XX (Δ = 0) Free-Fermion Quench Observables

This page documents the closed-form quench observables exposed by QAtlas.fetch(::XXZ1D, ::LoschmidtEcho{:rate}, ::Infinite; ...) introduced in issue #148 (phase 1).

Setup

The XXZ chain in the spin convention Sᵅ = σᵅ/2,

\[H_{\text{XX}}(J) = J \sum_i \bigl( S^x_i S^x_{i+1} + S^y_i S^y_{i+1} \bigr) ,\]

at Δ = 0 Jordan–Wigner-transforms (after the standard nearest-neighbour string cancellation) into a tight-binding fermion chain without pairing and at zero chemical potential,

\[H_{\text{JW}}(J) = \tfrac{J}{2} \sum_i \bigl( c^\dagger_i c_{i+1} + \text{h.c.} \bigr) , \qquad \varepsilon_J(k) = J \cos k .\]

The half-filled ground state is the Slater determinant of all modes with ε_J(k) < 0,

\[|\mathrm{GS}(J)\rangle = \prod_{k\,:\,J \cos k < 0} c^\dagger(k) \, |\varnothing\rangle .\]

Because there is no Bogoliubov pairing, H_XX(J₀) and H_XX(J_f) are diagonalised in the same plane-wave basis c(k) for every choice of J₀, J_f.

Loschmidt rate function

The Loschmidt rate of the quench H_XX(J₀) → H_XX(J_f) from the initial ground state |ψ₀⟩ = |GS(J₀)⟩ is, by definition,

\[\lambda(t) = -\lim_{N \to \infty} \frac{1}{N} \log \bigl| \langle \psi_0 | e^{-i H_f t} | \psi_0 \rangle \bigr|^2 .\]

In the diagonal basis the amplitude factorises mode by mode:

\[\langle \psi_0 | e^{-i H_f t} | \psi_0 \rangle = \prod_k \langle n_0(k) | e^{-i \varepsilon_{J_f}(k) t (\hat n_k - \tfrac12)} | n_0(k) \rangle = \prod_k \exp\!\left\{ -i \varepsilon_{J_f}(k) t \cdot \bigl( n_0(k) - \tfrac12 \bigr) \right\}\]

with n_0(k) = Θ(-ε_{J₀}(k)) the initial-state occupation. The modulus is identically 1, so

\[\lambda(t) \equiv 0 \qquad \text{whenever } \operatorname{sgn} J_0 = \operatorname{sgn} J_f .\]

Fermi-sea topology

The Fermi sea {k : J cos k < 0} depends on sgn J only. Three regimes appear:

The sign-flip case is the well-known orthogonality catastrophe: any two Slater determinants with complementary occupied-mode sets are exactly orthogonal in the thermodynamic limit, hence |⟨ψ₀ | ψ_f⟩| = 0 and λ(t) = +∞ for every t.

Why this is degenerate (and what's deferred)

The Calabrese–Essler–Fagotti analysis of XX quench dynamics (J. Stat. Mech. (2012) P07016) treats initial states like the Néel state or a dimerised state, which are not Gaussian in the same fermion basis as the post-quench Hamiltonian. Such states induce a non-trivial single-particle Bogoliubov rotation at the quench instant, and the Loschmidt amplitude becomes the textbook integral

\[\lambda(t) = -\int_0^\pi \frac{dk}{\pi} \log \!\left| \cos^2(\Delta\varphi_k) + \sin^2(\Delta\varphi_k)\, e^{2 i \varepsilon_{J_f}(k) t} \right|^2 .\]

The current XXZ1D model carries only (J, Δ) with no magnetic field, dimerisation, or staggered-state machinery, so the only XX → XX quench expressible at present is |GS(J₀)⟩ → e^{-iH_f t} |GS(J₀)⟩, which is the degenerate (λ ≡ 0 / +∞) case derived above. Phase 2 will lift this restriction by adding either

• a magnetic-field generalisation of XXZ1D (so H₀ and H_f differ in chemical potential), or
• a separate XYModel carrying the pairing γ (so the quench rotates the Bogoliubov modes),

at which point the closed-form Calabrese–Essler–Fagotti integral becomes the meaningful return value.

API

fetch(model_f::XXZ1D,
      ::LoschmidtEcho{:rate},
      ::Infinite;
      initial::XXZ1D,
      t::Real) -> Float64

A Δ ≠ 0 model on either side raises DomainError. The kwarg initial is the initial-state Hamiltonian whose ground state is taken as |ψ₀⟩; t is the real evolution time.

Example

julia> using QAtlas

julia> m_f = XXZ1D(; J=1.0, Δ=0.0);

julia> m_0 = XXZ1D(; J=0.5, Δ=0.0);

julia> fetch(m_f, LoschmidtEcho(; mode=:rate), Infinite();
             initial=m_0, t=1.0)
0.0

References

• P. Calabrese, F.H.L. Essler, M. Fagotti, J. Stat. Mech. (2012) P07016.
• M. Heyl, A. Polkovnikov, S. Kehrein, Phys. Rev. Lett. 110, 135704 (2013).
• F.H.L. Essler, M. Fagotti, J. Stat. Mech. (2016) 064002.