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arXiv:0709.0001 · Physical Review B

Highest weight state description of the isotropic spin-1 chain

Markus Andres, Imke Schneider, Sebastian Eggert

Published Jan 22, 2008• 3 citations
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Abstract

We introduce an overcomplete highest weight state basis as a calculational tool for the description of the isotropic spin-1 chain with bilinear exchange coupling ${J}_{1}$ and biquadratic coupling ${J}_{2}$. The ground state can be expressed exactly at the three special points in the phase diagram where the Hamiltonian corresponds to a sum of nearest neighbor total spin projection operators (${J}_{1}=0>{J}_{2}$, ${J}_{1}=\ensuremath{-}{J}_{2}<0$, and ${J}_{1}=\ensuremath{-}{J}_{2}∕3>0$). In particular, at the phase transition point ${J}_{1}=\ensuremath{-}{J}_{2}<0$, it is possible to exactly compute the ground states, excited states, expectation values, and correlation functions by using the new total spin basis.

DOI
10.1103/physrevb.77.014429

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The S = 1 quantum spin chain: some exact and numerical results

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Abstract There has been great interest recently in quantum spin chains with S = 1. Much of this work was stimulated by Haldane’s (1983a, b) discussion of the isotropic Heisenberg model for which he predicted the existence of a disordered ground state (i.e. one in which the correlations decay exponentially) and a gap to the first excited state at k = n. We consider a system with isotropic Heisenberg and biquadratic exchange, and a magnetic field, described by the Hamiltonian.

