We show that the Haldane phase of S=1 chains is characterized by a double degeneracy of the entanglement spectrum. The degeneracy is protected by a set of symmetries (either the dihedral group of $\pi$-rotations about two orthogonal axes, time-reversal symmetry, or bond centered inversion symmetry), and cannot be lifted unless either a phase boundary to another, "topologically trivial", phase is crossed, or the symmetry is broken. More generally, these results offer a scheme to classify gapped phases of one dimensional systems. Physically, the degeneracy of the entanglement spectrum can be observed by adiabatically weakening a bond to zero, which leaves the two disconnected halves of the system in a finitely entangled state.Comment: 11 pages, 4 figures, references added, minor corrections, meta data update

We discuss the characterization and stability of the Haldane phase in integer spin chains on the basis of simple, physical arguments. We find that an odd-S Haldane phase is a topologically non-trivial phase which is protected by any one of the following three global symmetries: (i) the dihedral group of π-rotations about x, y and z axes; (ii) time-reversal symmetry S x,y,z → −S x,y,z ; (iii) link inversion symmetry (reflection about a bond center), consistently with previous results [Phys. Rev. B 81, 064439 (2010)]. On the other hand, an even-S Haldane phase is not topologically protected (i.e., it is indistinct from a trivial, site-factorizable phase). We show some numerical evidence that supports these claims, using concrete examples.

The recent observation [Oosawa et al. J. Phys. : Condens. Matter 11, 265 (1999)] of the field-induced Néel ordering in a spin-gap magnetic compound TlCuCl3 is interpreted as a Bose-Einstein Condensation of magnons. A mean-field calculation based on this picture is shown to describe well the temperature dependence of the magnetization. The present system opens a new area for studying Bose-Einstein condensation of thermodynamically large number of particles in a grand-canonical ensemble.PACS number 75.10.Jm

We discuss zero-temperature quantum spin chains in a uniform magnetic field, with axial symmetry. For integer or half-integer spin, S, the magnetization curve can have plateaus and we argue that the magnetization per site m is topologically quantized as n͑S 2 m͒ integer at the plateaus, where n is the period of the ground state. We also discuss conditions for the presence of the plateau at those quantized values. For S 3͞2 and m 1͞2, we study several models and find two distinct types of massive phases at the plateau. One of them is argued to be a "Haldane gap phase" for half-integer S. [S0031-9007(97)02624-0] PACS numbers: 75.10.JmOne-dimensional antiferromagnets are expected not to have long-range magnetic order in general. It was argued by Haldane [1], in 1983, that for integer, but not half-integer spin, S, there is a gap to the excited states. In the presence of a magnetic field, the S 1͞2 Heisenberg antiferromagnetic (AF) chain remains gapless from zero field up to the saturation field, where the ground state is fully polarized [2]. For integer S, the gap persists up to a critical field, equal to the gap, where Bose condensation of magnons occurs [3]. The S 1 Heisenberg AF chain is known to be gapless from the critical field up to the saturation field [4]. Recently Hida observed that an S 1͞2 antiferromagnetic chain with period 3 exchange coupling shows a plateau in the magnetization curve at magnetization per site m 1͞6 (1͞3 of the full magnetization) [5]. Related works on bond-alternating chains have also been reported [6-9] including experimental observation [10].In this Letter, we consider the zero-temperature behavior of general quantum spin chains, including chains with periodic structures, in a uniform magnetic field pointing along the direction of the axial symmetry (z axis) (i.e., the total S z is conserved). We argue that, in quantum spin chains, there is a phenomenon which is strikingly analogous to the quantum Hall effect-topological quantization of a physical quantity under a changing magnetic field [11]. We first consider an extension of the Lieb-Schultz-Mattis (LSM) theorem [12] to the case with an applied field. This indicates that translationally invariant spin chains in an applied field can be gapful without breaking translation symmetry, only when the magnetization per spin, m, obeys S 2 m integer. We expect such gapped phases to correspond to plateaus at these quantized values of m. "Fractional quantization" can also occur, if accompanied by (explicit or spontaneous) breaking of the translational symmetry. The generalized LSM theorem does not prove the presence of the plateau, however. Thus we construct a corresponding argument using Abelian bosonization, which is in complete agreement with the generalized LSM theorem, and also gives a condition for the presence of the plateau. As simplest examples, we study translationally invariant S 3͞2 chains at m 1͞2. We present numerical diagonalization and density matrix renormalization group (DMRG) [13] calculations, which demonstrate the existen...

Topologically ordered phases are gapped states, defined by the properties of excitations when taken around one another. Here we demonstrate a method to extract the statistics and braiding of excitations, given just the set of ground-state wave functions on a torus. This is achieved by studying the Topological Entanglement Entropy (TEE) on partitioning the torus into two cylinders. In this setting, general considerations dictate that the TEE generally differs from that in trivial partitions and depends on the chosen ground state. Central to our scheme is the identification of ground states with minimum entanglement entropy, which reflect the quasi-particle excitations of the topological phase. The transformation of these states allows for a determination of the modular S and U matrices which encode quasi-particle properties. We demonstrate our method by extracting the modular S matrix of a chiral spin liquid phase using a Monte Carlo scheme to calculate TEE, and prove that the quasi-particles obey semionic statistics. This method offers a route to a nearly complete determination of the topological order in certain cases.