🔬 condensed matter

Discovery of a non-Hermitian phase transition in a bulk condensed-matter system

This paper reports the experimental realization of a non-Hermitian phase transition in the bulk ferromagnetic semiconductor EuO, where optical excitation induces a qualitative shift in relaxation dynamics from bi-exponential real to single-exponential complex decay near an exceptional point, offering a new mechanism to control bulk-dynamic properties in condensed-matter systems.

Original authors: Jingwen Li, Michael Turaev, Masakazu Matsubara, Kristin Kliemt, Cornelius Krellner, Shovon Pal, Manfred Fiebig, Johann Kroha

Published 2026-09-14

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CC BY 4.0

Original authors: Jingwen Li, Michael Turaev, Masakazu Matsubara, Kristin Kliemt, Cornelius Krellner, Shovon Pal, Manfred Fiebig, Johann Kroha

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 world of physics, materials usually settle into a state of balance. When a magnet cools down, its internal atoms align in a specific order, and the material changes its properties in a predictable way. This is a phase transition, a fundamental shift that happens while the system remains in thermal equilibrium, a state where energy is evenly distributed and the laws of physics look the same whether time moves forward or backward. For decades, scientists have studied these transitions to understand how matter behaves under different conditions, from the freezing of water to the superconductivity of metals. However, nature also holds states that are far from this calm balance. When a system is driven hard by an external force, such as a burst of light, it can enter a chaotic, non-equilibrium state where energy is constantly being pumped in and lost. In these extreme conditions, the usual rules of symmetry break down, opening the door to entirely new kinds of behavior that cannot exist in a quiet, resting system.

A team of researchers has now captured one of these elusive behaviors in a solid block of material, a discovery that bridges the gap between theoretical physics and the tangible world of condensed matter. Working with a crystal of europium oxide, a ferromagnetic semiconductor, the scientists used ultrafast laser pulses to jolt the material out of its equilibrium state. By carefully measuring how the crystal reflected light over time, they observed a dramatic shift in how the material relaxed back to rest. At certain temperatures, the material's response changed from a simple, predictable decay into a complex, oscillating pattern that defies the standard rules of equilibrium physics. This transition marks the arrival of a non-Hermitian phase transition, a phenomenon where the very nature of the material's dynamics changes at a critical point, creating a new state of matter defined by its motion rather than its static structure.

The experiment took place in a laboratory where a single crystal of europium oxide was cooled to various temperatures, ranging from near absolute zero to room temperature. The researchers fired a short, intense pulse of laser light at the crystal, which acted like a hammer striking a bell, knocking electrons out of their stable positions and sending them into a higher energy state. This created a temporary population of excited particles within the material. To see what happened next, they fired a second, weaker pulse of light at the crystal a tiny fraction of a second later, measuring how much light bounced back. By repeating this process with different time delays and at different temperatures, they built a movie of the material's recovery process.

At lower temperatures, the crystal behaved in a way that was familiar to physicists. The excited electrons relaxed back to their ground state through two distinct, independent pathways, causing the reflected light signal to fade away in a smooth, double-exponential curve. This is typical behavior for a system where energy dissipates in a straightforward manner. However, as the researchers raised the temperature, a strange transformation occurred. At a specific threshold of 84 Kelvin, the two separate decay pathways suddenly merged into one. Instead of fading smoothly, the signal began to oscillate, rising and falling in a complex pattern that could only be described by a single, complex decay rate. This critical point, where the two distinct modes of behavior collapsed into one, is known as an exceptional point, and it signals the onset of a non-Hermitian phase transition.

The researchers were careful to rule out other possible explanations for this strange behavior. They considered whether the negative signal they observed could be caused by common effects like the recombination of electrons and holes in a way that reduces the number of light-emitting particles, or by vibrations in the crystal lattice. They found that none of these standard mechanisms could explain the specific way the signal changed with temperature or the intensity of the laser pulse. The only model that fit the data perfectly was one that accounted for the unique interplay between the excited electrons and the magnetic moments of the europium atoms. In this model, the excited electrons can flip their spin, turning into a "dark" state that cannot emit light, and then flip back. The balance between these two states is tipped by the temperature, and at the critical point, the system loses its ability to distinguish between the two, leading to the observed complex dynamics.

This discovery is significant because it proves that these exotic phase transitions are not just theoretical curiosities confined to tiny, isolated systems like atoms in a vacuum or photons in a cavity. They can occur in a bulk solid, a material you could hold in your hand. The study shows that by driving a material far from equilibrium, scientists can access a new regime of physics where the dynamics of the system are governed by different rules. The transition happens at a temperature distinct from the material's usual magnetic ordering point, suggesting that these two phenomena, while related, are fundamentally different. The researchers found that the point where this transition occurs is not fixed; it can be shifted by changing the intensity of the laser pulse, offering a way to tune the material's properties with light.

The implications of this work extend beyond the specific crystal used in the experiment. The theoretical framework developed by the team suggests that similar transitions could exist in a wide variety of other materials that have coupled components, such as different types of excitations or spins. If these transitions can be controlled, they could provide a highly sensitive way to manipulate the dynamic properties of materials, potentially leading to new technologies for controlling light and magnetism. For now, the finding stands as a clear demonstration that the laws of physics governing the quiet, balanced world of equilibrium are only part of the story. In the chaotic, driven world of non-equilibrium, matter can take on new forms, revealing behaviors that are as rich and complex as the materials themselves.

