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arXiv:2511.06423

Phase transitions in the spin-1/2 Heisenberg antiferromagnet on the square lattice

Qiao, Jie, Zhang, Shu-Hao, Qin, Jing-Bo, Zhao, Xiao-Long, Cheng, Qiang

Published Nov 9, 2025• 0 citations
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Abstract

The nature of the intermediate ground-state phase in the spin-1/2 frustrated square lattice model has long been debated. Using cluster density matrix embedding theory, we investigate the phase diagram of this model. The Neel phase is directly identified for J2<0.45 and the collinear phase for J2>0.65 based on the ground state. Although no direct evidence of an internal phase transition is found within the intermediate phase from the ground state, analysis of the first excited state wave function reveals a continuous quantum phase transition in this region, with a critical point at J2=0.55. This critical point divides the intermediate phase into PVBS and CVBS.

DOI
10.48550/arxiv.2511.06423

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The nature of the intermediate ground-state phase in the spin-1/2 frustrated square lattice model has long been debated. Using cluster density matrix embedding theory, we investigate the phase diagram of this model. The Neel phase is directly identified for J2<0.45 and the collinear phase for J2>0.65 based on the ground state. Although no direct evidence of an internal phase transition is found within the intermediate phase from the ground state, analysis of the first excited state wave function reveals a continuous quantum phase transition in this region, with a critical point at J2=0.55. This critical point divides the intermediate phase into PVBS and CVBS.
The nature of the intermediate ground‐state phase in the spin‐1/2 frustrated square lattice model has long been debated. We investigate the phase diagram of this model using cluster density matrix embedding theory (CDMET). Two antiferromagnetic phases are directly identified from our analysis of the ground‐state wave function: the Néel antiferromagnetic phase for J 2 ≤ 0.45 and the stripe antiferromagnetic phase for J 2 ≥ 0.65. Although our analysis of the ground state finds no direct evidence of an internal phase transition within the intermediate phase, an examination of the first excited state reveals a continuous quantum phase transition therein, with a critical point at J 2 = 0.55 that separates the intermediate phase into a plaquette valence bond solid (PVBS) and a (P + C) valence bond solid (VBS).
We perform an in-depth investigation of the phase diagram of the ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model on the square lattice. We take advantage of density matrix renormalization group and fully augmented matrix product states methods and reach unprecedented accuracy with large bond dimensions. We utilize excited-level crossing analysis to pinpoint the phase transition points. It was believed before that there exists a narrow spin liquid phase sandwiched by the N\'eel antiferromagnetic (AFM) and valence bond solid (VBS) phases. Through careful finite-size scaling of the level crossing points, we find a direct phase transition between the N\'eel AFM and VBS phases at J2/J1=0.535(3){J}_{2}/{J}_{1}=0.535(3)J2​/J1​=0.535(3), suggesting the absence of an intermediate spin liquid phase. We also provide accurate results for ground-state energies for a variety of sizes, from which we find that the transition between the N\'eel AFM and VBS phases is continuous. These results indicate the existence of a deconfined quantum critical point at J2/J1=0.535(3){J}_{2}/{J}_{1}=0.535(3)J2​/J1​=0.535(3) in the model. From the crossing of the first derivative of the energies with J2{J}_{2}J2​ for different sizes, we also determine the precise location of the first-order phase transition between the VBS and stripe AFM phases at J2/J1=0.610(5){J}_{2}/{J}_{1}=0.610(5)J2​/J1​=0.610(5).
