Easy-Plane Magnets: Quantum & Topological Insights

• Easy-plane magnets are magnetic systems defined by anisotropy that confines spins within a crystallographic plane while suppressing out-of-plane magnetization.
• They exhibit rich phase behavior and excitation spectra, with theoretical models using bosonic mappings and quantum phase transition analysis.
• Their robust topological textures, such as bimerons and hopfions, underpin novel applications in spintronic devices and quantum simulators.

Easy-plane magnets are magnetic systems characterized by an anisotropy energy that favors alignment of spins within a particular crystallographic plane, suppressing out-of-plane magnetization components. This easy-plane anisotropy fundamentally modifies low-energy excitations, topological textures, collective dynamics, and phase behavior, enabling a diverse array of quantum and classical phenomena that distinguish these systems from their easy-axis counterparts. Easy-plane magnets arise both in bulk compounds with strong single-ion anisotropy and in low-dimensional engineered materials, including van der Waals magnets, ultrathin films, and cold atom quantum simulators.

1. Microscopic Hamiltonians and Theoretical Frameworks

The generic Hamiltonian for an easy-plane magnet involves isotropic exchange (Heisenberg), anisotropic exchange, and explicit single-ion anisotropy (or more general anisotropic terms): $$\mathcal{H} = \sum_{\langle i j \rangle} J_{ij} \mathbf{S}_i \cdot \mathbf{S}_j + D \sum_i (S_i^z)^2 + \cdots$$ where $D > 0$ (easy-axis) versus $D < 0$ (easy-plane). The sign convention can differ, but the essential ingredient is the $-K_u m_z^2$ (with $K_u < 0$) contribution in a continuum $\mathbf{m}(\mathbf{r})$ description which penalizes out-of-plane magnetization (Leonov et al., 2017). In honeycomb e.g. Kitaev–Heisenberg materials, the “easy-plane” limit refers to the restriction of exchange interactions to components within the magnetic plane (Maksimov et al., 2022).

For large $|D|$, perturbative and diagrammatic techniques become viable: integer-spin magnets with large single-ion easy-plane anisotropy can be mapped to effective dilute bosonic theories, with paramagnetic $S^z=0$ ground state and $S^z=\pm 1, \ldots$ excitations represented by bosonic operators $a_i$, $b_i$ under constraints enforcing physical Hilbert space occupancy (see section below) (Sizanov et al., 2011, Sizanov et al., 2011).

In itinerant and two-dimensional magnets, easy-plane anisotropy is realized by a strong suppression of $M_z$ either by intrinsic spin–orbit coupling, crystal fields, or demagnetization effects (shape anisotropy) (Montoya et al., 2023, Resch et al., 28 Apr 2025).

2. Magnetic Excitations and Quantum Phase Transitions

Bosonic Representation and Excitation Spectrum

The large-$D$ regime permits a bosonic expansion where the exchange acts as a perturbation:

• $S^z_i = n_{b,i} - n_{a,i}$
• $$S^+_i \approx c_1 (b_i + a_i^\dagger) - c_2 (b_i^\dagger n_{b,i} + n_{a,i} a_i)$$, with commutators and strong occupation constraints $n_{a,i}, n_{b,i} \leq S$, $n_{a,i} n_{b,i}=0$ (Sizanov et al., 2011).

The paramagnetic phase consists mainly of $|S^z=0\rangle$ states; $a_i^\dagger$ and $b_i^\dagger$ create local $S^z=-1$ and $+1$ excitations, respectively. Elementary excitation spectra and ground state energies can then be computed systematically in powers of $J/D$: $$\varepsilon_{\mathbf{p}}^2 = [\varepsilon_{1 \mathbf{p}} + \Sigma(\varepsilon_{1\mathbf{p}})]^2 - |\Pi(\varepsilon_{1\mathbf{p}})|^2$$ with diagrammatically computed self-energies and higher-order lattice sums giving high accuracy except near quantum critical points (Sizanov et al., 2011, Sizanov et al., 2011). These methods outperform RPA and self-consistent spin wave approaches, particularly in two dimensions.

