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Physical review. B, Condensed matter

Static and dynamical correlations in a spin-1/2 frustrated antiferromagnet

Didier Poilblanc, Eduardo R. Gagliano, Silvia Bacci, Elbio R. Dagotto

Published May 1, 1991• 43 citations
Publisher

Abstract

We study the frustrated spin-1/2 antiferromagnetic Heisenberg model on a square lattice of 16 sites, using a Lanczos algorithm and group-theoretical techniques. We investigate the competition between collinear, dimer, chiral, and spiral orders by calculating the dynamical and spatial correlation functions. Our numerical results strongly suggest the presence of long-range collinear order in the interval 0.55\ensuremath{\le}${\mathit{J}}_{2}$/${\mathit{J}}_{1}$\ensuremath{\le}0.9 (we cannot show numerically that this order exists for larger values of ${\mathit{J}}_{2}$/${\mathit{J}}_{1}$). Dimer- and chiral-order parameters both show increasing low-energy fluctuations and a longer range of the correlations in space around ${\mathit{J}}_{2}$/${\mathit{J}}_{1}$\ensuremath{\sim}0.55. We propose an order parameter for chiral order exhibiting the full symmetry of the cluster and hence being less sensitive to finite-size effects. The mean value of the square of this plaquette chiral-order parameter is enhanced for ${\mathit{J}}_{2}$/${\mathit{J}}_{1}$\ensuremath{\sim}0.55, where the characteristic chiral spin-fluctuation frequency is reduced, but no indications of long-range chiral order were found. Although the correlation function of the twist-order parameter is not a simple monotonically decreasing function of distance, little change is induced by frustration, suggesting that spiral order is not strongly favored. The energy gap corresponding to dimer order is smaller than those of spin waves, spiral order, and chiral order; thus, this state is the best candidate for an intermediate-(${\mathit{J}}_{2}$/${\mathit{J}}_{1}$) phase in the ${\mathit{J}}_{1}$-${\mathit{J}}_{2}$ model.

