Comment on "Extension of the adiabatic theorem"

This paper refutes a 2026 conjecture claiming that quantum quenches within the same phase maximize overlap with the post-quench ground state by presenting a specific gapped free-fermion counterexample where this condition fails despite symmetry preservation and a continuous thermodynamic-limit spectrum.

The Gist

The Big Picture: A Broken Promise

Imagine a group of physicists made a bold prediction (a "conjecture") about how nature behaves when you suddenly change the rules of a game. They claimed that if you have a system in a stable state (like a calm lake) and you suddenly change the environment (like turning on a storm), the system is most likely to end up in the new "calm" state, rather than in a chaotic, excited state.

In technical terms, they said: The overlap between the old calm state and the new calm state is always the biggest possible overlap.

This paper says: "Actually, that's not always true."

The author, Jie Gu, has built a specific mathematical model to prove that sometimes, the system is actually more likely to end up in a chaotic, excited state than in the new calm state.

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The Analogy: The "Switching Dance Floor"

To understand how this works, let's use an analogy of a dance floor with two types of dancers: Low-Energy Dancers (who stay on the floor) and High-Energy Dancers (who jump onto a balcony).

1. The Setup (The "Before" State)

Imagine a dance floor where everyone is dancing calmly on the ground level. This is our Initial Ground State ($|GS_i\rangle$). Everyone is happy and low-energy.

2. The Change (The "Quench")

Suddenly, the music changes. The rules of the dance floor shift. Now, the "calm" dance floor looks different.

• The New Ground State ($|GS_f\rangle$) is the new version of everyone dancing calmly on the ground.
• However, because the music changed, the old dancers don't know the new steps perfectly. They are confused.

3. The Physicists' Claim

The previous researchers claimed: "Because the old and new dance floors are in the same 'phase' (they look similar), the dancers will naturally try to find the new calm spot. So, the chance of everyone ending up in the New Calm State is the highest."

4. The Counter-Example (The Twist)

Jie Gu says, "Wait a minute. Let's look at a specific type of dance floor (a free-fermion model)."

He sets up a scenario where the "steps" change so drastically that the old dancers are actually more likely to jump onto the balcony (an excited state) than to stay on the ground.

• The Math Magic: The paper uses a variable called $\phi$ (phi) to control how much the music changes.
  • If the change is small, the dancers stay on the ground.
  • But if the change is large (specifically, if $\phi > \pi/2$), the "confusion" is so great that the probability of a dancer jumping to the balcony becomes higher than the probability of them staying on the ground.

5. The "All-or-Nothing" Result

The most surprising part is that it's not just one dancer jumping.

• If the change is big enough, the paper shows that the most likely outcome is actually for every single dancer to jump to the balcony at the same time.
• In this scenario, the "New Calm State" (everyone on the ground) is actually the least likely outcome, even though it's the "ground state" of the new rules.

Why Does This Matter?

Think of it like a weather forecast.

• The Old Theory: "If the climate changes slightly, the weather tomorrow will most likely be the 'new normal' (sunny)."
• This Paper: "Actually, if the climate shifts in a specific way, the weather tomorrow is more likely to be a massive hurricane (an excited state) than the new normal. The 'new normal' is actually the rarest outcome."

The Takeaway

The author isn't saying the old theory is useless. It works for some specific models (like the Transverse Field Ising Model mentioned in the paper). But the author proves that the general rule proposed by the other scientists is false.

You cannot assume that a system will always settle into its new "ground state" just because the change was smooth. Sometimes, the system prefers to jump into a highly excited, chaotic state instead.

In short: Nature doesn't always pick the "easiest" new path. Sometimes, it takes the most dramatic turn possible.

Technical Summary

Technical Summary: "Extension of the adiabatic theorem" by Jie Gu

1. Problem Statement

The paper addresses a conjecture proposed in a previous work (Ref. [1], Damerow and Kehrein, 2026) regarding the behavior of quantum systems undergoing sudden quenches (non-adiabatic changes) between Hamiltonians residing in the same phase.

The conjecture (Eq. 1 in the original text) posits that for such quenches, the squared overlap between the initial ground state $|GS_i\rangle$ and the post-quench ground state $|GS_f\rangle$ is the maximum possible overlap with any eigenstate of the final Hamiltonian. Mathematically, it claims:
$$\max_n |\langle \psi_n^{(f)} | GS_i \rangle|^2 = |\langle GS_f | GS_i \rangle|^2$$
where $\{|\psi_n^{(f)}\rangle\}$ are the eigenstates of the post-quench Hamiltonian. The author, Jie Gu, aims to test the validity of this conjecture, specifically within the context of local, translationally invariant, gapped free-fermion systems.

