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.
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.
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 ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ 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)-symmetric iPEPS ansatz (for a fixed bond dimension D) set a better upper bound, improving previous tensor-network-based results. By studying the finite-D scaling of the magnetically order parameter, we find a N\'eel phase for J2/J1<0.53. For 0.53<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), 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 and finally a first-order quantum phase transition takes place at 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.
Infinite Projected entangled-pair state (iPEPS) is a two-dimensional tensor network (TN) ansatz targeting the ground states of lattice models directly in the thermodynamic limit. It is the most successful among the true two-dimensional TN methods, challenging the more common density matrix renormalization group computations conducted on a finite-width cylinders. iPEPS, being a variational method does not suffer from the infamous sign problem, and hence could be readily applied to both fermionic problems off the half-filling and frustrated spin system. Using the imaginary-time optimization, the so called Full Update, led to the celebrated results of iPEPS such as the analysis of stripes in doped t-J and Hubbard models or the insight into the magnetization plateaus in Shastry-Sutherland model. However, the physics of the continuous phase transitions from the symmetry-broken Neel phase into the valence-bond solids or spin-liquids in frustrated magnets is still eluding iPEPS. The main difficulty lies in the optimization, since Full Update is just not precise enough. In the pursuit of the best variational iPEPS states, the gradient methods were put forward in 2016 by Corboz and Vanderstraeten. However, due to their complex nature, they have not been widely adopted especially for iPEPS ansatze with larger unit cells. We will introduce an alternative approach first applied in the context of iPEPS by Liao et al., based on the well established method of Algorithmic differentiation (AD) which allows us to evaluate gradients for iPEPS in a robust and conceptually simple way. Finally, we will present the results of our investigation into the physics of frustrated magnets putting AD to work.
We use the recently developed tensor network algorithm based on infinite projected entangled pair states (iPEPS) to study the phase diagram of frustrated antiferromagnetic J1-J2 Heisenberg model on a checkerboard lattice. The simulation indicates a N\'eel ordered phase when J2<0.88J1, a plaquette valence bond solid state when 0.88<J2/J1<1.11, and a stripe phase when J2>1.11J1, with two first-order transitions across the phase boundaries. The calculation shows that the cross dimer state proposed before is unlikely to be the ground state of the model, although such a state indeed arises as a metastable state in some parameter region.
The U(1)-symmetric iPEPS study of the spin-1/2 square J1-J2 Heisenberg model
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 J1-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)-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 J2c1=0.530(5), where the correlation length of the system diverges as ξ∼D1.2. The estimated critical anomalous exponents ηs=0.65(5) and ηd=1.9(1) for spin and dimer correlations respectively show strong deviation from those of the well-known J-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 J2 and finally a first-order quantum phase transition takes place at 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.
We study the spin-1 Heisenberg model on the square lattice with the antiferromagnetic nearest-neighbor J1 and the next-nearest-neighbor 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, 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 apply the plaquette renormalization scheme of tensor network states [Phys. Rev. E 83, 056703 (2011)] to study the spin-1/2 frustrated Heisenberg J1-J2 model on an $L\ifmmode\times\else\texttimes\fi{}L$ square lattice with L= 8, 16, and 32. By treating tensor elements as variational parameters, we obtain the ground states for different 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, 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.