Magnetoelectric coupling capable of electrically switching magnetic anisotropy is essential for the development of next-generation magnetic memory and spintronic devices. However, achieving strong magnetoelectric coupling in multiferroic heterostructures is often hindered by the weak and unstable ferroelectricity of conventional ferroelectrics at the ultrathin limit. Here, we propose a strategy based on bilayer aluminum nitride (AlN), a sliding ferroelectric in which opposite out-of-plane polarization states are realized in the AB- and AC-stacked configurations and can be reversibly switched via interlayer sliding, thereby overcoming the scaling limitations of conventional ferroelectrics. Moreover, we combine bilayer AlN with the ferromagnetic monolayer 1T-${\mathrm{MnS}}_2$ to construct an ${\mathrm{MnS}}_2$/bi-AlN multiferroic heterostructure. We find that polarization switching in bilayer AlN strongly modifies the magnetic anisotropy energy (MAE) of the ${\mathrm{MnS}}_2$ layer, where the reversal of AlN sliding ferroelectricity leads to a switching of the magnetic easy axis, while the MAE changes by more than 700%. Furthermore, orbital-projected analysis reveals that this strong magnetoelectric coupling originates from polarization-driven interfacial charge transfer, which modifies orbital hybridization and the crystal field environment at the magnetic ion sites. These results demonstrate that sliding-ferroelectric-based heterostructures provide an effective route toward strong magnetoelectric coupling and electrically switchable magnetic anisotropy, offering opportunities for low-power two-dimensional spintronic devices.

The pump-induced magnon mode (PIM) exhibits strong nonlinear coupling with the uniform Kittel mode (KM), offering a novel paradigm for cavity-free hybrid magnonics. However, the fundamental dynamic properties of the PIM, particularly its evolution under varying external magnetic fields, remain largely unexplored. Here, we systematically investigate the magnetic field dependence of the PIM-KM coupling in yttrium iron garnet (YIG) samples. Using the condition of exceptional points, we extract the PIM excitation threshold at different magnetic fields. Through comparative transmission measurements on a single-crystal YIG sphere and an in-plane magnetized YIG thin film, we quantify the field-driven evolution of the effective nonlinear coupling coefficient ($a_0$). We observe distinct field-dependent behaviors of $a_0$ in the two geometries. The evolution of $a_0$ shows a qualitative correlation with the field-dependent KM linewidth, suggesting that magnetic dissipation plays an important role in PIM dynamics.

With the advent of nonrelativistic spin splitting in collinear compensated antiferromagnets, several candidate materials have also been proposed, among which the family of transition-metal difluorides stands out as a prominent example. Within this family, most members exhibit planar $d$-wave spin splitting, whereas ${\mathrm{CuF}}_2$ shows bulk $d$-wave splitting. In this work, we show that this transition from planar to bulk $d$-wave splitting in ${\mathrm{CuF}}_2$ is primarily driven by the antipolar displacements of the F ions, which are absent in the tetragonal rutile structure of the other family members. Our calculations reveal that these additional structural distortions introduce an extra plane of anisotropic magnetization density, giving rise to an additional totally symmetric component of the magnetic octupole tensor. The $k$-space representation of this octupole component consequently dictates an additional direction of spin splitting, thereby transforming the $d$-wave spin splitting pattern from planar to bulklike. We further analyze the effect of spin-orbit coupling on the magnetic octupoles and the resulting spin splitting in the band structure. Our work highlights the possibility of controlling the pattern of nonrelativistic spin splitting through structural modifications, for example, via the application of external pressure.

Magnetic precursor temperature regimes in low-dimensional magnets provide a unique window into how spin, lattice, and electronic degrees of freedom become coupled before long-range spin order is established. Here we present a comprehensive investigation of the intertwined magnetic and lattice behavior above the Néel temperature ($T_{\mathrm{N}}$) in the van der Waals (vdW) antiferromagnet CrSBr. The results reveal a crossover from high-temperature two-dimensional (2D) XY behavior dominated by short-range ferromagnetic (FM) fluctuations to an intermediate three-dimensional (3D) state dominated by interlayer antiferromagnetic (AFM) correlations preceding the onset of long-range AFM order. The heat capacity shows only the anomaly associated with AFM ordering, further indicating that this intermediate regime does not correspond to an additional thermodynamic phase transition. Polarized Raman measurements reveal phonon self-energy anomalies and relative intensity changes above $T_{\mathrm{N}}$, indicating that the lattice dynamics have already renormalized in the magnetic precursor regime. These findings support the conclusion that dynamic spin-phonon coupling in CrSBr is already established prior to long-range AFM ordering and is mediated by short-range magnetic correlations.

