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arXiv:2310.09344

Thermal Phase Diagram of the Square Lattice Ferro-antiferromagnetic $J_1-J_2$ Heisenberg Model

Olivier Gauthé, Frédéric Mila

Published Oct 13, 2023• 0 citations
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Abstract

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 $J_1-J_2$ Heisenberg model on the square lattice with ferromagnetic $J_1 < 0$ and antiferromagnetic $J_2 > 0$. We localize the first order line originating from the first order zero temperature point as well as the second order thermal transition at large $J_2$ 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.

DOI
10.48550/arxiv.2310.09344

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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.
Using an SU(2) invariant finite-temperature tensor network algorithm, we provide strong numerical evidence in favor of an Ising transition in the collinear phase of the spin-1/2 J_{1}-J_{2} Heisenberg model on the square lattice. In units of J_{2}, the critical temperature reaches a maximal value of T_{c}/J_{2}≃0.18 around J_{2}/J_{1}≃1.0. It is strongly suppressed upon approaching the zero-temperature boundary of the collinear phase J_{2}/J_{1}≃0.6, and it vanishes as 1/log(J_{2}/J_{1}) in the large J_{2}/J_{1} limit, as predicted by Chandra et al., [Phys. Rev. Lett. 64, 88 (1990)PRLTAO0031-900710.1103/PhysRevLett.64.88]. Enforcing the SU(2) symmetry is crucial to avoid the artifact of finite-temperature SU(2) symmetry breaking of U(1) algorithms, opening new perspectives in the investigation of the thermal properties of quantum Heisenberg antiferromagnets.
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 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).
Phase diagram of the antiferromagnetic J1J_1J1​-J2J_2J2​ spin-111 pyrochlore Heisenberg model
We study the phase diagram of the antiferromagnetic J1J_1J1​-J2J_2J2​ Heisenberg model on the pyrochlore lattice with S=1S=1S=1 spins at zero and finite temperatures. We use a combination of complementary state-of-the-art quantum many-body approaches such as density matrix renormalization group (DMRG), density-matrix purification and pseudo-Majorana functional renormalization group (PMFRG). We present an efficient approach to preserve the applicability of the PMFRG for spin-1 systems at finite temperatures despite the inevitable presence of unphysical spin states. The good performance of our methods is first demonstrated for the nearest-neighbor pyrochlore Heisenberg model where the finite temperature behavior of the specific heat and uniform susceptibility show excellent agreement within PMFRG and density-matrix purification. Including an antiferromagnetic second neighbor coupling we find that the non-magnetic ground-state phase of the nearest neighbor model extents up to $J_2/J_1 \sim 0.02 $ within DMRG, beyond which magnetic k=0\boldsymbol{k}=0k=0 long-range order sets in. Our PMFRG calculations find the phase transition in a similar regime J2/J1∼0.035(8)J_2/J_1\sim 0.035(8)J2​/J1​∼0.035(8) which, together with the DMRG result, provides a strong argument for the existence of a small but finite non-magnetic ground-state phase in the spin-1 pyrochlore Heisenberg model. We also discuss the origin of discrepancies between different versions of the functional renormalization group concerning the location of this phase transition.
We study the phase diagram of the antiferromagnetic ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on the pyrochlore lattice with S=1S=1S=1 spins at zero and finite temperatures. We use a combination of complementary state-of-the-art quantum many-body approaches such as density matrix renormalization group (DMRG), density-matrix purification, and pseudo-Majorana functional renormalization group (PMFRG). We present an efficient approach to preserve the applicability of the PMFRG for spin-1 systems at finite temperatures despite the inevitable presence of unphysical spin states. The good performance of our methods is first demonstrated for the nearest-neighbor pyrochlore Heisenberg model where the finite-temperature behavior of the specific heat and uniform susceptibility show excellent agreement within PMFRG and density-matrix purification. Including an antiferromagnetic second-neighbor coupling we find that the nonmagnetic ground-state phase of the nearest-neighbor model extents up to ${J}_{2}/{J}_{1}\ensuremath{\sim}0.02$ within DMRG, beyond which magnetic $\mathbit{k}=0$ long-range order sets in. Our PMFRG calculations find the phase transition in a similar regime ${J}_{2}/{J}_{1}\ensuremath{\sim}0.035(8)$ which, together with the DMRG result, provides a strong argument for the existence of a small but finite nonmagnetic ground-state phase in the spin-1 pyrochlore Heisenberg model. We also discuss the origin of discrepancies between different versions of the functional renormalization group concerning the location of this phase transition.
We apply the plaquette renormalization scheme of tensor network states [Phys. Rev. E 83, 056703 (2011)] to study the spin-1/2 frustrated Heisenberg J1{J}_{1}J1​-J2{J}_{2}J2​ model on an $L\ifmmode\times\else\texttimes\fi{}L$ square lattice with LLL === 8, 16, and 32. By treating tensor elements as variational parameters, we obtain the ground states for different J2/J1{J}_{2}/{J}_{1}J2​/J1​ values, and investigate staggered magnetizations, nearest-neighbor spin-spin correlations, and plaquette order parameters. In addition to the well-known N\'eel order and collinear order at low and high J2/J1{J}_{2}/{J}_{1}J2​/J1​, we observe a plaquettelike order at ${J}_{2}/{J}_{1}\ensuremath{\approx}0.5$. A continuous transition between the N\'eel order and the plaquettelike order near ${J}_{2}^{{c}_{1}}\ensuremath{\approx}0.40{J}_{1}$ is observed. The collinear order emerges at ${J}_{2}^{{c}_{2}}\ensuremath{\approx}0.62{J}_{1}$ through a first-order phase transition.