🔬 materials science

Different critical exponents on two sides of the magnetic transition in two dimensions

This study experimentally demonstrates asymmetric two-dimensional critical behavior in a DMI-active hybrid perovskite, revealing distinct Ising-type and XY-type critical exponents below and above the magnetic transition temperature, respectively, a phenomenon attributed to symmetry-breaking Dzyaloshinskii-Moriya interactions.

Original authors: P. Biswal, Ruma Khatun, G. Tripathy, Diptikanta Swain, D. Samal

Published 2026-09-14

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CC BY 4.0

Original authors: P. Biswal, Ruma Khatun, G. Tripathy, Diptikanta Swain, D. Samal

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

Magnetism is a familiar force, the invisible pull that makes a compass needle point north or holds a refrigerator door shut. But when scientists look at how magnetism behaves as a material is heated or cooled, they find a hidden order that governs the transition. Imagine a crowd of people; at high temperatures, they move randomly, but as the room cools, they might suddenly organize into a line. In physics, this sudden organization is called a phase transition. For decades, the standard view was that the rules describing this change—the mathematical "fingerprint" of how the material organizes—were the same whether you were just above or just below the temperature where the change happens. However, a few theoretical ideas suggested that if the material had a specific kind of internal twist or imbalance, these rules could be different on either side of the transition. While this idea had been seen in three-dimensional materials, it remained a mystery whether such a split personality could exist in materials that are essentially flat, or two-dimensional.

A team of researchers in India has now found the first clear evidence that this asymmetry does occur in a two-dimensional world. They studied a specific type of crystal known as a hybrid perovskite, which is built from layers of atoms separated by organic molecules. This structure makes the material behave like a flat sheet of magnets, a state physicists call quasi-two-dimensional. The researchers focused on a version of this crystal made with bromine, which lacks a center of symmetry, allowing for a subtle interaction called the Dzyaloshinskii–Moriya interaction. This interaction acts like a gentle, directional push that prevents the magnetic spins from aligning perfectly straight, introducing a twist to the system. By carefully measuring how the magnetism changed as they warmed and cooled the crystal, the team discovered that the material follows one set of rules when it is cold and a completely different set when it is hot.

Below the critical temperature where the material becomes magnetic, the atoms behave as if they are locked into a rigid, one-dimensional line, following a pattern known as the two-dimensional Ising model. This is a state of strict order. However, as soon as the temperature rises above that critical point, the behavior shifts. The atoms no longer follow the rigid rules; instead, they adopt a more fluid, circular pattern known as the two-dimensional XY model. This means the material effectively changes its fundamental nature depending on which side of the temperature threshold it is on. To confirm that this strange behavior was caused by the specific twist in the bromine crystal, the researchers compared it to two other similar crystals. One was made with chlorine, which is lighter and creates a weaker twist, and the other was a bromine version that was perfectly symmetrical. In both of these control samples, the material behaved normally, following the same set of rules on both sides of the transition. This comparison proved that the asymmetry was not a fluke but a direct result of the specific, symmetry-breaking interaction present only in the bromine crystal.

The findings are significant because they show that in the thin, flat limit of two dimensions, even a very weak disturbance can have a dramatic effect, splitting the rules of physics into two different sets. This challenges the long-held assumption that a single set of rules must describe a material's critical behavior. The researchers observed that the difference between the low-temperature and high-temperature behavior in their flat crystal was even more pronounced than what has been seen in three-dimensional materials. This suggests that the two-dimensional world is uniquely sensitive to these kinds of internal twists. By demonstrating this phenomenon in a material that can be tuned and adjusted, the study opens a new door for understanding how magnetism works in the thinnest possible layers, providing a real-world example of a theoretical prediction that had remained unproven for nearly fifty years.

Technical Summary: Asymmetric 2D Criticality in a DMI-Active Hybrid Perovskite

Problem Statement
While theoretical frameworks proposed by Nelson (1976) and later generalized by Leonard and Delamotte (2015) predict that symmetry-breaking perturbations can induce distinct critical exponents on either side of a continuous phase transition (asymmetric criticality), experimental realization has been limited to three-dimensional (3D) systems. Examples include skyrmion-host magnets like Cu $_2$ OSeO $_3$ and other 3D materials. The existence of such asymmetric scaling in two-dimensional (2D) systems remains an open question. In 2D, enhanced fluctuations make the system highly sensitive to weak symmetry-breaking perturbations, yet it is unclear if these can drive different universality classes (UCs) above and below the critical temperature ( $T_C$ ).

