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Physical review. B, Condensed matter

Theory of the spin liquid state of the Heisenberg antiferromagnet

Vadim Kalmeyer, R. B. Laughlin

Published Jun 1, 1989• 188 citations
Publisher

Abstract

We propose that the disordered state of a two-dimensional spin-1/2 Heisenberg antiferromagnet is physically equivalent to the incompressible liquid state of the fractional quantum Hall system. The fractional quantum Hall state for bosons is shown to be an exact spin singlet and to possess a low variational energy for the near-neighbor Heisenberg model on a triangular lattice. Variational wave functions for neutral spin-1/2 excitations are constructed and shown to form an exact spin doublet. Variational energies of these states are calculated, and their spin density profiles are determined. We find that a localized spin-1/2 quaisparticle has a size comparable to a lattice bond length and an excitation energy \ensuremath{\Delta}=1.3J. The energy-momentum dispersion of quasiparticles and spin-1 collective modes, obtained variationally, supports the hypothesis that the spin liquid state has a finite energy gap. The 1/2 fractional statistics exhibited by the quasiparticle excitations is explicitly demonstrated.

DOI
10.1103/physrevb.39.11879

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•Nov 10, 2013•0 citations
We propose that the disordered state of a two-dimensional spin-1/2 Heisenberg antiferromagnet is physically equivalent to the incompressible liquid state of the fractional quantum Hall system. The fractional quantum Hall state for bosons is shown to be an exact spin singlet and to possess a low variational energy for the near-neighbor Heisenberg model on a triangular lattice. Variational wave functions for neutral spin-1/2 excitations are constructed and shown to form an exact spin doublet. Variational energies of these states are calculated, and their spin density profiles are determined. We find that a localized spin-1/2 quaisparticle has a size comparable to a lattice bond length and an excitation energy \ensuremath{\Delta}=1.3J. The energy-momentum dispersion of quasiparticles and spin-1 collective modes, obtained variationally, supports the hypothesis that the spin liquid state has a finite energy gap. The 1/2 fractional statistics exhibited by the quasiparticle excitations is explicitly demonstrated.
A disordered spin state, the nearest-neighbor virtual bonding state of Wigner-Jordan fermions, of the quantum antiferromagnetic Heisenberg model on a triangular lattice is found to have a zero-temperature energy -1.106J per spin, lower than that calculated from spin-wave theories, and in excellent agreement with that from numerical calculations, (-1.094\ifmmode\pm\else\textpm\fi{}0.018)J per spin. The excitations of the state are Wigner-Jordan fermions with a gapless spectrum. The low-temperature specific heat is predicted to be proportional to ${\mathit{T}}^{\ensuremath{\nu}}$, with \ensuremath{\nu} between 2/3 and 1, significantly deviating from the spin-wave prediction, T2{\mathit{T}}^{2}T2. At higher temperature, a broad peak in the specific heat near T/2J\ensuremath{\approxeq}0.75 is also predicted.
We study the spin-12\frac{1}{2}21​ Heisenberg model on the triangular lattice with antiferromagnetic first- (J1)({J}_{1})(J1​) and second- (J2)({J}_{2})(J2​) nearest-neighbor interactions using density matrix renormalization group. By studying the spin correlation function, we find a ${120}^{\ensuremath{\circ}}$ magnetic order phase for ${J}_{2}\ensuremath{\lesssim}0.07{J}_{1}$ and a stripe antiferromagnetic phase for ${J}_{2}\ensuremath{\gtrsim}0.15{J}_{1}$. Between these two phases, we identify a spin-liquid region characterized by exponential decaying spin and dimer correlations, as well as large spin singlet and triplet excitation gaps on finite-size systems. We find two near degenerating ground states with distinct properties in two sectors, which indicates more than one spin-liquid candidate in this region. While the sector with spinons is found to respect time reversal symmetry, the even sector without spinons breaks such a symmetry for finite-size systems. Furthermore, we detect the signature of the fractionalization by following the evolution of different ground states with inserting spin flux into the cylinder system. Moreover, by tuning the anisotropic bond coupling, we explore the nature of the spin-liquid phase and find the optimal parameter region for gapped Z2{Z}_{2}Z2​ spin liquids.