2. Methodology

To refute the conjecture, the author constructs a specific counterexample using a one-dimensional, translationally invariant, two-band free-fermion model. The methodology involves the following steps:

• Model Construction:

  • The system is defined by a Hamiltonian $H_\mu = \sum_k \Psi_k^\dagger h_\mu(k) \Psi_k$ in momentum space.
  • Pre-quench Hamiltonian ($H_i$): Defined by $h_i(k) = \epsilon(k)(\cos\phi \sigma_z + \sin\phi \sigma_x)$.
  • Post-quench Hamiltonian ($H_f$): Defined by $h_f(k) = \epsilon(k)\sigma_z$.
  • Dispersion Relation: $\epsilon(k) = m + t \cos k$, with $m > |t| > 0$, ensuring the system is gapped at half-filling.
  • Locality: The specific form of $\epsilon(k)$ (constant + cosine) ensures the Hamiltonians are local nearest-neighbor models in real space.

• Phase Verification:

  • The author demonstrates that $H_i$ and $H_f$ lie in the same phase by constructing an interpolation path $h_s(k)$ for $s \in [0,1]$.
  • This path preserves translation symmetry, particle-number conservation, and, crucially, maintains a finite energy gap (spectrum remains $\pm \epsilon(k)$) throughout the interpolation. This satisfies the premises of Ref. [1].

• Analytical Calculation:

  • The initial ground state $|GS_i\rangle$ is expressed in terms of the eigen-creation operators of the final Hamiltonian ($\beta_{k,\pm}^\dagger$).
  • The overlap between the initial ground state and arbitrary post-quench eigenstates $|S\rangle$ (characterized by a subset $S$ of momenta promoted to the upper band) is calculated analytically.
  • The ratio of the overlap of a single particle-hole excitation to the overlap of the final ground state is derived.

3. Key Results

The analytical derivation yields a general formula for the squared overlap of a state $|S\rangle$ with $|S|$ excited particles:
$$|\langle S | GS_i \rangle|^2 = \left(\cos^2 \frac{\phi}{2}\right)^{N-|S|} \left(\sin^2 \frac{\phi}{2}\right)^{|S|}$$
where $N$ is the number of occupied momenta.

• Violation Condition: The author shows that if the rotation angle $\phi > \pi/2$, then $\sin^2(\phi/2) > \cos^2(\phi/2)$.
• Specific Counterexample:
  • For a single particle-hole excitation ($|S|=1$), the ratio of overlaps is $\tan^2(\phi/2)$.
  • If $\phi = 3\pi/4$, then $\tan^2(3\pi/8) = (1+\sqrt{2})^2 > 1$.
  • Consequently, $|\langle q | GS_i \rangle|^2 > |\langle GS_f | GS_i \rangle|^2$, where $|q\rangle$ is an excited state.
• Maximal Overlap: In the regime where $\phi > \pi/2$, the overlap actually increases monotonically with the number of excitations $|S|$. The state with the maximum overlap is not the ground state, but the highly excited state where all particles are promoted to the upper band ($|S_{max}\rangle$).

4. Conclusion

The paper concludes that the conjecture from Ref. [1] is false in general. Even under strict conditions—locality, translational invariance, gapped spectra, and a symmetry-preserving gapped path connecting the initial and final Hamiltonians—the post-quench ground state does not necessarily have the largest overlap with the initial ground state.

5. Significance

• Refutation of a General Conjecture: This work provides a rigorous mathematical counterexample that invalidates a proposed generalization of the adiabatic theorem for quenches within the same phase.
• Clarification of Previous Results: It suggests that the positive results obtained in Ref. [1] for specific models (like the Transverse Field Ising Model and special cases of the ANNNI model) rely on additional structural features not captured by the broad conjecture.
• Implications for Quantum Dynamics: The findings highlight that in free-fermion systems with large parameter rotations ($\phi > \pi/2$), the system's memory of the initial state is maximized not in the ground state of the new Hamiltonian, but in highly excited states. This has implications for understanding quantum quenches, thermalization, and the validity of adiabatic approximations in non-equilibrium quantum statistical mechanics.

Note: The paper is dated April 2026 and cites a 2026 reference, indicating this is a prospective or future-dated theoretical work within the context of the provided text.