Easy-Plane XXZ Quantum Ferromagnetism

• Easy-plane XXZ quantum ferromagnetism is defined by anisotropic exchange interactions that favor in-plane (xy) spin alignment and induce U(1) symmetry breaking.
• The model’s Hamiltonian incorporates tunable anisotropy via exchange (Jxy and Jz) and single-ion terms, leading to diverse phases including TLL, dimerized, and symmetry-protected topological states.
• Experimental realizations in materials like monolayer CrCl₃ validate theoretical predictions through observable KT transitions, vortex dynamics, and scalable spin squeezing useful for quantum metrology.

Easy-plane XXZ quantum ferromagnetism encompasses a broad class of low-dimensional quantum spin models characterized by anisotropic exchange interactions that favor spin alignment within a specific plane—typically the $xy$-plane—over the out-of-plane ($z$) direction. Realizations span one- and two-dimensional systems with tunable anisotropy ratios, manifesting a range of collective quantum phenomena, including quantum phase transitions, Kosterlitz-Thouless (KT) topological order, nontrivial entanglement, and the emergence of both conventional and exotic spin textures.

1. Model Hamiltonians and Anisotropy Regimes

The canonical easy-plane XXZ Hamiltonian on a lattice of spin-$S$ moments is

$$H = -\sum_{\langle i,j \rangle} [ J_{xy}(S_i^x S_j^x + S_i^y S_j^y) + J_z S_i^z S_j^z ] - K \sum_i (S_i^z)^2,$$

where $J_{xy}>0$ sets the in-plane (XY) ferromagnetic exchange, $J_z$ quantifies the Ising-type anisotropy, and $K>0$ introduces single-ion easy-plane anisotropy. The ratio $\Delta \equiv J_z/J_{xy}$ or, equivalently, $\Delta \equiv \Delta^z/\Delta^{xy}$ for generalized XXZ models, defines the easy-plane regime as $0 \leq \Delta < 1$ (Sarıyer, 2018, Block et al., 2023).

Variants include frustrated chains with next-nearest-neighbor (NNN) exchange (Furukawa et al., 2010), honeycomb models with multi-neighbor couplings (Gu et al., 2023), and bond-operator representations for $S=1$ models with single-ion anisotropy (Carvalho et al., 2015). Easy-plane anisotropy arises both via exchange ($J_z<J_{xy}$) and through single-ion ($K>0$), the latter being especially relevant in actual 2D van der Waals magnets such as CrCl$_3$ (Bedoya-Pinto et al., 2020).

2. Ground State Order, Excitations, and Quantum Criticality

For quantum spins in the easy-plane regime, the classical ground state is ferromagnetic within the plane, breaking a continuous $U(1)$ symmetry (Block et al., 2023). At $T=0$, quantum fluctuations do not destroy this order. The low-energy excitation spectrum is dominated by gapless in-plane Goldstone spin waves with linear dispersion $\omega(k)\sim c|k|$, where $c$ is the spin-wave velocity (Sarıyer, 2018, Block et al., 2023).

In strictly one-dimensional systems, the inclusion of frustration (e.g., ferromagnetic NN $J_1<0$ and antiferromagnetic NNN $J_2>0$) produces a rich phase diagram. For the spin-½ chain

$$H = \sum_{j=1}^L \left\{ J_1 [ S_j^x S_{j+1}^x + S_j^y S_{j+1}^y + \Delta S_j^z S_{j+1}^z ] + J_2 \mathbf{S}_j \cdot \mathbf{S}_{j+2} \right\},$$

the ground state evolves through Tomonaga-Luttinger liquid (TLL), Neel, and dimerized phases as $J_2/|J_1|$ and $\Delta$ are tuned (Furukawa et al., 2010, Ueda et al., 2020). The dimerized phases in the easy-plane regime can be symmetry-protected topological (SPT) and exhibit string order (Ueda et al., 2020).

In two dimensions, true long-range order at finite $T$ is forbidden by the Mermin-Wagner theorem for strictly short-range couplings. However, algebraically ordered Kosterlitz-Thouless (KT or BKT) phases occur for $0<\Delta<1$ (Sarıyer, 2018, Bedoya-Pinto et al., 2020). The KT transition temperature vanishes logarithmically as $\Delta\to1^-$: $$T_c(\Delta)/J \approx A [\ln(B/(1-\Delta))]^{-1}, \quad A=1.7432,\, B=36.28$$ (Sarıyer, 2018).

3. Topological and Finite-Size Effects: KT/BKT Transitions and Vortex Physics

In the KT phase, the relevant physics is governed by binding/unbinding of topological vortex-antivortex pairs in the in-plane spin angle field $\theta$. The effective continuum action is (Bedoya-Pinto et al., 2020): $$F[\theta] = \frac{\rho_s}{2}\int d^2 r |\nabla\theta(r)|^2,$$ where $\rho_s$ is the spin stiffness. At $T<T_{BKT}$, vortex pairs are bound, and correlations decay algebraically; above, they unbind, destroying phase coherence and leading to exponential decay.

