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arXiv:1803.03101 · Physical review. B./Physical review. B

Rigorous decoupling between edge states in frustrated spin chains and ladders

Natalia Chepiga, Frédéric Mila

Published May 31, 2018• 7 citations
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Abstract

We investigate the occurrence of exact zero modes in one-dimensional quantum magnets of finite length that possess edge states. Building on conclusions first reached in the context of the spin-1/$2XY$ chain in a field and then for the spin-$1{J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model, we show that the development of incommensurate correlations in the bulk invariably leads to oscillations in the sign of the coupling between edge states, and hence to exact zero energy modes at the crossing points where the coupling between the edge states rigorously vanishes. This is true regardless of the origin of the frustration (e.g., next-nearest-neighbor coupling or biquadratic coupling for the spin-1 chain), of the value of the bulk spin (we report on spin-1/2, spin-1, and spin-2 examples), and of the value of the edge-state emergent spin (spin-1/2 or spin-1).

DOI
10.1103/physrevb.97.174434

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We investigate the occurrence of exact zero modes in one-dimensional quantum magnets of finite length that possess edge states. Building on conclusions first reached in the context of the spin-1/2XY2XY2XY chain in a field and then for the spin-1J1\ensuremath−J21{J}_{1}\ensuremath{-}{J}_{2}1J1​\ensuremath−J2​ Heisenberg model, we show that the development of incommensurate correlations in the bulk invariably leads to oscillations in the sign of the coupling between edge states, and hence to exact zero energy modes at the crossing points where the coupling between the edge states rigorously vanishes. This is true regardless of the origin of the frustration (e.g., next-nearest-neighbor coupling or biquadratic coupling for the spin-1 chain), of the value of the bulk spin (we report on spin-1/2, spin-1, and spin-2 examples), and of the value of the edge-state emergent spin (spin-1/2 or spin-1).
We show that the effective coupling between the spin-1/2 edge states of a spin-1 chain of finite length can be continuously tuned by frustration. For the ${J}_{1}\ensuremath{-}{J}_{2}$ model with nearest- and next-nearest-neighbor antiferromagnetic interactions, we show that the effective coupling in a chain of length LLL changes sign $N\ensuremath{\simeq}0.38L$ times in the window $0.28\ensuremath{\lesssim}{J}_{2}/{J}_{1}\ensuremath{\lesssim}0.75$ where the short-range correlations are incommensurate. This implies that there are NNN zero modes where the singlet and the triplet are strictly degenerate, i.e., NNN values of J2/J1{J}_{2}/{J}_{1}J2​/J1​ where the spin-1/2 edge states are completely decoupled. We argue that this effect must be generic for all incommensurate phases with localized edge states, and we briefly discuss a few experimental implications.
The ground state properties of an Ising chain with nearest (J1{J}_{1}J1​) and next-nearest neighbor (J2{J}_{2}J2​) interactions in a transverse field are investigated using the density matrix renormalization group and cluster mean-field theory methods. Its quantum phase diagram has two ordered regions (separated by a quantum disordered region), one of which has simple ferromagnetic or N\'eel order depending upon the sign of J1{J}_{1}J1​, and the other is the double-staggered antiferromagnetic phase. The presence of the edge modes in these ordered phases is inferred by calculating the end-to-end spin-spin correlations on open chains. It is found that there occur four edge modes in the double-staggered phase, while the ferromagnetic and N\'eel phases support two edge modes, except very near J1=0{J}_{1}=0J1​=0, where it seems they have four edge modes like the exact case at J1=0{J}_{1}=0J1​=0 itself.
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.
We study the ground-state properties of weakly frustrated Heisenberg ferrimagnetic chains with nearest- and next-nearest-neighbor antiferromagnetic exchange interactions and two types of alternating sublattice spins S1>S2,{S}_{1}>{S}_{2},S1​>S2​, using 1/S1/S1/S spin-wave expansions, density-matrix renormalization group, and exact diagonalization techniques. It is argued that the zero-point spin fluctuations completely destroy the classical commensurate-incommensurate continuous transition. Instead, the long-range ferrimagnetic state disappears through a discontinuous transition to a singlet state at a larger value of the frustration parameter. In the ferrimagnetic phase we find a disorder point marking the onset of incommensurate real-space short-range spin-spin correlations.
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.
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.