🔬 condensed matter

Logarithmic singularity in a dynamical quantum phase transition for free fermions

This paper investigates the logarithmic singularity in the dynamical free energy of non-interacting lattice fermions released from a double-domain-wall state, revealing that the dynamical quantum phase transition arises from a topological change in the dominant complex instanton configuration, a mechanism that can be further manipulated via post-selection to induce successive transitions.

Original authors: Yasser Bezzaz, Dimitri M. Gangardt, Pavel L. Krapivsky, Jean-Marc Luck, Kirone Mallick, Sylvain Prolhac

Published 2026-09-17

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CC BY 4.0

Original authors: Yasser Bezzaz, Dimitri M. Gangardt, Pavel L. Krapivsky, Jean-Marc Luck, Kirone Mallick, Sylvain Prolhac

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

In the microscopic world of quantum physics, particles do not always behave like the solid objects we see in daily life. Instead, they exist in a state of probability, described by a wave that can spread out, overlap, and interfere with itself. When a large group of these particles is suddenly disturbed—a process physicists call a "quench"—they begin to evolve in complex ways, trying to settle into a new state. Scientists are deeply interested in how these systems remember their past. If you watch a quantum system evolve and then ask, "How much does it still look like the way it started?", you are measuring something called the Loschmidt echo. This value tells you the probability of the system returning to its original configuration. For a long time, researchers believed that as the number of particles grew larger, this return probability would simply fade away smoothly, like a sound wave dying out in a quiet room. However, recent work suggests that under certain conditions, this fading is not smooth at all. Instead, the system undergoes a sudden, sharp change in its behavior, a moment where the rules of its evolution shift dramatically. This phenomenon, known as a dynamical quantum phase transition, is not about the system settling into a new temperature or arrangement, but about a fundamental change in how the entire collection of particles moves through time.

A team of researchers has now mapped out exactly how this happens in a system of non-interacting fermions, which are a type of particle that cannot occupy the same space at the same time. They studied a specific setup where a block of these particles, initially packed tightly together in a line, is released to spread out across an infinite grid. By tracking the system as it evolves, they discovered that the return probability does not just decay; it hits a critical moment where the mathematical description of the system's path undergoes a topological change. To understand this, imagine the particles as a crowd of people moving through a dark field. In the beginning, the crowd moves as a single, cohesive group along a specific path. As time passes, this path deforms and stretches. The researchers found that at a precise moment, this single path splits into two separate, disconnected loops. This is not a gradual merging or a slow drift; it is a sudden breaking of the path's shape, a transition from one connected form to two distinct forms. This event marks the dynamical quantum phase transition.

The study reveals that this transition is driven by a change in the "instanton" configuration, which is a fancy way of describing the most likely path the system takes through the complex landscape of possibilities. In the early stages of the evolution, the dominant path forms a single, heart-shaped loop in the mathematical space where the particles' momenta live. As time progresses, this heart shape deforms until, at a critical moment, it snaps. The single loop breaks apart into two separate, symmetric loops, resembling a broken heart. This splitting is the key to the transition. The researchers showed that this event causes a specific kind of mathematical singularity, a point where the rate of change in the system's behavior becomes infinitely sharp in a way that is slightly more severe than a standard second-order transition but less severe than a third-order one. They call this a transition of order "two plus zero," a unique signature of this specific type of quantum event.

What makes this discovery particularly significant is that it challenges a common assumption in physics: that real-time evolution can be understood simply by looking at imaginary-time evolution and rotating the clock. In many quantum problems, physicists use a mathematical trick called a Wick rotation to switch between real time and imaginary time, treating them as two sides of the same coin. However, this paper demonstrates that for this specific system, the two are fundamentally different. In imaginary time, the transition involves a gap opening up within a single loop, leaving the overall shape connected but with a hole. In real time, the transition is much more dramatic: the loop actually breaks into two pieces. This topological difference means that the behavior of the system in real time cannot be predicted by simply rotating the imaginary-time solution. The researchers had to develop a new approach, mapping the problem to a model of charged particles repelling each other, to correctly identify the two distinct phases and the moment they switch.

