A series of symmetry-protected topological(SPT) and trivial spin-gap phases in the spin-1/2 ferromagnetic--antiferromagnetic alternating Heisenberg chain with alternating next-nearest-neighbour interaction are investigated using two kinds of entanglement spectra defined by different divisions of the whole chain. In the case one of the next-nearest-neighbor interaction vanishes, the model reduces to the
Δ-chain in which a series of spin-gap phases are found in J. Phys. Soc. Jpn. 77, 044707 (2008). From the degeneracy of the entanglement spectra, these phases are identified as the SPT and trivial phases. It is found that the ground state phases boundaries are insensitive to the strength of the alternation in next-nearest neighbor interaction. These results are consistent with the analysis based on the nonlinear
σ model and exact solution on the ferromagnetic-nonmagnetic phase boundary.
The spin-1/2 ferromagnetic-antiferromagnetic alternating Heisenberg chain with ferromagnetic next-nearest-neighbour (NNN) interaction is investigated. The ground state is the Haldane phase for weak NNN interaction, and is the ferromagnetic phase for weak antiferromagnetic interaction. We find a series of topologically distinct spin-gap phases with various magnitudes of edge spins for strong NNN interaction. The phase boundaries between these phases are determined on the basis of the DMRG calculation with additional spins that compensate the edge spins. It is found that each of the exact solutions with short-range antiferromagnetic correlation on the ferromagnetic-nonmagnetic phase boundary is representative of each spin gap phase.
Abstract A Heisenberg spin- s chain with alternating ferromagnetic (FM) ( − J 1 F < 0 ) and antiferromagnetic ( J 1 A > 0 ) nearest-neighbor (NN) interactions, exhibits the dimer and spin-2 s Haldane phases in the limits J 1 F / J 1 A → 0 and J 1 F / J 1 A → ∞ respectively. These two phases are understood to be topologically equivalent. Induction of the frustration through the next NN FM interaction ( − J 2 F < 0 ) produces a very rich quantum phase diagram. With frustration, the whole phase diagram is divided into a FM and a nonmagnetic (NM) phase. For s = 1/2, the full NM phase is seen to be of Haldane–dimer type, but for s > 1/2, a spiral phase comes between the FM and the Haldane–dimer phases. The study of a suitably defined string-order parameter and spin-gap at the phase boundary indicates that the Haldane–dimer and spiral phases have different topological characters. We also find that, along the J 2 F = 1 2 J 1 F line in the NM phase, an NN dimer state is the exact groundstate, provided J 1 A > J C = κ J
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 study Heisenberg spin-1/2 and spin-1 chains with alternating ferromagnetic (
J1F) and antiferromagnetic (
J1A) nearest-neighbor interactions and ferromagnetic next-nearest-neighbor interaction (
J2F). In this model frustration is induced by the non-zero
J2F and the effect of dimerization is realized by changing the strength of
J1A for a fixed value of
J1F. The model with site spin
s behaves like a Haldane spin chain with site spin 2
s in the limit of vanishing
J2F and large
J1F/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−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 Heisenberg spin-1/2 and spin-1 chains with alternating ferromagnetic (J(1)(F)) and antiferromagnetic (J(1)(A)) nearest-neighbor interactions and a ferromagnetic next-nearest-neighbor interaction (J(2)(F)). In this model frustration is present due to the non-zero J(2)(F). The model with site spin s behaves like a Haldane spin chain, with site spin 2s in the limit of vanishing J(2)(F) and large J(1)(F)/J(1)(A). We show that the exact ground state of the model can be found along a line in the parameter space. For fixed J(1)(F), the phase diagram in the space of J(1)(A)-J(2)(F) is determined using numerical techniques complemented by analytical calculations. A number of quantities, including the structure factor, energy gap, entanglement entropy and zero temperature magnetization, are studied to understand the complete phase diagram. An interesting and potentially important feature of this model is that it can exhibit a macroscopic magnetization jump in the presence of a magnetic field; we study this using an effective Hamiltonian.