Altermagnetism and bond-nematicity in the spin-$1/2$ square lattice $J_1-J_2-δ$ model

Abstract: We study appearance of bond-nematicity in insulating altermagnetic materials induced by increased frustration and quantum-fluctuations driven melting of the altermagnetic order. Using novel machine learning approach that combines symmetry enhanced neural network architectures and variational Monte Carlo we consider the spin-$1/2$ square lattice $J_1-J_2-δ$ model known to have altermagnetic ordering in the regime of small geometric frustration and a gapless spin liquid phase in the regime of strong frustration and small exchange interaction modulation parameter $δ$. In the regime where exchange modulation is relatively large, resulting in the significant splitting of the magnon modes with different chiralities in the altermagnetic regime, we find that melting of the altermagnetic order by increased frustration leads to an intriguing phase that hosts coexisting symmetry protected topological valence bond solid and bond-nematic orders. The phase is characterized by condensation of magnon pairs that results in bond-nematicity, breaking of U(1) spin rotation and $\mathbb{Z}_2$ spin inversion symmetries and chiral splitting of the triplon-like energy levels in the excitation spectrum. Whilst numerous recent studies address non-trivial impact of altermagnetic moments on various properties in altermagnetic materials, like electronic band structure and superconductivity, influence of strong quantum fluctuations and phases that can result from melting of the altermagnetic order are much less explored. Our study therefore presents an important step in identifying exotic phases of matter that can emerge in vicinity of the altermagnetic order.

• The paper introduces a GCNN-based neural network quantum state method within VMC to capture altermagnetic order and bond-nematic phases in the spin-1/2 J1-J2-δ model.
• It reveals chiral magnon splitting and scaling of staggered magnetization, highlighting differences between conventional AFM and altermagnetic regimes.
• Results emphasize the interplay of symmetry breaking, quantum fluctuations, and topologically nontrivial orders with implications for spintronics applications.

Altermagnetism and Bond-Nematicity in the Spin-1/2 Square Lattice $J_1$-$J_2$-$\delta$ Model

Background and Motivation

This work analyzes the magnetic phases in the spin-1/2 square lattice $J_1$-$J_2$-$\delta$ model, with a focus on altermagnetic and magnetically disordered regimes. Altermagnetism, defined by non-collinear spin order not reducible to conventional AFM or FM phases, deviates from zero total magnetization due to combined action of time reversal and nontrivial point-group symmetries. Recent interest in altermagnets stems from their atypical band structure, chiral magnon splitting, and potential impact in spintronics, as well as their distinctive symmetry properties (2606.14101).

Despite substantial research on band structures and superconducting responses in altermagnetic materials, the effects of strong quantum fluctuations and frustration—especially the exotic phases resulting from order melting—remain underexplored. This manuscript offers an in-depth study of the $J_1$-$J_2$-$\delta$ model, employing symmetry-enhanced neural network quantum state (NQS) ansätze within a variational Monte Carlo (VMC) framework powered by group equivariant convolutional neural networks (GCNNs).

Model Specification and Computational Approach

The $J_1$-$J_2$0-$J_2$1 Hamiltonian is

$J_2$2

where $J_2$3 modulates NNN exchange, mimicking inequivalent environments and geometric frustration. This model is experimentally relevant in iron oxychalcogenides and can be realized in ultracold atomic systems.

A NQS ansatz based on GCNNs captures both lattice and point-group symmetries ($J_2$4), as well as $J_2$5 spin parity. Parameter optimization leverages VMC combined with stochastic reconfiguration (SR) and natural gradient descent (NGD). The GCNN-VMC formalism efficiently models large lattices and treats quantum fluctuations non-perturbatively.

Altermagnetic Phase and Chiral Magnon Splitting

In the weak frustration regime (e.g., $J_2$6, $J_2$7), ground states correspond to the $J_2$8 irrep at $J_2$9 with spin parity $\delta$0. Order is determined via finite-size scaling of staggered magnetization $\delta$1, static spin structure factors, and energy extrapolation.

