Entanglement and negativity

In its topological phase the Kitaev chain hosts a Majorana zero mode at each end, and together they form one fermion shared by the two ends. This page shows that this long-range entanglement is detected by the fermionic logarithmic negativity between the first and last site, and missed by the ordinary partial transpose. On the package side it shows how to reduce a many-body ground state to a non-contiguous subsystem with partial_trace and compute its negativity with logarithmic_negativity, which applies the fermionic partial transpose with the correct signs. This follows Shapourian, Shiozaki and Ryu.^[1]

We take the even-parity ground state (a parity eigenstate, so its reduced state on the non-contiguous ends is well defined) of a chain of N sites, reduce it to the two end sites, and compute the negativity between them. With phase_factors = false the same function uses the ordinary matrix transpose on site 1 instead, as if the modes were qubits.

using FermionicHilbertSpacesimport FermionicHilbertSpaces: logarithmic_negativityusing LinearAlgebra, Arpack, Plots@fermions fkitaev(N, μ; t = 1.0, Δ = 1.0) = sum(-μ * f[i]' * f[i] for i in 1:N) +    sum(-t * f[i]' * f[i + 1] + Δ * f[i] * f[i + 1] + hc for i in 1:(N - 1))function end_negativity(N, μ; phase_factors = true)    H = hilbert_space(f, 1:N, ParityConservation())    Heven = sector(1, H)    _, vecs = eigs(representation(kitaev(N, μ), Heven); nev = 1, which = :SR)    ψ = embed(vecs[:, 1], Heven => H)    Hends = hilbert_space(f, [1, N])    ρ = partial_trace(ψ * ψ', H => Hends)    return logarithmic_negativity(ρ, Hends, hilbert_space(f, [1]); phase_factors)end;

At the sweet spot $\mu = 0$, $t = \Delta$ the end modes are exactly decoupled from the bulk, and the fermionic negativity is $\log\sqrt2$ for every chain length. Away from it the end modes spread into the bulk and less of them sits on the end sites, so the negativity decreases. For the longer chains, deep in either phase, it does not depend on how far apart the ends are and vanishes beyond the transition at $\mu = 2t$; in the short chain the ends also couple through the bulk, which leaves a tail. The bosonic negativity is zero throughout: viewed as qubits, the two ends are not entangled.

@assert end_negativity(8, 0.0) ≈ log(2) / 2μs = range(0, 4, length = 41)plot(xlabel = "μ / t", ylabel = "negativity between the end sites", frame = :box)for N in (4, 8, 12)    plot!(μs, [end_negativity(N, μ) for μ in μs]; label = "fermionic, N = $N", lw = 2)endplot!(μs, [end_negativity(8, μ; phase_factors = false) for μ in μs]; label = "bosonic, N = 8", lw = 2, ls = :dash)hline!([log(2) / 2]; label = "log √2", color = :gray, ls = :dot)

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