Bilayers composed of two ferromagnetic Chern-insulating ${\mathrm{V}}_2{\mathrm{WS}}_4$ monolayers coupled antiferromagnetically between the layers have recently been identified as a promising platform for realizing an altermagnetic state characterized by a $d$-wave spin texture. In this work, we investigate the impact of spin-orbit coupling (SOC) on the altermagnetic spin texture in bilayer ${\mathrm{V}}_2{\mathrm{WS}}_4$ using first-principles calculations. In the absence of SOC, our nonrelativistic calculations reproduce a distinct altermagnetic state characterized by a $d$-wave spin texture, with a strictly vanishing net magnetization and a momentum-dependent altermagnetic spin polarization. Upon inclusion of SOC, however, the characteristic nonrelativistic $d$-wave spin texture undergoes a continuous SOC-driven reconstruction into a relativistic noncollinear momentum-space spin texture. Our fully relativistic calculations reveal that, although the net magnetization remains exactly zero and the momentum-dependent spin polarization persists, owing to the underlying antiunitary symmetries, the spin texture becomes noncollinear with finite $S_x$ and $S_y$ components beyond the nonrelativistic $d$-wave form. These results demonstrate that, while the symmetry-protected altermagnetic characteristics in bilayer ${\mathrm{V}}_2{\mathrm{WS}}_4$ are robust against SOC, the specific nonrelativistic $d$-wave spin texture is reconstructed by relativistic effects. Our study therefore provides a material-specific first-principles analysis of the SOC-driven reconstruction of the altermagnetic spin texture in bilayer ${\mathrm{V}}_2{\mathrm{WS}}_4$ and highlights how relativistic effects modify its electronic and spin structures.

Nonlinear transport has emerged as a sensitive probe of quantum geometry beyond the Berry-curvature physics of linear response. However, the intrinsic second-order dc response remains conceptually subtle: Different quantum and semiclassical formulations can appear to give different static limits, with different assignments of Fermi sea and Fermi surface contributions. Here we resolve this ambiguity by developing a gauge-consistent density-matrix theory of intrinsic nonlinear conductivity in both the length gauge, where the electric field couples through the position operator, and the velocity gauge, where it enters through the vector potential. We show that the two gauges give the same adiabatic dc response when the same retarded continuation is used for all external frequencies and when the velocity gauge current includes all field-dependent vertices. The apparent Fermi sea terms cancel in the full expression, leaving a Fermi surface quantum geometric contribution determined by the band-normalized quantum metric. This result implies that a fully gapped insulator has no residual dc nonlinear Hall current in the adiabatic clean limit. The reactive part of the Fermi surface term agrees with the original semiclassical Berry-connection-polarizability response, while the dissipative Ohmic sector requires a more careful treatment of relaxation and impurity scattering. Our work establishes the length–velocity gauge equivalence for adiabatic dc quantum geometric nonlinear response and provides a foundation for using nonlinear transport to probe magnetic quantum geometry, especially in $\mathcal{PT}$-symmetric antiferromagnets.

The superconducting (SC) pairing symmetry and its link to time-reversal symmetry breaking (TRSB) in the vanadium-based kagome superconductors remain unresolved, with ambiguities stemming from sublattice interference and charge-density-wave (CDW) entanglement with superconductivity. Using two representative SC pairings, i.e., the conventional on-site $s$ wave and the TRSB $d_{x^2-y^2}+i{d}_{xy}$ wave, as a model study, we theoretically show that while single magnetic impurity yield qualitatively identical spectral behavior of local density of states (LDOS) for these two symmetries, two magnetic impurities give rise to distinct LDOS patterns. For the conventional on-site $s$-wave pairing, time-reversal symmetry (TRS) enforces equivalent forward and backward scattering between two impurities across all impurity configurations, leading to near disappearance of a Yu-Shiba-Rusinov (YSR) state pair along the line connecting the two impurities. However, for the TRSB $d_{x^2-y^2}+i{d}_{xy}$-wave pairing, this scattering equivalence holds only for inversion-symmetric impurity configurations, with a pair of YSR disappearance restricted to this case. These distinct spectral features are resolvable in scanning tunneling microscopy (STM) experiments, providing a direct avenue to discriminate TRSB and non-TRSB SC pairing symmetries in kagome superconductors and an alternative method to probe SC nonreciprocity that circumvents the ambiguities of conventional critical current-based techniques.

