Haldane gap and quantum spin ladder

Hiroshi K. Watanabe

Xiaoqun Wang

Eugene H. Kim, Jenő Sólyom

We study the opening of the Haldane gap in two-leg and four-leg anisotropic spin ladders using bosonization and renormalization group methods, and we determine the phase diagram as a function of the interchain coupling and relative anisotropy ${J}^{z}{/J}^{\mathrm{xy}}.$ It is found that the opening of the Haldane gap is qualitatively different for the two cases considered. For the two-leg ladder the Haldane gap opens for arbitrarily small interchain coupling, independent of ${J}^{z}{/J}^{\mathrm{xy}},$ and the Haldane phase exists in a large region of parameter space. For the four-leg ladder the opening of the Haldane gap is strongly dependent on both the interchain coupling as well as ${J}^{z}{/J}^{\mathrm{xy}},$ and the Haldane phase exists only in a narrow region about the isotropic antiferromagnet.

Jenő Sólyom

Spin chains and spin ladders can have a variety of gapless (critical) or gapped (massive) phases depending on the length of the spin and the type of coupling.A brief review of the results on some simple models is given with emphasis on the generation of the Haldane gap in anisotropic spin ladders.

M. Maslen

Raul A. Santos, Chao‐Ming Jian, Rex Lundgren

We study the bulk entanglement of a series of gapped ground states of spin ladders, representative of the Haldane phase. These ground states of spin $S/2$ ladders generalize the valence bond solid ground state. In the case of spin 1/2 ladders, we study a generalization of the Affleck-Kennedy-Lieb-Tasaki and Nersesyan-Tsvelik states and fully characterize the bulk entanglement Hamiltonian. In the case of general spin $S$, we argue that in the Haldane phase the bulk entanglement spectrum of a half-integer ladder is either gapless or possess a degenerate ground state. For ladders with integer valued spin particles, the generic bulk entanglement spectrum should have an entanglement gap. Finally, we give an example of a series of trivial states of higher spin $S>1$ in which the bulk entanglement Hamiltonian is critical, signaling that the relation between topological states and a critical bulk entanglement Hamiltonian is not unique to topological systems.