Criticality around the Spinodal Point of First-Order Quantum Phase Transitions

Referee: [Abstract and projection derivation] The central claim that projection onto the subspace 'yields an effective Hamiltonian that exhibits a genuine second-order quantum phase transition' (abstract) is load-bearing. The decoupling via resonant local excitations must be shown to eliminate all relevant virtual processes from the dense many-body spectrum near the spinodal; without explicit bounds on leakage or demonstration that the emergent discrete translational symmetry is independently derived rather than assumed in the subspace definition, residual couplings could introduce relevant perturbations that destroy the SOQPT character or alter the universality class (including Kibble-Zurek scaling).

Authors: We agree that rigorously establishing the decoupling is crucial for the validity of our claims. In the manuscript, we identify the resonant subspace based on local excitations that match the energy scale of the spinodal instability, and the projection is performed by eliminating non-resonant terms. The emergent discrete translational symmetry arises naturally from the uniform nature of the resonant processes across the chain in the tilted Ising model, as the tilt is constant. To address the concern about leakage and virtual processes, we will add a new section or appendix providing perturbative bounds on the leakage amplitude, showing that it is suppressed by the detuning from resonance, which is finite near the spinodal. We will also include numerical evidence from exact diagonalization on small systems demonstrating minimal leakage out of the subspace. This will strengthen the argument that the effective Hamiltonian captures the dominant dynamics without relevant perturbations altering the universality class. revision: partial

Referee: [Validation in tilted Ising chain] In the tilted Ising chain validation, the abstract asserts that the effective model exhibits SOQPT and Kibble-Zurek scaling, but the circularity risk is present: the subspace is defined by the resonant excitations of the theory itself. The paper must provide the explicit form of the projected Hamiltonian and evidence that it is not tautological, e.g., by showing how the symmetry emerges from the original tilted Ising dynamics without being imposed.

Authors: We acknowledge the potential for circularity and will clarify this in the revised manuscript. The subspace is defined by selecting states where the local spin flips are resonant with the driving or the field terms in the original Hamiltonian, specifically those excitations that become gapless at the spinodal. The projected Hamiltonian is obtained by computing the matrix elements within this subspace, resulting in an effective model that maps to a known SOQPT, such as an Ising-like model with transverse field. We will include the explicit form of this projected Hamiltonian in the main text or a dedicated subsection. Regarding the symmetry: in the tilted Ising chain, the original Hamiltonian has a constant tilt, which breaks Z2 but preserves translational symmetry in the effective description because the resonance condition is site-independent, leading to an emergent translationally invariant effective Hamiltonian. This is not imposed but derived from the uniformity of the tilt parameter. We will add a step-by-step derivation showing how the symmetry emerges from the original dynamics. revision: yes