The interplay between geometric frustration, spin-orbit coupling, and structural disorder can give rise to unconventional magnetic ground states in rare-earth triangular-lattice magnets. We report low-temperature magnetic and ${}^7\mathrm{Li}$ NMR investigations of the disordered triangular-lattice compound ${\mathrm{LiCaYb}}_5({\mathrm{BO}}_3{)}_6$ (LCYBO). Rietveld refinement confirms a hexagonal $P{6}_{5}22$ structure with partial Ca/Yb antisite disorder and fractional Li occupancy. Magnetic susceptibility and magnetization measurements indicate a well-isolated effective $J_{\mathrm{eff}}=1/2$ Kramers doublet with weak antiferromagnetic interactions $({\theta }_{\mathrm{CW}}\approx -0.54\mathrm{K})$. The specific heat reveals a weak anomaly near 0.43 K, suggesting the development of short-range correlated magnetism rather than conventional long-range order. ${}^7\mathrm{Li}$ NMR spectra broaden strongly upon cooling, consistent with increasingly inhomogeneous internal magnetic fields. The spin-lattice relaxation exhibits two relaxation components consistent with disorder-induced distributions of local magnetic environments arising from antisite disorder and competing exchange pathways. Our results establish LCYBO as a structurally disordered frustrated triangular-lattice magnet with correlated low-energy spin dynamics.

Magnetoelectric (ME) effects in antiferromagnets provide a fertile platform for exploring symmetry-driven cross-correlated responses. However, their microscopic origin remains elusive and is often obscured in simplified low-energy descriptions. In this study, we revisit the microscopic mechanism of the ME effect in a collinear antiferromagnetic zigzag chain by employing a multiorbital tight-binding model that explicitly includes both $s$- and $p$-orbital degrees of freedom. Using analytical and numerical calculations based on the Kubo formula, we demonstrate that the ME response is governed by orbital degrees of freedom activated through $s\text{−}p$ hybridization, while the spin contribution vanishes because of spin conservation. To elucidate the low-energy description, we derive an effective Hamiltonian projected onto the $s$-orbital subspace using the Schur complement. We show that a naive application of the Kubo formula within this effective model fails to capture the ME response. This issue is resolved by systematically incorporating vertex corrections in terms of orbital hybridization into the response functions. Furthermore, by introducing a quasiparticle renormalization scheme, we formulate a renormalized Kubo formula that preserves conservation laws and accurately reproduces the full multiorbital results. Our analysis revisits the conventional low-energy perspective and reveals that the ME effect originates from virtual interorbital processes encoded in vertex corrections, rather than from the bare low-energy Hamiltonian. The effective framework developed here provides a unified microscopic understanding of orbital-driven ME responses and offers a systematic route to incorporating hybridization effects beyond simple low-energy models.

We introduce a method for constructing global approximations to correlation functions of strongly interacting quantum field theories, starting from perturbative results. The key idea is to employ an interpolation method, such as the two-point Padé expansion, to interpolate between the weak- and strong-coupling expansions of the correlation function. We benchmark this many-body interpolation approach on two prototypical models: the lattice ${\varphi }^4$ field theory and the two-dimensional Hubbard model. For the ${\varphi }^4$ theory, the resulting two-point Padé approximants exhibit uniform and global convergence to the exact correlation function. For the Hubbard model, we show that even at second order, the Padé approximant already provides a reasonable characterization of the Matsubara Green's function for a wide range of parameters. Finally, we offer a heuristic explanation for these convergence properties based on analytic function theory.

Two-dimensional multiferroics that host nontrivial topological bands offer a rich platform for correlated and tunable quantum phenomena, yet such materials remain rare. Here, using first-principles calculations, we reveal that monolayer ${\mathrm{CuVP}}_2{\mathrm{Se}}_6$ unites a tunable triple-well ferroelectric transition with a spin-polarized Chern flat band. The ferroelectric and paraelectric phases are close in energy and can be reversibly switched by moderate strain or an electric field. During the transition, a kagomelike flat band emerges near the Fermi level, which we describe via a minimal three-orbital tight-binding model on a triangular lattice. Furthermore, the system exhibits sizable magnetic anisotropy and a magnetization-dependent Chern insulating state: The Chern number is $C=\pm 1$ for out-of-plane magnetization but becomes trivial when the moments rotate in-plane. These findings establish ${\mathrm{CuVP}}_2{\mathrm{Se}}_6$ as a promising candidate for exploring electrically tunable flat-band correlations and topological magnetism in a multiferroic monolayer.

