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arXiv:2604.16439

Comment on "Extension of the adiabatic theorem"

Jie Gu

cond-mat.stat-mechquant-phSubmitted Apr 7, 2026• Announced Apr 8, 2026• 0 citations• Revised Apr 21, 2026 (v2)
v1 · Apr 7, 2026v2 · Apr 21, 2026
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Abstract

Phys. Rev. B 113, 165102 (2026) proposed the conjecture that, for quantum quenches within the same phase, the overlap between the initial ground state and postquench eigenstates is maximal for the postquench ground state. We show that this conjecture is not valid in general. An explicit local, translationally invariant, gapped free-fermion counterexample exists even though the pre- and postquench Hamiltonians are connected by a symmetry-preserving gapped path and the thermodynamic-limit spectrum is continuous.

Journal reference
Phys. Rev. B 114, 177101 (2026)
DOI
10.1103/8dmf-hrs2
Comments
Comment on arXiv:2505.06029

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Phys. Rev. B 113, 165102 (2026) proposed the conjecture that, for quantum quenches within the same phase, the overlap between the initial ground state and postquench eigenstates is maximal for the postquench ground state. We show that this conjecture is not valid in general. An explicit local, translationally invariant, gapped free-fermion counterexample exists even though the pre- and postquench Hamiltonians are connected by a symmetry-preserving gapped path and the thermodynamic-limit spectrum is continuous.
It was recently conjectured and verified for the transverse-field Ising model [Phys. Rev. B 113, 165102 (2026)] that, after a sudden quench within the same equilibrium phase, the initial ground state has its largest overlap with the final ground state. We show that this phase-based criterion is generally false, even in translationally invariant free-fermion systems. For Hamiltonians that factorize into independent 2×22\times 22×2 momentum sectors, we derive the exact necessary-and-sufficient condition for ground-state overlap dominance: the initial and final sector Bloch vectors must have positive dot product for every momentum. This result proves the conjecture in classes where same-phase quenches enforce this geometric condition, but gives explicit same-phase counterexamples in Kitaev chains, where excited final eigenstates can dominate the overlap distribution. We further show that the same obstruction controls real-time Fisher-zero crossings, allowing dynamical quantum phase transitions without crossing an equilibrium phase boundary.
We show that recent results on adiabatic theory for interacting gapped many-body systems on finite lattices remain valid in the thermodynamic limit. More precisely, we prove a generalised super-adiabatic theorem for the automorphism group describing the infinite volume dynamics on the quasi-local algebra of observables. The key assumption is the existence of a sequence of gapped finite volume Hamiltonians which generates the same infinite volume dynamics in the thermodynamic limit. Our adiabatic theorem holds also for certain perturbations of gapped ground states that close the spectral gap (so it is an adiabatic theorem also for resonances and in this sense `generalised'), and it provides an adiabatic approximation to all orders in the adiabatic parameter (a property often called `super-adiabatic'). In addition to existing results for finite lattices, we also perform a resummation of the adiabatic expansion and allow for observables that are not strictly local. Finally, as an application, we prove the validity of linear and higher order response theory for our class of perturbations also for infinite systems. While we consider the result and its proof as new and interesting in itself, they also lay the foundation for the proof of an adiabatic theorem for systems with a gap only in the bulk, which will be presented in a follow-up article.
We show that recent results on adiabatic theory for interacting gapped many-body systems on finite lattices remain valid in the thermodynamic limit. More precisely, we prove a generalized super-adiabatic theorem for the automorphism group describing the infinite volume dynamics on the quasi-local algebra of observables. The key assumption is the existence of a sequence of gapped finite volume Hamiltonians, which generates the same infinite volume dynamics in the thermodynamic limit. Our adiabatic theorem also holds for certain perturbations of gapped ground states that close the spectral gap (so it is also an adiabatic theorem for resonances and, in this sense, “generalized”), and it provides an adiabatic approximation to all orders in the adiabatic parameter (a property often called “super-adiabatic”). In addition to the existing results for finite lattices, we also perform a resummation of the adiabatic expansion and allow for observables that are not strictly local. Finally, as an application, we prove the validity of linear and higher order response theory for our class of perturbations for infinite systems. While we consider the result and its proof as new and interesting in itself, we also lay the foundation for the proof of an adiabatic theorem for systems with a gap only in the bulk, which will be presented in a follow-up article.
