Significance and Sensor Utility of Phase in Quantum Localization Transition

The degree of localization of the Harper-Hofstadter model is shown to display striking periodic dependence on phase degrees of freedom, which can depend on the nature of the boundary condition, reminiscent of the Aharonov-Bohm effect. In the context of implementation in a finite ring-shaped lattice structure, this phase dependence can be utilized as a fundamentally different principle for precision sensing of rotation and magnetic fields based on localization rather than on interferometry.

Figure 1

Counterclockwise from top center, with $\theta$ in units of $2\pi$ in all figures. (a) Schematic of a discrete ring-shaped lattice with eight sites shown as gray circles at the bottom. Cosine modulation at a different period (five full oscillations around, shown in red) shifts the energies at the sites, with the resulting lattice shown above as blue circles (the lines are for visual guidance). (b) The ground state IPR transitions around $\gamma =J_2/J_1=2$ from unity, indicating localization of the medium, to a low value, indicating an extended state. But, in the localized regime, there is partial delocalization near certain values of the phase, $\theta$, that manifests as narrow dips. (d) The top view of the same. (f) The dual Hamiltonian shows the opposite trend, the ground state IPR transitions from a value of unity for $\gamma <2$ to a low value for $\gamma >2$, and now there is partial localization near the same values of the phase, marked by narrow ridges in the plot. (c),(e) Top view contour plots corresponding to sections of (b) and (f), respectively, for extended values of $\gamma$ showing progressive narrowing of the delocalization dips and localization ridges.

Figure 2

Features illustrated with the discrete model for a ring lattice with $q=5$ sites and $\alpha =3/5$, all for the Harper equation Eq. (1) except where mentioned. (a) The IPR averaged over all states in the lowest band displays the localization transition and dependence on the phase $\theta$ as for the ground state in Fig. 1. (b) The dependence of the IPR on the phase $\theta$ at fixed $\gamma =5$ is shown for each state in the lowest band, labeled by eigenenergies $E_i$ in ascending value. (c) With a box boundary condition, the phase dependence of the IPR, shown for the ground state, persists for the Harper equation although distorted, but for the dual case Eq. (3) that dependence vanishes altogether. (d) Thouless exponent ${\lambda }_0$ for the lowest band, referenced to the ground state, displays the same phase dependence as the IPR. (e) The ground state IPR reveals an asymmetry with respect to the sign of $\gamma$. (f) This asymmetry is conspicuous when $J_1$ and $J_2$ are varied for fixed $\theta =0$.

Figure 3

(a) Schematic of the continuum model with two sinusoidal potentials of commensurate periodicities, $p=3$ and $q=5$, in a ring with $\alpha =3/5$. (b) The IPR computed with Hamiltonian Eq. (4) displays behavior analogous to its discrete counterpart [Fig. 2] with matching narrow dips. (c) The continuum dual Hamiltonian, Eq. (5), also reproduces the behavior of its discrete counterpart, with localization ridges now as function of gauge potential $\mathrm{\Omega }$ (units of $2\pi /a$), occurring in the extended regime $\gamma >2$. (d) The Fisher information plotted for the discrete Harper equation, Eq. (1), displays high sensitivity to the phase $\theta$ at the location of the dips in the IPR. (e) Sensitivity to $\mathrm{\Omega }$ also spikes at the position of the partial localization ridges for the continuum dual Hamiltonian, Eq. (5), but the log scale reveals more structure.

Figure 4

The Bose-Hubbard model [16] is used to simulate effects of interparticle interactions in the dual case, shown here for $N=7$ particles in $q=5$ sites for $\alpha =3/5$. The left panels compute the IPR with the ground state and the right panels utilize the single particle density matrix. (a),(b) The IPR computed with no interaction, $U=0$. (c),(d) The IPR computed with interaction $U=1$. (e),(f) The line plots display how the sharpness of a partial localization ridge varies with the interaction strength $U$ at fixed $\gamma =4$.