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arXiv:cond-mat/0210567 · Physica B Condensed Matter

Identification of topologically different valence bond states in spin ladders

Masaaki Nakamura

Published May 1, 2003• 36 citations
arXivHTMLPDF
DOI
10.1016/s0921-4526(02)02180-4

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We consider various two-leg ladder models exhibiting gapped phases. All of these phases have short-ranged valence bond ground states, and they all exhibit string order. However, we show that short-ranged valence bond ground states divide into two topologically distinct classes, and as a consequence, there exist two topologically distinct types of string order. Therefore, not all gapped phases belong to the same universality class. We show that phase transitions occur when we interpolate between models belonging to different topological classes, and we study the nature of these transitions.
We show that spin-S ladders undergo, for antiferromagnetic rung coupling, a series of 4S topological phase transitions when the ratio of the leg to rung coupling changes from strongly antiferromagnetic to strongly ferromagnetic. These phase transitions are characterized by a change of Berry phase on the rungs, and, with appropriate boundary conditions, by the appearance and disappearance of edge states without the bulk gap ever closing. The various phases can be distinguished by the number of valence bond singlets on the rungs, which decreases from 2S for strong rung coupling to 0 for dominant leg coupling. In particular, the isotropic spin-3/2 ladder with equal leg and rung coupling is explicitly shown to be in the valence bond solid phase with one singlet per bond.
We propose an order parameter to characterize valence-bond-solid (VBS) states in quantum spin systems, given by the ground-state expectation value of a unitary operator appearing in the Lieb-Schultz-Mattis argument. We show that the order parameter changes the sign according to the configuration of the valence bonds. This allows us to determine the phase transition point in between different VBS states accurately. We demonstrate this theory in the bond-alternating Heisenberg chain and in the frustrated spin ladder.