Many-body systems with spurious modular commutators

Abstract

Recently, it was proposed that the chiral central charge of a gapped, two-dimensional quantum many-body system is proportional to a bulk ground state entanglement measure known as the modular commutator. While there is significant evidence to support this relation, we show in this paper that it is not universal. We give examples of lattice systems that have vanishing chiral central charge which nevertheless give nonzero "spurious" values for the modular commutator for arbitrarily large system sizes, in both one and two dimensions. Our examples are based on cluster states and utilize the fact that they can generate nonlocal modular Hamiltonians.

Figures (5)

• Figure 1: Geometry used to compute the modular commutator in Eq. (\ref{['eq:CCC']}).
• Figure 2: Depiction of our 1D example violating Eq. (\ref{['eq:CCC']}). Region $ABC$ consists of $2N+1$ even-numbered qubits, with region $B$ containing qubits ranging from $-2M$ to $2M$. Each vertex corresponds to a qubit while the black and blue edges represent a depth-two circuit that creates our state from a product state.
• Figure 3: Translationally invariant 2D system that violates Eq. (\ref{['eq:CCC']}) for the $ABC$ configuration shown. The trapezoidal region can be made arbitrarily large, while a protrusion at the bottom is drawn so that the outer boundary intersects a single blue edge.
• Figure 4: Reduction of $|\psi\rangle$ to equivalent states that consist of (a) chains that have support crossing a boundary of $ABC$ and (b) chains that have support on all regions $A, B, C, D$.
• Figure 5: (a) Cluster state operator $h_i$ for a general graph. (b) One-dimensional cluster state with open boundary conditions for $N=3$. The even-numbered $h_i$ multiply to give a term completely supported below the entanglement cut.