We study the spin-1 Heisenberg model on the square lattice with the antiferromagnetic nearest-neighbor J1{J}_{1}J1​ and the next-nearest-neighbor J2{J}_{2}J2​ couplings by using the infinite projected entangled-pair state (iPEPS) ansatz and the density matrix renormalization group (DMRG) calculation. The iPEPS simulation, which studies the model directly in the thermodynamic limit, finds a crossing of the ground state from the N\'eel magnetic state to the stripe magnetic state at ${J}_{2}/{J}_{1}\ensuremath{\simeq}0.549$, showing a direct phase transition. In the finite-size DMRG calculation on the cylinder geometry up to the cylinder width Ly=10{L}_{y}=10Ly​=10, we find that around the same critical point the N\'eel and the stripe orders are strongly suppressed, which implies the absence of an intermediate phase. Both calculations identify that the stripe order comes with a first-order transition at ${J}_{2}/{J}_{1}\ensuremath{\simeq}0.549$. Our results indicate that unlike the spin-1/2 ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ square model, quantum fluctuations in the spin-1 model may not be strong enough to stabilize an intermediate nonmagnetic phase.
We use the density matrix renormalization group method to calculate several energy eigenvalues of the frustrated S=1/2 square-lattice J_{1}-J_{2} Heisenberg model on 2L×L cylinders with L≤10. We identify excited-level crossings versus the coupling ratio g=J_{2}/J_{1} and study their drifts with the system size L. The lowest singlet-triplet and singlet-quintuplet crossings converge rapidly (with corrections ∝L^{-2}) to different g values, and we argue that these correspond to ground-state transitions between the Néel antiferromagnet and a gapless spin liquid, at g_{c1}≈0.46, and between the spin liquid and a valence-bond solid at g_{c2}≈0.52. Previous studies of order parameters were not able to positively discriminate between an extended spin liquid phase and a critical point. We expect level-crossing analysis to be a generically powerful tool in density matrix renormalization group studies of quantum phase transitions.
The ground-state (GS) phase diagram of the frustrated spin- s J 1 -J 2 -J 3 Heisenberg antiferromagnet on the honeycomb lattice is studied using the coupled cluster method implemented to high orders of approximation, for spin quantum numbers s = 1, 3/2, 2 , 5/2. The model has antiferromagnetic (AFM) nearest-neighbour, next-nearest-neighbour and next-next-nearest-neighbour exchange couplings (with strength J 1 > 0, J 2 > 0 and J 3 > 0, respectively). We specifically study the case J 3 = J 2 = κJ 1 , in the range 0 < κ < 1 of the frustration parameter, which includes the point of maximum classical (s → ∞ ) frustration, viz., the classical critical point at κ cl = 1/2 , which separates the Neel phase for κ < κ cl and the collinear striped AFM phase for κ > κ cl . Results are presented for the GS energy, magnetic order parameter and plaquette valence-bond crystal (PVBC) susceptibility. For all spins s > 3/2 we find a quantum phase diagram very similar to the classical one, with a direct first-order transition between the two collinear AFM states at a value κ c (s) which is slightly greater than κ c l [e.g., κ c (3/2) ≈ 0.53(1)] and which approaches it monotonically as s → ∞ . By contrast, for the case s = 1 the transition is split into two such that the stable GS phases are one with Néel AFM order for κ < κ c1 = 0.485(5) and one with striped AFM order for κ > κ c2 = 0.528(5), just as in the case s = 1/2 (for which κ c1 ≈ 0.47 and κ c2 ≈ 0.60). For both the s = 1/2 and s = 1 models the transition at κ c2 appears to be of first-order type, while that at κ c1 appears to be continuous. However, whereas in the s = 1/2 case the intermediate phase appears to have PVBC order over the entire range κ c1 < κ < κ c2 , in the s = 1 case PVBC ordering either exists only over a very small part of the region or, more likely, is absent everywhere.
Thermal Phase Diagram of the Square Lattice Ferro-antiferromagnetic J1−J2J_1-J_2J1​−J2​ Heisenberg Model
Using state of the art tensor network computations combined with spin wave theory, we compute the finite temperature phase diagram of the spin 1/2 J1−J2J_1-J_2J1​−J2​ Heisenberg model on the square lattice with ferromagnetic J1<0J_1 < 0J1​<0 and antiferromagnetic J2>0J_2 > 0J2​>0. We localize the first order line originating from the first order zero temperature point as well as the second order thermal transition at large J2J_2J2​ associated with spontaneous lattice rotation symmetry breaking. This second order line terminates at a critical end point where it meets the first order line. We found no evidence for an intermediate phase.