Critical Properties and Bose-Einstein Condensation

In a transverse field $h \sim D$, the field acts as a chemical potential; the closing of the gap at a critical field $h_{c1}(T)$ signifies a transition from a quantum paramagnet to a phase with long-range magnetic order, with analytic expressions for the phase boundary: $$h_{c1}(T) = \varepsilon(\mathbf{p}_0) + 4\Gamma_a(0,0) M(h,T)$$ and corresponding effective boson–boson interaction $v_0 = 2\Gamma_a(0,0)$. Near the QCP, thermodynamic quantities and ordering follow 3D BEC universality with $M, C \propto T^{3/2}$ scaling (Sizanov et al., 2011). These analytic theories correctly describe experimental signatures in DTN (NiCl₂-4SC(NH₂)₂), and inclusion of secondary exchange couplings (e.g., between sublattices) is crucial for quantitative agreement (Sizanov et al., 2011, Sizanov et al., 2011).

3. Topological Textures: Bimerons, Skyrmions, and Hierarchical Solitons

Bimerons and Merons

Bimerons are topological textures unique to easy-plane magnets, composed of a bound vortex–antivortex pair (merons) each carrying (approximately) half a unit of topological charge: $$Q = \frac{1}{4\pi} \int_{\mathbb{R}^2} \mathbf{m} \cdot (\partial_x \mathbf{m} \times \partial_y \mathbf{m})\, dx dy$$ Easy-plane symmetry allows for the existence and stability of both bimerons ($Q = -1$) and antibimerons ($Q = +1$) at zero field, which are degenerate due to invariance under global in-plane spin rotations and $m \to -m$ (Göbel et al., 2018, Goerzen et al., 11 Sep 2025).

Unlike skyrmions, bimerons in easy-plane magnets exhibit strong spatial anisotropy in their profiles: algebraic decay along directions where spin rotates inside the plane, and exponential decay in directions with out-of-plane variation (Goerzen et al., 11 Sep 2025). The absence of out-of-plane magnetization in the ground state leads to gapless Goldstone modes associated with spontaneous breaking of U(1) symmetry, impacting excitation lifetimes and entropy (Deng et al., 13 Jun 2025, Goerzen et al., 11 Sep 2025).

Entropic Effects and Lifetime

The lifetime $\tau$ of bimerons is described by an entropy-sensitive Arrhenius law: $$\tau = \Gamma_0^{-1} \exp\left(\frac{\Delta E}{k_B T}\right)$$ with the attempt frequency $\Gamma_0$ determined by the configuration space volume associated with translation and rotation of the soliton (resulting in a nearly temperature-independent $\Gamma_0$ as $T \to 0$). Bimerons’ lifetimes are entropically stabilized, and their collapse is controlled by anisotropic shrinking paths, contrasting with radially symmetric skyrmion annihilation (Goerzen et al., 11 Sep 2025).

Hierarchical Topological Structures

Easy-plane anisotropy permits not only isolated merons, bimerons, and their crystals (Göbel et al., 2018), but also more complex three-dimensional textures:

• Hopfions: closed loops of skyrmion strings with Hopf invariant

$$H = -\int \mathbf{B}(\mathbf{r}) \cdot \mathbf{A}(\mathbf{r})\, d\mathbf{r}$$

assembled into robust superstructures when thread by meron strings, leading to $N_{\rm sk}=2$ on any 2D cut and opening avenues for nontrivial transport phenomena (Kasai et al., 1 Nov 2024).

• Metastable meron–antimeron pairs and superstructures appear in chiral, frustrated, or easy-plane magnets, with their stability and size tunable by the strength of DMI, easy-plane anisotropy, and external fields (Lin et al., 2014, Bachmann et al., 2023, Deng et al., 13 Jun 2025).

4. Emergent Phases and Quantum Simulations

Quantum Simulators and Exotic Correlated Phases

Easy-plane models play a central role in realizing quantum simulators, especially with cold atoms in optical lattices. The mapping of the hard-core boson model to the quantum XY (easy-plane) spin Hamiltonian

$$H_\text{boson} = J \sum_{\langle i,j \rangle}(b^\dagger_i b_j + h.c.) + \frac{U}{2} \sum_i n_i(n_i-1)$$

allows for direct engineering of easy-plane magnetism, where on-site repulsion $U$ enforces the hard-core constraint (Läuchli et al., 2015).

These systems naturally realize emergent U(1) gauge fields, deconfined spin liquids, resonating valence bond (RVB) states, and magnetization plateaux in geometries such as the checkerboard and kagome lattices, with direct access to frustration and dynamical probes (Läuchli et al., 2015). Numerical studies using loop algorithms for easy-plane SU($N$) models reveal first-order superfluid-to-VBS transitions for moderate $N$ and an instability of deconfined quantum criticality in the easy-plane limit (D'Emidio et al., 2015).

