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:2508.05133 · Physical review. B./Physical review. B

Symmetry breaking and competing valence bond states in the star lattice Heisenberg antiferromagnet

Pratyay Ghosh, Jan Alexander Koziol, Samuel Nyckees, Kai Phillip Schmidt, Frédéric Mila

Published Sep 24, 2025• 3 citations
arXivHTMLPDF

Abstract

We investigate the ground state phase diagram of the spin-$1/2$ antiferromagnetic Heisenberg model on the star lattice using infinite projected entangled pair states (iPEPS) and high-order series expansions. The model includes two distinct couplings: ${J}_{d}$ on the dimer bonds and ${J}_{t}$ on the trimer bonds. While it is established that the system hosts a valence bond solid (VBS) phase for ${J}_{d}\ensuremath{\ge}{J}_{t}$, the ground state phase diagram for ${J}_{d}<{J}_{t}$ has remained unsettled. Our iPEPS simulations uncover a first-order phase transition at ${J}_{d}/{J}_{t}\ensuremath{\approx}0.18$, significantly lower than previously reported estimates. Beyond this transition, we identify a close competition between two valence bond crystal (VBC) states; a columnar VBC and a $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ VBC, with the latter consistently exhibiting lower energy across all finite bond dimensions. The high-order series expansion supports this by finding that the $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ VBC state indeed becomes energetically favorable, but only at sixth order in perturbation theory, revealing the subtle nature of the competition between candidate states.

DOI
10.1103/nbjt-hlvx

Related papers

Quantum phase diagram of the spin- 1 2 Heisenberg antiferromagnet on the square-kagome lattice: A tensor network study

Saeed S. Jahromi, Yasir Iqbal

We study the ground-state phase diagram of the spin-$1/2$ antiferromagnetic Heisenberg model on the square-kagome lattice using infinite projected entangled-pair states (iPEPS). By systematically varying the ratio of exchange couplings on triangular and square plaquettes, we establish a complete quantum phase diagram in the thermodynamic limit. In the intermediate-coupling regime, we identify four distinct nonmagnetic phases that are unambiguously characterized as valence-bond crystals (VBCs) by their symmetry-inequivalent patterns of strong and weak spin-spin correlations. These include a plaquette crossed-dimer VBC, a loop-six VBC stabilized around the isotropic point, a generalized pinwheel VBC with reduced rotational symmetry, and a decorated loop-six VBC proximate to ferrimagnetic order. We determine the phase boundaries using a combination of bond-resolved correlation functions, entanglement entropy, and magnetization. For transitions not accompanied by sharp entanglement signatures, we extract the spin gap from finite-field simulations, allowing us to distinguish gapped and gapless VBC phases. At larger coupling ratios, the system undergoes transitions into imperfect and perfect ferrimagnetic states. Our results resolve long-standing ambiguities in the square-kagome Heisenberg model and provide a quantitatively reliable reference for future theoretical and experimental studies of frustrated quantum magnets.

arXiv:2503.07689•Physical review. B./Physical review. B•Mar 10, 2026•1 citation

Competing quantum paramagnetic ground states of the Heisenberg antiferromagnet on the star lattice

Bohm‐Jung Yang, Arun Paramekanti, Yong Baek Kim

We investigate various competing paramagnetic ground states of the Heisenberg antiferromagnet on the two-dimensional star lattice which exhibits geometric frustration. Using slave particle mean-field theory combined with a projective symmetry group analysis, we examine a variety of candidate spin liquid states on this lattice, including chiral spin liquids, spin liquids with Fermi surfaces of spinons, and nematic spin liquids which break lattice rotational symmetry. Motivated by connection to large-$N$ $\text{SU}(N)$ theory as well as numerical exact diagonalization studies, we also examine various valence-bond solid (VBS) states on this lattice. Based on a study of energetics using Gutzwiller projected states, we find that a fully gapped spin liquid state is the lowest-energy spin liquid candidate for this model. We also find from a study of energetics using Gutzwiller projected wave functions and bond operator approaches that this spin liquid is unstable toward two different VBS states---a VBS state which respects all the Hamiltonian symmetries and a VBS state which exhibits $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ order---depending on the ratio of the Heisenberg exchange couplings on the two inequivalent bonds of the lattice. We compute the triplon dispersion in both VBS states within the bond operator approach and discuss possible implications of our work for future experiments on candidate materials.