The familiar spin-12\frac{1}{2}21​ quantum Heisenberg antiferromanget in a two-dimensional (2D) square lattice is shown, within the nonlinear sigma model approximations, to be another state of matter that has excitations with fractional quantum numbers above a finite energy gap. The 1-skyrmion with an energy $\ensuremath{\approx}2\ensuremath{\pi}J$ is shown to be made of two ``deconfined spinons'' or ``O(3) vortices.'' The many skyrmion operator and the wave functions that we have found are strikingly similar to quantum Hall quasiparticle operators and wave functions. We also predict the presence of finite energy ``spin-SSS spinon'' for a general spin S Heisenberg antiferromagnets in 2D. Some consequences are briefly discussed.
It is shown that one of the class of variational spin-liquid wave functions recently derived by Wen, Wilczek, and Zee [Phys. Rev. B 39, 11413 (1989)] is identical to the fractional-quantum-Hall analog state proposed by Kalmeyer and Laughlin [Phys. Rev. Lett. 59, 2095 (1987); Phys. Rev. B 39, 11 879 (1989)] and Laughlin [Ann. Phys. (N.Y.) 191, 163 (1989)], and that the neutral spin-1/2 excitations of the two states are also the same. The 1/2 fractional statistics obeyed by these particles is demonstrated explicitly. The spin-spin correlation function and quasiparticle profile for the Wen-Wilczek-Zee states are calculated both numerically and by a hypernetted-chain procedure. Both are consistent with the idea that spontaneous breaking of time-reversal symmetry is an essential feature of any spin state lacking magnetic order. A version of this state for a three-dimensional spin system is reported and shown to have similar properties. A case is made that the neutral spin-1/2 excitations of the three-dimensional spin liquid behave like anisotropic monopoles.
We investigate the quantum dynamics of the spins on different Heisenberg antiferromagnetic spin lattice systems. Firstly, we applied the coupled-cluster method to the spin-1/2 antiferromagnetic \emph{XXZ} model on a square lattice by employing an approximation which contains two-body long-range correlations and high-order four-body local correlations. Improvement is found for the ground-state energy, sublattice magnetization, and the critical anisotropy when comparing with the approximation including the two-body correlations alone. We also obtain the full excitation spectrum which is in good agreement with the quantum Monte Carlo results and the high-order spin-wave theory.Secondly, we study the longitudinal excitations of quantum antiferromagnets on a triangular lattice by a recently proposed microscopic many-body approach based on magnon-density waves. We calculate the full longitudinal excitation spectra of the antiferromagnetic Heisenberg model for a general spin quantum number in the isotropic limit. Similar to the square lattice model, we find that, at the center of the first hexagonal Brillouin zone Γ(q=0)\Gamma(\mathbf q=0)Γ(q=0) and at the magnetic ordering wavevectors ±[Q=(4π/3,0)]\pm[\mathbf Q= (4\pi/3,0)]±[Q=(4π/3,0)], the excitation spectra become gapless in the thermodynamic limit, due to the slow, logarithmic divergence of the structure factor. However, these longitudinal modes on two-dimensional models may be considered as quasi-gapped, as any finite-size effect or small anisotropy will induce a large energy gap, when compared with the counterpart of the transverse spin-wave excitations. We have also investigated the excited states of the quasi-one-dimensional quantum antiferromagnets on hexagonal lattices, including the longitudinal modes based on the magnon-density waves. A model Hamiltonian with a uniaxial single-ion anisotropy is first studied by a spin-wave theory based on the one-boson method; the ground state thus obtained is employed for the study of the longitudinal modes. The full energy spectra of both the transverse modes (i.e., magnons) and the longitudinal modes are obtained as functions of the nearest-neighbor coupling and the anisotropy constants. We have found two longitudinal modes due to the non-collinear nature of the triangular antiferromagnetic order, similar to that of the phenomenological field theory approach by Affleck. The excitation energy gaps due to the anisotropy and the energy gaps of the longitudinal modes without anisotropy are then investigated. We then compares our results for the longitudinal energy gaps at the magnetic wavevectors with the experimental results for several antiferromagnetic compounds with both integer and non-integer spin quantum numbers, and we find good agreements after the higher-order contributions are included in our calculations.