Entanglement negativity in free-fermion systems

Hassan Shapourian, Ken Shiozaki, Shinsei Ryu

Viktor Eisler, Zoltán Zimborás

Yanbai Zhang, Huichao Li, Long-Hui Sheng +2 more

Po-Yao Chang, Xueda Wen

In this paper, we calculate the entanglement negativity in free-fermion systems by use of the overlap matrices. For a tripartite system, if the ground state can be factored into triples of modes, we show that the partially transposed reduced density matrix can be factorized and the entanglement negativity has a simple form. However, the factorability of the ground state in a tripartite system does not hold in general. In this situation, the partially transposed reduced density matrix can be expressed in terms of the Kronecker product of matrices. We explicitly compute the entanglement negativity for the Su-Schrieffer-Heeger model, the integer Quantum Hall state, and a homogeneous one-dimensional chain. We find that the entanglement negativity for the integer quantum Hall states shows an area law behavior. For the entanglement negativity of two adjacent intervals in a homogeneous one-dimensional gas, we find agreement with the conformal field theory. Our method provides a numerically feasible way to study the entanglement negativity in free-fermion systems.

Hassan Shapourian, Shinsei Ryu

Abstract The entanglement entropy of free fermions with a Fermi surface is known to obey a logarithmic scaling and violate the area law in all dimensions. Here, we would like to see how temperature affects the logarithmic scaling behavior. To this end, we compute the entanglement negativity of free fermions using the fermionic partial transpose developed in our earlier paper (Shapourian et al 2017 Phys. Rev . B 95 165101). In one dimension, we analytically derive the leading order term in the finite-temperature entanglement negativity and show how the entanglement negativity indicates a crossover from a quantum entangled state to a classical thermal state, where the entanglement is completely lost. We explain how the one-dimensional result can be generalized to codimension-one Fermi surface of arbitrary shape in higher dimensions. In both one and two dimensions, we check that our analytical results agree with the numerical simulation of free fermions on a lattice.

Sara Murciano, Riccarda Bonsignori, Pasquale Calabrese

We consider the problem of symmetry decomposition of the entanglement negativity in free fermionic systems. Rather than performing the standard partial transpose, we use the partial time-reversal transformation which naturally encodes the fermionic statistics. The negativity admits a resolution in terms of the charge imbalance between the two subsystems. We introduce a normalised version of the imbalance resolved negativity which has the advantage to be an entanglement proxy for each symmetry sector, but may diverge in the limit of pure states for some sectors. Our main focus is then the resolution of the negativity for a free Dirac field at finite temperature and size. We consider both bipartite and tripartite geometries and exploit conformal field theory to derive universal results for the charge imbalance resolved negativity. To this end, we use a geometrical construction in terms of an Aharonov-Bohm-like flux inserted in the Riemann surface defining the entanglement. We interestingly find that the entanglement negativity is always equally distributed among the different imbalance sectors at leading order. Our analytical findings are tested against exact numerical calculations for free fermions on a lattice.