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Physical Review Letters

Phase diagram of the frustrated spin-1/2 Heisenberg antiferromagnet in 2 dimensions

Elbio R. Dagotto, Adriana Moreo

Published Nov 6, 1989• 329 citations
Publisher

Abstract

Using a Lanczos technique we study the frustrated spin-(1/2 Heisenberg model on square lattices of 16 and 20 sites. Frustration is introduced by an interaction along the diagonals of the plaquettes with coupling ${J}_{2}$\ensuremath{\ge}0. For large ${J}_{2}$ we found that the ground state breaks (spontaneously) the lattice rotational symmetry. For intermediate values of ${J}_{2}$, the squares of order parameters associated with spin-Peierls and ``twisted'' states have a peak while a similar quantity for a chiral state shows no interesting structure.

DOI
10.1103/physrevlett.63.2148

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Using a Lanczos technique we study the frustrated spin-(1/2 Heisenberg model on square lattices of 16 and 20 sites. Frustration is introduced by an interaction along the diagonals of the plaquettes with coupling J2{J}_{2}J2​\ensuremath{\ge}0. For large J2{J}_{2}J2​ we found that the ground state breaks (spontaneously) the lattice rotational symmetry. For intermediate values of J2{J}_{2}J2​, the squares of order parameters associated with spin-Peierls and ``twisted'' states have a peak while a similar quantity for a chiral state shows no interesting structure.
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-12\frac{1}{2}21​ J1{J}_{1}J1​-J2{J}_{2}J2​ model on the square lattice with plaquette structure. We consider antiferromagnetic (J1>0{J}_{1}>0J1​>0) as well as ferromagnetic (J1<0{J}_{1}<0J1​<0) nearest-neighbor interactions together with frustrating antiferromagnetic next-nearest-neighbor interaction J2>0{J}_{2}>0J2​>0. The strength of interplaquette interaction $\ensuremath{\lambda}$ varies between $\ensuremath{\lambda}=1$ (which corresponds to the uniform J1{J}_{1}J1​-J2{J}_{2}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{J}_{1}J1​) exhibit the same ground-state behavior, the ground-state phase diagram differs basically for the quantum case s=1/2s=1/2s=1/2. For the antiferromagnetic case (J1>0{J}_{1}>0J1​>0) N\'eel antiferromagnetic long-range order at small J2/J1{J}_{2}/{J}_{1}J2​/J1​ and $\ensuremath{\lambda}\ensuremath{\gtrsim}0.47$ as well as collinear striped antiferromagnetic long-range order at large J2/J1{J}_{2}/{J}_{1}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{J}_{1}<0J1​<0) we have the trivial ferromagnetic ground state at small J2/∣J1∣{J}_{2}/|{J}_{1}|J2​/∣J1​∣. By increasing J2{J}_{2}J2​ this classical phase gives way for a semiclassical plaquette phase, where the plaquette block spins of length s=2s=2s=2 are antiferromagnetically long-range ordered. Further increasing of J2{J}_{2}J2​ then yields collinear striped antiferromagnetic long-range order for $\ensuremath{\lambda}\ensuremath{\gtrsim}0.38$, but a nonmagnetic quantum paramagnetic phase $\ensuremath{\lambda}\ensuremath{\lesssim}0.38$.
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The spin-(1/2 frustrated Heisenberg model is studied on a square lattice using the modified Lanczos technique. Frustration is introduced in the system through an additional coupling along the diagonals of the plaquettes. On a 4\ifmmode\times\else\texttimes\fi{}4 lattice the ground state smoothly interpolates between two asymptotic N\'eel-like limits. However, on a narrow region of parameter space we find that the lowest-lying levels above the ground state are singlets rather than triplets, suggesting the existence of a new phase in the thermodynamic limit.
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.