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arXiv:1711.07584 · Physical review. B./Physical review. B

U(1) -symmetric infinite projected entangled-pair states study of the spin-1/2 square J1−J2 Heisenberg model

Reza Haghshenas, D. N. Sheng

Published May 8, 2018• 87 citations
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Abstract

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 ${J}_{2}/{J}_{1}<0.53$. For $0.53<{J}_{2}/{J}_{1}<0.61$, 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 ${J}_{2}^{{c}_{1}}/{J}_{1}=0.530(5)$, 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 ${J}_{2}/{J}_{1}$ and finally a first-order quantum phase transition takes place at ${J}_{2}^{{c}_{2}}/{J}_{1}=0.610(2)$ to the conventional Stripe phase. We compare our results with earlier DMRG and PEPS studies and suggest future directions for resolving remaining issues.

DOI
10.1103/physrevb.97.174408

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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
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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.
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 use the recently developed tensor network algorithm based on infinite projected entangled pair states (iPEPS) to study the phase diagram of frustrated antiferromagnetic J1{J}_{1}J1​-J2{J}_{2}J2​ Heisenberg model on a checkerboard lattice. The simulation indicates a N\'eel ordered phase when J2<0.88J1{J}_{2}<0.88{J}_{1}J2​<0.88J1​, a plaquette valence bond solid state when 0.88<J2/J1<1.110.88<{J}_{2}/{J}_{1}<1.110.88<J2​/J1​<1.11, and a stripe phase when J2>1.11J1{J}_{2}>1.11{J}_{1}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.
Motivated by two-dimensional bilayer systems we numerically study an anti-ferromagnetic spin-1/21/21/2 Heisenberg model on the honeycomb lattice with a nearest-neighbour exchange coupling J1J_1J1​, and a next-nearest-neighbour exchange coupling J2′J_2'J2′​ for the B sublattice sites only. Using infinite Projected Entangled-Pair States (iPEPS), variational uniform matrix-product states on cylinders, and exact diagonalization, we find an extended symmetric regime 0.4≲J2′/J1≲0.60.4\lesssim J'_2/J_1\lesssim0.60.4≲J2′​/J1​≲0.6, in which local observables show no magnetic, valence-bond, or chiral spin order. At J2′/J1=0.5J'_2/J_1=0.5J2′​/J1​=0.5, the PEPS correlation length grows systematically with bond dimension, and an inverse-correlation-length extrapolation favors a vanishing limit. We also find that various observables scale algebraically with the finite bond-dimension-induced correlation length, which points to a gapless spin liquid ground state. We propose a Z2\mathbb{Z}_2Z2​ Dirac spin liquid parton state, with Dirac points that are protected by translation, time-reversal and three-fold rotation symmetry, as a promising candidate state to explain the numerical results. We also discuss the possibility that the symmetric ground state is a featureless, short-range entangled state with a small gap.
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.