The team also explored what happens if the system is observed in a more general setting, where time is treated as a complex number with both real and imaginary parts. This extension is not just a mathematical exercise; it corresponds to a physical scenario called post-selection, where researchers filter the results of an experiment to focus on specific outcomes. In this complex time landscape, the researchers found that the system can undergo two separate transitions. First, the single heart-shaped path splits into two, just as in the real-time case. But as the observation continues, one of these two paths eventually shrinks and disappears, leaving a single path again, though this time it is different from the original one. This second event, the "death" of a cut, is another dynamical phase transition, also marked by the same sharp, logarithmic singularity. This means that by carefully tuning how the system is observed, one can drive it through a sequence of topological changes, watching the dominant path of the system break and then reform in a new shape.

The findings are supported by both analytical calculations and numerical simulations. The researchers calculated the behavior for a large number of particles and confirmed that the transition occurs at a specific rescaled time, approximately 0.331 times the number of particles. They verified their theoretical predictions by simulating the system with finite numbers of particles and observing the order parameter, a value that tracks the shape of the path, as it moved from the single-loop phase to the two-loop phase and back. The data showed a clear convergence toward the predicted critical point, confirming that the transition is a robust feature of the system. The study also clarified the nature of the singularity at the transition point. While the first and second derivatives of the system's free energy remain continuous, the third derivative diverges, creating a logarithmic spike. This specific behavior distinguishes it from other known phase transitions and highlights the unique nature of dynamical quantum phase transitions in free fermion systems.

Ultimately, this work provides a clear, detailed picture of how a quantum system can undergo a sudden, structural change in its evolution. It shows that the path a quantum system takes is not always a smooth, continuous curve but can undergo abrupt topological shifts, breaking and reforming in response to the passage of time. By identifying the exact moment and nature of these shifts, the researchers have opened a new window into understanding the complex dynamics of many-body quantum systems. The ability to predict and characterize these transitions, especially in the context of post-selected measurements, offers a deeper insight into the fundamental laws governing quantum evolution. The study confirms that even in a simple system of non-interacting particles, the collective behavior can give rise to rich and surprising phenomena, where the very shape of the system's history changes in a sudden, dramatic way.

Technical Summary: Logarithmic Singularity in a Dynamical Quantum Phase Transition for Free Fermions

Problem Statement
The paper investigates the dynamical quantum phase transition (DQPT) in a system of $N$ non-interacting spinless fermions on a one-dimensional infinite lattice. The system is initialized in a double-domain-wall (DDW) configuration, where $N$ adjacent sites are occupied, and evolves under a nearest-neighbor tight-binding Hamiltonian. The primary observable is the Loschmidt echo, $L_N(t) = |\langle \Psi_0 | e^{-iHt} | \Psi_0 \rangle|^2$ , which measures the return probability to the initial state.

While previous studies established that the return probability decays algebraically for large times and exhibits Gaussian decay for short times, the nature of the transition between these regimes remained unresolved. Specifically, a DQPT was previously identified at a critical rescaled time $\tau_c \approx 0.331$ (where $\tau = t/N$ ) via analytic continuation of the Gross-Witten-Wadia (GWW) model, but the dominant saddle-point configuration and the precise nature of the non-analyticity in real time were undetermined. The authors aim to resolve the dominant instanton configuration on both sides of the transition and characterize the singularity of the dynamical free energy.

Methodology
The authors employ a large- $N$ saddle-point analysis by mapping the Loschmidt amplitude to a Coulomb gas problem in the complex plane.

Integral Representation: The Loschmidt amplitude is expressed as an integral over $N$ particle momenta. In the large- $N$ limit with fixed $\tau = t/N$ , this is reformulated as a functional integral over charge densities $\rho(z)$ on a contour $\gamma$ in the complex plane ( $z = e^{iq}$ ).
Coulomb Gas Mapping: The problem is treated as a system of $N$ charges interacting via a repulsive logarithmic potential, subject to a complex external potential $V(z) = \log z + i\tau(z + 1/z)$ . Unlike standard random matrix theory problems where charges are confined to the unit circle, the complex nature of the potential in real time allows the dominant saddle-point configuration to move off the unit circle into the complex plane.
Electrostatic Equilibrium: The dominant configuration is determined by minimizing the action, leading to an electrostatic equilibrium condition where the real part of the potential is constant on the support $\gamma$ . This is solved using the resolvent method and loop equations, reducing the problem to finding the roots of a quartic polynomial $P(z, c)$ , where $c$ serves as an order parameter.
Complex Time Extension: The analysis is extended to complex time $t = |t|e^{i\phi}$ to interpret the Loschmidt amplitude in the context of post-selected work statistics. This involves tracking the evolution of the saddle-point topology across the complex $\tau$ -plane.