Altermagnetic order is evidenced by:

• Chiral splitting of magnon excitations: In NQS-VMC calculations, magnon modes with opposite chiralities (distinguished by $\delta$2, $\delta$3, $\delta$4, $\delta$5 irreps) exhibit energy splitting at $\delta$6, absent in conventional AFM phases.
• Sharp structure factor peaks: $\delta$7 and $\delta$8 show definitive maxima at $\delta$9 consistent with long-range order.
• Broken SU(2) symmetry: In VMC-NQS, symmetry breaking occurs even for finite clusters, mimicking thermodynamic limit behavior, with ground states generally not SU(2)-symmetric.

Energy scaling matches the expected $J_1$0 for AFM-type order. Staggered magnetization scales as $J_1$1, consistent with long-range order. Notably, $J_1$2 values are reduced compared to iPEPS due to NQS capturing stronger quantum renormalizations.

Magnon spectrum analysis reveals LSWT-consistent chiral splitting, with maximum separation at $J_1$3 and degeneracy at high-symmetry points in the thermodynamic limit.

Frustrated Regime: Bond-Nematic and SPT VBS Coexistence

In the strongly frustrated regime ($J_1$4, $J_1$5), quantum fluctuations melt altermagnetic order, yielding a magnetically disordered phase characterized by:

• Vanishing magnetic order: $J_1$6 scales critically as $J_1$7 with $J_1$8, confirming absence of long-range magnetization.
• Bond-nematicity and SPT VBS: Structure factors for nematic and dimer correlators exhibit broad, s-wave symmetry peaks at $J_1$9 and sharp VBS-associated peaks at $J_2$0 and $J_2$1. Nematic order parameters $J_2$2, $J_2$3 and $J_2$4 extrapolate to nonzero values in the thermodynamic limit, with scaling behavior indicative of a symmetry-protected topological (SPT) phase. Clusters with $J_2$5 and $J_2$6 demonstrate distinct order parameter scaling, hallmark of SPT VBS states.
• Magnon pair condensation: The ground state supports two-magnon bound pair condensation (operator $J_2$7), breaking U(1) spin rotation symmetry and generating nematic Goldstone modes.
• Broken $J_2$8 spin inversion symmetry: Finite $J_2$9 indicates spontaneous breaking of $\delta$0 symmetry; energy level splitting for $\delta$1 also confirms this.

Energy spectra validate coexistence, with nematic Goldstone modes corresponding to $\delta$2, $\delta$3 irreps (not fully symmetric under $\delta$4) and degenerate ground state manifolds.

Implications and Theoretical Perspective

The identification of a magnetically disordered phase hosting both bond-nematicity and SPT VBS order advances understanding of quantum frustrated magnetism:

• Exotic phase formation: Demonstrates that melting altermagnetic order via frustration can yield nontrivial, symmetry-broken, topologically nontrivial quantum phases beyond spin liquids.
• Interplay of symmetry and topology: Coexistence of nematic and SPT VBS orders highlights complex symmetry fractionalization and topological sector separation, especially manifested in cluster-size dependent scaling.
• Low-energy excitation complexity: Emergence of nematic Goldstone and chiral triplon-like modes refines classification of excitations in these systems, relevant for both theoretical analysis and experimental probes.
• Quantum simulation and spintronics: The phases studied here have implications for simulating gravitational analogs and enhancing functionalities in spintronic devices utilizing altermagnets.

Future Directions

• Quantum criticality: Characterization of phase transitions, especially potential deconfined quantum critical points between Dirac spin liquid and bond-nematic/SPI VBS states.
• Superconductivity upon doping: Exploration of superconducting orders in the model via fermionic NQS approaches.
• Model extensions: Analysis of spin-1 versions and generalizations to other lattices and symmetry classes.

Conclusion

This study establishes that the spin-1/2 $\delta$5-$\delta$6-$\delta$7 square lattice model supports altermagnetic order with chiral magnon splitting in weakly frustrated regimes and a magnetically disordered phase exhibiting coexisting bond-nematic and SPT VBS orders in the highly frustrated regime. The application of symmetry-adapted GCNN NQS within VMC defines a robust computational protocol for probing emergent quantum phases. These findings elucidate the nature of quantum melting in altermagnets and pave the way for future theoretical and experimental investigations into nontrivial quantum order and criticality in frustrated magnets (2606.14101).