⚛️ high-energy theory

Dynamical Quantum Phase Transitions in a Pseudo-Hermitian Hamiltonian: The Imbalanced-Pairing Kitaev Model

This paper investigates the nonequilibrium dynamics of the imbalanced-pairing Kitaev model, a pseudo-Hermitian system, and demonstrates that dynamical quantum phase transitions under a linear chemical potential ramp occur exclusively when the post-ramp Hamiltonian possesses a real energy spectrum, with their critical times and velocity thresholds strongly dependent on the non-Hermiticity parameter.

Original authors: R. Jafari, Alireza Akbari, Shukhrat Mardonov, H. Yavartanoo

Published 2026-07-28

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: R. Jafari, Alireza Akbari, Shukhrat Mardonov, H. Yavartanoo

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, invisible orchestra. Usually, when physicists study how this orchestra plays, they assume every instrument follows strict, predictable rules where energy is always conserved and nothing is lost to the void. This is the world of "Hermitian" physics, the standard textbook view. But in recent years, scientists have started peeking behind the curtain to see what happens when the orchestra is allowed to leak sound, or when the instruments are slightly out of tune in a very specific, magical way. This is the realm of "non-Hermitian" physics, where systems can gain or lose energy, and where the rules of reality get a little wobbly.

In this strange new world, there's a concept called a "phase transition." Think of it like water turning into ice: the material suddenly changes its nature. But what if you could watch that change happen in real-time, not just by cooling water down, but by watching the atoms dance? That's where "Dynamical Quantum Phase Transitions" (DQPTs) come in. They are like sudden, dramatic "snap" moments in time where a quantum system's behavior flips, even though nothing in the environment has changed yet. The big question scientists are asking is: What happens to these dramatic snaps when the system is allowed to be "leaky" or "unbalanced"? Do the snaps still happen, or does the whole show fall apart?

This paper dives into that exact question using a specific, imaginary model of a superconductor called the "imbalanced-pairing Kitaev model." The researchers, led by R. Jafari and colleagues, decided to simulate what happens when they slowly change the "chemical potential" (a fancy knob that controls how many particles are in the system) over time. They wanted to see if the system would still experience those dramatic "snap" moments (DQPTs) when the physics is slightly broken or unbalanced.

Here is what they found, and it's a bit of a twist in the tale. First, they discovered a strict rule: these dramatic snaps only happen if the system's energy levels remain "real" numbers. If the energy levels turn into complex, imaginary numbers (which happens when the system gets too unbalanced), the snaps vanish completely. It's as if the orchestra stops playing a recognizable tune and just makes noise; without a clear tune, there's no rhythm to snap to.

When the system does keep its real energy levels (which happens when a specific "non-Hermiticity" parameter, called $\gamma$ , is positive), the story gets interesting. If the researchers slowly turned the knob to cross just one critical point (one specific setting where the system changes), a single family of "snap" moments appeared, just like in the normal, balanced world. The unbalanced nature of the system didn't stop the snaps; it just shifted when they happened, like a drummer slightly changing the tempo but keeping the beat.

However, the plot thickens when they turned the knob to cross two critical points at once. In the normal world, this might still produce snaps, but in this unbalanced world, the rules are stricter. The researchers found that if the knob is turned too fast, the snaps disappear entirely. There is a "critical speed" limit. If you turn the knob faster than this limit, the system is too confused to snap. Even more surprisingly, as they made the system more unbalanced (lowering $\gamma$ toward -1), this speed limit got lower and lower. At a specific point where the system is perfectly "staggered" (where $\gamma = -1$ ), the speed limit drops to zero. This means that for any speed at all, the snaps are completely suppressed. The system simply refuses to perform the dramatic transition.

The team used a special mathematical toolkit called the "biorthogonal framework" to figure this out. You can think of this as a pair of glasses that lets them see both the "right" and "left" versions of the quantum states simultaneously, ensuring they don't miss any hidden details in this weird, unbalanced world. They also tracked a "topological order parameter" (a kind of quantum winding number), which acts like a counter that jumps up or down every time a snap happens. They found that when there is one critical point, the counter just jumps up in a steady rhythm. But when there are two critical points, the counter jumps up and then drops down, creating a zig-zag pattern that reveals the complex dance of the two critical points fighting for control.

In short, this paper shows that while unbalanced quantum systems are weird, they aren't chaotic. They have strict boundaries. If the system's energy stays real, the dramatic "snap" transitions can still happen, but they are incredibly sensitive to how fast you change the system and how unbalanced the system is. If you push the system too hard or make it too unbalanced, the magic snaps disappear, leaving the system in a quiet, unchanging state. This helps scientists understand the limits of quantum control in future technologies, like quantum computers, where keeping these delicate "snaps" under control might be the key to making them work.

