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Journal of Applied Physics

Spontaneously dimerized ground states in 1D frustrated Heisenberg antiferromagnets

Alberto Pimpinelli

Published Apr 15, 1991• 2 citations
Publisher

Abstract

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.

DOI
10.1063/1.347887

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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.
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.
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.
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.
We study Heisenberg spin-1/2 and spin-1 chains with alternating ferromagnetic (J1FJ_1^FJ1F​) and antiferromagnetic (J1AJ_1^AJ1A​) nearest-neighbor interactions and ferromagnetic next-nearest-neighbor interaction (J2FJ_2^FJ2F​). In this model frustration is induced by the non-zero J2FJ_2^FJ2F​ and the effect of dimerization is realized by changing the strength of J1AJ_1^AJ1A​ for a fixed value of J1FJ_1^FJ1F​. The model with site spin sss behaves like a Haldane spin chain with site spin 2sss in the limit of vanishing J2FJ_2^FJ2F​ and large J1F/J1AJ_1^F/J_1^AJ1F​/J1A​. We show that the exact ground state of the model can be found along a line in the parameter space. The phase diagram in the space of J1A−J2FJ_1^A-J_2^FJ1A​−J2F​ is determined using numerical techniques complemented by some analytical calculations. A number of quantities, including the ground state spin, energy gap, structure factor, entanglement entropy, and zero temperature magnetization, are studied to understand the complete phase diagram. An interesting and application-wise important feature of this model is that it exhibits a macroscopic magnetization jump in the presence of a magnetic field; we study this using an effective Hamiltonian.
Coexistence of valence-bond formation and topological order in the Frustrated Ferromagnetic J1J_1J1​-J2J_2J2​ Chain
Frustrated one-dimensional (1D) magnets are known as ideal playgrounds for new exotic quantum phenomena to emerge. We consider an elementary frustrated 1D system: the spin- \frac{1}{2} 1 2 ferromagnetic ( J_1 J 1 ) Heisenberg chain with next-nearest-neighbor antiferromagnetic ( J_2 J 2 ) interactions. On the basis of density-matrix renormalization group calculations we show the existence of a finite spin gap at J_2/|J_1|>1/4 J 2 / | J 1 | > 1 / 4 and we find the ground state in this region to be a valence bond solid (VBS) with spin-singlet dimerization between third-neighbor sites. The VBS is the consequence of spontaneous symmetry breaking through order by disorder. Quite interestingly, this VBS state has a Affleck-Kennedy-Lieb-Tasaki-type topological order. This is the first example of a frustrated spin chain in which quantum fluctuations induce gapped topological order.
We report on a detailed investigation of the spin-$1 {J}_{1}\text{\ensuremath{-}}{J}_{2}\text{\ensuremath{-}}{J}_{3}$ Heisenberg model, a frustrated model with nearest-neighbor coupling J1{J}_{1}J1​, next-nearest neighbor coupling J2{J}_{2}J2​, and a three-site interaction ${J}_{3}\left[({\mathbf{S}}_{i\ensuremath{-}1}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{S}}_{i})({\mathbf{S}}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{S}}_{i+1})+\mathrm{H}.\mathrm{c}.\right]$ previously studied in [Phys. Rev. B 93, 241108(R) (2016)]. Using density matrix renormalization group (DMRG) and exact diagonalizations, we show that the phase boundaries between the Haldane phase, the next-nearest neighbor Haldane phase, and the dimerized phase can be very accurately determined by combining the information deduced from the dimerization, the ground-state energy, the entanglement spectrum and the Berry phase. By a careful investigation of the finite-size spectrum, we also show that the transition between the next-nearest neighbor Haldane phase and the dimerized phase is in the Ising universality class all along the critical line. Furthermore, we justify the conformal embedding of the SU(2)2\mathrm{SU}{(2)}_{2}SU(2)2​ Wess-Zumino-Witten conformal field theory in terms of a boson and an Ising field, and we explicitly derive a number of consequences of this embedding for the spectrum along the SU(2)2\mathrm{SU}{(2)}_{2}SU(2)2​ transition line between the Haldane phase and the dimerized phase. We also show that the solitons along the first-order transition line between the Haldane phase and the dimerized phase carry a spin-1/2, while the domain walls between different dimerization domains inside the dimerized phase carry a spin 1. Finally, we show that short-range correlations change character in the Haldane and dimerized phases through disorder and Lifshitz lines, as well as through the development of short-range dimer correlations in the Haldane phase, leading to a remarkably rich phase diagram.