•Mar 21, 1991•0 citations
We introduce an overcomplete highest weight state basis as a calculational tool for the description of the isotropic spin-1 chain with bilinear exchange coupling J1{J}_{1}J1​ and biquadratic coupling J2{J}_{2}J2​. The ground state can be expressed exactly at the three special points in the phase diagram where the Hamiltonian corresponds to a sum of nearest neighbor total spin projection operators (J1=0>J2{J}_{1}=0>{J}_{2}J1​=0>J2​, J1=\ensuremath−J2<0{J}_{1}=\ensuremath{-}{J}_{2}<0J1​=\ensuremath−J2​<0, and J1=\ensuremath−J2∕3>0{J}_{1}=\ensuremath{-}{J}_{2}∕3>0J1​=\ensuremath−J2​∕3>0). In particular, at the phase transition point J1=\ensuremath−J2<0{J}_{1}=\ensuremath{-}{J}_{2}<0J1​=\ensuremath−J2​<0, it is possible to exactly compute the ground states, excited states, expectation values, and correlation functions by using the new total spin basis.
The spin-1/2 chain with isotropic exchange J1, J2 > 0 between first and second neighbors is frustrated for either sign of J1 and has a singlet ground state (GS) for J1/J2 ⩾ -4. Its rich quantum phase diagram supports gapless, gapped, commensurate (C), incommensurate (IC) and other phases. Critical points J1/J2 are evaluated using exact diagonalization and density matrix renormalization group calculations. The wave vector qG of spin correlations is related to GS degeneracy and obtained as the peak of the spin structure factor S(q). Variable qG indicates IC phases in two J1/J2 intervals, [-4, - 1.24] and [0.44, 2], and a C-IC point at J1/J2 = 2. The decoupled C phase in [-1.24, 0.44] has constant qG = π/2, nondegenerate GS, and a lowest triplet state with broken spin density on sublattices of odd and even numbered sites. The lowest triplet and singlet excitations, E m and Eσ, are degenerate in finite systems at specific frustration J1/J2. Level crossing extrapolates in the thermodynamic limit to the same critical points as qG. The S(q) peak diverges at qG = π in the gapless phase with J1/J2 > 4.148 and quasi-long-range order (QLRO(π)). S(q) diverges at ±π/2 in the decoupled phase with QLRO(π/2), but is finite in gapped phases with finite-range correlations. Numerical results and field theory agree at small J2/J1 but disagree for the decoupled phase with weak exchange J1 between sublattices. Two related models are summarized: one has an exact gapless decoupled phase with QLRO(π/2) and no IC phases; the other has a single IC phase without a decoupled phase in between.
We study the phase diagram, ground-state properties, and excitation gaps of a frustrated spin-Peierls chain, i.e., a spin-Peierls chain with both nearest-neighbor exchange J1{\mathit{J}}_{1}J1​ and next-nearest-neighbor exchange J2{\mathit{J}}_{2}J2​. The phase diagram is calculated using a bosonization renormalization-group method. We propose a bond-operator mean-field approximation to calculate the ground-state properties and excitation gaps.
Exact diagonalization of finite spin-1/2 chains with periodic boundary conditions is applied to the ground state (GS) of chains with ferromagnetic (F) exchange J1<0{J}_{1}<0J1​<0 between first neighbors, antiferromagnetic (AF) exchange ${J}_{2}=\ensuremath{\alpha}{J}_{1}>0$ between second neighbors, and axial anisotropy $0\ensuremath{\le}\ensuremath{\Delta}\ensuremath{\le}1$. In zero field, the GS is in the Sz=0{S}_{z}=0Sz​=0 sector for the relevant parameters and is doubly degenerate at multiple points ${\ensuremath{\gamma}}_{m}=({\ensuremath{\alpha}}_{m},{\ensuremath{\Delta}}_{m})$ in the $\ensuremath{\alpha}$, $\ensuremath{\Delta}$ plane. Degeneracy under inversion at sites or spin-parity or both leads, respectively, to a bond order wave (BOW), to staggered magnetization, or to vector chiral (VC) order. Exact results up to N=28N=28N=28 spins directly yield order parameters and spin correlation functions whose weak NNN dependencies allow inferences about infinite chains. The high-spin GS at J2=0{J}_{2}=0J2​=0 changes discontinuously at ${\ensuremath{\gamma}}_{1}=(\ensuremath{-}1/4,1)$ to a singlet in the isotropic ($\ensuremath{\Delta}=1$) chain. The transition from high to low spin $S(\ensuremath{\alpha},\ensuremath{\Delta})$ is continuous for $\ensuremath{\Delta}<{\ensuremath{\Delta}}_{B}=0.95\ifmmode\pm\else\textpm\fi{}0.01$ on the degeneracy line ${\ensuremath{\alpha}}_{1}(\ensuremath{\Delta})$. The GS has staggered magnetization between ${\ensuremath{\Delta}}_{A}=0.72$ and ${\ensuremath{\Delta}}_{B}$ and a BOW for $\ensuremath{\Delta}<{\ensuremath{\Delta}}_{A}$. When both inversion and spin-parity are reversed at ${\ensuremath{\gamma}}_{m}$, the correlation functions C(p)C(p)C(p) for spins separated by ppp sites are identical. C(p)C(p)C(p) minima are shifted by $\ensuremath{\pi}/2$ from the minima of VC order parameters at separation ppp, consistent with right- and left-handed helices along the zzz axis and spins in the xyxyxy plane. Degenerate GSs of finite chains are related to quantum phase diagrams of extended $\ensuremath{\alpha}$, $\ensuremath{\Delta}$ chains, with good agreement for order parameters along the line ${\ensuremath{\alpha}}_{1}(\ensuremath{\Delta})$. Degenerate GSs limit a VC phase to intermediate $\ensuremath{\alpha}$ and $\ensuremath{\Delta}$ where $S(\ensuremath{\alpha},\ensuremath{\Delta})$ varies rapidly but continuously, in contrast to many-body treatments in which VC phases extend over a larger range of parameters.
We calculate the quantum phase diagram of an XXZ chain with nearest-neighbor (NN) J1{J}_{1}J1​ and next-NN exchange J2{J}_{2}J2​ with anisotropies ${\ensuremath{\Delta}}_{1}$ and ${\ensuremath{\Delta}}_{2},$ respectively. In particular we consider the case ${\ensuremath{\Delta}}_{1}=\ensuremath{-}{\ensuremath{\Delta}}_{2},$ to interpolate between the XX chain $({\ensuremath{\Delta}}_{i}=0)$ and the isotropic model with ferromagnetic phase J2.{J}_{2}.J2​. For ${\ensuremath{\Delta}}_{1}<\ensuremath{-}1,$ a ferromagnetic phase and two antiferromagnetic phases exist. For $|{\ensuremath{\Delta}}_{i}|<1,$ the boundary between the dimer and spin fluid phases is determined by the method of crossing of excitation spectra. For large J2/J1,{J}_{2}{/J}_{1},J2​/J1​, this method seems to indicate the existence of a second spin fluid critical phase. However, an analysis of the spin stiffness and magnetic susceptibility for ${\ensuremath{\Delta}}_{1}={\ensuremath{\Delta}}_{2}=1$ suggests that a small gap is present.
Quantum phase diagram of a frustrated spin-12\frac{1}{2}21​ J1J_{1}J1​--J2J_{2}J2​--J1⊥J_{1}^{\perp}J1⊥​ Heisenberg model on an AAAAAA-stacked bilayer honeycomb lattice
The zero-temperature quantum phase diagram of the spin-12\frac{1}{2}21​ J1J_{1}J1​--J2J_{2}J2​--J1⊥J_{1}^{\perp}J1⊥​ model on an AAAAAA-stacked bilayer honeycomb lattice is investigated using the coupled cluster method (CCM). The model comprises two monolayers in each of which the spins, residing on honeycomb-lattice sites, interact via both nearest-neighbor (NN) and frustrating next-nearest-neighbor isotropic antiferromagnetic (AFM) Heisenberg exchange iteractions, with respective strengths J1>0J_{1} > 0J1​>0 and J2≡κJ1>0J_{2} \equiv \kappa J_{1}>0J2​≡κJ1​>0. The two layers are coupled via a comparable Heisenberg exchange interaction between NN interlayer pairs, with a strength J1⊥≡δJ1J_{1}^{\perp} \equiv \delta J_{1}J1⊥​≡δJ1​. The complete phase boundaries of two quasiclassical collinear AFM phases, namely the Neel and Neel-II phases, are calculated in the κδ\kappa \deltaκδ half-plane with κ>0\kappa > 0κ>0. Whereas on each monolayer in the Neel state all NN pairs of spins are antiparallel, in the Neel-II state NN pairs of spins on zigzag chains along one of the three equivalent honeycomb-lattice directions are antiparallel, while NN interchain spins are parallel. We calculate directly in the thermodynamic (infinite-lattice) limit both the magnetic order parameter MMM and the excitation energy Δ\DeltaΔ from the sTz=0s^{z}_{T}=0sTz​=0 ground state to the lowest-lying ∣sTz∣=1|s^{z}_{T}|=1∣sTz​∣=1 excited state (where sTzs^{z}_{T}sTz​ is the total zzz component of spin for the system as a whole, and where the collinear ordering lies along the zzz direction) for both quasiclassical states used (separately) as the CCM model state, on top of which the multispin quantum correlations are then calculated to high orders (n≤10n \leq 10n≤10) in a systematic series of approximations involving nnn-spin clusters. The sole approximation made is then to extrapolate the sequences of nnnth-order results for MMM and Δ\DeltaΔ to the exact limit, n→∞n \to \inftyn→∞.