Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2606.14101

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

Tanja Đurić, Shenlong Yu, Pinaki Sengupta

Published Jun 12, 2026• 0 citations
arXivHTMLPDF

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.

DOI
10.48550/arxiv.2606.14101

Related papers

Nematic spin liquid phase in a frustrated spin-1 system on the square lattice

Wen-Jun Hu, Shou-Shu Gong, Hsin-Hua Lai +3 more

Frustration in quantum spin systems promote a variety of novel quantum phases. An important example is the frustrated spin-1 model on the square lattice with the nearest-neighbor bilinear (${J}_{1}$) and biquadratic (${K}_{1}$) interactions. We provide strong evidence for a nematic spin liquid phase in a range of ${K}_{1}/{J}_{1}$ near the SU(3)-symmetric point (${J}_{1}={K}_{1}$), based on the linear flavor-wave theory and extensive density matrix renormalization group calculation. This phase displays no spin dipolar or quadrupolar order, preserves translational symmetry but spontaneously breaks ${C}_{4}$ lattice rotational symmetry, and possesses fluctuations peaked at the wave vector $(\ensuremath{\pi},2\ensuremath{\pi}/3)$. The spin excitation gap drops rapidly with system size and appears to be gapless, and the nematic order is attributed to the dominant $(\ensuremath{\pi},2\ensuremath{\pi}/3)$ fluctuations. Our results provide a novel mechanism for electronic nematic order and, more generally, open up a new avenue to explore frustration-induced exotic ground states.

arXiv:1711.06523•Physical review. B./Physical review. B•Oct 28, 2019•16 citations

Quantum dynamics in a spin-1/2 square lattice $J_{1}$-$J_{2}$-$δ$ altermagnet

Yang Liu, Shiqi Shao, Saisai He +4 more

A key feature of the newly discovered altermagnet is that its spin degeneracy is lifted, although it has an antiferromagnetic order and zero net magnetization. In this work, we investigate a frustrated spin-1/2 $J_1$-$J_2$-$δ$ Heisenberg model on the square lattice by the tensor network methodin combination with the linear spin-wave theory, with our focus on both the magnon excitations and longitudinal excitations.For a small $J_2$ and a finite range of $δ$ we demonstrate that such a model hosts an altermagnetic ground state. Its magnon spectrum is split into two branches and the largest splitting occurs at $\left(\pmπ/2, \pmπ/2\right)$ in the Brillouin zone. The magnitudes of splitting in the two magnon modes are equal with respect to the case of $δ=0$. Dynamical spin structure factors show that the low-energy peak in the longitudinal spectral weight around $(π/2, π/2)$ is also split, and thus the relative positions of the magnon modes and longitudinal modes in energy may change in the presence of a finite $δ$. These findings demonstrate that the altermagnets harbor more complex quantum dynamics than the conventional collinear antiferromagnets.

arXiv:2410.06955•Oct 9, 2024•0 citations

Ground state phase diagram and the exotic phases in spin- 1 2 square lattice J 1 − J 2 − J χ model

Jianwei Yang, Zhao Liu, Ling Wang

The intricate interplay between frustration and spin chirality, induced by the Dzyaloshinskii-Moriya interaction, holds promise for unveiling novel phases in frustrated quantum magnets. This study investigates the ground state phase diagram of the spin-$\frac{1}{2}$ square lattice ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model upon incorporation of the chiral interaction ${J}_{\ensuremath{\chi}}$. Employing exact diagonalization (ED) with full lattice symmetries, we analyze the phase evolution as a function of ${J}_{2}$ at fixed ${J}_{\ensuremath{\chi}}$, utilizing highly symmetric energy levels. Critical level crossings and ground state fidelity susceptibility (FS) techniques aid in pinpointing phase boundaries: Magnetic to nonmagnetic phase transitions are identified through critical level crossings between gapless magnetic excitations and quasidegenerate ground states of nonmagnetic phases. Direct transitions between two nonmagnetic phases are characterized by FS peaks due to avoided ground state level crossing. Based on these observations, we identify an anticipated chiral spin liquid (CSL) state and an adjacent nematic spin liquid (NSL) phase within a significant range of ${J}_{\ensuremath{\chi}}$, demarcated by a nearly vertical boundary line at ${J}_{2}\ensuremath{\approx}0.65$. This critical line terminates at the lower boundary in ${J}_{\ensuremath{\chi}}$ of a magnetic ordered chiral spin solid phase, which gains prominence with increasing ${J}_{\ensuremath{\chi}}$ from both CSL and NSL phases. The topological nature of the CSL is confirmed using the modular $\mathcal{S}$ matrix of minimally entangled states on a torus and the entanglement spectra of even and odd sectors on a cylinder, employing ED and $\mathrm{SU}(2)$-symmetric density matrix renormalization group method, respectively. Furthermore, a comprehensive discussion on the nature of the NSL is provided, exploring aspects such as ground state degeneracy, local bond energy landscape, and singlet and triplet gaps on various tori, offering substantial evidence supporting the nematic nature of the NSL.

