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Physical Review

Long-Range Order of the Linear Antiferromagnetic Chain

Nick Karayianis, Carole A. Morrison, D. E. Wortman

Published May 15, 1962• 2 citations
Publisher

Abstract

The long-range order in the ground state of the linear antiferromagnetic chain is investigated. A chain of $N$ identical spin-\textonehalf{} particles coupled by the anisotropic Hamiltonian $\mathcal{H}=2J\ensuremath{\Sigma}\stackrel{N}{j=1}{{S}_{j}}^{z}{{S}_{j+1}}^{z}+bJ\ensuremath{\Sigma}\stackrel{N}{j=1}({{S}_{j}}^{+}{{S}_{j+1}}^{\ensuremath{-}}+{{S}_{j}}^{\ensuremath{-}}{{S}_{j+1}}^{+})$ is considered, and the periodic boundary condition, ${S}_{i+N}\ensuremath{\equiv}{S}_{i}$, is imposed so that the chain may be considered a loop of particles. The operator $\mathfrak{M}$, which rotates the particles in the loop so that each of the particles moves to the next highest site number, is introduced. It is proved that simultaneous eigenstates of $\mathcal{H}$ and $\mathfrak{M}$ have no long-range order as measured by ${\ensuremath{\sigma}}^{*}$ of Kasteleijn. The ground state of the Hamiltonian for $b\ensuremath{\ne}0$, $N<\ensuremath{\infty}$ is shown to be nondegenerate. It is therefore necessarily an eigenstate of $\mathfrak{M}$ and as such has no long-range order.