DOI
10.1103/physrevb.43.10970

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We study the frustrated spin-1/2 antiferromagnetic Heisenberg model on a square lattice of 16 sites, using a Lanczos algorithm and group-theoretical techniques. We investigate the competition between collinear, dimer, chiral, and spiral orders by calculating the dynamical and spatial correlation functions. Our numerical results strongly suggest the presence of long-range collinear order in the interval 0.55\ensuremath{\le}J2{\mathit{J}}_{2}J2​/J1{\mathit{J}}_{1}J1​\ensuremath{\le}0.9 (we cannot show numerically that this order exists for larger values of J2{\mathit{J}}_{2}J2​/J1{\mathit{J}}_{1}J1​). Dimer- and chiral-order parameters both show increasing low-energy fluctuations and a longer range of the correlations in space around J2{\mathit{J}}_{2}J2​/J1{\mathit{J}}_{1}J1​\ensuremath{\sim}0.55. We propose an order parameter for chiral order exhibiting the full symmetry of the cluster and hence being less sensitive to finite-size effects. The mean value of the square of this plaquette chiral-order parameter is enhanced for J2{\mathit{J}}_{2}J2​/J1{\mathit{J}}_{1}J1​\ensuremath{\sim}0.55, where the characteristic chiral spin-fluctuation frequency is reduced, but no indications of long-range chiral order were found. Although the correlation function of the twist-order parameter is not a simple monotonically decreasing function of distance, little change is induced by frustration, suggesting that spiral order is not strongly favored. The energy gap corresponding to dimer order is smaller than those of spin waves, spiral order, and chiral order; thus, this state is the best candidate for an intermediate-(J2{\mathit{J}}_{2}J2​/J1{\mathit{J}}_{1}J1​) phase in the J1{\mathit{J}}_{1}J1​-J2{\mathit{J}}_{2}J2​ model.
We study the ground-state (GS) properties of the frustrated spin-12\frac{1}{2}21​ J1{J}_{1}J1​--J2{J}_{2}J2​--J3{J}_{3}J3​ Heisenberg model on the two-dimensional honeycomb lattice with ferromagnetic nearest-neighbor (${J}_{1}=\ensuremath{-}1$) exchange and frustrating antiferromagnetic next-nearest-neighbor (J2>0{J}_{2}>0J2​>0) and next-next-nearest-neighbor (J3>0{J}_{3}>0J3​>0) exchanges, for the case J3=J2{J}_{3}={J}_{2}J3​=J2​. We use the coupled-cluster method implemented to high orders of approximation, complemented by the Lanczos exact diagonalization of a large finite lattice with 32 sites, in order to calculate the GS energy, magnetic order parameter, and spin-spin correlation functions. In one scenario we find a quantum phase transition point between regions characterized by ferromagnetic order and a form of antiferromagnetic (``striped'') collinear order at ${J}_{2}^{c}\ensuremath{\approx}0.1095\ifmmode\pm\else\textpm\fi{}0.0005$, which is below the corresponding hypothetical transition point at J2cl=17{J}_{2}^{\mathrm{cl}}=\frac{1}{7}J2cl​=71​ ($\ensuremath{\approx}0.143$) for the classical version of the model, in which we momentarily ignore the intervening noncollinear spiral phase in the region 110<J2<15\frac{1}{10}<{J}_{2}<\frac{1}{5}101​<J2​<51​. Hence we see that quantum fluctuations appear to stabilize somewhat the collinear antiferromagnetic order in preference to the ferromagnetic order in this model. We compare results for the present ferromagnetic case (with ${J}_{1}=\ensuremath{-}1$) to previous results for the corresponding antiferromagnetic case (with J1=+1{J}_{1}=+1J1​=+1). The magnetic order parameter is found to behave similarly for the ferromagnetic and the antiferromagnetic models for large values of the frustration parameter J2{J}_{2}J2​. However, there are considerable differences in the behavior of the order parameters for the two models for ${J}_{2}/|{J}_{1}|\ensuremath{\lesssim}0.6$. For example, the quasiclassical collinear magnetic long-range order for the antiferromagnetic model (with J1=+1{J}_{1}=+1J1​=+1) breaks down at ${J}_{2}^{{c}_{2}}\ensuremath{\approx}0.60$, whereas the ``equivalent'' point for the ferromagnetic model (with ${J}_{1}=\ensuremath{-}1$) occurs at ${J}_{2}^{c}\ensuremath{\approx}0.11$. Unlike in the antiferromagnetic model (with J1=+1{J}_{1}=+1J1​=+1), where a plaquette valence-bond crystal phase intrudes between the two corresponding quasiclassical antiferromagnetic phases (with N\'eel and striped order) for J2c1<J2<J2c2{J}_{2}^{{c}_{1}}<{J}_{2}<{J}_{2}^{{c}_{2}}J2c1​​<J2​<J2c2​​, with ${J}_{2}^{{c}_{1}}\ensuremath{\approx}0.47$, we find no clear indications at all in the ferromagnetic model for an intermediate magnetically disordered phase between the corresponding phases exhibiting ferromagnetic and striped order. Instead the evidence for the ferromagnetic model (with ${J}_{1}=\ensuremath{-}1$) points to one of two scenarios: either there is a direct first-order transition between the two magnetically ordered phases, as mentioned above; or there exists an intervening phase between them in the very narrow range $0.10\ensuremath{\lesssim}{J}_{2}\ensuremath{\lesssim}0.12$, which is probably a remnant of the spiral phase that exists in the classical counterpart of the model over the larger range 110<J2<15\frac{1}{10}<{J}_{2}<\frac{1}{5}101​<J2​<51​.
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We examine the J1{\mathit{J}}_{1}J1​-J2{\mathit{J}}_{2}J2​, spin-1/2 Heisenberg model on a square lattice with 16 and 20 sites. We evaluate the ground-state correlations of an operator, measuring the handiness of plaquettes. We find a strong, physically relevant enhancement in these chiral correlations for intermediate values of J2{\mathit{J}}_{2}J2​/J1{\mathit{J}}_{1}J1​. We compare with the known results for correlations between column-wise-ordered singlets. We find both types of correlations to be viable candidates for the ground-state correlations in the thermodynamic limit.
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