Physical review. B./Physical review. B•Dec 2, 2024•2 citations

Novel quantum phases in correlated electron systems

Kyusung Hwang

The quantum nature of electrons emerges in various exotic ways when they are put together with quantum many-body interactions. In Mott insulating systems with strong electron correlation, frustration and strong quantum fluctuations promote localized electron spins to form nontrivial quantum paramagnetic phases such as quantum spin liquid and valence bond solid preventing the development of conventional magnetic orders. In this thesis, we study such exotic quantum phases in various different frustrated quantum magnets. First, encouraged by the spin liquid phenomenology observed in Ba3XSb2O9 (X=Cu, Ni), we investigate Z2 spin liquids on a three dimensional triangular lattice by employing the Schwinger boson mean-field theory with projective symmetry group analysis. It is found that a nematic spin liquid is energetically favorable on this lattice structure rather than symmetric spin liquids. Next, we study various valence bond solid phases that can appear in kagome lattice antiferromagnets by generalizing the recently proposed phenomenological Z2 gauge theory for the topological Z2 spin liquid state found in the nearest-neighbor kagome lattice Heisenberg model. This work provides a bridge between the topological spin liquid and valence bond solid phases previously proposed for kagome lattice antiferromagnets via vison condensation in the gauge theory. Then, we consider optical conductivity in Mott insulators as an experimental probe to detect quantum paramagnetic phases like valence bond solid. Applying this idea to deformed kagome lattice antiferromagnet Rb2Cu3SnF12 with a 12-site valence bond solid state, we estimate the size of optical conductivity in this quantum magnet. After this, another quantum paramagnet BiCu2PO6 is carefully investigated to clarify its puzzling ground state nature and quasiparticle excitations. Utilizing recent high resolution neutron scattering data, we construct a microscopic spin model and also an effective theory for the low energy spin excitations in the compound. We find significant influences of magnetic anisotropies on the ground state and quasiparticle dynamics. Lastly, we investigate emergent quantum phases in the thin film geometries of the pyrochlore iridates, where a number of exotic quantum ground states are proposed to occur in bulk materials as a result of the interplay between electron correlation and strong spin-orbit coupling. The fate of these bulk phases, as well as the novel quantum states that may arise only in a thin film setting, are studied via a theoretical model that allows layer-dependent magnetic structures.

•Nov 1, 2016•0 citations

Ground state phase diagram and the exotic phases in the spin-1/2 square lattice J1-J2-Jx model

