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arXiv:2606.31021

Investigation of the $J_1$-$J_2$ Heisenberg model on the triangular lattice: A study with projected entangled-pair states

L R Ma, Wei-Lin Tu, Didier Poilblanc, Ji-Yao Chen

Published Jun 30, 2026• 0 citations
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Abstract

The nature of the quantum spin liquid (QSL) phase in the frustrated $J_1$-$J_2$ Heisenberg model on the triangular lattice remains an open and actively debated problem. In this work, we employ the infinite projected entangled-pair state (PEPS) to systematically investigate the model under different symmetry constraints. Our simulations reveal a direct transition from the $120^\circ$ Néel state to a putative QSL at $J_2/J_1\approx 0.08$, signaled by the collapse of magnetic order. We further show that, through either an appropriate unitary rotation or spontaneous spin long-range order, the stripe antiferromagnetic phase can also be accurately captured within the infinite PEPS framework. A central focus of our study is the role played by the PEPS symmetry in approximating the QSL ground-state sandwiched between the two magnetic phases. We first found that a fully-symmetric topological $\mathbb{Z}_2$ Resonating Valence Bond state, which can be written as a simple PEPS with bond dimension $D=3$, exhibits a reasonably good variational energy. Motivated by this finding, we have further constructed generic $\mathbb{Z}_2$-symmetric PEPS of larger bond dimension (up to $D=7$). We found that, under wavefunction optimization, spinons condense and, simultaneously, topological vison excitations get confined, hence precluding $\mathbb{Z}_2$ topological order. This strongly indicates the gapless (or critical) nature of the QSL phase, which is most naturally consistent with a U(1) Dirac spin liquid scenario.

