A quantum critical Bose gas of magnons in the quasi-two-dimensional antiferromagnet YbCl₃ under magnetic fields

Bose-Einstein condensation (BEC) is a fundamental quantum phenomenon where a large number of bosons occupy the lowest energy state at low temperatures, exhibiting macroscopic coherence. This phenomenon has been extensively studied in various systems, including ultracold atomic gases and superfluids. In the realm of quantum magnetism, BEC has been observed in three-dimensional (3D) antiferromagnets under magnetic fields. In these systems, the application of a magnetic field polarizes the spins, leading to the emergence of magnons—collective spin excitations—that can undergo BEC. However, the behavior of magnons in lower dimensions, particularly in strictly 2D systems, presents a unique challenge. The Mermin-Wagner theorem rigorously states that long-range order cannot exist in 2D systems with continuous symmetry at any finite temperature. This theoretical constraint casts doubt on the possibility of observing a true BEC in strictly 2D systems. However, realistic magnetic materials often exhibit quasi-2D structures, consisting of weakly coupled layers. In such systems, a small interlayer coupling can stabilize BEC, although the underlying physics is still expected to be dominated by the 2D character of the layers. While a few quasi-2D materials have shown evidence of BEC at very high magnetic fields, the experimental accessibility of these high fields often presents significant difficulties. This paper investigates the quasi-two-dimensional (quasi-2D) antiferromagnet YbCl₃, a material initially of interest due to its potential to exhibit Kitaev quantum spin liquid behavior. The authors have studied the material's behavior under relatively low in-plane magnetic fields, observing a magnetic-field-induced transition to a fully polarized state. This transition is investigated as a potential realization of a quantum critical BEC in the 2D limit.

Previous studies have successfully described the magnetic-field-induced transition in 3D antiferromagnets as a Bose-Einstein condensation (BEC) of magnons. Examples include systems like TiCuCl₃, BaCuSi₂O₆, and NiCl₂·4SC(NH₂)₂, where the emergence of BEC-like behavior under specific field conditions has been demonstrated. Theoretical frameworks based on mapping the spin Hamiltonian onto a bosonic system provide a foundation for understanding these observations. The effective Hamiltonian for interacting bosons incorporates the energy dispersion of the magnons and an effective chemical potential. The application of an external magnetic field modifies this effective chemical potential, driving the system towards BEC. In the case of strictly 2D systems, however, the Mermin-Wagner theorem prohibits the existence of long-range order at finite temperatures. Despite this theoretical limitation, the study of quasi-2D systems remains relevant because they offer a unique opportunity to investigate the interplay between 2D and 3D physics. The Berezinskii-Kosterlitz-Thouless (BKT) transition, a hallmark of 2D systems, predicts a transition to quasi-long-range order characterized by algebraically decaying correlations. Experimental investigations of quasi-2D systems have shown signatures consistent with BKT behavior, particularly in low-dimensional magnetic systems near the saturation field.

Single crystals of YbCl₃ were grown using a chemical vapor transport technique. Magnetization (M) measurements below 3 K were performed using a homemade Faraday magnetometer within a ³He-⁴He dilution refrigerator. A careful calibration process ensured accurate determination of the magnetization values, taking into account potential oxidation of the samples. Specific heat (C) measurements were conducted using relaxation calorimetry on aligned crystals, with appropriate corrections for the contribution from the Apiezon-N grease used to protect the samples. Thermal conductivity (κ) measurements were performed using a conventional steady-state method within a ³He-⁴He dilution refrigerator. The thermal conductivity was measured in a temperature range of 0.1–3 K, and the results were corroborated by measurements in a ⁴He cryostat to validate the accuracy in the overlapping temperature range. The high quality of the crystals was confirmed by the negligibly small paramagnetic impurity contribution in magnetization and by the large boundary-limited thermal conductivity. X-ray diffraction was used to determine the crystallographic axis. In all measurements, the magnetic field was applied parallel to the in-plane *a*-axis.

The study reveals that YbCl₃ exhibits a magnetic-field-induced transition to a fully polarized state at a critical field H_c ≈ 5.93 T. Near H_c, the system displays clear signatures of a 2D Bose gas in the dilute limit. The critical exponents observed for the specific heat C(T) and magnetization M(T) strongly suggest a BEC-quantum critical point (QCP) in the 2D limit. This is contrary to the expected behavior in 3D systems. Specifically: C(T) shows a linear temperature dependence at H_c, and M(T) exhibits a linear decrease with temperature from the saturation moment. The temperature of the transition T_c scales linearly with H-H_c near H_c, again consistent with 2D behavior. The effective boson-boson interaction U_eff is estimated to be approximately 0.2J at the QCP, significantly smaller than the values found in typical 3D BEC systems. This reduction is attributed to the logarithmic renormalization of the boson-boson scattering, a characteristic feature of 2D systems. Furthermore, analysis of the thermal conductivity data reveals a significant enhancement near H_c, suggesting a highly mobile nature of the 2D Bose gas. The exceptionally large mean free path of the 2D bosons at the QCP (estimated to be at least 0.3 μm) provides further evidence of the system's unusual properties. The observed BEC behavior at H < H_c is quantitatively described by incorporating a very weak interlayer coupling J_⊥ ≈ 10⁻⁵J, indicating that YbCl₃ is exceptionally close to the 2D limit.

The findings of this study provide strong evidence for a novel type of quantum criticality in a 2D limit. The observed critical exponents for specific heat and magnetization are in excellent agreement with theoretical predictions for a BEC-QCP in 2D. The remarkably low value of the effective boson-boson interaction, along with the high mobility of the Bose gas, highlights the unique nature of this 2D system. The small interlayer coupling plays a crucial role in stabilizing the 3D long-range order below T_c, but its influence is subtle enough to allow the 2D character to dominate the behavior near the QCP. This system provides a unique opportunity to probe the fundamental physics of interacting bosons in the 2D limit, potentially offering deeper insights into phenomena such as the BKT transition.

This research establishes YbCl₃ as a model system for studying quantum critical BEC in the 2D limit. The observed linear temperature dependence of specific heat and magnetization at the QCP, along with the significantly reduced boson-boson interaction and high mobility, are compelling evidence for a highly mobile dilute 2D Bose gas. The weak interlayer coupling stabilizes the 3D long-range order, yet the system remains remarkably close to the 2D limit, providing a valuable platform for future investigations into the fundamental properties of 2D interacting bosons. Further studies could focus on exploring the low-temperature behavior of the system to better understand the interplay between 2D and 3D effects and the precise nature of the low-temperature fluctuations.

The lowest temperature achieved in the experiments (50 mK) is still significantly higher than the energy scale of the interlayer coupling (estimated to be on the order of 0.1 mK). Therefore, a full understanding of the low-temperature behavior and the contribution of the interlayer coupling requires further investigation at even lower temperatures. Additionally, the quantitative analysis of thermal conductivity is challenging due to the difficulty in separating the contributions from phonons and magnetic excitations. More advanced theoretical models incorporating the effects of both 2D and 3D characteristics may be needed for a more complete description of the system's behavior.