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arXiv:cond-mat/9810216 · Physical review. B, Condensed matter

Excitation spectrum of the homogeneous spin liquid

R. Eder

Published Jun 1, 1999• 5 citations
arXivHTMLPDF

Abstract

We discuss the excitation spectrum of a disordered, isotropic, and translationally invariant spin state in the two-dimensional Heisenberg antiferromagnet. The starting point is the nearest-neighbor resonating-valence-bond state which plays the role of the vacuum of the theory, in a similar sense as the N\'eel state is the vacuum for antiferromagnetic spin-wave theory. We discuss the elementary excitations of this state and show that these are not Fermionic spin-1/2 ``spinons'' but spin-1 excited dimers which must be modeled by bond bosons. We derive an effective Hamiltonian describing the excited dimers which is formally analogous to spin-wave theory. Condensation of the bond bosons at zero temperature into the state with momentum $(\ensuremath{\pi},\ensuremath{\pi})$ is shown to be equivalent to antiferromagnetic ordering. The latter is a key ingredient for a microscopic interpretation of Zhang's SO(5) theory of cuprate superconductivity.

DOI
10.1103/physrevb.59.13810

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We discuss the excitation spectrum of a disordered, isotropic, and translationally invariant spin state in the two-dimensional Heisenberg antiferromagnet. The starting point is the nearest-neighbor resonating-valence-bond state which plays the role of the vacuum of the theory, in a similar sense as the N'eel state is the vacuum for antiferromagnetic spin-wave theory. We discuss the elementary excitations of this state and show that these are not Fermionic spin-1/2 ``spinons'' but spin-1 excited dimers which must be modeled by bond bosons. We derive an effective Hamiltonian describing the excited dimers which is formally analogous to spin-wave theory. Condensation of the bond bosons at zero temperature into the state with momentum (\ensuremathπ,\ensuremathπ)(\ensuremath{\pi},\ensuremath{\pi})(\ensuremathπ,\ensuremathπ) is shown to be equivalent to antiferromagnetic ordering. The latter is a key ingredient for a microscopic interpretation of Zhang's SO(5) theory of cuprate superconductivity.
We study the electron spectral function of various zero-temperature spin-charge separated phases in two dimensions. In these phases, the electron is not a fundamental excitation of the system, but rather ``decays'' into a spin-1/2 chargeless fermion (the spinon) and a spinless charge e boson (the chargon). Using low-energy effective theories for the spinons (d-wave pairing plus possible N\'eel order) and the chargons (condensed or quantum-disordered bosons), we explore three phases of possible relevance to the cuprate superconductors: (1) AF∗,{\mathrm{AF}}^{*},AF∗, a fractionalized antiferromagnet where the spinons are paired into a state with long-ranged N\'eel order and the chargons are 1/2-filled and (Mott) insulating; (2) the nodal liquid, a fractionalized insulator where the spinons are d-wave paired and the chargons are uncondensed; and (3) the d-wave superconductor, where the chargons are condensed and the spinons retain a d-wave gap. Working within the Z2{Z}_{2}Z2​ gauge theory of such fractionalized phases, our results should be valid at scales below the energy gap of the vison---the basic vortex excitation in the theory. However, on a phenomenological level, our results should apply to any spin-charge separated system where the excitations have these low-energy effective forms. Comparison with angle-resolved photoemission spectroscopy data in the undoped, pseudogapped, and superconducting regions is made.
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 ground-state phase diagram of the spin-1/21/21/2${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model on the square lattice with an accurate bosonic resonating-valence-bond (RVB) wave function. In contrast to the RVB ansatz based on Schwinger fermions, the representation based on Schwinger bosons, supplemented by a variational Monte Carlo technique enforcing the exact projection onto the physical subspace, is able to describe a fully gapped spin liquid in the strongly frustrated regime. In particular, a fully symmetric Z2{Z}_{2}Z2​ spin liquid is stable between two antiferromagnetic phases; a continuous transition at J2=0.4J1{J}_{2}=0.4{J}_{1}J2​=0.4J1​, when the Marshall sign rule begins to be essentially violated, and a first-order transition around J2=0.6J1{J}_{2}=0.6{J}_{1}J2​=0.6J1​ are present. Most importantly, the triplet gap is found to have a nonmonotonic behavior, reaching a maximum around J2=0.51J1{J}_{2}=0.51{J}_{1}J2​=0.51J1​, when the lowest spinon excitation moves from the $\ensuremath{\Gamma}$ to the MMM point, i.e., $k=(\ensuremath{\pi},0)$.
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.
The large-N limit of a nearest-neighbor SU(N) antiferromagnet on a bipartite lattice exhibits in dimensions d\ensuremath{\ge}2 a zero-temperature phase transition between a N\'eel ordered state and a resonating-valence-bond state. Here it is shown in d=1,2 that topological effects produce spin-Peierls or valence-bond-solid order in the non-N\'eel phase with a ground-state degeneracy which varies periodically with ``spin'' for fixed N with periodicity given by the coordination number of the lattice. Thus a non-N\'eel phase of the spin-(1/2 Heisenberg model on a square lattice would be a spin-Peierls state with a fourfold degeneracy due to broken lattice rotational symmetry.