We use the finite projected entangled pair state (PEPS) method to investigate the global phase diagram of the spin-$\frac{1}{2}$ square-lattice ${J}_{1}\text{\ensuremath{-}}{J}_{2}\text{\ensuremath{-}}{J}_{3}$ antiferromagnetic (AFM) Heisenberg model. The ground-state phase diagram is established with a rich variety of phases: the N\'eel AFM, gapless quantum spin liquid, valence-bond solid (VBS), stripe AFM, and incommensurate spiral phases. The nature of the VBS region is revealed, which contains a plaquette VBS and a mixed columnar-plaquette VBS, with the emergence of short-range incommensurate spin correlation in some region. The long-range incommensurate magnetic phase is also explicitly characterized as a planar spiral with incommensurate spatial periodicities. Most interestingly, there exist several multicritical points connecting different phases. These findings elucidate the true nature of the long-standing square-lattice ${J}_{1}\text{\ensuremath{-}}{J}_{2}\text{\ensuremath{-}}{J}_{3}$ antiferromagnet at zero temperature. Our results also pave the way to accurately simulate complex two-dimensional quantum spin systems that may host nonuniform features by means of the finite PEPS.
We use the finite projected entangled pair state (PEPS) method to investigate the global phase diagram of the spin-21 square-lattice J1\ensuremath-J2\ensuremath-J3 antiferromagnetic (AFM) Heisenberg model. The ground-state phase diagram is established with a rich variety of phases: the N'eel AFM, gapless quantum spin liquid, valence-bond solid (VBS), stripe AFM, and incommensurate spiral phases. The nature of the VBS region is revealed, which contains a plaquette VBS and a mixed columnar-plaquette VBS, with the emergence of short-range incommensurate spin correlation in some region. The long-range incommensurate magnetic phase is also explicitly characterized as a planar spiral with incommensurate spatial periodicities. Most interestingly, there exist several multicritical points connecting different phases. These findings elucidate the true nature of the long-standing square-lattice J1\ensuremath-J2\ensuremath-J3 antiferromagnet at zero temperature. Our results also pave the way to accurately simulate complex two-dimensional quantum spin systems that may host nonuniform features by means of the finite PEPS.
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), 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) in the model. From the crossing of the first derivative of the energies with 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).
We use the coupled-cluster method in high orders of approximation to make a comprehensive study of the ground-state (GS) phase diagram of the spin-1/2 J1-J2-J3 model on a two-dimensional honeycomb lattice with antiferromagnetic (AFM) interactions up to third-nearest neighbors. Results are presented for the GS energy and the average local onsite magnetization. With the nearest-neighbor coupling strength ${J}_{1}\ensuremath{\equiv}1$, we find four magnetically ordered phases in the parameter window ${J}_{2},{J}_{3}\ensuremath{\in}[0,1]$, namely, the N\'eel, striped, and N\'eel-II collinear AFM phases, plus a spiral phase. The N\'eel-II phase appears as a stable GS phase in the classical version of the model only for values J3<0. Each of these four ordered phases shares a boundary with a disordered quantum paramagnetic (QP) phase, and at several widely separated points on the phase boundaries the QP phase has an infinite susceptibility to plaquette valence-bond crystalline order. We identify all of the phase boundaries with good precision in the parameter window studied, and we find three tricritical quantum critical points therein at (a) $({J}_{2}^{{c}_{1}},{J}_{3}^{{c}_{1}})=(0.51\ifmmode\pm\else\textpm\fi{}0.01,0.69\ifmmode\pm\else\textpm\fi{}0.01)$ between the N\'eel, striped, and QP phases; (b) $({J}_{2}^{{c}_{2}},{J}_{3}^{{c}_{2}})=(0.65\ifmmode\pm\else\textpm\fi{}0.02,0.55\ifmmode\pm\else\textpm\fi{}0.01)$ between the striped, spiral, and QP phases; and (c) $({J}_{2}^{{c}_{3}},{J}_{3}^{{c}_{3}})=(0.69\ifmmode\pm\else\textpm\fi{}0.01,0.12\ifmmode\pm\else\textpm\fi{}0.01)$ between the spiral, N\'eel-II, and QP phases.
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.
Using the coupled cluster method (CCM) we study the full (zero-temperature) ground-state (GS) phase diagram of a spin-half (s=21) J1-J2 Heisenberg model on a cross-striped square lattice. Each site of the square lattice has four nearest-neighbor exchange bonds of strength J1 and two next-nearest-neighbor (diagonal) bonds of strength J2. The J2 bonds are arranged so that the basic square plaquettes in alternating columns have either both or no J2 bonds included. The classical ($s\ensuremath{\rightarrow}\ensuremath{\infty}$) version of the model has four collinear phases when J1 and J2 can take either sign. Three phases are antiferromagnetic (AFM), showing so-called N\'eel, double N\'eel, and double columnar striped order, respectively, while the fourth is ferromagnetic. For the quantum s=21 model we use the three classical AFM phases as CCM reference states, on top of which the multispin-flip configurations arising from quantum fluctuations are incorporated in a systematic truncation hierarchy. Calculations of the corresponding GS energy, magnetic order parameter, and the susceptibilities of the states to various forms of valence-bond crystalline (VBC) order are thus carried out numerically to high orders of approximation and then extrapolated to the (exact) physical limit. We find that the s=21 model has five phases, which correspond to the four classical phases plus a new quantum phase with plaquette VBC order. The positions of the five quantum critical points are determined with high accuracy. While all four phase transitions in the classical model are first order, we find strong evidence that three of the five quantum phase transitions in the s=21 model are of continuous deconfined type.
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.
The nature of quantum spin liquids is studied for the spin-1/2 antiferromagnetic Heisenberg model on a square lattice containing exchange interactions between nearest-neighbor sites, J1, and those between next-nearest-neighbor sites, J2. 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≤0.6 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≤0.5, while the gapped valence-bond-crystal (VBC) phase is stabilized for 0.5<J2/J1≤0.6. The VBC order is likely to have the columnar symmetry with a spontaneous symmetry breaking of the 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 at a long distance r appears to increase from z+η∼1 at J2/J1∼0.4 toward the continuous transition to the VBC phase at J1/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.