Quantum Quench — Transverse-Field Ising Chain

Trotterized time evolution, Loschmidt echo, and dynamical quantum phase transitions

Spin expectation values ⟨σᶻᵢ⟩ along the chain

Loschmidt echo |⟨ψ(0)|ψ(t)⟩|² — dips signal dynamical phase transitions

Energy density after quench

Parameters

System size N 8 Pre-quench field h₀ 0.30 Post-quench field h₁ 1.50 Trotter step δt 0.05

State

Time t0.00

Echo |⟨ψ(0)|ψ(t)⟩|²1.000

DQPT dips0

h_c = J = 1.0critical

Transverse-field Ising model: H = −J Σᵢ σᶻᵢσᶻᵢ₊₁ − h Σᵢ σˣᵢ. Critical point at h_c = J = 1. A quantum quench suddenly changes h₀→h₁ across h_c. Loschmidt echo: L(t) = |⟨ψ₀|e^{−iH₁t}|ψ₀⟩|². Non-analytic dips in rate function λ(t) = −(1/N)ln L(t) signal dynamical quantum phase transitions (DQPT). Trotterization: e^{−iHδt} ≈ e^{−iH_ZZδt}e^{−iH_Xδt} with error O(δt²).