Quasiperiodic Quantum Ising Transitions in 1D

Unlike random potentials, quasiperiodic modulation can induce localization-delocalization transitions in one dimension. In this Letter, we analyze the implications of this for symmetry breaking in the quasiperiodically modulated quantum Ising chain. Although weak modulation is irrelevant, strong modulation induces new ferromagnetic and paramagnetic phases which are fully localized and gapless. The quasiperiodic potential and localized excitations lead to quantum criticality that is intermediate to that of the clean and randomly disordered models with exponents of $\nu =1^{+}$ (exact) and $z\approx 1.9$, ${\mathrm{\Delta }}_{\sigma }\approx 0.16$, and ${\mathrm{\Delta }}_{\gamma }\approx 0.63$ (up to logarithmic corrections). Technically, the clean Ising transition is destabilized by logarithmic wandering of the local reduced couplings. We conjecture that the wandering coefficient $w$ controls the universality class of the quasiperiodic transition and show its stability to smooth perturbations that preserve the quasiperiodic structure of the model.

Figure 1

Phase diagram. The hatched region defines the weakly modulated regime with no weak coupling [$J(Qi)$, $\mathrm{\Gamma }(Qi)>0\forall i$]. The usual gapped ferromagnetic (blue) and paramagnetic (green) phases appear in this regime, separated by a continuous transition in the clean Ising class (segment $AB$). At stronger modulation, we find two new modulated gapless phases: the QP-PM (yellow) and the QP-FM (red). The continuous transitions out of these phases (double and dashed lines) are in the new QP Ising class.

Figure 2

Localization properties of the excitation spectrum on line $ABC$. The low-energy excitations on the segment $AB$ are extended (red) up to a finite energy cutoff $\mathrm{\Lambda }$, above which they are localized (blue). The cutoff $\mathrm{\Lambda }$ vanishes at the multicritical point $B$ so that all finite energy excitations are localized on the segment $BC$. The color quantifies the scaling of the inverse participation ratio $I=\sum _i|{\psi }_i{|}^4\sim q^{-a}$; $a=0$ (1) for localized (extended) states. Parameters: $q=233$, $\mathrm{\Delta }=42\pi /233$, and $\varphi =\sqrt{2}$.

Figure 3

Dynamical scaling at the QP Ising transition. (Left) The maximum energy of the lowest band ${[{\epsilon }_0]}_{\varphi ,\mathrm{\Delta }}$ scales as a power law $q^{-z}$ over 5 orders of magnitude (mean over 5000 $\varphi$, $\mathrm{\Delta }$ samples at each Fibonacci length). (Inset) Least squares fit exponent $z$ as a function of the parameter along the phase boundary. (Center) The integrated density of states $[n(\epsilon ){]}_{\varphi ,\mathrm{\Delta }}\sim {\epsilon }^{1/z}$ over 7 orders of magnitude in energy at the largest size available ($q=4181$). (Right) The inverse localization length $[1/\xi (\epsilon ){]}_{\varphi ,\mathrm{\Delta }}$ is linearly proportional to $n(\epsilon )$, consistent with single-parameter scaling. The deviations from the central trend show sharp features at the log-periodically spaced convergents of the golden ratio $\tau$ (vertical dashed lines). In all panels, the measurements are shown at five different values of $\overline{\mathrm{\Gamma }}/A_{\mathrm{\Gamma }}$ on segment $BC$ of Fig. 1, indicating universality. Standard errors are smaller than the point size; deviations from power law trends are deterministic and due to the QP modulation.

Figure 4

Finite-size scaling of spin correlations. (Main) The average spin correlations $q^{2{\mathrm{\Delta }}_{\sigma }}[⟨{\sigma }_i^x{\sigma }_{i+r}^x⟩{]}_{i,\varphi ,\mathrm{\Delta }}$ collapse when plotted versus fractional separation $r/q$ for critical dimension ${\mathrm{\Delta }}_{\sigma }\approx 0.16$ at $(\overline{J}=\overline{\mathrm{\Gamma }})/A_J=0.5$. (Inset) Least median deviation fit exponent ${\mathrm{\Delta }}_{\sigma }$ is stable along segment $BC$ consistent with universality.