Entropic Spectroscopy of Spatiotemporal Replica Defects in (1 + 1)D Quantum Ising Dynamics

Abstract

Generalised temporal entropies extend entanglement-based diagnostics from spatial subsystems to cuts through real-time quantum evolution. This thesis develops a spatiotemporal Rényi-2 construction that combines a crossed temporal replica geometry with spatial SWAP readouts, and applies it to the finite open one-dimensional quantum Ising chain. A general Boolean/Möbius framework organises replica-defect geometries, while the selected protocol is formulated as a crossed-replica spacetime tensor network. On the integrable transverse-field line h = 0, an exact Gaussian–Pfaffian evaluator is constructed and benchmarked against DMRG/TEBD; the same protocol is then studied at h = 0.5 with TEBD. At g = 0.5, 1.0, 1.5, the integrable entropy dynamics show phase- and geometry-dependent real-space structure within an exactly known ballistic envelope, and site-resolved Fourier analysis resolves the freefermion energy-difference and two-quasiparticle continua. Breaking integrability reorganises these spectra qualitatively: g = 0.5 exhibits confinement-scale localisation and mesonic structure, g = 1.0 develops a finite lowest branch consistent with the lightest magnetic-Ising E8 scale, and g = 1.5 reveals a mobile interacting branch with two-excitation kinematic structure. Complementary symmetric two-region Rényi-2 mutual-information diagnostics are also evaluated for disconnected spatial supports. In the mobile mixed-field cases, fixed local entropies remain bounded and oscillatory over the accessible T = 8 window while their spatial support continues to expand. These results establish spatiotemporal entropy and entropic spectroscopy as controlled probes of locality, integrability breaking, interacting spectral organisation, and dynamical complexity