Key Contributions and Results

Topological Interpretation of DQPT: The authors identify the DQPT at $\tau_c$ as a topological change in the dominant instanton configuration.
- Short-time regime ( $\tau < \tau_c$ ): The "unbroken-heart" phase. The charge support forms a single, ungapped, heart-shaped contour in the complex plane. The dynamical free energy is $S(\tau) = \tau^2$ .
- Long-time regime ( $\tau > \tau_c$ ): The "broken-heart" phase. The support splits into two symmetric, disconnected cuts. The order parameter $c(\tau)$ is determined numerically by enforcing real filling fractions and electrostatic equilibrium between the cuts.
- Transition Mechanism: The transition corresponds to the "birth of a cut," a phenomenon known in random matrix theory where the genus of the Riemann surface increases by one. This is distinct from the imaginary-time GWW transition, which involves the opening of a gap within a single connected support (genus remains unchanged).
Nature of the Singularity: The paper derives the asymptotic behavior of the dynamical free energy $S(\tau)$ near the critical point $\tau_c$ .
- The first and second derivatives of $S(\tau)$ are continuous at $\tau_c$ .
- The third derivative diverges logarithmically as $\tau \to \tau_c^+$ . Specifically, the singular correction to the derivative scales as $(\tau - \tau_c) \log(\tau - \tau_c)$ , and the second derivative of this singular correction vanishes as $1/\log(\tau - \tau_c)$ .
- The authors classify this as a phase transition of order $2+0$ . It is infinitesimally weaker than a standard second-order transition (where the second derivative would diverge) but more singular than the third-order GWW transition (where the third derivative has a finite discontinuity).
Complex Time and Post-Selection: Extending the analysis to complex time reveals a richer phase structure governed by the post-selected work distribution.
- Three distinct phases emerge: an ungapped one-cut phase, an intermediate two-cut phase, and a gapped one-cut phase.
- The phase diagram features two critical lines: $\tau_{c1}(\phi)$ (birth of a cut) and $\tau_{c2}(\phi)$ (death of a cut).
- The second critical line, $\tau_{c2}(\phi)$ , corresponds to a "death of a cut" transition where one of the two cuts in the intermediate phase becomes completely depleted and collapses, reducing the support to a single gapped cut. This transition is accompanied by the same logarithmic singularity and is therefore also of order $2+0$ .
- On the purely imaginary time axis ( $\phi = \pi/2$ ), the two critical lines merge, and the transition reduces to the standard third-order GWW transition, recovering the known results where logarithmic corrections vanish.
Numerical Validation: The analytical results for the order parameter $c(\tau)$ and the phase boundaries are supported by numerical computations of finite- $N$ Loschmidt echoes using Toeplitz determinants, showing excellent convergence to the large- $N$ saddle-point solutions.

Significance and Claims
The paper claims to provide the first complete analytical characterization of the dominant instanton configuration for the DQPT in free fermions with a DDW initial state. By explicitly solving the electrostatic problem in the complex plane, the authors demonstrate that the DQPT is driven by a topological change (birth of a cut) rather than a simple gap opening. This distinction explains the emergence of a logarithmic singularity, leading to a transition of order $2+0$ , which is more singular than the previously known third-order transitions in imaginary time.

Furthermore, the work clarifies the limitations of analytic continuation from imaginary to real time. While analytic continuation correctly captures the short-time behavior and the imaginary-time phase structure, it fails to predict the intermediate two-cut regime and the specific nature of the real-time singularity. The extension to complex time provides a unified framework linking DQPTs to post-selected work statistics, revealing that post-selection can drive the system through two successive DQPTs associated with the birth and death of cuts in the complex plane.

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