arXiv:0911.2702•Physical Review B•Apr 15, 2010•35 citations

Closely competing valence bond crystal orders in the ground state of the spin-$\frac{1}{2}$ antiferromagnetic Heisenberg model on the pyrochlore lattice: a large scale unrestricted variational study

Cheng, Rong, Tao Li

The spin-$\frac{1}{2}$ antiferromagnetic Heisenberg model on the pyrochlore lattice(PAFH) is arguably the most well known strongly frustrated quantum magnet in three spatial dimension. However, due to the rapid scaling of Hilbert space with the linear size of such a three dimensional system, the nature of its ground state in the thermodynamic limit remains elusive after about 30 years' intense debate. Here we apply a recently developed powerful algorithm to perform large scale unrestricted variational optimization of the ground state of the spin-$\frac{1}{2}$ PAFH from the resonating valence bond(RVB) theory perspective. We find a highly competitive candidate ground state of the system. This novel state features a maximally resonating valence bond crystal(VBC) pattern with $2\vec{a}_{1}\times2\vec{a}_{2}\times2\vec{a}_{3}$ periodicity. There are at least four levels of hierarchical structure in such a VBC state, with the first and the second level of hierarchy related to the breaking of the inversion and the translational symmetry. Intriguingly, we find that within the RVB framework such a maximally resonating VBC state is almost degenerate with a recently proposed VBC state that is obtained from dressing hard hexagon covering of the pyrochlore lattice, although they have very different structures. We also find that further symmetry breaking will occur in the dressed hard hexagon VBC state under unrestricted optimization, which results in strong disparity in $\langle \hat{\mathbf{S}}^{2}_{u} \rangle$ for up and down tetrahedrons as we observe in the maximally resonating VBC state. We show that the maximally resonating VBC state found here will be favored by a tiny next-neighboring exchange coupling over the dressed hard hexagon covering VBC state.

arXiv:2509.13746•Sep 17, 2025•0 citations

The $U(1)$-symmetric iPEPS study of the spin-$1/2$ square $J_{1}$-$J_{2}$ Heisenberg model

Reza Haghshenas, D. N. Sheng

We develop an improved variant of $U(1)$-symmetric infinite projected entangled-pair states (iPEPS) ansatz to investigate the ground state phase diagram of the spin-$1/2$ square $J_{1}$-$J_{2}$ Heisenberg model. In order to improve the accuracy of the ansatz, we discuss a simple strategy to select automatically relevant symmetric sectors and also introduce an optimization method to treat second-neighbor interactions more efficiently. We show that variational ground-state energies of the model obtained by the $U(1)$-symmetric iPEPS ansatz (for a fixed bond dimension $D$) set a better upper bound, improving previous tensor-network-based results. By studying N\'{e}el order parameter and correlation function in the large-$D$ limit, we find that a deconfined quantum-critical point occurs at $J^{c_1}_{2}=0.530(5)$, where the correlation length of the system diverges as $\xi \sim D^{1.2}$. The estimated critical anomalous exponents $\eta_{s}=0.65(5)$ and $\eta_{d}=1.9(1)$ for spin and dimer correlations respectively show strong deviation from those of the well-known $\mathcal{J}$-$\mathcal{Q}$ model, suggesting possibly a different universality class. The pattern of local bond energy for the intermediate phase is compatible with the columnar valence bond solid (VBS) state, with a quite competitive variational ground-state energy in comparison with plaquette VBS one predicted by the density matrix renormalization group. We show that the associated VBS order parameter first monotonically increases with $J_2$ and finally a first-order quantum phase transition takes place at $J^{c_2}_{2}=0.610(3)$ to the conventional Stripe phase. We compare our results with earlier DMRG and PEPS studies and suggest future directions for resolving remaining issues.

•Nov 21, 2017•0 citations

Competition between spin liquids and valence-bond order in the frustrated spin- 12 Heisenberg model on the honeycomb lattice