Quantum Correlations: Spin Squeezing

Spin squeezing, a manifestation of entanglement in collective spin systems, can be achieved in easy-plane settings using short-range contact interactions (superexchange). The anisotropic XXZ Hamiltonian generates entangled states with reduced quantum projection noise: $$\xi^2 = \frac{N \min_\theta \mathrm{Var}[S^\theta]}{\langle S^x \rangle^2}$$ Easy-plane (XY-like) interactions enable scalable squeezing in quasi-1D systems, but in 3D, hole densities and coupling between spin and density reduce achievable squeezing, revealing the crucial role of spin–density interplay in such quantum simulators (Lee et al., 25 Sep 2024).

5. Experimental Systems and Signatures

Crystalline and van der Waals Magnets

ErB₂ is a canonical example of a bulk easy-plane ferromagnet, with:

• Second-order ordering at $T_c = 14$ K,
• Strong basal-plane magnetocrystalline anisotropy (Curie–Weiss $\theta_{CW} \approx +19$ K in-plane, $-30$ K out-of-plane),
• Field-polarization via a spin-flip transition at $\mu_0 H_c = 12$ T along the hard axis,
• Evolution of anisotropic fluctuations below a crossover temperature $T_x \approx 50$ K in the paramagnetic state (Resch et al., 28 Apr 2025).

Ultrathin polar magnets and van der Waals heterostructures (e.g., FGT/CGT) provide control over anisotropy and symmetry at the atomic level, supporting coexisting degenerate bimerons and antibimerons at zero field, with robust topological lifetimes and tunable nonlinear interactions (Goerzen et al., 11 Sep 2025).

Engineered Nanostructures and Device Applications

Microwave nano-oscillators exploiting engineered easy-plane anisotropy in nanowires (Pt|Co|Ni) show large-amplitude precessional dynamics and high output, with the easy-plane tailored by balancing shape and interfacial perpendicular anisotropies (Montoya et al., 2023). This design principle enables energy-efficient spintronic devices, including spiking neuron emulators and enhanced-recording elements.

Topological textures (bimerons, hopfions) in easy-plane magnets present opportunities for information storage and unconventional computing; their entropy-enhanced lifetimes, nonlinear interactions, and tunability far exceed those of traditional skyrmion-based architectures, especially for neuromorphic reservoir computing or quantum information science (Goerzen et al., 11 Sep 2025, Kasai et al., 1 Nov 2024).

6. Topology, Symmetry, and Field-Induced Phenomena

Easy-plane magnets exhibit unique topological and symmetry properties absent in easy-axis materials:

• The unbroken continuous U(1) in-plane symmetry gives rise to gapless magnons and entropic stabilization of solitons; this influences transport responses and thermal activation processes.
• The relative shift and anisotropy of skyrmion, meron, and antiskyrmion cores under tilted fields is dictated by crystal symmetry (e.g., $C_{nv}$ or $D_{2d}$), the sign of DMI, and field orientation (Leonov et al., 2017).
• Field–tilting and varying anisotropy yield rich phase diagrams, encompassing transitions between skyrmion lattices, vortex–antivortex superlattices, and canted or field-polarized states, with easy-plane anisotropy enhancing the stability region for the skyrmion lattice under axial field (Leonov et al., 2017, Lin et al., 2014).
• Conformal analysis for minimizers with prescribed topological degree $Q = \pm 1$ reveals that bimeron localization occurs on the scale $\sim 1/|\ln \sigma^2|$, with Möbius map structure, contrasting markedly with easy-axis systems (Deng et al., 13 Jun 2025).

7. Open Problems and Outlook

The broad richness of solitonic and collective phenomena in easy-plane magnets, rooted in their unique symmetry, leads to fundamentally different responses to thermal fluctuations, field control, and disorder, as compared to easy-axis models. Outstanding challenges and frontiers include:

• Realizing scalable, stable, high-density bimeron-based devices;
• Exploring hierarchy and interactions of complex topological textures (e.g., hopfion–meron superstructures) and their impact on emergent electrodynamics (Kasai et al., 1 Nov 2024);
• Controlling quantum correlations for quantum metrology and simulation in low-entropy regimes, particularly in the presence of finite density defects (Lee et al., 25 Sep 2024);
• Developing mathematical and computational tools to rigorously treat the conformal regime, anisotropy effects, and the nontrivial impact of DMI and boundary conditions in reduced dimensions (Deng et al., 13 Jun 2025, Bachmann et al., 2023).

Current research continues to expand the taxonomy of topological magnetic solitons, dynamical modes, and entangled quantum phases that are unique to easy-plane magnets, driving both fundamental understanding and innovative applications in spintronics, quantum information, and beyond.