Methodology
The authors investigated the critical behavior of three quasi-2D organic–inorganic hybrid perovskites (OIHPs) using critical scaling analysis of magnetic measurements:

Samples:
- (C $_7$ H $_9$ NBr) $_2$ CuBr $_4$ (NCS-Br): A non-centrosymmetric (NCS) compound with Dzyaloshinskii–Moriya interaction (DMI) active.
- (C $_7$ H $_9$ NCl) $_2$ CuCl $_4$ (NCS-Cl): An isostructural NCS analog with a lighter Cl ligand, expected to have weaker DMI.
- (C $_7$ H $_{10}$ N) $_2$ CuBr $_4$ (CS-Br): A centrosymmetric (CS) analog where inversion symmetry is preserved, suppressing DMI.
Experimental Techniques: Magnetic measurements were performed on single crystals using a SQUID magnetometer. Field-dependent magnetization $M(H)$ was measured near the transition region.
Analysis: The authors employed Modified Arrott Plots (MAPs) based on the Arrott-Noakes relation ( $(H/M)^{1/\gamma} = a\epsilon + bM^{1/\beta}$ ) to extract critical exponents $\beta$ (order parameter), $\gamma$ (susceptibility), and $\delta$ (critical isotherm). They utilized an iterative method to optimize exponents, ensuring the MAPs at $T_C$ pass through the origin and yield parallel straight lines. They also tested for data collapse using renormalized magnetization ( $m = M|\epsilon|^{-\beta}$ ) and field ( $h = H|\epsilon|^{-(\beta+\gamma)}$ ) to verify universality.

Key Results

NCS-Br (DMI-Active): The system exhibits asymmetric critical behavior.
- Below $T_C$ ( $T < 12.5$ K): The critical exponents are $\beta_- = 0.125$ and $\gamma_- = 1.047$ , corresponding to the 2D Ising universality class.
- Above $T_C$ ( $T > 12.5$ K): The critical exponents shift to $\beta_+ = 0.19$ and $\gamma_+ = 1.575$ , corresponding to the 2D XY universality class (specifically the 2D XY/ $h_4$ limit).
- The critical isotherm exponent $\delta$ was experimentally determined to be $9.17(1)$ , consistent with the Widom scaling relation ( $\delta = 1 + \gamma/\beta$ ) for both sides.
- The renormalized $m$ vs. $h$ data collapses into two distinct branches below and above $T_C$ , confirming the asymmetry.
NCS-Cl (Weaker DMI): This compound displays predominantly symmetric critical behavior consistent with the 2D Ising universality class, though with a noted weak asymmetry. The exponents are $\beta \approx 0.144$ and $\gamma \approx 1.429$ . While the normalized slope (NS) curve fits the NS=1 line well below $T_C$ , it deviates slightly from the NS=1 line above $T_C$ . This slight asymmetry is attributed to non-negligible but weaker DMI in this compound compared to NCS-Br.
CS-Br (Centrosymmetric): This compound exhibits symmetric critical behavior with exponents $\beta \approx 0.128$ and $\gamma \approx 1.25$ , characteristic of the 2D Ising universality class throughout the transition.

Key Contributions

First Experimental Realization in 2D: This study provides the first experimental evidence of asymmetric critical exponents in a two-dimensional magnetic system, a scenario previously only established in 3D.
Role of DMI: By comparing NCS-Br with its centrosymmetric and weaker-DMI analogs, the authors isolate the Dzyaloshinskii–Moriya interaction as the specific symmetry-breaking perturbation responsible for the asymmetric scaling. The DMI acts as a "dangerously irrelevant" perturbation that couples differently above and below $T_C$ , altering the effective universality class.
Dimensionality Effects: The study suggests that reduced dimensionality (2D) may provide a more favorable regime for observing the effects of such perturbations, as the difference between low- $T$ and high- $T$ exponents in NCS-Br is larger than in previously reported 3D cases.

Significance and Claims
The paper claims to have experimentally realized a theoretically predicted but previously unobserved phenomenon: asymmetric 2D criticality driven by DMI. The authors assert that the structural flexibility of hybrid perovskites allows for the tuning of dimensionality and symmetry, making them ideal platforms for studying such effects. They conclude that the DMI in the non-centrosymmetric NCS-Br compound breaks the symmetry sufficiently to drive the system from a 2D Ising regime (below $T_C$ ) to a 2D XY-like regime (above $T_C$ ). This finding challenges the conventional expectation of a single set of critical exponents for a given phase transition and highlights the crucial role of symmetry-breaking interactions in determining magnetic criticality in low-dimensional systems. The results are intended to motivate further theoretical and numerical studies on asymmetric criticality in quasi-2D magnets.

Technical Summary: Asymmetric 2D Criticality in a DMI-Active Hybrid Perovskite

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