Affleck–Kennedy–Lieb–Tasaki Model

Abstract

In 1987, Affleck, Kennedy, Lieb, and Tasaki proposed a one-dimensional \(S=1\) antiferromagnetic quantum spin model, now called the AKLT model, whose ground state can be written down explicitly. It was shown that the energy spectrum of the model has a finite gap above the ground state energy, and the ground state correlation function decays exponentially [4, 5]. These results agree with Haldane’s conclusion for the integer spin antiferromagnetic Heisenberg chain. But note that this does not prove the Haldane “conjecture” since the model is different from the Heisenberg model. Nevertheless this work proves that there exists an \(S=1\) chain that has a unique disordered ground state with a gap, and hence provides a strong support to Haldane’s conclusion. It also provides a starting point of further studies of various phenomena and concepts related to the Haldane gap, as we shall see in Chap. 8. Now it is commonly believed that the AKLT model is (in some sense) at the “center” of the Haldane phase. In Sect. 7.1 we give a detailed discussion of the main results about the AKLT model. In Sect. 7.2 we study some unexpected features of the model, which will turn out to be essential for the study of Haldane phenomena. In Sect. 7.3 we discuss extensions and closely related models.