Entanglement negativity for a free scalar chiral current

TL;DR

This work computes the entanglement negativity for the free scalar chiral current in 1+1 dimensions, highlighting the role of Haag duality violations in regions with nontrivial topology. By exploiting the replica approach and the known entanglement spectrum of the chiral current, the authors derive an analytic expression for the negativity of two disjoint intervals valid for all cross-ratio values η ∈ (0,1), and they show an exponential decay at large separations and a double-logarithmic topological term at small separations. A key finding is that the topological contribution to negativity, analogous to that in mutual information, persists in the incomplete neutral algebra, with ΔI = ΔE in the pure-state adjacent limit and a universal behavior for the difference with the fermionic completion. The results are corroborated by detailed lattice simulations, demonstrating excellent agreement with the continuum predictions and illustrating the impact of symmetries on entanglement structure in quantum field theories with nontrivial topology.

Abstract

We study the entanglement negativity for the free, scalar chiral current in two spacetime dimensions, which is a simple model violating the Haag duality in regions with nontrivial topology. For the ground state of the system, both on the line and on the circle, we consider the setups given by two intervals, either adjacent or disjoint. We find analytic expressions for the moments of the partial transpose of the reduced density matrix and the logarithmic negativity. In the limit of small separation distance, this expression yields the same subleading topological contribution occurring in the mutual information. In the limit of large separation distance between the two intervals, the exponential decay of the logarithmic negativity is obtained from its analytic expression. The analytic formulas are checked against exact numerical results from a bosonic lattice model, finding a perfect agreement. We observe that, since the chiral current generates the neutral subalgebra of the full chiral Dirac fermion theory, this analysis highlights how symmetries produce nontrivial features in the entanglement structure that are analogue to those ones already observed in the mutual information for regions with nontrivial topology.

Figures (14)

• Figure 1: A configuration in two spatial dimensions where $V_1$ and $V_2$ are nearly complementary.
• Figure 2: The subsystem $V=V_1\cup V_2$ and its complement $V^{\prime} =V_3\cup V_4$, either on the line (left) or on the circle (right).
• Figure 3: The function $D_{1/2}$, from \ref{['Dn-integral-ACHP']}.
• Figure 4: Comparison of $S^{(1/2)}$ and $S$ for a subsystem given by either one block (left) or two equal blocks (right) in the circle made by $100$ sites.
• Figure 5: Logarithmic negativity for two disjoint intervals on the line. The solid cyan line corresponds to (\ref{['log-neg-from-AbelPlana']}). The data points are obtained from the lattice model, as discussed in Sec. \ref{['renyisvscorrelators']}, for two equal blocks (each of them made by $N_1$ consecutive sites) with varying distance. The inset highlights the large separation regime $\eta \to 0^+$, where an exponential decay \ref{['eq:EN_decay_exp']} occurs (magenta dashed line).