⚛️ quantum physics

Fate of dynamical quantum phase transitions from sudden quench to slow quench limit

This paper demonstrates that while dynamical quantum phase transitions (DQPTs) persist under slow quenching only when crossing an equilibrium quantum phase transition, accidental DQPTs within the same phase are filtered out, thereby establishing slow quenches as a robust dynamical probe for identifying equilibrium quantum critical points.

Original authors: Xiang Zhang, Fuxiang Li

Published 2026-07-20

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CC BY 4.0

Original authors: Xiang Zhang, Fuxiang Li

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, cosmic orchestra. Usually, when we think of music changing, we imagine a conductor slowly waving their baton, guiding the musicians from a gentle lullaby into a thunderous crescendo. This is how things change in our everyday world: gradually, smoothly, and predictably. But in the strange, tiny world of quantum physics—the realm of atoms and subatomic particles—things can change in a heartbeat. Scientists call this a "quench." It's like snapping your fingers and instantly swapping the entire orchestra's sheet music from a jazz tune to a heavy metal riff. The musicians (the particles) are left scrambling to figure out what to play next, creating a chaotic, out-of-sync performance.

For a long time, physicists have been fascinated by what happens during these sudden, jarring changes. They discovered something called a "Dynamical Quantum Phase Transition" (DQPT). Think of this as a moment of pure chaos in the orchestra's performance where, for a split second, the music completely loses its rhythm and the players seem to forget their tune entirely. It's a dramatic event that happens in real-time, not because the temperature changed, but because the rules of the game changed instantly. Scientists care about this because understanding these chaotic moments might help us build better quantum computers, which rely on controlling these very same tiny, jittery particles. But there's a catch: sometimes, the music seems to break even when the conductor didn't actually change the genre of the song. Is the chaos real, or just a fluke?

This is the puzzle that Xiang Zhang and Fuxiang Li set out to solve. They wanted to know: if we slow down the conductor's snap, turning that sudden finger-snap into a slow, gentle wave of the baton, does the chaos still happen? Their research, which uses detailed computer simulations of three different quantum "orchestras," reveals a fascinating truth. They found that the dramatic "chaos" (the DQPT) is actually a reliable signal of a real, fundamental change in the system's nature only when the change crosses a major boundary. However, if the system stays in the same "genre" before and after the change, the chaos that appeared during the sudden snap turns out to be a fake-out—a glitch that disappears when you slow things down.

The authors tested this idea using three specific models: the XY chain (a line of tiny magnets), the Aubry-André model (a chain of particles in a wobbly, repeating landscape), and the trimer Su-Schrieffer-Heeger model (a more complex chain with three types of links). In their simulations, they first performed "sudden quenches," where they instantly switched the settings of the system. They confirmed that even when the system stayed in the same phase (like staying in the ferromagnetic phase for the XY chain or the extended phase for the Aubry-André model), the chaotic DQPTs still popped up. These were the "accidental" glitches.

Then, they switched to "slow quenches." Instead of snapping their fingers, they slowly turned the dial over time. When they did this, the accidental glitches vanished. The DQPTs that appeared when the system stayed in the same phase simply disappeared, as if the orchestra had finally calmed down and found its rhythm. But here is the exciting part: when they slowly changed the settings so that the system crossed a major boundary (like moving from a ferromagnetic phase to a paramagnetic phase, or from an extended state to a localized one), the DQPTs remained robust. They didn't go away. In fact, the timing of these events followed a precise mathematical rule that matched the known properties of the equilibrium phase transition.

The paper also looked at a special case in the trimer Su-Schrieffer-Heeger model where the system has "edge states" (special particles stuck at the ends of the chain). They found that if the change didn't involve the "bulk gap" (the main energy gap in the middle of the system) closing, the DQPTs were just accidental and disappeared under slow quenching. But if the bulk gap did close, the DQPTs stayed, proving they were a true signature of the phase transition.

In short, Zhang and Li's work suggests that if you want to use these chaotic quantum moments to identify a real, fundamental change in a material, you shouldn't just snap your fingers and hope for the best. You need to slow down. By moving slowly, you can filter out the noise and the accidental glitches, leaving behind only the genuine, deep-seated signals of a quantum phase transition. This turns the DQPT from a confusing, sometimes misleading event into a powerful, reliable tool—a "dynamical probe"—that can help scientists pinpoint exactly where and how quantum materials change their nature.

Technical Summary: Fate of Dynamical Quantum Phase Transitions from Sudden Quench to Slow Quench Limit

Problem Statement
Dynamical Quantum Phase Transitions (DQPTs) have been extensively studied as non-equilibrium counterparts to equilibrium phase transitions, characterized by non-analyticities in the rate function of the Loschmidt echo. However, previous investigations have predominantly focused on sudden quench protocols. A significant ambiguity exists regarding the relationship between DQPTs and equilibrium quantum phase transitions (EQPTs). While early studies suggested a direct correspondence when a quench crosses a phase boundary, subsequent research revealed that DQPTs can also occur "accidentally" when the initial and final Hamiltonians reside within the same equilibrium phase. These accidental DQPTs, which may depend on specific initial states or microscopic details, obscure the intrinsic connection between dynamical criticality and equilibrium criticality. The central problem addressed is whether a universal relationship exists between DQPTs and EQPTs and how this relationship can be isolated from accidental dynamical features.