We present a comparative Ni core level photoemission study of ${\mathrm{La}}_3{\mathrm{Ni}}_2{\mathrm{O}}_7, {\mathrm{Nd}}_3{\mathrm{Ni}}_2{\mathrm{O}}_7$, and ${\mathrm{LaNiO}}_3$ using both the $\mathrm{Ni}2p$ and the $\mathrm{Ni}1s$. We address the challenges in analyzing the widely investigated $\mathrm{Ni}2p$ spectra arising from the substantial overlap in energy of the $\mathrm{Ni}2p$ with the La $3d$. We show that, on the other hand, the deep $\mathrm{Ni}1s$ core level does provide a clean view on the intrinsic electronic excitations and we highlight its potential to resolve detailed differences in the electronic structure within the strongly correlated Ruddlesden-Popper series ${\mathrm{La}}_{n+1}{\mathrm{Ni}}_n{\mathrm{O}}_{3n+1}$.

We present a comprehensive investigation of the magnetic, thermodynamic, and transport properties of ${\mathrm{Fe}}_{0.925}{\mathrm{Cr}}_{0.075}\mathrm{Si}$ to clarify the origin of its low-temperature metallic behavior. Although Cr substitution introduces a small number of itinerant $d$ electrons and modestly enhances the high-temperature electronic specific heat, the conduction-electron density of states inferred from ${\chi }_0$ and ${\gamma }_{\mathrm{el}}$ remains low and nearly temperature independent, indicating that impurity-band filling plays only a minor role in establishing metallicity. At elevated temperatures the system behaves as a narrow-gap semiconductor, whereas below 150 K it enters a metal-insulator crossover regime characterized by a competition between a thermally activated insulating channel and an unconventional metallic channel. Magnetic susceptibility, heat capacity, and electrical resistivity consistently demonstrate that this emergent metallic state is driven not by enhanced itinerancy but by strong, predominantly finite-$q$ spin fluctuations with a characteristic scale $T_{\mathrm{SF}}\approx 20\mathrm{K}$. The large low-temperature enhancement of $\gamma$ combined with only a weak increase in the uniform susceptibility yields a Wilson ratio and Stoner enhancement factor far below unity, underscoring the non-Pauli, fluctuation-dominated nature of the metallic state. At the lowest temperatures ($<4\mathrm{K}$), the resistivity upturn is well described by quantum-interference corrections in the dirty-metal limit, indicating that coherence effects develop atop a fluctuation-renormalized diffusive electronic background. These results establish ${\mathrm{Fe}}_{0.925}{\mathrm{Cr}}_{0.075}\mathrm{Si}$ as a rare example of a correlated semiconductor in which a spin-fluctuation-induced metallic phase dominates over impurity-band contributions and evolves into a quantum-corrected regime at ultralow temperatures.

We show that a $d$-wave altermagnet can be transformed into a Chern insulator by irradiating it with elliptically polarized light from a high-frequency photon beam. We further explore the intrinsic anomalous thermoelectric and thermal Hall effects in light-irradiated altermagnets. At low temperatures, the thermoelectric Hall coefficient exhibits a linear temperature dependence but vanishes within the energy gap between the conduction and valence bands near the $M$ point. However, it displays pronounced peaks and dips at the gap boundaries near both the $M$ and $\mathrm{\Gamma }$ points, suggesting that thermoelectric Hall conductivity is a sensitive probe for these gapped regions. Similarly, the low-temperature thermal Hall coefficient, which also shows a linear temperature dependence, becomes quantized across the bandwidth, reflecting the underlying topological character of the light-induced Chern insulating phase. These results establish thermoelectric and thermal Hall transports as powerful signatures of topology in driven altermagnetic systems.

