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Physical review. B, Condensed matter

Spontaneous dimerization in the S = 1 2 Heisenberg antiferromagnetic chain with competing interactions

F. D. M. Haldane

Published Apr 1, 1982• 523 citations
Publisher

Abstract

Spontaneous dimerization is found in the $S=\frac{1}{2}$ isotropic Heisenberg antiferromagnetic chain with competing nearest- and next-nearest-neighbor exchange, $\frac{{J}_{2}}{{J}_{1}}\ensuremath{\gtrsim}\frac{1}{6}$, and results from the same umklapp processes that lead to the N\'eel state when easy-axis exchange anisotropy is present. Spontaneous and externally induced dimerizations are contrasted.

DOI
10.1103/physrevb.25.4925

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Spontaneous dimerization is found in the S=12S=\frac{1}{2}S=21​ isotropic Heisenberg antiferromagnetic chain with competing nearest- and next-nearest-neighbor exchange, J2J1\ensuremath≳16\frac{{J}_{2}}{{J}_{1}}\ensuremath{\gtrsim}\frac{1}{6}J1​J2​​\ensuremath≳61​, and results from the same umklapp processes that lead to the N'eel state when easy-axis exchange anisotropy is present. Spontaneous and externally induced dimerizations are contrasted.
Starting from the exact result of Majumdar and Ghosh for the dimerized ground state of a Heisenberg linear-chain antiferromagnet with competing next-nearest-neighbor interaction J2{\mathit{J}}_{2}J2​=1/2J1{\mathit{J}}_{1}J1​, it is shown that this state is an exact eigenstate of the same Hamiltonian with couplings extended up to fourth-nearest neighbors, if the relations J1{\mathit{J}}_{1}J1​-2J2{\mathit{J}}_{2}J2​+J3{\mathit{J}}_{3}J3​=0 and J3{\mathit{J}}_{3}J3​=2J4{\mathit{J}}_{4}J4​ hold. Furthermore, it is proved to all orders in perturbation theory that this state is the ground state of this Hamiltonian in a finite interval of values of the parameter \ensuremath{\lambda}==1/2-j2{\mathit{j}}_{2}j2​ around \ensuremath{\lambda}=0. The actual extension of this interval is computed numerically for finite chains and is shown to be significant.
We perform a systematic DMRG investigation of the two natural spin-1 generalizations of the spin-1/2 Majumdar-Ghosh chain, the spin-1 ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg chain, where J2{J}_{2}J2​ is a next-nearest-neighbor Heisenberg coupling, and the spin-1 ${J}_{1}\text{\ensuremath{-}}{J}_{3}$ model, where J3{J}_{3}J3​ refers to a three-site interaction defined by ${J}_{3}\left[({\mathbf{S}}_{i\ensuremath{-}1}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{S}}_{i})({\mathbf{S}}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{S}}_{i+1})+\mathrm{H}.\mathrm{c}.\right]$. Although both models are rigorously equivalent to the Majumdar-Ghosh chain for spin 1/2, their physics appears to be quite different for spin 1. Indeed, when all couplings are antiferromagnetic, the spin-1 ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model undergoes an effective decoupling into two next-nearest-neighbor (NNN) Haldane chains upon increasing J2{J}_{2}J2​, while the ${J}_{1}\text{\ensuremath{-}}{J}_{3}$ chain undergoes a spontaneous dimerization similar to the spin-1/2 Majumdar-Ghosh chain upon increasing J3{J}_{3}J3​. By extending the phase diagram to all signs of the couplings, we show that both the dimerized and the NNN-Haldane phase are actually present in the ${J}_{1}\text{\ensuremath{-}}{J}_{3}$ model, the former one adjacent to the Haldane one, the latter to the ferromagneric one, with an Ising transition between them. By contrast, the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ chain only has a NNN-Haldane phase between the Haldane phase and the ferromagnetic phase for positive J2{J}_{2}J2​. In both cases, our DMRG data are consistent with a continuous Kosterlitz-Thouless transition between the NNN-Haldane and the ferromagnetic phases.