Jianwei Yang, Zhao Liu, Ling Wang

The intricate interplay between frustration and spin chirality has the potential to give rise to unprecedented phases in frustrated quantum magnets. We examine the ground state phase diagram of the spin-1/2 square lattice J1-J2-Jx model by employing critical level crossings and ground state fidelity susceptibility (FS) using exact diagonalization (ED) with full lattice symmetries. Our analysis reveals the evolution of highly symmetric energy levels as a function of J2 at fixed Jx. During a magnetic to non-magnetic phase transition, the precise identification of the phase boundary is achieved through critical level crossings between the gapless excitation of a magnetic phase and the quasi-degenerate ground state of a non-magnet phase. Conversely, a direct transition between two non-magnetic phases is characterized by a FS peak accompanied by an avoided ground state level crossing, serving as a distinctive signal. Within a substantial range of Jx, we identify an anticipated chiral spin liquid (CSL) state and an adjacent nematic spin liquid (NSL) phase with a degeneracy of two on a cylinder. These two phases are demarcated by a nearly vertical boundary line at J2 = 0.65. This critical line terminates at the lower boundary of a magnetic ordered chiral spin solid (CSS) phase. We validate the topological nature of the CSL using the modular S matrix of the minimum entangled states (MES) on a torus, along with the entanglement spectra (ES) of even and odd sectors on a cylinder, employing an SU(2)-symmetric density matrix renormalization group (DMRG) method. Furthermore, we delve into a comprehensive discussion on the nature of the NSL, exploring aspects such as ground state degeneracy, the local bond energy landscape, and the singlet and triplet gaps on various tori. These analysis provide substantial evidence supporting the nematic nature of the NSL.

arXiv:2401.03434•Jan 7, 2024•1 citation

Machine learning quantum criticality in the spin-1/2 quantum antiferromagnets on the square lattice with plaquette structure

Tanja Đurić

The power of machine learning algorithms to automatically classify different phases of matter and detect quantum phase transitions without necessity to characterize phases by various quantities like local order parameters or topological invariants as in conventional approaches defined machine learning phases of matter as a new research frontier and basic research tool in condensed matter and statistical physics. We study quantum criticality in the spin-1/2 square-lattice J1-J2 model with additional plaquette structure by combination of reinforcement and supervised machine learning techniques. In our calculations the ground-state spin-spin correlation matrices for several system sizes are first found by restricted Boltzmann machine based variational Monte Carlo method, equivalent to reinforcement learning, and then used as a training data for convolutional neural network based supervised machine learning algorithm for phases classification. The model exhibits a quantum phase transition from paramagnetic plaquette resonating valence bond state to an antiferromagnetic state and has been a topic of great interest because of its close connection to cuprate superconductors and possibility of realization in cold atoms experiments. We consider both frustrated and unfrustrated regimes and compare our results with the results obtained previously with other methods. We find that our results are in good agreement with the results obtained by coupled cluster and real space renormalization group methods for both frustrated and unfrustrated regimes. The quantum Monte Carlo and finite-size scaling result for the unfrustrated case however slightly differs from our result for the critical value of the inter-plaquette coupling strength, although the results are still in reasonable agreement.