We conducted a comprehensive and comparative muon-spin relaxation and rotation (μSR) investigation on two fcc-lattice metallic compounds, ${\mathrm{HoCdCu}}_4$ $(T_{\mathrm{N}}\approx 8\mathrm{K})$ and ${\mathrm{HoInCu}}_4$ $(T_{\mathrm{N}}\approx 0.76\mathrm{K})$, to elucidate the nature of their magnetic ground states and the role of frustration in stabilizing them. Our μSR results reveal that, in contrast to ${\mathrm{HoCdCu}}_4$, strong magnetic frustration exists in ${\mathrm{HoInCu}}_4$. Notably, in ${\mathrm{HoInCu}}_4$, the majority of the Ho-moments exhibit dynamic correlations and persistent spin dynamics down to 0.3 K, resembling a quantum spin-liquid (QSL) behavior, and the remaining Ho-moments show static correlation. By contrast, in ${\mathrm{HoCdCu}}_4$, all the Ho-moments contribute to the magnetic order below $T_{\mathrm{N}}$. Furthermore, in ${\mathrm{HoInCu}}_4$, the temperature dependence of the relaxation rate indicates the presence of quantum critical fluctuations in the paramagnetic state near $T_{\mathrm{N}}$, suggesting the proximity to a quantum critical point (QCP). These observations suggest that the ground state of ${\mathrm{HoInCu}}_4$ is a proximate quantum spin liquid (PQSL), hitherto unobserved in frustrated metallic systems. Our μSR findings are further corroborated by recent inelastic neutron results on ${\mathrm{HoInCu}}_4$, which show similarities to other insulating PQSL candidates, thus reinforcing our conclusions.

The interplay of strong atomic spin-orbit coupling with Jahn-Teller (JT) lattice distortions is an important theme in quantum materials hosting heavy atoms. A prototypical example of such a system is a degenerate $j=3/2$ multiplet coupled to local phonon modes. Here, we study the multipolar ground states of this system as realized in Mott insulators, explore its thermal phase diagram via an $SU\left(4\right)$ spin Monte Carlo approach coupled to JT phonons, and study the resulting temperature dependent spectral distribution of the $j=3/2$ multiplet. Our work sheds light on coexisting distinct multipolar orders engendered by phonon coupling, role of thermal JT fluctuations, and the entropy of the coupled multipole-phonon system. We discuss broad implications for double perovskites ${\mathrm{Ba}}_2{\mathrm{MgReO}}_6, {\mathrm{Ba}}_2{\mathrm{NaOsO}}_6$ and lacunar spinels such as ${\mathrm{GaTa}}_4{\mathrm{Se}}_8$ and ${\mathrm{GaNb}}_4{\mathrm{Se}}_8$.

This study resolves a previously reported mechanical instability of the experimentally observed superconducting P4/ncc phase of ${\mathrm{Mg}}_4{\mathrm{AlSi}}_3$ using first-principles calculations. Previous theoretical work reported a negative ${\mathrm{C}}_{66}$ elastic constant for the P4/ncc structure and consequently proposed a lower symmetry Pccn structure as the mechanically stable phase. Here, we reinvestigate the elastic properties of P4/ncc ${\mathrm{Mg}}_4{\mathrm{AlSi}}_3$ using both energy-strain and stress-strain approaches. We find that all independent elastic constants are positive and satisfy the mechanical stability criteria, with the two approaches yielding consistent results. The absence of imaginary modes in the calculated phonon dispersion further establishes the dynamical stability of the phase, with the negative formation enthalpy indicating its thermodynamic stability. We further investigate for the first time the microscopic origin of its superconductivity through the electron-phonon coupling calculations based on Migdal-Eliashberg theory. The calculated electron-phonon coupling constant $\lambda$ of 0.618 and a logarithmic average frequency of 205.174 K, lead to a superconducting transition temperature $T_c$ of 5.123 K, in close agreement with the experimental value of 5.2 K. Therefore our results resolve the previous discrepancy concerning the mechanical stability of the experimentally realized P4/ncc phase and provide a consistent microscopic description of its lattice stability and phonon-mediated superconductivity.