Francesco Ferrari, Samuel Bieri, Federico Becca

Using variational wave functions and Monte Carlo techniques, we study the antiferromagnetic Heisenberg model with first-neighbor ${J}_{1}$ and second-neighbor ${J}_{2}$ antiferromagnetic couplings on the honeycomb lattice. We perform a systematic comparison of magnetically ordered and nonmagnetic states (spin liquids and valence-bond solids) to obtain the ground-state phase diagram. N\'eel order is stabilized for small values of the frustrating second-neighbor coupling. Increasing the ratio ${J}_{2}/{J}_{1}$, we find strong evidence for a continuous transition to a nonmagnetic phase at ${J}_{2}/{J}_{1}\ensuremath{\approx}0.23$. Close to the transition point, the Gutzwiller-projected uniform resonating valence-bond state gives an excellent approximation to the exact ground-state energy. For $0.23\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\lesssim}0.36$, a gapless ${Z}_{2}$ spin liquid with Dirac nodes competes with a plaquette valence-bond solid. In contrast, the gapped spin liquid considered in previous works has significantly higher variational energy. Although the plaquette valence-bond order is expected to be present as soon as the N\'eel order melts, this ordered state becomes clearly favored only for ${J}_{2}/{J}_{1}\ensuremath{\gtrsim}0.3$. Finally, for $0.36\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\le}0.5$, a valence-bond solid with columnar order takes over as the ground state, being also lower in energy than the magnetic state with collinear order. We perform a detailed finite-size scaling and standard data collapse analysis, and we discuss the possibility of a deconfined quantum critical point separating the N\'eel antiferromagnet from the plaquette valence-bond solid.

arXiv:1706.07810•Physical review. B./Physical review. B•Sep 1, 2017•46 citations

Emergent spin-1 trimerized valence bond crystal in the spin- 12 Heisenberg model on the star lattice

Shi-Ju Ran, Wei Li, Shou-Shu Gong +3 more

We explore the frustrated spin-$\frac{1}{2}$ Heisenberg model on the star lattice with antiferromagnetic (AF) couplings inside each triangle and ferromagnetic (FM) intertriangle couplings (${J}_{e}<0$), and calculate its magnetic and thermodynamic properties. We show that the FM couplings do not sabotage the magnetic disordering of the ground state due to the frustration from the AF interactions inside each triangle, but trigger a fully gapped inversion-symmetry-breaking trimerized valence bond crystal (TVBC) with emergent spin-1 degrees of freedom. We discover that with strengthening ${J}_{e}$, the system exhibits a universal scaling behavior either with or without a magnetic field $h$: the order parameter, the five critical fields that separate the ${J}_{e}\text{\ensuremath{-}}h$ ground-state phase diagram into six phases, and the excitation gap obtained by low-temperature specific heat, all depend exponentially on ${J}_{e}$. Our work implies that the spin-1 VBCs can be stabilized by introducing small FM couplings in the geometrically frustrated spin-$\frac{1}{2}$ systems.