arXiv:2103.07236•Mar 12, 2021•0 citations
Altermagnetism and bond-nematicity in the spin-1/21/21/2 square lattice J1−J2−δJ_1-J_2-δJ1​−J2​−δ model
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-1/21/21/2 square lattice J1−J2−δJ_1-J_2-δJ1​−J2​−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 δδδ. 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 Z2\mathbb{Z}_2Z2​ 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.
Frustration in quantum spin systems promote a variety of novel quantum phases. An important example is the frustrated spin-1 model on the square lattice with the nearest-neighbor bilinear (J1{J}_{1}J1​) and biquadratic (K1{K}_{1}K1​) interactions. We provide strong evidence for a nematic spin liquid phase in a range of K1/J1{K}_{1}/{J}_{1}K1​/J1​ near the SU(3)-symmetric point (J1=K1{J}_{1}={K}_{1}J1​=K1​), based on the linear flavor-wave theory and extensive density matrix renormalization group calculation. This phase displays no spin dipolar or quadrupolar order, preserves translational symmetry but spontaneously breaks C4{C}_{4}C4​ lattice rotational symmetry, and possesses fluctuations peaked at the wave vector $(\ensuremath{\pi},2\ensuremath{\pi}/3)$. The spin excitation gap drops rapidly with system size and appears to be gapless, and the nematic order is attributed to the dominant $(\ensuremath{\pi},2\ensuremath{\pi}/3)$ fluctuations. Our results provide a novel mechanism for electronic nematic order and, more generally, open up a new avenue to explore frustration-induced exotic ground states.
Quantum dynamics in a spin-1/2 square lattice J1J_{1}J1​-J2J_{2}J2​-δδδ altermagnet
A key feature of the newly discovered altermagnet is that its spin degeneracy is lifted, although it has an antiferromagnetic order and zero net magnetization. In this work, we investigate a frustrated spin-1/2 J1J_1J1​-J2J_2J2​-δδδ Heisenberg model on the square lattice by the tensor network methodin combination with the linear spin-wave theory, with our focus on both the magnon excitations and longitudinal excitations.For a small J2J_2J2​ and a finite range of δδδ we demonstrate that such a model hosts an altermagnetic ground state. Its magnon spectrum is split into two branches and the largest splitting occurs at (±π/2,±π/2)\left(\pmπ/2, \pmπ/2\right)(±π/2,±π/2) in the Brillouin zone. The magnitudes of splitting in the two magnon modes are equal with respect to the case of δ=0δ=0δ=0. Dynamical spin structure factors show that the low-energy peak in the longitudinal spectral weight around (π/2,π/2)(π/2, π/2)(π/2,π/2) is also split, and thus the relative positions of the magnon modes and longitudinal modes in energy may change in the presence of a finite δδδ. These findings demonstrate that the altermagnets harbor more complex quantum dynamics than the conventional collinear antiferromagnets.
The intricate interplay between frustration and spin chirality, induced by the Dzyaloshinskii-Moriya interaction, holds promise for unveiling novel phases in frustrated quantum magnets. This study investigates the ground state phase diagram of the spin-12\frac{1}{2}21​ square lattice ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model upon incorporation of the chiral interaction ${J}_{\ensuremath{\chi}}$. Employing exact diagonalization (ED) with full lattice symmetries, we analyze the phase evolution as a function of J2{J}_{2}J2​ at fixed ${J}_{\ensuremath{\chi}}$, utilizing highly symmetric energy levels. Critical level crossings and ground state fidelity susceptibility (FS) techniques aid in pinpointing phase boundaries: Magnetic to nonmagnetic phase transitions are identified through critical level crossings between gapless magnetic excitations and quasidegenerate ground states of nonmagnetic phases. Direct transitions between two nonmagnetic phases are characterized by FS peaks due to avoided ground state level crossing. Based on these observations, we identify an anticipated chiral spin liquid (CSL) state and an adjacent nematic spin liquid (NSL) phase within a significant range of ${J}_{\ensuremath{\chi}}$, demarcated by a nearly vertical boundary line at ${J}_{2}\ensuremath{\approx}0.65$. This critical line terminates at the lower boundary in ${J}_{\ensuremath{\chi}}$ of a magnetic ordered chiral spin solid phase, which gains prominence with increasing ${J}_{\ensuremath{\chi}}$ from both CSL and NSL phases. The topological nature of the CSL is confirmed using the modular S\mathcal{S}S matrix of minimally entangled states on a torus and the entanglement spectra of even and odd sectors on a cylinder, employing ED and SU(2)\mathrm{SU}(2)SU(2)-symmetric density matrix renormalization group method, respectively. Furthermore, a comprehensive discussion on the nature of the NSL is provided, exploring aspects such as ground state degeneracy, local bond energy landscape, and singlet and triplet gaps on various tori, offering substantial evidence supporting the nematic nature of the NSL.
