sample sizes were doubled. Error bars represent one standard deviation. Entropy has logarithm with base 2.

Figure 5

Plot of the generalized-CGP measures, (a) $\langle f_B^{\left(\text{time-avg}\right)}\left(X_{{\mathcal{V}}_W}\right)\rangle$, (b) $\langle f_B^{\left(\text{det}\right)}\left(X_{{\mathcal{V}}_W}\right)\rangle$, and (c) $\langle f_B^{\left(\infty \right)}\left(X_{{\mathcal{V}}_W}\right)\rangle$ as a function of the system size $L$ for different values of the disorder strength $W$. The disorder values displayed are $W=0.4,1.0,1.4,1.8,2.5,3.1,3.7,5.0,7.0,9.0$ [monotonically from the top to bottom for (a) and bottom to top for (b)] for $L=4,6,\cdots ,12$ with sample sizes $20000,20000,20000,8000,2000$, except at $W=3.7$, where the sample sizes were doubled. Error bars represent one standard deviation.