arXiv:1508.03451•Physical review. B./Physical review. B•Feb 26, 2018•23 citations
We investigate the ground state phase diagram of the spin-1/21/21/2 antiferromagnetic Heisenberg model on the star lattice using infinite projected entangled pair states (iPEPS) and high-order series expansions. The model includes two distinct couplings: Jd{J}_{d}Jd​ on the dimer bonds and Jt{J}_{t}Jt​ on the trimer bonds. While it is established that the system hosts a valence bond solid (VBS) phase for Jd\ensuremath≥Jt{J}_{d}\ensuremath{\ge}{J}_{t}Jd​\ensuremath≥Jt​, the ground state phase diagram for Jd<Jt{J}_{d}<{J}_{t}Jd​<Jt​ has remained unsettled. Our iPEPS simulations uncover a first-order phase transition at Jd/Jt\ensuremath≈0.18{J}_{d}/{J}_{t}\ensuremath{\approx}0.18Jd​/Jt​\ensuremath≈0.18, significantly lower than previously reported estimates. Beyond this transition, we identify a close competition between two valence bond crystal (VBC) states; a columnar VBC and a 3\ifmmode×\else\texttimes\fi3\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}3​\ifmmode×\else\texttimes\fi3​ VBC, with the latter consistently exhibiting lower energy across all finite bond dimensions. The high-order series expansion supports this by finding that the 3\ifmmode×\else\texttimes\fi3\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}3​\ifmmode×\else\texttimes\fi3​ VBC state indeed becomes energetically favorable, but only at sixth order in perturbation theory, revealing the subtle nature of the competition between candidate states.
We study the ground-state phase diagram of the spin-1/21/21/2 antiferromagnetic Heisenberg model on the square-kagome lattice using infinite projected entangled-pair states (iPEPS). By systematically varying the ratio of exchange couplings on triangular and square plaquettes, we establish a complete quantum phase diagram in the thermodynamic limit. In the intermediate-coupling regime, we identify four distinct nonmagnetic phases that are unambiguously characterized as valence-bond crystals (VBCs) by their symmetry-inequivalent patterns of strong and weak spin-spin correlations. These include a plaquette crossed-dimer VBC, a loop-six VBC stabilized around the isotropic point, a generalized pinwheel VBC with reduced rotational symmetry, and a decorated loop-six VBC proximate to ferrimagnetic order. We determine the phase boundaries using a combination of bond-resolved correlation functions, entanglement entropy, and magnetization. For transitions not accompanied by sharp entanglement signatures, we extract the spin gap from finite-field simulations, allowing us to distinguish gapped and gapless VBC phases. At larger coupling ratios, the system undergoes transitions into imperfect and perfect ferrimagnetic states. Our results resolve long-standing ambiguities in the square-kagome Heisenberg model and provide a quantitatively reliable reference for future theoretical and experimental studies of frustrated quantum magnets.
We investigate various competing paramagnetic ground states of the Heisenberg antiferromagnet on the two-dimensional star lattice which exhibits geometric frustration. Using slave particle mean-field theory combined with a projective symmetry group analysis, we examine a variety of candidate spin liquid states on this lattice, including chiral spin liquids, spin liquids with Fermi surfaces of spinons, and nematic spin liquids which break lattice rotational symmetry. Motivated by connection to large-NNN SU(N)\text{SU}(N)SU(N) theory as well as numerical exact diagonalization studies, we also examine various valence-bond solid (VBS) states on this lattice. Based on a study of energetics using Gutzwiller projected states, we find that a fully gapped spin liquid state is the lowest-energy spin liquid candidate for this model. We also find from a study of energetics using Gutzwiller projected wave functions and bond operator approaches that this spin liquid is unstable toward two different VBS states---a VBS state which respects all the Hamiltonian symmetries and a VBS state which exhibits $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ order---depending on the ratio of the Heisenberg exchange couplings on the two inequivalent bonds of the lattice. We compute the triplon dispersion in both VBS states within the bond operator approach and discuss possible implications of our work for future experiments on candidate materials.
Closely competing valence bond crystal orders in the ground state of the spin-12\frac{1}{2}21​ antiferromagnetic Heisenberg model on the pyrochlore lattice: a large scale unrestricted variational study
The spin-12\frac{1}{2}21​ antiferromagnetic Heisenberg model on the pyrochlore lattice(PAFH) is arguably the most well known strongly frustrated quantum magnet in three spatial dimension. However, due to the rapid scaling of Hilbert space with the linear size of such a three dimensional system, the nature of its ground state in the thermodynamic limit remains elusive after about 30 years' intense debate. Here we apply a recently developed powerful algorithm to perform large scale unrestricted variational optimization of the ground state of the spin-12\frac{1}{2}21​ PAFH from the resonating valence bond(RVB) theory perspective. We find a highly competitive candidate ground state of the system. This novel state features a maximally resonating valence bond crystal(VBC) pattern with 2a⃗1×2a⃗2×2a⃗32\vec{a}_{1}\times2\vec{a}_{2}\times2\vec{a}_{3}2a1​×2a2​×2a3​ periodicity. There are at least four levels of hierarchical structure in such a VBC state, with the first and the second level of hierarchy related to the breaking of the inversion and the translational symmetry. Intriguingly, we find that within the RVB framework such a maximally resonating VBC state is almost degenerate with a recently proposed VBC state that is obtained from dressing hard hexagon covering of the pyrochlore lattice, although they have very different structures. We also find that further symmetry breaking will occur in the dressed hard hexagon VBC state under unrestricted optimization, which results in strong disparity in ⟨S^u2⟩\langle \hat{\mathbf{S}}^{2}_{u} \rangle⟨S^u2​⟩ for up and down tetrahedrons as we observe in the maximally resonating VBC state. We show that the maximally resonating VBC state found here will be favored by a tiny next-neighboring exchange coupling over the dressed hard hexagon covering VBC state.
