Topological Quantum Criticality in Quasiperiodic Ising Chain

Abstract

Topological classifications of quantum critical systems have recently attracted growing interest, as they go beyond the traditional paradigms of condensed matter and statistical physics. However, such classifications remain largely unexplored at critical points in aperiodic environments, particularly under quasiperiodic modulations. In this Letter, we uncover a novel class of topological quasiperiodic fixed points that are intermediate between the clean and infinite-randomness limits. By exactly solving the quasiperiodic cluster-Ising chain, we unambiguously demonstrate that all phase boundaries separating quasiperiodically modulated phases are governed by a new family of topological Ising-like fixed points unique to strongly modulated quasiperiodic systems: Despite exhibiting indistinguishable bulk critical properties, these fixed points host robust topological edge degeneracies and are therefore topologically distinct from previously recognized quasiperiodic universality classes, as further supported by complementary lattice simulations.

Figures (16)

• Figure 1: (a-b) Spatial profiles of couplings $J_{i}$ and $g_{i}$ for two representative points marked in (c) (black square and red star). Parameters: fixed $\phi_{1} = 1/10$ and $\phi_{2} = 1$. (c) Ground-state phase diagram of the QP cluster-Ising model containing FM, QP-FM, SPT, and QP-SPT phases. The vertical black and red lines represent the topological Ising and QP-Topological Ising critical lines, respectively. Symbols indicate points analyzed in later figures.
• Figure 2: (a) Logarithmic-law behavior of the half-chain entanglement entropy $\overline{S_{\rm vN}} = \frac{c_\text{eff}}{3} \log(q) + {\rm const}$. (b) Energy gap $\overline{\delta_\text{e}}$ versus $q$ for strongly and weakly modulated QCP. The dashed lines represent fits to the scaling form $\overline{\delta_\text{e}}(q) \sim 1/q^{z}$. (c-d) Variance of the wandering $\sigma^{2}(S_{l})$ as a function of $l$ for weakly and strongly modulated QCPs, respectively. The blue and green lines are Cesàro means and the red dashed lines are fits of $\sigma^{2}(S_{l}) \propto w \log(l)$; the markers represent the data used in fits. Parameters: $\bar{J} = \bar{g} = 1/2, h_{J} = h_{g} = 1$ [the red star in Fig. \ref{['fig:phase_diagram']}(c)] for strongly modulated QCP and $\bar{J} = \bar{g} = 1/2, h_{J} = h_{g} = 1/3$ [the black square in Fig. \ref{['fig:phase_diagram']}(c)] for the weakly modulated QCP. The parameter $q$ ranges from $13$ to $610$ under PBC for (a) and (b). In (c) and (d), we have fixed $\phi_{1} = 1/10$ and $\phi_{2} = 1$.
• Figure 3: Data collapse of (a) bulk-bulk and (b) boundary-bulk connected correlation functions $\overline{C_\text{FM}(r)}$ for QP-Topological Ising and QP-Ising transitions. The dashed lines are $\propto 1/(r/q)^{2\Delta_{\sigma}^\text{bulk}}$ or $\propto 1/(2r/N)^{\Delta_{\sigma}^\text{bulk} + \Delta_{\sigma}^\text{bdy}}$ for comparison with the data collapse. To achieve a better visualization, we have multiplied an additional global factor for the correlations of QP-Ising as indicated by "$\times 2$" and "$\times 200$". (c) The power-law behaviors of SPT/PM string order; the corresponding scaling dimensions are extracted via data collapse shown in SM Sec. II . (d) The bulk many-body entanglement spectrum $\overline{\xi_{n}}$. Parameters: $\bar{J} = \bar{g} = 1/2, h_{J} = h_{g} = 1$. The parameter $q$ ranges from $13$ to $377$ under PBC for (a) and $N$ ranges from $100$ to $600$ under OBC for (b). In (c) and (d), we have chosen $q = 377$ and $610$, respectively, under PBC.
• Figure 4: (a) Energy gap $\overline{\delta_\text{e}}$ versus $1/q$ for QP-SPT and cluster SPT phases. The black dashed line represents a fit to the scaling form $\overline{\delta_\text{e}}(q) \sim A/q^{\omega} + \overline{\delta_\text{e}}(q \to \infty)$, yielding $\omega \approx 1.00(12)$ with $\overline{\delta_\text{e}}(q \to \infty) \approx 0.002(2)$ . (b) Area-law scaling of the half-chain entanglement entropy $\overline{S_{\rm vN}}$. (c) Scaling behavior of the SPT string order parameter $\overline{O_{\rm SPT}(r)}$. (d) The bulk many-body entanglement spectrum $\overline{\xi_{n}}$. All results are obtained under PBC. Parameters: $\bar{J} = \bar{g} = 1/2, h_{J}= 1, h_{g} = 2$ [the left-pointing triangle in Fig. \ref{['fig:phase_diagram']}(c)] for QP-SPT and $\bar{J} = \bar{g} = 1/2, h_{J} = 1/2, h_{g} = 1/3$ [the right-pointing triangle in Fig. \ref{['fig:phase_diagram']}(c)] for cluster SPT . The parameter $q$ ranges from $13$ to $610$ for (a,b) and is fixed at $q = 377$ and $610$, respectively, for (c) and (d).
• Figure S1: Data collapse of (a) bulk-bulk and (b) boundary-bulk energy-energy correlation functions $\overline{C_\text{EN}(r)}$ for QP-Ising and QP-Topological Ising transitions. The dashed lines are $\propto 1/(r/q)^{2\Delta_{\epsilon}^\text{bulk}}$ or $\propto 1/(2r/N)^{\Delta_{\epsilon}^\text{bulk} + \Delta_{\epsilon}^\text{bdy}}$ for comparison with the data collapse. To achieve a better visualization, we have multiplied an additional global factor for the correlations of QP-Ising or QP-Topological Ising as indicated by "$\times 200$". Parameters: $\bar{J} = \bar{g} = 1/2, h_{J} = h_{g} = 1$ [the red star in Fig. 1(c) in the main text]. The parameter $q$ ranges from $13$ to $377$ in (a) under PBC, and $N$ ranges from $100$ to $600$ in (b) under OBC.