Problem and Context
Phase transitions are traditionally understood as qualitative changes in the static ground-state properties of a system occurring while it remains in thermodynamic equilibrium. While ultrafast techniques have allowed for the manipulation of magnetic phases on femtosecond timescales, these processes generally remain within the realm of equilibrium physics. A distinct class of phenomena, non-Hermitian phase transitions (NHPTs), arises when a system is driven far from equilibrium. In such systems, the breaking of time-reversal symmetry by excitation and dissipation leads to non-Hermitian dynamics where eigenvalues become complex. At an "exceptional point" (EP), two real eigenvalues coalesce into a single complex-conjugate pair, causing a qualitative change in dynamical behavior rather than static properties. While NHPTs have been theoretically studied and experimentally realized in nanoscopic, photonic, and ultracold atomic systems, their existence in bulk condensed-matter systems has remained unexplored, largely due to the difficulty of establishing the required extreme non-equilibrium states in such materials.

Methodology
The authors experimentally investigated the bulk ferromagnetic semiconductor europium monoxide (EuO). They utilized a pump-probe spectroscopy technique to drive the system out of equilibrium and monitor its relaxation dynamics.

Sample Preparation: High-quality EuO single crystals were grown via the gradient cooling technique and prepared with [100]-oriented surfaces.
Excitation: A 120-fs pump pulse (1.55 eV) excited electrons from the Eu 4f valence band to the 5d(t2g) conduction band, creating a transient population of charge carriers.
Detection: Time-delayed probe pulses (1.31 eV) measured the change in reflectivity ( $\Delta R/R$ ) resonant with the 5d(t2g) $\to$ 4f transition.
Variable Parameters: Experiments were conducted across a temperature range of 10 K to 296 K and with varying pump fluences to map the relaxation dynamics.
Theoretical Modeling: The authors developed a theoretical model based on the Lindblad formalism. This model describes the coupled dynamics of "bright" (optically active, spin-0) and "dark" (optically inactive, spin-1) excitons. The coupling between these states is mediated by the exchange interaction ( $J_{df}$ ) between conduction electrons and Eu 4f local moments, with phonon-assisted spin flips introducing dissipation and breaking time-reversal symmetry.

Key Results
The study reveals a distinct transition in the relaxation dynamics of photoexcited EuO:

Dynamical Transition: At low temperatures (below a critical temperature $T^* \approx 84$ K), the reflectivity decay follows a bi-exponential behavior with two real relaxation rates. Above $T^*$ , the dynamics collapse into a single mode characterized by a complex decay rate, manifesting as a transition from purely positive decay to a sign-changing (oscillatory-like) decay.
Identification of the Exceptional Point: The critical temperature $T^* \approx 84$ K is identified as the exceptional point (EP) where the two real eigenvalues of the relaxation matrix coalesce. This temperature is distinct from and higher than the Curie temperature ( $T_C = 69$ K) of the equilibrium ferromagnetic phase transition.
Scaling Laws: The imaginary part of the relaxation rate ( $\gamma''$ ), which is zero below $T^*$ and non-zero above it, exhibits a square-root dependence on the distance from the critical point: $\gamma'' \sim \sqrt{T - T^*}$ with respect to temperature and $\gamma'' \sim \sqrt{P - P^*}$ with respect to pump fluence ( $P$ ). These scaling behaviors are unique signatures of an NHPT and rule out alternative explanations such as Auger recombination or band-filling effects.
Coupling of Transitions: Theoretical analysis shows that both the equilibrium ferromagnetic transition and the non-Hermitian phase transition originate from the same microscopic coupling mechanism ( $J_{df}$ ), yet they represent fundamentally different phenomena occurring at different critical temperatures. The model successfully reproduces the cusp-like features in relaxation times observed near $T_C$ and the emergence of the EP at $T^*$ .

Significance and Claims
The paper claims the first experimental realization of a non-Hermitian phase transition in a bulk condensed-matter system. The authors demonstrate that EuO supports a new dynamical state of matter characterized by complex relaxation rates, distinct from its ground-state magnetic order.

Fundamental Insight: The work establishes that NHPTs are not limited to engineered nanoscale or photonic systems but can occur in standard bulk materials when driven sufficiently far from equilibrium.
Mechanism: It provides a concrete physical mechanism where the interplay between a Hermitian phase transition (ferromagnetism) and non-Hermitian dynamics (exciton coupling and dissipation) can be disentangled and observed.
Control and Tunability: The authors propose that the critical temperature $T^*$ of the NHPT can be tuned via pump fluence. They suggest the possibility of tuning $T^*$ to coincide with $T_C$ , potentially creating a critical EP with enhanced sensitivity to external parameters, although this specific realization requires further experimental verification of the necessary laser fluence.
Generality: The theoretical framework suggests that similar non-Hermitian phase transitions could exist in a broad class of condensed-matter systems featuring two-component dynamics coupled by a third mode.

The authors conclude that these findings open a pathway to sensitively controlling bulk-dynamic properties in condensed-matter systems through the exploitation of non-Hermitian phase transitions.

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