DOI
10.1103/physrev.126.1443

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Journal of the Physical Society of Japan•Jan 15, 1989•27 citations
The long-range order in the ground state of the linear antiferromagnetic chain is investigated. A chain of NNN identical spin-\textonehalf{} particles coupled by the anisotropic Hamiltonian H=2J\ensuremathΣj=1NSjzSj+1z+bJ\ensuremathΣj=1N(Sj+Sj+1\ensuremath−+Sj\ensuremath−Sj+1+)\mathcal{H}=2J\ensuremath{\Sigma}\stackrel{N}{j=1}{{S}_{j}}^{z}{{S}_{j+1}}^{z}+bJ\ensuremath{\Sigma}\stackrel{N}{j=1}({{S}_{j}}^{+}{{S}_{j+1}}^{\ensuremath{-}}+{{S}_{j}}^{\ensuremath{-}}{{S}_{j+1}}^{+})H=2J\ensuremathΣj=1N​Sj​zSj+1​z+bJ\ensuremathΣj=1N​(Sj​+Sj+1​\ensuremath−+Sj​\ensuremath−Sj+1​+) is considered, and the periodic boundary condition, Si+N\ensuremath≡Si{S}_{i+N}\ensuremath{\equiv}{S}_{i}Si+N​\ensuremath≡Si​, is imposed so that the chain may be considered a loop of particles. The operator M\mathfrak{M}M, which rotates the particles in the loop so that each of the particles moves to the next highest site number, is introduced. It is proved that simultaneous eigenstates of H\mathcal{H}H and M\mathfrak{M}M have no long-range order as measured by \ensuremathσ∗{\ensuremath{\sigma}}^{*}\ensuremathσ∗ of Kasteleijn. The ground state of the Hamiltonian for b\ensuremath≠0b\ensuremath{\ne}0b\ensuremath=0, N<\ensuremath∞N<\ensuremath{\infty}N<\ensuremath∞ is shown to be nondegenerate. It is therefore necessarily an eigenstate of M\mathfrak{M}M and as such has no long-range order.
The exact solution is given for a linear chain of NNN atoms of spin \textonehalf{} coupled together by the anisotropic Hamiltonian $\mathcal{H}=2J\ensuremath{\Sigma}\stackrel{N}{i=1}[{{S}_{i}}^{z}{{S}_{i+1}}^{z}+(1\ensuremath{-}\ensuremath{\alpha})({{S}_{i}}^{x}{{S}_{i+1}}^{x}+{{S}_{i}}^{y}{{S}_{i+1}}^{y})].$ The energy of the antiferromagnetic ground state is computed and comparison is made with a variational method. The parameter $\ensuremath{\alpha}$ is allowed to vary between 0 and 1, regulating the relative amount of Ising anisotropy. The short-range order, $\ensuremath{\Sigma}{i}^{}{{S}_{i}}^{z}{{S}_{i+1}}^{z}$, is calculated exactly from the variation of the ground-state energy with $\ensuremath{\alpha}$. It is shown that a kink in the short-range order curve calculated using the variational method is fictitious, and the associated discontinuity in $\frac{{\ensuremath{\partial}}^{2}E}{\ensuremath{\partial}{\ensuremath{\alpha}}^{2}}$ is nonexistent. A discussion is given of long-range order and criticisms are presented regarding the predictions of the variational method.
The quantum phases of one-dimensional spin s=1/2s=1/2s=1/2 chains are discussed for models with two parameters, frustrating exchange g=J2>0g={J}_{2}>0g=J2​>0 between second neighbors and normalized nonfrustrating power-law exchange with exponent $\ensuremath{\alpha}$ and distance dependence ${r}^{\ensuremath{-}\ensuremath{\alpha}}$. The ground state (GS) at g=0g=0g=0 has a long-range order (LRO) for $\ensuremath{\alpha}<2$ and long-range spin fluctuations for $\ensuremath{\alpha}>2$. The models conserve total spin S=SA+SBS={S}_{A}+{S}_{B}S=SA​+SB​, have singlet GS for any ggg, $\ensuremath{\alpha}\ensuremath{\ge}0$ and decouple at 1/g=01/g=01/g=0 to linear Heisenberg antiferromagnets on sublattices AAA and BBB of odd- and even-numbered sites. Exact diagonalization of finite chains gives the sublattice spin $\ensuremath{\langle}{S}_{A}^{2}\ensuremath{\rangle}$, the magnetic gap Em{E}_{m}Em​ to the lowest triplet state, and the excitation ${E}_{\ensuremath{\sigma}}$ to the lowest singlet with opposite inversion symmetry to the GS. An analytical model that conserves sublattice spin has a first-order quantum transition at gc=1/4ln2{g}_{c}=1/4\mathrm{ln}2gc​=1/4ln2 from a GS with perfect LRO to a decoupled phase with SA=SB=0{S}_{A}={S}_{B}=0SA​=SB​=0 for $g\ensuremath{\ge}4/{\ensuremath{\pi}}^{2}$ and no correlation between spins in different sublattices. The model with $\ensuremath{\alpha}=1$ has a first-order transition to a decoupled phase that closely resembles the analytical model. The bond order wave (BOW) phase and continuous quantum phase transitions of finite models with $\ensuremath{\alpha}\ensuremath{\ge}2$ are discussed in terms of GS degeneracy where ${E}_{\ensuremath{\sigma}}(g)=0$, excited state degeneracy where ${E}_{\ensuremath{\sigma}}(g)={E}_{m}(g)$, and $\ensuremath{\langle}{S}_{A}^{2}\ensuremath{\rangle}$. The decoupled phase at large frustration has nondegenerate GS for any exponent $\ensuremath{\alpha}$ and excited states related to sublattice excitations.
We give a rigorous proof of the existence of long-range antiferromagnetic order in the ground state of several two-dimensional spin-1/2 Heisenberg systems.Three types of systems are considered.The first type consists of an even number, N , of coupled square lattices with antiferromagnetic nearest-neighbor Heisenberg interactions.In this case, we prove the existence of long-range order (LRO) in the ground state for an interplane to inplane coupling ratio r in the range 0.16 ≤ r ≤ 2.1 for N ≥ 4. Further, we prove that the antiferromagnetic bilayer with ferromagnetic next-nearest-neighbor (nnn) interplane couplings possesses LRO in the ground state for 0.21 ≤ r < 1/4, where r is the absolute value of the ratio of the ferromagnetic and antiferromagnetic couplings.The final example consists of two antiferromagnetic spin-1/2 square lattices that are coupled via an antiferromagnetic nnn interplane coupling r.For r = 1, this system is effectively a spin-1 square lattice and has an ordered ground state.We show that LRO exists in the region 0.85 ≤ r ≤ 1. * ) Note that E r q in the notation of KLS corresponds to 2 + r -E q in our notation.
The methods of Bethe and Hulth\'en are used to build spin-wave states for the antiferromagnetic linear chain. These states, of spin 1 and translational quantum number kkk, are eigenstates of the Hamiltonian $H=\ensuremath{\Sigma}{j}^{}{\mathrm{S}}_{j}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathrm{S}}_{j+1}$ with periodic boundary conditions. For an infinite chain, their spectrum is ${\ensuremath{\epsilon}}_{k}=(\frac{\ensuremath{\pi}}{2})|sink|$, whereas Anderson's spin-wave theory gives ${\ensuremath{\epsilon}}_{k}=|sink|$. For finite chains it has been verified by numerical computation that these states are the lowest states of given kkk, but no rigorous proof has been given for an infinite chain.
The existence of long-range order is proved under certain conditions for the antiferromagnetic XYZ model on the simple cubic or the square lattice. In particular, the spin-1/2 XXZ model on the square lattice is shown to have ground-state long-range order if the exchange anisotropy Δ satisfies 0≦ Δ <0.20 or Δ >1.72, which improves the result of Kubo and Kishi. The existence of long-range order of the z -component of the spin operator is proved for the XXZ model with XY -like anisotropy (0≦ Δ ≦1) under certain conditions. A similar result is shown to hold for the long-range order in the x -direction for the Ising-like model ( Δ ≧1). The XXZ model on the two-dimensional hexagonal lattice is proved to have finite ground-state long-range order for any value of Δ (≧0) if S ≧1 and for Δ >2.55 when S =1/2.