DOI
10.48550/arxiv.2606.31021

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Investigation of the J1J_1J1​-J2J_2J2​ Heisenberg model on the triangular lattice: A study with projected entangled-pair states
The nature of the quantum spin liquid (QSL) phase in the frustrated J1J_1J1​-J2J_2J2​ Heisenberg model on the triangular lattice remains an open and actively debated problem. In this work, we employ the infinite projected entangled-pair state (PEPS) to systematically investigate the model under different symmetry constraints. Our simulations reveal a direct transition from the 120∘120^\circ120∘ Néel state to a putative QSL at J2/J1≈0.08J_2/J_1\approx 0.08J2​/J1​≈0.08, signaled by the collapse of magnetic order. We further show that, through either an appropriate unitary rotation or spontaneous spin long-range order, the stripe antiferromagnetic phase can also be accurately captured within the infinite PEPS framework. A central focus of our study is the role played by the PEPS symmetry in approximating the QSL ground-state sandwiched between the two magnetic phases. We first found that a fully-symmetric topological Z2\mathbb{Z}_2Z2​ Resonating Valence Bond state, which can be written as a simple PEPS with bond dimension D=3D=3D=3, exhibits a reasonably good variational energy. Motivated by this finding, we have further constructed generic Z2\mathbb{Z}_2Z2​-symmetric PEPS of larger bond dimension (up to D=7D=7D=7). We found that, under wavefunction optimization, spinons condense and, simultaneously, topological vison excitations get confined, hence precluding Z2\mathbb{Z}_2Z2​ topological order. This strongly indicates the gapless (or critical) nature of the QSL phase, which is most naturally consistent with a U(1) Dirac spin liquid scenario.
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 ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ 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)U(1)U(1)-symmetric iPEPS ansatz (for a fixed bond dimension D)D)D) set a better upper bound, improving previous tensor-network-based results. By studying the finite-DDD scaling of the magnetically order parameter, we find a N\'eel phase for J2/J1<0.53{J}_{2}/{J}_{1}<0.53J2​/J1​<0.53. For 0.53<J2/J1<0.610.53<{J}_{2}/{J}_{1}<0.610.53<J2​/J1​<0.61, a nonmagnetic columnar valence bond solid (VBS) state is established as observed by the pattern of local bond energy. The divergent behavior of correlation length $\ensuremath{\xi}\ensuremath{\sim}{D}^{1.2}$ and vanishing order parameters are consistent with a deconfined N\'eel-to-VBS transition at J2c1/J1=0.530(5){J}_{2}^{{c}_{1}}/{J}_{1}=0.530(5)J2c1​​/J1​=0.530(5), where estimated critical anomalous exponents are ${\ensuremath{\eta}}_{s}\ensuremath{\sim}0.6$ and ${\ensuremath{\eta}}_{d}\ensuremath{\sim}1.9$ for spin and dimer correlations, respectively. We show that the associated VBS order parameter monotonically increases with J2/J1{J}_{2}/{J}_{1}J2​/J1​ and finally a first-order quantum phase transition takes place at J2c2/J1=0.610(2){J}_{2}^{{c}_{2}}/{J}_{1}=0.610(2)J2c2​​/J1​=0.610(2) to the conventional Stripe phase. We compare our results with earlier DMRG and PEPS studies and suggest future directions for resolving remaining issues.
The nature of quantum spin liquids is studied for the spin-1/21/21/2 antiferromagnetic Heisenberg model on a square lattice containing exchange interactions between nearest-neighbor sites, J1J_1J1​, and those between next-nearest-neighbor sites, J2J_2J2​. We perform variational Monte Carlo simulations together with the quantum-number-projection technique and clarify the phase diagram in the ground state together with its excitation spectra. We obtain the nonmagnetic phase in the region 0.4<J2/J1≤0.60.4< J_2/J_1\le 0.60.4<J2​/J1​≤0.6 sandwiched by the staggered and stripe antiferromagnetic (AF) phases. Our direct calculations of the spin gap support the notion that the triplet excitation from the singlet ground state is gapless in the range of 0.4<J2/J1≤0.50.4 < J_2/J_1 \le 0.50.4<J2​/J1​≤0.5, while the gapped valence-bond-crystal (VBC) phase is stabilized for 0.5<J2/J1≤0.60.5 < J_2/J_1 \le 0.60.5<J2​/J1​≤0.6. The VBC order is likely to have the columnar symmetry with a spontaneous symmetry breaking of the C4vC_{4v}C4v​ symmetry. The power-law behaviors of the spin-spin and dimer-dimer correlation functions in the gapless region are consistent with the emergence of the algebraic quantum-spin-liquid phase (critical phase). The exponent of the spin correlation ⟨S(0)S(r)⟩∝1/rz+η\langle S(0)S(r)\rangle \propto 1/r^{z+η}⟨S(0)S(r)⟩∝1/rz+η at a long distance rrr appears to increase from z+η∼1z+η\sim 1z+η∼1 at J2/J1∼0.4J_2/J_1\sim0.4J2​/J1​∼0.4 toward the continuous transition to the VBC phase at J1/J1∼0.5J_1/J_1\sim0.5J1​/J1​∼0.5. Our results, however, do not fully exclude the possibility of a direct quantum transition between the staggered AF and VBC phases with a wide critical region and deconfined criticality.
Dynamics and stability of U(1) spin liquids beyond mean-field theory: Triangular-lattice J1J_1J1​-J2J_2J2​ Heisenberg model
Quantum spin liquids (QSLs) are long-range entangled phases of frustrated magnets exhibiting fractionalised spin excitations. In two dimensions, there is limited analytical understanding of their excitation spectra beyond parton mean-field theories, which fail to capture many features of the finite frequency dynamical response. We use a self-consistent mean-field theory combined with the random phase approximation (RPA) for the J1J_1J1​-J2J_2J2​ Heisenberg model on the triangular lattice to describe the strong spinon-spinon interactions of the U(1) Dirac QSL. We obtain quantitative results for the dynamical spin structure factor and phase diagram compatible with comprehensive numerical efforts. We show the continuum response at the Brillouin zone corners is described by a spinon-exciton hybridised with monopole gauge excitations. We further show that the transition from the QSL to 120-degree coplanar order is in the QED3_33​ O(3)-GN universality class. We extend the method to chiral QSLs and XXZ anisotropy, and discuss its broad range of applicability to other models and for describing inelastic neutron scattering experiments in KYbSe2_22​.
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.
The ground state phase of spin-1/2 J_1-J_2 antiferromagnetic Heisenberg model on square lattice in the maximally frustrated regime (J_2 ~ 0.5J_1) has been debated for decades. Here we study this model by using a recently proposed novel numerical method - the cluster update algorithm for tensor product states (TPSs). The ground state energies at finite sizes and in the thermodynamic limit (with finite size scaling) are in good agreement with the state of art exact diagonalization study, and the energy differences between these two studies are of the order of 10^{-3}J_1 per site. At the largest bond dimension available D (D=9), we find a paramagnetic ground state without any valence bond solid order in the thermodynamic limit in the range of 0.5 < J_2/J_1 < 0.6, which implies the emergence of a spin-liquid phase. Furthermore, we investigate the topologically ordered nature of such a spin-liquid phase by measuring a nonzero topological entanglement entropy.
The recent progress in the optimization of two-dimensional tensor networks [H.-J. Liao, J.-G. Liu, L. Wang, and T. Xiang, Phys. Rev. X 9, 031041 (2019)] based on automatic differentiation opened the way towards precise and fast optimization of such states and, in particular, infinite projected entangled-pair states (iPEPS) that constitute a generic-purpose Ansatz for lattice problems governed by local Hamiltonians. In this work, we perform an extensive study of a paradigmatic model of frustrated magnetism, the J_1-J_2 J 1 − J 2 Heisenberg antiferromagnet on the square lattice. By using advances in both optimization and subsequent data analysis, through finite correlation-length scaling, we report accurate estimations of the magnetization curve in the N'eel phase for J_2/J_1 \le 0.45 J 2 / J 1 ≤ 0.45 . The unrestricted iPEPS simulations reveal an U(1) U ( 1 ) symmetric structure, which we identify and impose on tensors, resulting in a clean and consistent picture of antiferromagnetic order vanishing at the phase transition with a quantum paramagnet at J_2/J_1 \approx 0.46(1) J 2 / J 1 ≈ 0.46 ( 1 ) . The present methodology can be extended beyond this model to study generic order-to-disorder transitions in magnetic systems.