We investigate how long-range power-law interactions ($V\propto r^{-\alpha }$) influence the finite-size stability of many-body localization against boundary-induced thermalization in a one-dimensional disordered Heisenberg spin chain. Whereas previous studies have primarily examined avalanche instabilities through perturbative analyses or short-range models, we adopt an open-system approach that directly probes the dynamical response of a localizedlike regime to a controlled boundary thermalizing perturbation. By combining exact diagonalization of static properties with Lindblad master-equation simulations, we systematically examine the interplay among the interaction exponent $\alpha$, disorder strength $W$, and system size $L$. A finite-size scaling analysis of the half-chain entanglement entropy identifies an effective crossover near $\alpha \simeq 2$, separating a strongly long-range regime exhibiting enhanced delocalizing tendencies from a short-range-like regime with more robust localization signatures. To characterize avalanchelike dynamics, we couple one boundary of the chain to an infinite-temperature bath that serves as a minimal thermalizing seed and monitor the resulting bath-induced decay of the staggered magnetization imbalance. Within the accessible system sizes and simulation times, the characteristic threshold time is well described by the empirical scaling relation $T_{r_{\mathrm{th}}}\sim exp\left[\kappa \left(\alpha \right)LW\right]$, where the coefficient $\kappa \left(\alpha \right)$ increases monotonically with increasing interaction exponent. This trend demonstrates that shorter-range interactions provide substantially stronger resistance to boundary-induced thermalization in the strongly disordered regime. Motivated by this scaling behavior, we introduce an interaction-dependent disorder scale $W_{\mathrm{stab}}\left(\alpha \right)$ as an operational measure of avalanche resistance. Rather than representing a sharp thermodynamic phase boundary, this quantity provides a conservative, protocol-dependent characterization of finite-size dynamical stability. Our results show that long-range interactions facilitate bath-induced thermalization, whereas increasing the interaction exponent significantly enhances the robustness of localizedlike behavior against boundary perturbations.

The magnetoelectric (ME) effect, the induction of electric polarization ($P$) by an applied magnetic field or magnetization ($M$) by an electric field, provides a pathway for low-power-consumption data memory devices. In this work, we present a systematic experimental investigation of the magnetism, heat capacity, and anisotropic ME effect in $\mathrm{BaN}{\mathrm{i}}_2{\left(\mathrm{P}{\mathrm{O}}_4\right)}_2$ single crystals. Our results reveal long-range antiferromagnetic (AFM) ordering below the Néel temperature $T_N=24\mathrm{K}$, where the $\mathrm{N}{\mathrm{i}}^{2+}$ spins are AFM aligned along the $x$ // [1-10] axis. Detailed ME measurements reveal that $\mathrm{BaN}{\mathrm{i}}_2{\left(\mathrm{P}{\mathrm{O}}_4\right)}_2$ exhibits dominant out-of-plane electric polarization and significant in-plane electric polarization along the $y$ // [110] direction. The largest magnetically induced electric polarization of 128 $\mu \mathrm{C}/{\mathrm{m}}^2$ along the $z$ // [001] axis is observed at $T =2$ K under the magnetic field of ${\mu }_{0}H=9$ T applied along the $x$ direction, with a ME coefficient of α $\sim 16.8$ ps/m. The angular-dependent measurement demonstrates that the induced in-plane electric polarization $P_y$ changes its direction by −2θ around the $z$ axis upon rotating the magnetic field $H$ by an angle θ. Based on the crystal and magnetic symmetry of the honeycomb layer, the microscopic origin of in-plane electric polarization $P_x$ and $P_y$ under $H$ // $z$ and $P_y$ under $H$ // $x$ and $H$ // $y$ can be attributed to the inverse Dzyaloshinskii-Moriya mechanism and the spin-dependent $p\text{−}d$ hybridization mechanism, respectively. Thus, in this work, we provide insights for understanding the microscopic origin of the ME coupling in linear ME materials.

Infrared singularities of gapless Goldstone modes preclude magnetic long-range order at finite temperature in conventional two-dimensional systems. By studying the spin-$S$ Heisenberg model on regular tilings of the hyperbolic plane, we show that this obstruction is absent in negatively curved space. Using spin-wave theory, we find that the zero-energy collective modes required by symmetry carry vanishing local spectral weight and are separated from the thermodynamic bulk magnon continuum by a finite gap in the bulk local spectral density. As a result, local transverse correlations remain short-ranged, with a finite correlation length, despite the presence of Goldstone modes associated with the broken $\mathrm{SO}\left(3\right)$ spin-rotation symmetry. Stronger negative curvature is found to suppress quantum fluctuations in bulk thermodynamic quantities, pushing the ordered state toward “mean-field-like” behavior. We further estimate the ordering temperature from the thermal spin-wave correction to the ordered moment. These results establish hyperbolic geometry as a route to finite-temperature magnetic order that circumvents the Mermin-Wagner obstruction without breaking or modifying the continuous symmetry.