Starting from the exact result of Majumdar and Ghosh (1969) for the dimerized ground state of a Heisenberg S=1/2 antiferromagnet with competing next-nearest-neighbour interaction J2+1/2J1 on a linear chain, it is shown that this state is the ground state of the same kind of Hamiltonian, if couplings are extended up to 2n-nearest-neighbours, and the relation J1=2n, J2=2n-1, ..., J2n=1 holds. Furthermore, it is proved rigorously that this state is the ground state when n=2 and J1-2J2+J3=0, J3=2J4, in a finite interval of values of the parameter J4. A rigorous lower limit for the extension of this interval is found to be 0<or=J4<or=1/4J1. A comparison with ground-state configurations of the equivalent classical Heisenberg and Ising model is discussed.
The J1-J3 Heisenberg spin models with nearest-neighbor (J1) and additional isotropic three-site (J3) spin interactions remain relatively less explored, although such types of competing exchange terms can naturally emerge from different sources, including the strong-coupling expansion of the multiorbital Hubbard model. Below we present a short survey of the recently published research in this field, the emphasis being on the characteristics of the variety of quantum phases supported by a few generic uniform- and alternating-spin J1-J3 Heisenberg chains. For the reason that the positive (J3>0) three-site couplings tend towards the formation of local quantum dimers, the J1-J3 spin models typically experience some spontaneous dimerization upon increasing J3. Actually, it occurred that the established dimer phases in spin-S J1-J3 Heisenberg chains (S>1/2) serve as complete analogues of the famous gapped Majumdar-Ghosh dimer phase in the spin-1/2 Heisenberg chain with next-nearest-neighbor couplings. The same dimerizations have been observed in the alternating-spin (S,σ) J1-J3 chains (S>σ), provided that the cell spin S+σ= integer, whereas for half-integer cell spin, the local dimer formation produces gapless spin-liquid ground states. The alternating-spin J1-J3 chains also provide some typical examples of spin models supporting the so-called non-Lieb-Mattis magnetic phases.
We give an explicit proof that the completely dimerized (’Kekulè or nearest-neighbor resonant valence bond) state is the degenerate ground state of a linear chain S =1/2 Heisenberg antiferromagnet, with competing couplings extending up to 2n nearest neighbors, if the relation J1 = 2n, J2 = 2n − 1,...,J2n = 1, holds between the various exchange integrals. The previously known result for J2 =1/2 J1 is then just a special case corresponding to n=1. We study in detail n=2 (fourth-nearest neighbors) and show that this state is the ground state of the Hamiltonian when J1 − 2J2 + J3 = 0, J3 = 2J4, in a finite interval of values of the parameter J4. A rigorous lower limit for the extension of this interval is found to be 0 ≤ J4 ≤1/4 J1. A comparison with ground-state configurations of the equivalent classical Heisenberg and Ising models is made.
The ground-state phase diagram of a spin-12\frac{1}{2}21​ XXZ chain with competing ferromagnetic nearest-neighbor (J1<0{J}_{1}<0J1​<0) and antiferromagnetic second-neighbor (J2>0{J}_{2}>0J2​>0) exchange couplings is studied by means of the infinite time evolving block decimation algorithm and effective field theories. For the SU(2)-symmetric (Heisenberg) case, we show that the nonmagnetic phase in the range $\ensuremath{-}4<{J}_{1}/{J}_{2}<0$ has a small but finite ferromagnetic dimer order. We argue that this spontaneous dimer order is associated with effective spin-1 degrees of freedom on dimerized bonds, which collectively form a valence bond solid state as in the spin-1 antiferromagnetic Heisenberg chain (the Haldane spin chain). We thus call this phase the Haldane dimer phase. With easy-plane anisotropy, the model exhibits a variety of phases including the vector chiral phase with gapless excitations and the even-parity dimer and N\'eel phases with gapped excitations, in addition to the Haldane dimer phase. Furthermore, we show the existence of gapped phases with coexisting orders in narrow regions that intervene between the gapless chiral phase and any one of Haldane dimer, even-parity dimer, and N\'eel phases. Possible implications for quasi-one-dimensional edge-sharing cuprates are discussed.