The quantum nature of electrons emerges in various exotic ways when they are put together with quantum many-body interactions. In Mott insulating systems with strong electron correlation, frustration and strong quantum fluctuations promote localized electron spins to form nontrivial quantum paramagnetic phases such as quantum spin liquid and valence bond solid preventing the development of conventional magnetic orders. In this thesis, we study such exotic quantum phases in various different frustrated quantum magnets. First, encouraged by the spin liquid phenomenology observed in Ba3XSb2O9 (X=Cu, Ni), we investigate Z2 spin liquids on a three dimensional triangular lattice by employing the Schwinger boson mean-field theory with projective symmetry group analysis. It is found that a nematic spin liquid is energetically favorable on this lattice structure rather than symmetric spin liquids. Next, we study various valence bond solid phases that can appear in kagome lattice antiferromagnets by generalizing the recently proposed phenomenological Z2 gauge theory for the topological Z2 spin liquid state found in the nearest-neighbor kagome lattice Heisenberg model. This work provides a bridge between the topological spin liquid and valence bond solid phases previously proposed for kagome lattice antiferromagnets via vison condensation in the gauge theory. Then, we consider optical conductivity in Mott insulators as an experimental probe to detect quantum paramagnetic phases like valence bond solid. Applying this idea to deformed kagome lattice antiferromagnet Rb2Cu3SnF12 with a 12-site valence bond solid state, we estimate the size of optical conductivity in this quantum magnet. After this, another quantum paramagnet BiCu2PO6 is carefully investigated to clarify its puzzling ground state nature and quasiparticle excitations. Utilizing recent high resolution neutron scattering data, we construct a microscopic spin model and also an effective theory for the low energy spin excitations in the compound. We find significant influences of magnetic anisotropies on the ground state and quasiparticle dynamics. Lastly, we investigate emergent quantum phases in the thin film geometries of the pyrochlore iridates, where a number of exotic quantum ground states are proposed to occur in bulk materials as a result of the interplay between electron correlation and strong spin-orbit coupling. The fate of these bulk phases, as well as the novel quantum states that may arise only in a thin film setting, are studied via a theoretical model that allows layer-dependent magnetic structures.
The intricate interplay between frustration and spin chirality has the potential to give rise to unprecedented phases in frustrated quantum magnets. We examine the ground state phase diagram of the spin-1/2 square lattice J1-J2-Jx model by employing critical level crossings and ground state fidelity susceptibility (FS) using exact diagonalization (ED) with full lattice symmetries. Our analysis reveals the evolution of highly symmetric energy levels as a function of J2 at fixed Jx. During a magnetic to non-magnetic phase transition, the precise identification of the phase boundary is achieved through critical level crossings between the gapless excitation of a magnetic phase and the quasi-degenerate ground state of a non-magnet phase. Conversely, a direct transition between two non-magnetic phases is characterized by a FS peak accompanied by an avoided ground state level crossing, serving as a distinctive signal. Within a substantial range of Jx, we identify an anticipated chiral spin liquid (CSL) state and an adjacent nematic spin liquid (NSL) phase with a degeneracy of two on a cylinder. These two phases are demarcated by a nearly vertical boundary line at J2 = 0.65. This critical line terminates at the lower boundary of a magnetic ordered chiral spin solid (CSS) phase. We validate the topological nature of the CSL using the modular S matrix of the minimum entangled states (MES) on a torus, along with the entanglement spectra (ES) of even and odd sectors on a cylinder, employing an SU(2)-symmetric density matrix renormalization group (DMRG) method. Furthermore, we delve into a comprehensive discussion on the nature of the NSL, exploring aspects such as ground state degeneracy, the local bond energy landscape, and the singlet and triplet gaps on various tori. These analysis provide substantial evidence supporting the nematic nature of the NSL.
The power of machine learning algorithms to automatically classify different phases of matter and detect quantum phase transitions without necessity to characterize phases by various quantities like local order parameters or topological invariants as in conventional approaches defined machine learning phases of matter as a new research frontier and basic research tool in condensed matter and statistical physics. We study quantum criticality in the spin-1/2 square-lattice J1-J2 model with additional plaquette structure by combination of reinforcement and supervised machine learning techniques. In our calculations the ground-state spin-spin correlation matrices for several system sizes are first found by restricted Boltzmann machine based variational Monte Carlo method, equivalent to reinforcement learning, and then used as a training data for convolutional neural network based supervised machine learning algorithm for phases classification. The model exhibits a quantum phase transition from paramagnetic plaquette resonating valence bond state to an antiferromagnetic state and has been a topic of great interest because of its close connection to cuprate superconductors and possibility of realization in cold atoms experiments. We consider both frustrated and unfrustrated regimes and compare our results with the results obtained previously with other methods. We find that our results are in good agreement with the results obtained by coupled cluster and real space renormalization group methods for both frustrated and unfrustrated regimes. The quantum Monte Carlo and finite-size scaling result for the unfrustrated case however slightly differs from our result for the critical value of the inter-plaquette coupling strength, although the results are still in reasonable agreement.