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).
The spin-
1/2 J1-
J2 Heisenberg model on the square lattice represents one of the simplest examples in which the effects of magnetic interactions may suppress magnetic order, eventually leading to a pure quantum phase with no local order parameters. This model has been extensively studied in the last three decades, with conflicting results. Here, by using Gutzwiller-projected wave functions and recently developed methods to assess the low-energy spectrum, we show the existence of a level crossing between the lowest-energy triplet and singlet excitations for ${J}_{2}/{J}_{1}\ensuremath{\approx}0.54$. This fact supports the existence of a phase transition between a gapless spin liquid (which is stable for $0.48\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\lesssim}0.54$) and a valence-bond solid (for $0.54\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\lesssim}0.6$), even though no clear sign of dimer order is visible in the correlations functions. These results, which confirm recent density-matrix renormalization calculations on cylindrical clusters [L. Wang and A. W. Sandvik, Phys. Rev. Lett. 121, 107202 (2018)], reconcile the contradicting results obtained within different approaches over the years.
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.
The intricate interplay between frustration and spin chirality has the potential to give rise to unprecedented phases in frustrated quantum magnets. We examine the ground state phase diagram of the spin-1/2 square lattice J1-J2-Jx model by employing critical level crossings and ground state fidelity susceptibility (FS) using exact diagonalization (ED) with full lattice symmetries. Our analysis reveals the evolution of highly symmetric energy levels as a function of J2 at fixed Jx. During a magnetic to non-magnetic phase transition, the precise identification of the phase boundary is achieved through critical level crossings between the gapless excitation of a magnetic phase and the quasi-degenerate ground state of a non-magnet phase. Conversely, a direct transition between two non-magnetic phases is characterized by a FS peak accompanied by an avoided ground state level crossing, serving as a distinctive signal. Within a substantial range of Jx, we identify an anticipated chiral spin liquid (CSL) state and an adjacent nematic spin liquid (NSL) phase with a degeneracy of two on a cylinder. These two phases are demarcated by a nearly vertical boundary line at J2 = 0.65. This critical line terminates at the lower boundary of a magnetic ordered chiral spin solid (CSS) phase. We validate the topological nature of the CSL using the modular S matrix of the minimum entangled states (MES) on a torus, along with the entanglement spectra (ES) of even and odd sectors on a cylinder, employing an SU(2)-symmetric density matrix renormalization group (DMRG) method. Furthermore, we delve into a comprehensive discussion on the nature of the NSL, exploring aspects such as ground state degeneracy, the local bond energy landscape, and the singlet and triplet gaps on various tori. These analysis provide substantial evidence supporting the nematic nature of the NSL.
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.
Based on a special variant of plaquette expansion, an operator is constructed whose eigenvalues give the low-energy singlet spectrum of spin-
21 Heisenberg antiferromagnet on square lattice with nearest- and frustrating next-nearest-neighbor exchange couplings
J1 and
J2. It is well known that a non-magnetic phase arises in this model at
0.4≲J2/J1≲0.6 sandwiched by two Neel ordered phases. In agreement with previous results, we observe a first-order quantum phase transition (QPT) at
J2≈0.64J1 from the non-magnetic phase to the Neel one. Large gap (
≳0.4J1) is found in the singlet spectrum at
J2<0.64J1 that excludes a gapless spin-liquid state at
0.4≲J2/J1≲0.6 and the deconfined quantum criticality scenario for the QPT to another Neel phase. We observe a first-order QPT at
J2≈0.55J1 presumably between two non-magnetic phases.