Gapped ground states of quantum spin systems have been referred to in the physics\n literature as being `in the same phase' if there exists a family of Hamiltonians H(s), with\n finite range interactions depending continuously on sin[0,1]s \\in [0,1]sin[0,1], such that for each sss,\n H(s) has a non-vanishing gap above its ground state and with the two initial states being\n the ground states of H(0) and H(1), respectively. In this work, we give precise conditions\n under which any two gapped ground states of a given quantum spin system that 'belong to the\n same phase' are automorphically equivalent and show that this equivalence can be\n implemented as a flow generated by an sss-dependent interaction which decays faster than\n any power law (in fact, almost exponentially). The flow is constructed using Hastings'\n 'quasi-adiabatic evolution' technique, of which we give a proof extended to\n infinite-dimensional Hilbert spaces. In addition, we derive a general result about the\n locality properties of the effect of perturbations of the dynamics for quantum systems with\n a quasi-local structure and prove that the flow, which we call the {\\em spectral flow},\n connecting the gapped ground states in the same phase, satisfies a Lieb-Robinson bound. As\n a result, we obtain that, in the thermodynamic limit, the spectral flow converges to a\n co-cycle of automorphisms of the algebra of quasi-local observables of the infinite spin\n system. This proves that the ground state phase structure is preserved along the curve of\n models H(s),0leqsleq1H(s), 0\\leq s\\leq 1H(s),0leqsleq1.
In a recent work by Gong and Ueda [Phys. Rev. Lett. 121, 250601 (2018)], the classification of (1+1)-dimensional quench dynamics for the ten Altland-Zirnbauer classes is achieved, and entanglement-spectrum crossings of the time-dependent states for the topological classes (AIII, DIII, CII, BDI, and D) are discovered as a consequence of the bulk-edge correspondence. We note that their classification scheme for the (1+1)-dimensional bulk topology as well as the adoption of entanglement-spectrum crossings as a diagnosis of topological edge states focus on the limit that the spectrum of the postquench Hamiltonian is flat. When band dispersion of the postquench Hamiltonian is considered, even though the bulk topology is still stable, the entanglement-spectrum crossing at the edge might be unstable due to the fact that the band dispersion can be regarded as a ``symmetry-breaking disorder'' in the frequency domain. Such a symmetry-breaking disorder lowers the symmetry for the edge side, and the conventional bulk-edge correspondence could not be applied directly for the quench dynamics. We show that because of the reduction of symmetry by finite energy dispersion the gapless entanglement-spectrum crossing in the flatband limit in classes AIII, DIII, and CII is unstable and could be gapped without closing the bulk gap. The entanglement-spectrum crossing in classes BDI and D is still stable against energy dispersion. We show that the quench process for classes BDI and D can be understood as a Z2{\mathbb{Z}}_{2}Z2​ fermion parity pump, and the entanglement-spectrum crossing for this case is protected by the conservation of fermion parity.
In this paper, we study the relation between wave-function overlap and adiabatic continuity in gapped quantum systems. We show that for two band insulators, a scalar function can be defined in the momentum space, which characterizes the wave-function overlap between Bloch states in the two insulators. If this overlap is nonzero for all momentum points in the Brillouin zone, these two insulators are adiabatically connected, i.e., we can deform one insulator into the other smoothly without closing the band gap. In addition, we further prove that this adiabatic path preserves all the symmetries of the insulators. The existence of such an adiabatic path implies that two insulators with nonzero wave-function overlap belong to the same topological phase. This relation, between adiabatic continuity and wave-function overlap, can be further generalized to correlated systems. The generalized relation cannot be applied to study generic many-body systems in the thermodynamic limit, because of the orthogonality catastrophe. However, for certain interacting systems (e.g., quantum Hall systems), the quantum wave-function overlap can be utilized to distinguish different quantum states. Experimental implications are also discussed.