Methodology
The authors investigate the fate of DQPTs by systematically transitioning from sudden quench protocols to the slow quench limit. The study employs a general theoretical framework based on two-band models and extends to multiband systems.

Theoretical Framework: The authors analyze the Loschmidt amplitude $G(t)$ $G (t)$ and the associated rate function $r(t)$ $r (t)$ . They derive the conditions for DQPTs (non-analyticities in $r(t)$ $r (t)$ ) in both sudden and slow quench regimes.
- For sudden quenches, the condition for a DQPT at critical momentum $k_c$ requires the initial ground state to be an equal-weight superposition of the final Hamiltonian's eigenstates ( $|c_{k_c,1}|^2 = |c_{k_c,2}|^2$ ).
- For slow quenches, the system evolves under a time-dependent Hamiltonian $H_k(t)$ . The authors utilize the Landau-Zener-Stückelberg-Majorana formalism to analyze transition probabilities between instantaneous eigenstates. They establish that the condition for DQPTs in the slow limit depends on the product of occupation amplitudes and transition coefficients ( $|p_{k_c,1}||c_{k_c,1}| = |p_{k_c,2}||c_{k_c,2}|$ ).
Numerical Models: The theoretical arguments are validated through comprehensive numerical simulations on three distinct models:
- The XY Chain: A spin-1/2 model exhibiting ferromagnetic and paramagnetic phases.
- The Aubry-André (AA) Model: A quasiperiodic lattice model demonstrating a delocalization-localization transition.
- The Trimer Su-Schrieffer-Heeger (SSH3) Model: A three-band topological model featuring two types of phase transitions (one with bulk gap closing, one without).
Protocols: For each model, the authors compare sudden quenches (instantaneous parameter switch) with slow linear quenches (parameter $h(t) = h_i + \beta t$ ) across both intra-phase (same phase) and inter-phase (crossing a critical point) scenarios.

Key Results

Filtering of Accidental DQPTs: The study demonstrates that DQPTs occurring when the quench remains within a single equilibrium phase (accidental DQPTs) are suppressed in the slow quench limit.
- In the XY chain, DQPTs observed in the ferromagnetic phase under sudden quenches disappear under slow quenches.
- In the AA model, energy-dependent DQPTs (sensitive to the initial state) observed within the extended phase under sudden quenches are eliminated under slow quenches.
Robustness of Genuine DQPTs: Conversely, when the quench protocol crosses an equilibrium quantum phase transition, DQPTs remain robust even in the slow quench regime.
- In both the XY chain and AA model, crossing the critical point (e.g., from ferromagnetic to paramagnetic, or delocalized to localized) results in persistent DQPTs under slow quenching.
Role of Bulk Gap Closure: The connection between slow-quench DQPTs and equilibrium criticality is fundamentally tied to the presence of a bulk gap closure.
- In the SSH3 model, DQPTs are observed under slow quenches only when the transition involves a bulk gap closing (mirror-symmetric case, $t_1 = t_2$ ).
- When the transition involves topological edge states emerging without a bulk gap closing (non-mirror-symmetric case, $t_1 \neq t_2$ ), DQPTs are absent in the slow quench limit, despite appearing in the sudden quench limit.
Scaling Laws: For robust DQPTs (those crossing a gap-closing transition), the first critical time $\tau_c$ exhibits a universal scaling law with the quench rate $\beta$ : $\tau_c \sim \beta^{-\sigma}$ . The exponent $\sigma$ is determined by the critical exponents of the underlying equilibrium phase transition ( $\sigma = z\nu / (1+z\nu)$ ). This scaling holds for both two-band (XY, AA) and multiband (SSH3) models.

Significance and Claims
The paper claims that slow quench protocols serve as a filter to distinguish between accidental dynamical features and intrinsic connections to equilibrium criticality.

Intrinsic Connection: The authors establish that slow quenches reveal a robust, intrinsic link between DQPTs and equilibrium quantum phase transitions, provided the transition involves a bulk gap closure.
Dynamical Probe: Consequently, DQPTs in the slow quench limit can serve as a dynamical probe for identifying equilibrium quantum critical points and extracting their critical exponents.
Limitations: The study modestly notes that this correspondence relies on the presence of bulk gap closure. In its absence (as seen in certain topological transitions without gap closing), dynamical signatures in the slow limit fail to reliably capture the underlying equilibrium critical behavior.

The work does not propose new experimental setups but leverages existing theoretical frameworks and numerical methods to clarify the theoretical landscape of DQPTs, suggesting that the "accidental" nature of many previously observed DQPTs is an artifact of the sudden quench approximation.

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