Chirality reversal at finite magnetic impurity strength and local signatures of a topological phase transition

TL;DR

The paper investigates chirality reversal in a Dirac-cone system perturbed by a dilute Haldane hexagon magnetic impurity on a honeycomb lattice. It integrates defect-superlattice T-matrix analysis, real-space probes (local Chern marker/orbital magnetization), and low-energy current measurements to reveal a robust chirality reversal near $h_c\approx0.9$--$1$ and a zero-density critical point linked to a pole in the impurity-projected T-matrix. A defect-core 6-site toy model captures a local reversal at $h'_c=\sqrt{3}/2$, while a full T-matrix calculation yields a related but slightly shifted critical point $|h_c|\approx0.948$, showing a deep connection between local defect structure and global topology. The results highlight how microscopic impurity details can influence topological transitions and suggest experimental pathways using local magnetometry to detect impurity-induced topological sign changes in Dirac systems.

Abstract

We study the honeycomb lattice with a single magnetic impurity modeled by adding imaginary next-nearest-neighbor hopping ih on a single hexagon. This Haldane defect gives a topological mass term to the gapless Dirac cones and generates chirality. For a small density of defects Neehus et al [arXiv:2405.19289] found that the system's chirality reverses at a critical hc ~ 0.95 associated with an unexpected tri-critical point of Dirac fermions at zero defect density. We investigate this zero-density limit by analyzing a single defect and computing two experimentally relevant measures of chirality: (1) orbital magnetization via local Chern marker, a bulk probe of all occupied states; and (2) electronic currents of low-energy states. Both probes show a chirality reversal at a critical hc ~ 0.9--1. Motivated by this consistency we propose a defect-scale toy model whose low energy states reverse their chirality at hc' ~ 0.87. Remarkably, the same pair of zero energy bound states also generate the critical point hc in the full impurity projected T-matrix. Our results show how the chirality reversal produced by an impurity can be observed either in local probes or in the global topology and suggest a possible role of the microscopic defect structure at the critical point.

Figures (10)

• Figure 1: Chirality reversal transition observable as a sign change of current circulation around a Haldane hexagon defect. (a) The model. Honeycomb lattice with a single Haldane hexagon defect consisting of imaginary hopping $i h$ (oriented counterclockwise, red arrows) on second-neighbor bonds. Circulating current $I_{\text{circ}}$ is defined as the total current flow across the horizontal grey dashed line $+\hat{x}$. (b) Sign flip of $I_{\text{circ}}$ at the critical point $h=h_c$. Main panel: $I_{\text{circ}}$ as a function of impurity scattering strength $h$ showing a sign flip of $I_{\text{circ}}$ at $h_c \approx 0.9$ independent of system size $L$. Current is computed from low energy states (Gaussian energy filter width $\sigma=\alpha/L$ with $\alpha=2.0$). Inset: spatial distribution of currents for $L=25$ at $h$ below and above the transition ($0.7$ and $1.3$; see Appendix \ref{['appendix:visualization']} for additional values). While currents on most lattice bonds (red arrows) clearly show chirality reversal, currents on the core defect bonds (thin black arrows) show more complicated behavior which is analyzed further below.
• Figure 2: Chirality transitions for $l \times l$ defect superlattices. Top: Chern number topological invariant showing a transition as a function of $h$ near $h_c\approx0.95$. Bottom: the energy gap also shows the transition. (a) Left: $l\neq6\mathcal{Z}$ arrays ($\mathcal{Z}$ integer) with distinct $K$, $K'$ Dirac cones show the chirality reversal transition. (b) Right: $l=6\mathcal{Z}$ arrays with Dirac cones overlapping at $\Gamma$ show a chirality extinction. The chirality reversal is the generic case.
• Figure 3: Reversal of orbital magnetization as measured by the integrated local marker $M$. Main panel: $M$ as a function of $h$ over different system sizes. $\text{edge}=0.2$ indicates that boundary strips of width 20% are taken in both directions, and the sites belonging to the edge region are excluded from $M$ integration. Bottom-left: A zoomed-in view around the chirality reversal transition point $h_c$. Top-right: Comparison of $h_c$ obtained from the two different probes: from current flow $I_{\text{circ}}$ (upper/blue lines) and from integrated local Chern marker $M$ (lower/red lines). Dark blue and dark red solid lines correspond to the main parameter values used ($\alpha=2$ and $\text{edge}=0.2$). The surrounding dashed lines are results under other parameter choices ($\alpha=0.5,1,4$; $\text{edge}=0.3,0.4,0.5$). The bulk probe $M$ gives slightly higher values of $h_c$ and shows larger finite size effects compared to $I_{\text{circ}}$.
• Figure 4: Localized and extended currents and their chirality reversals. (a) Distribution of $\hat{\theta}$ component of bond current, $\mathcal{J}(r)$ (Eq.\ref{['eq:Jr']} ), as a function of radial distance $r$ for two values of $h$, below and above $h_c$. The smallest $r$ points, $a,b,c,d,e$, are shown for clarity. Current at $a$ ($b$) is always negative (positive) respectively (see right panel). Bond $c$ is radial hence has $\mathcal{J}=0$. The $\hat{\theta}$ current at $d$,$e$ flips sign between $h=0.7$ and $h=1.3$. Results are calculated for $L=25$, $\alpha=2$. The low energy current is primarily concentrated near the central defect area and decays with distance. The sign flip indicating chirality reversal is visually apparent for $r>1$. (b) Locality of sign flip of low energy circulating current $I_\text{circ}$. $I_\text{circ}$ across the full system (blue line) and $I_\text{circ}$ on bonds of the central hexagon defect contributing to $I_\text{circ}$ (highlighted in inset): two next-nearest-neighbor (NNN) imaginary-hopping bonds (dashed green line); nearest-neighbor (NN) real-hopping bond (dotted green line); and their sum (solid green line). The near cancellation of NNN negative currents and NN positive currents enables the net NN+NNN current on defect core bonds to show a chirality reversal at $h\approx 0.85$, near the total current reversal at $h\approx 0.9$.
• Figure 5: Single hexagon toy model of low energy wavefunction phase windings. Left: Counterclockwise labeling of sites A, B, C on the Haldane hexagon defect. Right: Current $J$ on NN bonds ($J_{AB}$) and NNN bond ($J_{AC}$), and resulting NN+NNN $I_\text{circ}^\hexagon = J_{AB}+ 2J_{AC}$, as functions of wavefunction phase difference $\theta$ between neighboring sites. Currents are expressed in units of $|\psi|^2$. Note that $J_{AC}$ involves the imaginary hopping $h$. The low energy states have $\theta=4\pi/3,5\pi/3$, giving a sign flip for $I_\text{circ}'$ at a critical $h_c^\hexagon=\sqrt{3}/2$.