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arXiv:cond-mat/0402055 · Physical Review Letters

Dimerized Phase and Transitions in a Spatially Anisotropic Square Lattice Antiferromagnet

Oleg A. Starykh, Leon Balents

Published Sep 15, 2004• 128 citations
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Abstract

We investigate the spatially anisotropic square lattice quantum antiferromagnet. The model describes isotropic spin-1/2 Heisenberg chains (exchange constant J) coupled antiferromagnetically in the transverse (J( perpendicular )) and diagonal (J(x)), with respect to the chain, directions. Classically, the model admits two ordered ground states-with antiferromagnetic and ferromagnetic interchain spin correlations-separated by a first-order phase transition at J( perpendicular )=2J(x). We show that in the quantum model this transition splits into two, revealing an intermediate quantum-disordered columnar dimer phase, both in two dimensions and in a simpler two-leg ladder version. We describe quantum-critical points separating this spontaneously dimerized phase from classical ones.

DOI
10.1103/physrevlett.93.127202

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We investigate the spatially anisotropic square lattice quantum antiferromagnet. The model describes isotropic spin-1/2 Heisenberg chains (exchange constant J) coupled antiferromagnetically in the transverse (J( perpendicular )) and diagonal (J(x)), with respect to the chain, directions. Classically, the model admits two ordered ground states-with antiferromagnetic and ferromagnetic interchain spin correlations-separated by a first-order phase transition at J( perpendicular )=2J(x). We show that in the quantum model this transition splits into two, revealing an intermediate quantum-disordered columnar dimer phase, both in two dimensions and in a simpler two-leg ladder version. We describe quantum-critical points separating this spontaneously dimerized phase from classical ones.
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The S=1/2S=1/2S=1/2 and S=1S=1S=1 two-dimensional quantum Heisenberg antiferromagnets on the anisotropic dimerized square lattice are investigated by the quantum Monte Carlo method. By finite-size-scaling analyses on the correlation lengths, the ground-state phase diagram parametrized by strengths of the dimerization and of the spatial anisotropy is determined much more accurately than the previous works. It is confirmed that the quantum critical phenomena on the phase boundaries belong to the same universality class as that of the classical three-dimensional Heisenberg model. Furthermore, for S=1,S=1,S=1, we show that all the spin-gapped phases, such as the Haldane and dimer phases, are adiabatically connected in the extended-parameter space, though they are classified into different classes in terms of the string order parameter in the one-dimensional, i.e., the zero-interchain-coupling, case.
We study the spin-1/2 and spin-1 J1-J2 Heisenberg antiferromagnets (HAFs) on an in–nite, anisotropic, two-dimensional triangular lattice, using the coupled cluster method. With respect to an underlying square–lattice geometry the model contains antiferromag- netic (J1 > 0) bonds between nearest neighbours and competing (J02 > 0) bonds between next-nearest-neighbours across only one of the diagonals of each square plaquette, the same diagonal in each square. In a topologically equivalent triangular-lattice geometry the model has two sorts of nearest-neighbour bonds, with bonds along parallel chains and with J1 bonds providing an interchain coupling. The model thus interpolates between an isotropic HAF on the square lattice at one extreme and a set of de- coupled chains at the other, with the isotropic HAF on the triangular lattice in between at k = 1. For the spin-1/2 J1-J2 2 model, we a weakly order (or possibly second-order) quantum phase transition from a eel-ordered state to a helical state at critical point at and a second critical point at where a transition occurs between the helical state and a collinear stripe- ordered state. For the corresponding spin-1 model we an analogous transition of the second-order type at between states with and helical ordering, but we no evidence of a further transition in thiSocase to a stripe-ordered phase.
We study the stability region for the columnar dimer state proposed as a candidate ground state for the square-lattice quantum antiferromagnet with first- and second-neighbor antiferromagnetic couplings (J1{\mathit{J}}_{1}J1​-J2{\mathit{J}}_{2}J2​ model). We use a boson representation of the spin operators suited to the perturbative expansion around a dimer ground state. At lowest order, the columnar dimer is found to be stable only at the classical critical value J2{\mathit{J}}_{2}J2​/J2{\mathit{J}}_{2}J2​=1/2. However, we show that the leading anharmonic corrections stabilize the dimerized phase in a region of a finite width around J2{\mathit{J}}_{2}J2​/J1{\mathit{J}}_{1}J1​=1/2. A comparison of the ground-state energies shows that among the possible dimerized states the columnar dimer is the most favorable candidate to separate the two ordered states in the S=1/2 antiferromagnetic with first- and second-neighbor exchange.
Dimerized spin-1/2 ladders exhibit a variety of phase structures, which depend on the intra-chain and inter-chain spin exchange energies as well as on the dimerization pattern of the ladder. Using the density matrix renormalization group (DMRG) algorithm, we study critical properties of the bond-alternating two-leg Heisenberg spin ladder with diagonal interaction J ×. Two types of spin systems, staggered dimerized antiferromagnetic ladder and columnar dimerized ferro-antiferromagnetic couplings ladder, are investigated. To clarify the phase transition behaviors, we simultaneously analyze the string order parameter (SOP), the twisted order parameter (TOP), as well as a measurement of the quantum information analysis. Based on measuring this different observables, we establish the phase diagram accurately and give the fitting functions of the phase boundaries. In addition, the phase transition of cross-coupled spin ladder (in the absence of intrinsic dimerization) is also discussed.
The phase diagram of a spin-1/2 $J\ensuremath{-}{J}^{\ensuremath{'}}\ensuremath{-}{J}_{2}$ model is investigated by means of exact diagonalizations on finite samples. This model is a generalization of the J1−−J2{J}_{1}--{J}_{2}J1​−−J2​ model on the square lattice with two different nearest-neighbor couplings ${J,J}^{\ensuremath{'}}$ and may also be viewed as an array of coupled Heisenberg chains. The results suggest that the resonating valence bond state predicted by Nersesyan and Tsvelik [Phys. Rev. B 67, 024422 (2003)] for ${J}_{2}{=0.5J}^{\ensuremath{'}}\ensuremath{\ll}J$ 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 sides by a columnar dimer long range order. This columnar order could extend for ${J}^{\ensuremath{'}}\ensuremath{\rightarrow}J$ which correspond to the J1−−J2{J}_{1}--{J}_{2}J1​−−J2​ model.