Technical Summary: Dynamical Quantum Phase Transitions in a Pseudo-Hermitian Hamiltonian

Problem Statement
While Parity-Time (PT) symmetric Hamiltonians are known to possess real energy spectra, they represent only a subclass of the broader class of pseudo-Hermitian Hamiltonians, which also guarantee real spectra. A significant gap exists in understanding the nonequilibrium dynamics of these pseudo-Hermitian systems, particularly under continuous driving protocols (ramps) rather than sudden quenches. Specifically, the behavior of Dynamical Quantum Phase Transitions (DQPTs)—characterized by nonanalyticities in the dynamical free-energy density—remains largely unexplored in non-Hermitian systems that do not conserve particle number. This paper addresses the dynamics of the imbalanced-pairing Kitaev model, a prototypical pseudo-Hermitian system with time-reversal symmetry, subjected to a linearly time-dependent chemical potential.

Methodology
The authors employ a biorthogonal framework to analyze the nonequilibrium dynamics of an integrable system that decomposes into independent two-level Hamiltonians for each momentum mode $k$ .

Biorthogonal Formalism: Unlike Hermitian systems, the right eigenstates of the non-Hermitian Hamiltonian are not orthogonal. The study utilizes a biorthogonal basis consisting of left and right eigenvectors to define a normalized biorthogonal Loschmidt echo ( $L_k(t)$ ) and a corresponding return rate ( $g(t)$ ).
Ramp Protocol: The system is initialized in the ground state of a Hamiltonian with chemical potential $\mu_i$ . A linear ramp, $\mu(t) = \mu_f + vt$ , drives the system to a final value $\mu_f$ at a sweep velocity $v$ .
Analytical Solution: The time-dependent Schrödinger equation for each momentum sector is solved exactly using the Landau-Zener formalism adapted for non-Hermitian systems. This yields exact expressions for the excitation probabilities ( $p_k$ ) and the biorthogonal Loschmidt amplitude.
Topological Characterization: The study calculates the Dynamical Topological Order Parameter (DTOP), defined via the Pancharatnam geometric phase of the biorthogonal Loschmidt amplitude, to provide a topological marker for the transitions.

Key Contributions and Results
The paper establishes the conditions under which DQPTs occur in pseudo-Hermitian systems and characterizes their dependence on the non-Hermiticity parameter $\gamma$ and the sweep velocity $v$ .

Necessity of Real Spectra: The authors demonstrate that DQPTs under a linear ramp protocol occur only when the post-ramp Hamiltonian possesses a real energy spectrum. If the system enters a regime with complex eigenvalues (broken time-reversal symmetry), DQPTs are suppressed.
Single Critical Point Ramps ( $\gamma > 0$ ):
- For ramps crossing a single quantum critical point (where the gap closes at $k=\pi$ ), a unique critical momentum $k^*$ satisfying $p_{k^*} = 1/2$ always exists.
- This results in a single family of critical times ( $t^*_n$ ) and a sequence of DQPTs.
- The non-Hermiticity parameter $\gamma$ continuously shifts the location of $k^*$ but does not suppress the DQPTs or generate additional critical modes, provided $\gamma > 0$ .
Two Critical/Exceptional Point Ramps:
- When the ramp crosses two critical points (for $\gamma > 0$ ) or two exceptional points (for $\gamma < 0$ ), the dynamics are governed by the gap-closing modes at $k=0$ and $k=\pi$ , where $p_k=1$ .
- DQPTs occur only if the minimum excitation probability (near $k=\pi/2$ ) falls below $1/2$ . This condition is met only for sweep velocities $v$ below a critical threshold $v_c$ .
- The critical velocity $v_c$ is derived analytically as $v_c = \frac{\pi \gamma \Delta^2}{\ln(1+\gamma)}$ .
- As the non-Hermiticity parameter $\gamma$ decreases toward the staggered-pairing limit ( $\gamma = -1$ ), $v_c$ decreases and eventually vanishes. For $\gamma < -1$ , DQPTs are completely suppressed for any sweep velocity.
Topological Signatures:
- In the single-critical-point case, the DTOP exhibits successive jumps (or drops) of one unit, indicating a single critical mode.
- In the two-critical-point case, the DTOP shows alternating jumps and drops, reflecting the presence of two critical momenta with opposite slopes in the excitation probability.
- The authors note that critical times extracted from the biorthogonal return rate occur systematically earlier than those derived from conventional (Hermitian conjugate) definitions.

Significance
The paper claims to provide a general analytical framework for describing DQPTs in non-Hermitian superconductors with particle-number non-conserving interactions. By mapping the problem to an effective non-Hermitian Landau-Zener problem, the authors establish a direct connection between pseudo-Hermiticity, exceptional-point physics, and dynamical quantum phase transitions under continuous driving. The results highlight that while the qualitative behavior of DQPTs in the time-reversal symmetric regime ( $\gamma > 0$ ) resembles the Hermitian case, the non-Hermitian parameter critically controls the existence of these transitions in regimes involving multiple critical points. The analytical framework developed is presented as applicable to other integrable non-Hermitian systems, offering a platform for exploring nonequilibrium topology in engineered quantum systems.

More like this