Probing chiral topological states with permutation defects

TL;DR

This work introduces topological multi-entropy measures built from replica-permutation defects to access chirality directly from bulk ground-state wavefunctions. By mapping these measures to partition functions on manifolds with high-genus edge surfaces, the authors separate bulk TQFT data from edge CFT contributions, revealing the chiral central charge $c_-$ (and, in charged cases, the Hall conductance $σ_{xy}$) through universal edge-phase factors. They provide a unified calculation framework using Fenchel–Nielsen coordinates, validate the approach with free-fermion and interacting chiral states, and demonstrate that a finite replica limit suffices to extract key topological data, enabling Monte Carlo and quantum-device studies. The results offer a practical path to quantify chirality from bulk wavefunctions and open avenues for probing edge anomalies and higher central charges in 1+1D and experimental settings.

Abstract

The hallmark of two-dimensional chiral topological phases is the existence of anomalous gapless modes at the spatial boundary. Yet, the manifestation of this edge anomaly within the bulk ground-state wavefunction itself remains only partially understood. In this work, we introduce a family of multipartite entanglement measures that probe chirality directly from the bulk wavefunction. Our construction involves applying different permutations between replicas of the ground state wavefunction in neighboring spatial regions, creating "permutation defects" at the boundaries between these regions. We provide general arguments for the robustness of these measures and develop a field-theoretical framework to compute them systematically. While the standard topological field theory prescription misses the chiral contribution, our method correctly identifies it as the chiral conformal field theory partition function on high-genus Riemann surfaces. This feature is a consequence of the bulk-edge correspondence, which dictates that any regularization of the theory at the permutation defects must introduce gapless boundary modes. We numerically verify our results with both free-fermion and strongly-interacting chiral topological states and find excellent agreement. Our results enable the extraction of the chiral central charge and the Hall conductance using a finite number of wavefunction replicas, making these quantities accessible to Monte-Carlo numerical techniques and noisy intermediate-scale quantum devices.

Figures (18)

• Figure 1: The geometry considered in this work. Three permutation operators are applied on three regions $A,B,C$ of the plane.
• Figure 2: (a) Permutation defect (black) and the enclosing regularization surface $Y$ (purple). The handles of $Y$ have circumferences $\epsilon \ll L$. Cutting along the red lines yields four three-punctured spheres, one around each vertex, here depicted as $Y_{v_i}$. (b) Cross section of the tube on the boundary of region $C$, in the example of $J_1$. Unfolding the permutations around this tube creates two disjoin tubes, one corresponding to replicas 1,2 and one corresponding to replica 3.
• Figure 3: Calculation of the $n$th Rényi entropy using the extended Hilbert space approach. Gluing $n$ copies of $\rho$ results in the partition function on a single copy of $S^3$ with a solid torus excised.
• Figure 4: The structure of $\Sigma_v$ and its mapping to $Y_v$. Here we used the example of $v_1$ and the permutations giving $J_1$, with 3 replicas. Each of the surfaces $Y_v$ is a three-punctured sphere, drawn here as the complex plane with two holes (ramification points) at $z=0$,$1$, and a hole at infinity (blue curves). There are also two additional holes in $\Sigma$ with no ramification (i.e., $f'(z)\neq 0$), which will be ignored later in the paper. The horizontal line is the $t=0$ line, which divides the $t<0$ side of the surface, labeled with the replica index $i$, and the $t>0$ side, labeled with $i^*$. The covering surface $\Sigma_v$ covers a generic point of $Y_v$$R$ times, and is obtained from the permutation operators between the replicas. The black lines in $\Sigma_v$ map to the $t=0$ line in $Y_v$. We indicated with red labels which replica permutation is acted by when the $t=0$ lines are crossed.
• Figure 5: (a) Using seams to measure the twist angles $\tau_i$. The seam (purple line) is first homotoped such that it agrees with the geodesics (blue lines) inside each of the POPs. The angle $\tau_i$ is then the angle traveled by the seam between the two geodesics. (b) In our cases of interest, we might have four different "seams" between the two geodesics. Among them, two will measure the same value of $\tau_i$ (the purple lines), and two (the red lines, the solid one of which is on the front and the dashed one on the back) will measure the values $\tau_i\pm\pi$. We take the value as measured by the two seams that agree.