The U(1)U(1)U(1)-symmetric iPEPS study of the spin-1/21/21/2 square J1J_{1}J1​-J2J_{2}J2​ Heisenberg model
We develop an improved variant of U(1)U(1)U(1)-symmetric infinite projected entangled-pair states (iPEPS) ansatz to investigate the ground state phase diagram of the spin-1/21/21/2 square J1J_{1}J1​-J2J_{2}J2​ Heisenberg model. In order to improve the accuracy of the ansatz, we discuss a simple strategy to select automatically relevant symmetric sectors and also introduce an optimization method to treat second-neighbor interactions more efficiently. We show that variational ground-state energies of the model obtained by the U(1)U(1)U(1)-symmetric iPEPS ansatz (for a fixed bond dimension DDD) set a better upper bound, improving previous tensor-network-based results. By studying N\'{e}el order parameter and correlation function in the large-DDD limit, we find that a deconfined quantum-critical point occurs at J2c1=0.530(5)J^{c_1}_{2}=0.530(5)J2c1​​=0.530(5), where the correlation length of the system diverges as ξ∼D1.2\xi \sim D^{1.2}ξ∼D1.2. The estimated critical anomalous exponents ηs=0.65(5)\eta_{s}=0.65(5)ηs​=0.65(5) and ηd=1.9(1)\eta_{d}=1.9(1)ηd​=1.9(1) for spin and dimer correlations respectively show strong deviation from those of the well-known J\mathcal{J}J-Q\mathcal{Q}Q model, suggesting possibly a different universality class. The pattern of local bond energy for the intermediate phase is compatible with the columnar valence bond solid (VBS) state, with a quite competitive variational ground-state energy in comparison with plaquette VBS one predicted by the density matrix renormalization group. We show that the associated VBS order parameter first monotonically increases with J2J_2J2​ and finally a first-order quantum phase transition takes place at J2c2=0.610(3)J^{c_2}_{2}=0.610(3)J2c2​​=0.610(3) to the conventional Stripe phase. We compare our results with earlier DMRG and PEPS studies and suggest future directions for resolving remaining issues.
Using variational wave functions and Monte Carlo techniques, we study the antiferromagnetic Heisenberg model with first-neighbor J1{J}_{1}J1​ and second-neighbor J2{J}_{2}J2​ antiferromagnetic couplings on the honeycomb lattice. We perform a systematic comparison of magnetically ordered and nonmagnetic states (spin liquids and valence-bond solids) to obtain the ground-state phase diagram. N\'eel order is stabilized for small values of the frustrating second-neighbor coupling. Increasing the ratio J2/J1{J}_{2}/{J}_{1}J2​/J1​, we find strong evidence for a continuous transition to a nonmagnetic phase at ${J}_{2}/{J}_{1}\ensuremath{\approx}0.23$. Close to the transition point, the Gutzwiller-projected uniform resonating valence-bond state gives an excellent approximation to the exact ground-state energy. For $0.23\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\lesssim}0.36$, a gapless Z2{Z}_{2}Z2​ spin liquid with Dirac nodes competes with a plaquette valence-bond solid. In contrast, the gapped spin liquid considered in previous works has significantly higher variational energy. Although the plaquette valence-bond order is expected to be present as soon as the N\'eel order melts, this ordered state becomes clearly favored only for ${J}_{2}/{J}_{1}\ensuremath{\gtrsim}0.3$. Finally, for $0.36\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\le}0.5$, a valence-bond solid with columnar order takes over as the ground state, being also lower in energy than the magnetic state with collinear order. We perform a detailed finite-size scaling and standard data collapse analysis, and we discuss the possibility of a deconfined quantum critical point separating the N\'eel antiferromagnet from the plaquette valence-bond solid.
We explore the frustrated spin-12\frac{1}{2}21​ Heisenberg model on the star lattice with antiferromagnetic (AF) couplings inside each triangle and ferromagnetic (FM) intertriangle couplings (Je<0{J}_{e}<0Je​<0), and calculate its magnetic and thermodynamic properties. We show that the FM couplings do not sabotage the magnetic disordering of the ground state due to the frustration from the AF interactions inside each triangle, but trigger a fully gapped inversion-symmetry-breaking trimerized valence bond crystal (TVBC) with emergent spin-1 degrees of freedom. We discover that with strengthening Je{J}_{e}Je​, the system exhibits a universal scaling behavior either with or without a magnetic field hhh: the order parameter, the five critical fields that separate the ${J}_{e}\text{\ensuremath{-}}h$ ground-state phase diagram into six phases, and the excitation gap obtained by low-temperature specific heat, all depend exponentially on Je{J}_{e}Je​. Our work implies that the spin-1 VBCs can be stabilized by introducing small FM couplings in the geometrically frustrated spin-12\frac{1}{2}21​ systems.