Modular Commutator and Chiral Topological Order

Overview

Implementation and numerical study of the modular commutator, an entanglement-based diagnostic for chiral topological order in two-dimensional gapped quantum phases. This project underlies “Geometric additivity of the modular commutator for multipartite entanglement”, published with Isaac H. Kim and Eun-Gook Moon in Physical Review B 111, 075167 (2025).

Motivation

The modular commutator detects chirality directly from a many-body ground state’s entanglement structure, without relying on edge-mode counting or an explicit measurement of Hall conductance. It was originally defined for tripartite entanglement; extending it to multipartite regions raises a natural question — does the quantity compose additively across geometrically separated regions, and what does that additivity reveal about the underlying topological data of the phase?

Method

  • Defines the modular commutator from the modular Hamiltonians of a ground state’s reduced density matrices, and proves it decomposes additively over a partition with multiple tri-junctions — the commutator of a coarser region reduces to a sum of commutators over simplified regions, plus a residual term that vanishes for invertible (non-anyonic) states.
  • Applies the additivity formula to “pizza” partitions of both the bulk and the physical edge, deriving closed-form multiples of the chiral central charge for different tri-junction arrangements, plus a complementary identity for incomplete junctions that extracts a half-quantized value at smaller subsystem sizes.
  • Verifies the additivity numerically on the Haldane model on a honeycomb lattice, computing the modular commutator directly from ground-state correlation functions across its topological phase transitions.

Results

  • Confirms geometric additivity for invertible states: each complete tri-junction contributes (π/3)c₋, so a bulk pizza partition gives twice the single-tri-junction value while the corresponding physical-edge partition gives twice the value with the opposite sign — even though the edge modular Hamiltonians involved act on disconnected intervals.
  • Generalizes this to a single relation, where a “geometric integer” characterizing the tri-junction arrangement sets the multiple of the chiral central charge on both the bulk and the edge.
  • Numerically reproduces these predictions on the Haldane model to high precision away from its critical points, unchanged under added on-site disorder, and finds qualitatively consistent higher geometric integers on a π-flux square-lattice model.