The phase diagram of a spin-1/2
J−J′−J2 model is investigated by means of exact diagonalizations on finite samples. This model is a generalization of the
J1−J2 model on the square lattice with two different nearest-neighbor couplings
J,J′ and may be also viewed as an array of coupled Heisenberg chains. The results suggest that the resonnating valence bond state predicted by Nersesyan and Tsvelik [Phys. Rev. B {\bf 67}, 024422 (2003)] for
J2=0.5J′≪J is realized and extends beyond the limit of small interchain coupling along a curve nearly coincident with the line where the energy per spin is maximum. This line is likely bordered on both side by a columnar dimer long range order. This columnar order could extends for
J′→J which correspond to the
J1−J2 model.
Using the coupled cluster method for high orders of approximation and complementary exact diagonalization studies we investigate the ground state properties of the spin-1/2
J1−−J2 frustrated Heisenberg antiferromagnet on the square lattice. We have calculated the ground state energy, the magnetic order parameter, the spin stiffness, and several generalized susceptibilities to probe magnetically disordered quantum valence-bond phases. We have found that the quantum critical points for both the N\'eel and collinear orders are ${J}_{2}^{c1}\ensuremath{\approx}(0.44\ifmmode\pm\else\textpm\fi{}0.01){J}_{1}$ and ${J}_{2}^{c2}\ensuremath{\approx}(0.59\ifmmode\pm\else\textpm\fi{}0.01){J}_{1}$, respectively, which are in good agreement with the results obtained by other approximations. In contrast to the recent study by [Sirker et al. Phys. Rev. B 73, 184420 (2006)], our data do not provide evidence for the transition from the N\'eel to the valence-bond solid state to be first order. Moreover, our results are in favor of the deconfinement scenario for that phase transition. We also discuss the nature of the magnetically disordered quantum phase.
The ground state phase of a spin-$\frac{1}{2} {J}_{1}\text{\ensuremath{-}}{J}_{2}$ antiferromagnetic Heisenberg model on a square lattice around the maximally frustrated regime $({J}_{2}\ensuremath{\sim}0.5{J}_{1})$ has been debated for decades. Here we study this model using the cluster update algorithm for tensor-product states (TPSs). The ground state energies at finite sizes and in the thermodynamic limit (with finite size scaling) are in good agreement with exact diagonalization study. Through finite size scaling of the spin correlation function, we find the critical point
J2c1=0.572(5)J1 and critical exponents $\ensuremath{\nu}=0.50(8),\phantom{\rule{0.16em}{0ex}}{\ensuremath{\eta}}_{s}=0.28(6)$. In the range of $0.572<{J}_{2}/{J}_{1}\ensuremath{\le}0.6$ we find a paramagnetic ground state with an exponentially decaying spin-spin correlation. Up to a $24\ifmmode\times\else\texttimes\fi{}24$ system size, we observe power law decaying dimer-dimer and plaquette-plaquette correlations with an anomalous plaquette scaling exponent ${\ensuremath{\eta}}_{p}=0.24(1)$ and an anomalous columnar scaling exponent ${\ensuremath{\eta}}_{c}=0.28(1)$ at
J2/J1=0.6. These results are consistent with a potential gapless
U(1) spin-liquid phase. However, since the
U(1) spin liquid is unstable due to the instanton effect, a valence bond solid order with very small amplitude might develop in the thermodynamic limit. Thus, our numerical results strongly indicate a deconfined quantum critical point at
J2c1. Remarkably, all the observed critical exponents are consistent with the $J\text{\ensuremath{-}}Q$ model.
We study the spin 1/2 J1-J2 antiferromagnetic Heisenberg model on square lattice by using a recently proposed cluster update method for tensor product states (TPSs). The ground state energies in the thermodynamic limit are in good agreement with the state of art exact diagonalization study, and the energy differences between these two studies are of the order of 10J1 per site. For small bond dimension D (D < 5), we find a paramagnetic ground state with mixed columnar and staggered dimer orders of a magnitude 10−2 in the range of 0.47 < J2/J1 < 0.6, which agrees with previous results. For large bond dimension D (D ≥ 5), we observe all these orders vanish, implying the emergence of a spin liquid phase.
Using the coupled-cluster method for high orders of approximation and Lanczos exact diagonalization we study the ground-state phase diagram of a quantum spin-
21 J1-
J2 model on the square lattice with plaquette structure. We consider antiferromagnetic (
J1>0) as well as ferromagnetic (
J1<0) nearest-neighbor interactions together with frustrating antiferromagnetic next-nearest-neighbor interaction
J2>0. The strength of interplaquette interaction $\ensuremath{\lambda}$ varies between $\ensuremath{\lambda}=1$ (which corresponds to the uniform
J1-
J2 model) and $\ensuremath{\lambda}=0$ (which corresponds to isolated frustrated 4-spin plaquettes). While on the classical level ($s\ensuremath{\rightarrow}\ensuremath{\infty}$) both versions of models (i.e., with ferro- and antiferromagnetic
J1) exhibit the same ground-state behavior, the ground-state phase diagram differs basically for the quantum case
s=1/2. For the antiferromagnetic case (
J1>0) N\'eel antiferromagnetic long-range order at small
J2/J1 and $\ensuremath{\lambda}\ensuremath{\gtrsim}0.47$ as well as collinear striped antiferromagnetic long-range order at large
J2/J1 and $\ensuremath{\lambda}\ensuremath{\gtrsim}0.30$ appear which correspond to their classical counterparts. Both semiclassical magnetic phases are separated by a nonmagnetic quantum paramagnetic phase. The parameter region, where this nonmagnetic phase exists, increases with decreasing $\ensuremath{\lambda}$. For the ferromagnetic case (
J1<0) we have the trivial ferromagnetic ground state at small
J2/∣J1∣. By increasing
J2 this classical phase gives way for a semiclassical plaquette phase, where the plaquette block spins of length
s=2 are antiferromagnetically long-range ordered. Further increasing of
J2 then yields collinear striped antiferromagnetic long-range order for $\ensuremath{\lambda}\ensuremath{\gtrsim}0.38$, but a nonmagnetic quantum paramagnetic phase $\ensuremath{\lambda}\ensuremath{\lesssim}0.38$.
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.