Experimental realization in a monolayer CrCl$_3$ shows remanent magnetization $M_r(T)\sim (1-T/T_C)^\beta$ with $\beta=0.227\pm0.021$, matching the 2D-XY value $\beta_{XY}=0.231$, and susceptibility exponent $\gamma\simeq2.2$, consistent with 2D-XY values (Bedoya-Pinto et al., 2020). Finite grain size and substrate coupling round the phase transition, and magnetization scaling collapses as $L$ (system size) or temperature vary.

4. Quantum Effects: Entanglement, Spin Squeezing, and Symmetry Protection

Easy-plane XXZ ferromagnets exhibit nontrivial quantum entanglement, measurable through concurrence, entanglement of formation, and quantum discord built from nearest-neighbor correlations (Sarıyer, 2018). Finite values persist at low $T$ for $0\leq \Delta <1$, vanishing in the fully polarized Ising regime.

A salient attribute is their ability to support scalable spin squeezing at finite temperature for quantum-enhanced metrology (Block et al., 2023). The presence of U(1) symmetry breaking below $T_c$ leads to macroscopic quantum Fisher information $F_Q \sim N^2$ (Heisenberg scaling). The phase diagram has a sharp transition between scalable and non-scalable squeezing, matching the equilibrium XY ordering boundary. The optimal squeezing parameter scales as $\xi^2_{\mathrm{opt}}\sim N^{-2/5}$—intermediate between standard quantum limit and all-to-all one-axis twisting scaling.

In one dimension, SPT transitions between dimerized phases map onto effective spin-1 chains with Haldane string order, protected by time-reversal, bond inversion, and $Z_2\times Z_2$ symmetries (Ueda et al., 2020).

5. Frustration, Higher-Order Couplings, and Emergent Phases

Frustration, through competing $J_2$ (or $J_3$) terms or bond alternation, generates a sequence of phases and critical points not present in unfrustrated models. For the frustrated $S=1/2$ XXZ chain, as $\Delta\to1^{-}$, alternate Neel and dimer ordered lobes accumulate, stabilized in the quantum case by emergent trimer correlations (three-spin bound states) rather than classical spin patterns (Furukawa et al., 2010).

In higher dimensions, stability of unconventional ground states such as multi-$\mathbf{q}$ (double-$\mathbf{q}$ or triple-$\mathbf{q}$) textures in honeycomb cobaltates requires higher-order (ring- or biquadratic) couplings in addition to the easy-plane XXZ terms (Gu et al., 2023). These interactions stabilize noncollinear in-plane magnetic structures resilient to symmetry reduction and yield strong quantum reduction of ordered moments (up to 40%).

Bond-operator mean-field theory for $S=1$ cubic models with easy-plane single-ion anisotropy maps out second- and first-order quantum phase transitions between ferromagnetic, collinear antiferromagnetic, and disordered (spin-liquid) regions: the latter can be realized without single-ion anisotropy, solely from frustrated exchange (Carvalho et al., 2015).

6. Experimental Realizations and Prospective Applications

Monolayer CrCl$_3$ epitaxially deposited on graphene provides a prime example of a large-area, nearly ideal easy-plane XXZ ferromagnet (Bedoya-Pinto et al., 2020). Key parameters are $J_{xy}\approx 0.6$ meV, $J_z\simeq J_{xy}$, and single-ion anisotropy $K\approx 0.08-0.10$ meV per Cr$^{3+}$, producing a spin gap $\Delta_z\approx 2S K\approx 0.3$ meV for out-of-plane fluctuations. DFT and cluster models confirm the dominance of single-ion anisotropy over exchange anisotropy.

The system demonstrates observable 2D-XY scaling, a rounded BKT transition, and robustness to finite size, supporting avenues for atomistic meron/half-vortex imaging and superfluid spin transport.

In the context of quantum information and metrology, easy-plane XXZ models provide a generic Hamiltonian class enabling scalable spin squeezing by virtue of U(1) symmetry breaking at finite $T$ and associated enhanced quantum Fisher information (Block et al., 2023). This constrains the design of metrologically useful states and excludes short-range two-axis twisting models from yielding scalable gain.

7. Summary Table: Key Regimes and Physical Characteristics

Regime / Model	Order at $T=0$	Critical Behavior (2D)	Notable Excitations / Features
2D easy-plane XXZ ($0\leq\Delta<1$)	In-plane FM, KT phase	KT transition, BKT scaling	Goldstone modes, vortex/antivortex
1D frustrated chain ($J_1<0,J_2>0$)	TLL, Neel, dimer phases	Gaussian/cascade of critical lines	Emergent trimers, SPT transitions
S=1 cubic w/ easy-plane ($D>0$)	FM, CAF, SL regions	2nd/1st order quantum lines	Gapped/disordered phases, spin liquid
Honeycomb multi-q (XXZ + 4th order)	Double-q in-plane order	Multi-q stability	Strong quantum reduction, noncollinear

These entries catalogue the principal phases, phase transitions, and emergent physics as dictated by the easy-plane XXZ quantum ferromagnetism paradigm in contemporary theoretical and experimental settings (Bedoya-Pinto et al., 2020, Sarıyer, 2018, Block et al., 2023, Furukawa et al., 2010, Carvalho et al., 2015, Gu et al., 2023, Ueda et al., 2020).