Macrostates versus Microstates in the Classical Simulation of Critical Phenomena in Quench Dynamics of 1D Ising Models

We study critical phenomena in the quench dynamics of one-dimensional (1D) Ising models using truncated Matrix Product States (MPS). While accurate calculation of the full many-body state (microstate) is typically intractable due to the volume-law growth of entanglement, when simulating phases of matter associated with the macrostates of many-body systems, a precise specification of an exact microstate is rarely required. Here we simulate the critical behavior of a Z₂ symmetry breaking dynamical quantum phase transition for a nonintegrable transverse field Ising model with long-range interactions. Our simulations show that even when high-fidelity simulation of the full many-body state is intractable due to exponential scaling with system size, macroscopic quantities like order parameters, the critical point, and critical exponents of a phase transition can be efficiently simulated. We also estimate long-time correlation lengths of the integrable 1D nearest-neighbor transverse field Ising model, finding that properties like long-time correlation lengths, that depend on the exact microstate can also be efficiently simulated because they can be extracted from the short duration behavior of the dynamics. The tractability of simulation using truncated MPS is explained based on quantum chaos and equilibration in the model. We find a counterintuitive inverse relationship, whereby local expectation values are most easily approximated for the most chaotic systems whose exact many-body state is most intractable.