Infinite Matrix Product States for long range SU(N) spin models

We construct 1D and 2D long-range SU(N) spin models as parent Hamiltonians associated with infinite matrix product states. The latter are constructed from correlators of primary fields in the SU(N) level 1 WZW model. Since the resulting groundstates are of Gutzwiller-Jastrow type, our models can be regarded as lattice discretizations of fractional quantum Hall systems. We then focus on two specific types of 1D spin chains with spins located on the unit circle, a uniform and an alternating arrangement. For an equidistant distribution of identical spins we establish an explicit connection to the SU(N) Haldane-Shastry model, thereby proving that the model is critical and described by a SU(N) level 1 WZW model. In contrast, while turning out to be critical as well, the alternating model can only be treated numerically. Our numerical results rely on a reformulation of the original problem in terms of loop models.Comment: 37 pages, 6 figure

Topological phases of fermions in one dimension

In this paper we show how the classification of topological phases in insulators and superconductors is changed by interactions, in the case of 1D systems. We focus on the TR-invariant Majorana chain (BDI symmetry class). While the band classification yields an integer topological index $k$, it is known that phases characterized by values of $k$ in the same equivalence class modulo 8 can be adiabatically transformed one to another by adding suitable interaction terms. Here we show that the eight equivalence classes are distinct and exhaustive, and provide a physical interpretation for the interacting invariant modulo 8. The different phases realize different Altland-Zirnbauer classes of the reduced density matrix for an entanglement bipartition into two half-chains. We generalize these results to the classification of all one dimensional gapped phases of fermionic systems with possible anti-unitary symmetries, utilizing the algebraic framework of central extensions. We use matrix product state methods to prove our results.Comment: 14 pages, 3 figures, v2: references adde

Novel families of SU(N){\rm SU}(N) AKLT states with arbitrary self-conjugate edge states

Using the Matrix Product State framework, we generalize the Affleck-Kennedy-Lieb-Tasaki (AKLT) construction to one-dimensional spin liquids with global color ${\rm SU}(N)$ symmetry, finite correlation lengths, and edge states that can belong to any self-conjugate irreducible representation (irrep) of ${\rm SU}(N)$. In particular, ${\rm SU}(2)$ spin-$1$ AKLT states with edge states of arbitrary spin $s=1/2,1,3/2,\cdots$ are constructed, and a general formula for their correlation length is given. Furthermore, we show how to construct local parent Hamiltonians for which these AKLT states are unique ground states. This enables us to study the stability of the edge states by interpolating between exact AKLT Hamiltonians. As an example, in the case of spin-$1$ physical degrees of freedom, it is shown that a quantum phase transition of central charge $c = 1$ separates the Symmetry Protected Topological (SPT) phase with spin-$1/2$ edge states from a topologically trivial phase with spin-$1$ edge states. We also address some specificities of the generalization to ${\rm SU}(N)$ with $N>2$, in particular regarding the construction of parent Hamiltonians. For the AKLT state of the ${\rm SU}(3)$ model with the $3$-box symmetric representation, we prove that the edge states are in the $8$-dimensional adjoint irrep, and for the ${\rm SU}(3)$ model with adjoint irrep at each site, we are able to construct two different reflection-symmetric AKLT Hamiltonians, each with a unique ground state which is either even or odd under reflection symmetry and with edge states in the adjoint irrep. Finally, examples of two-column and adjoint physical irreps for ${\rm SU}(N)$ with $N$ even and with edge states living in the antisymmetric irrep with $N/2$ boxes are given, with a conjecture about the general formula for their correlation lengths.Comment: 37 pages, 14 figures, 4 table

Topological Many-Body States in Quantum Antiferromagnets via Fuzzy Super-Geometry

Recent vigorous investigations of topological order have not only discovered new topological states of matter but also shed new light to "already known" topological states. One established example with topological order is the valence bond solid (VBS) states in quantum antiferromagnets. The VBS states are disordered spin liquids with no spontaneous symmetry breaking but most typically manifest topological order known as hidden string order on 1D chain. Interestingly, the VBS models are based on mathematics analogous to fuzzy geometry. We review applications of the mathematics of fuzzy super-geometry in the construction of supersymmetric versions of VBS (SVBS) states, and give a pedagogical introduction of SVBS models and their properties [arXiv:0809.4885, 1105.3529, 1210.0299]. As concrete examples, we present detail analysis of supersymmetric versions of SU(2) and SO(5) VBS states, i.e. UOSp(N|2) and UOSp(N|4) SVBS states whose mathematics are closely related to fuzzy two- and four-superspheres. The SVBS states are physically interpreted as hole-doped VBS states with superconducting property that interpolate various VBS states depending on value of a hole-doping parameter. The parent Hamiltonians for SVBS states are explicitly constructed, and their gapped excitations are derived within the single-mode approximation on 1D SVBS chains. Prominent features of the SVBS chains are discussed in detail, such as a generalized string order parameter and entanglement spectra. It is realized that the entanglement spectra are at least doubly degenerate regardless of the parity of bulk (super)spins. Stability of topological phase with supersymmetry is discussed with emphasis on its relation to particular edge (super)spin states.Comment: Review article, 1+104 pages, 37 figures, published versio

Spinor bosons realization of the SU(3) Haldane phase with adjoint representation

The SU(3) Haldane phase with adjoint representation provides the simplest non-trivial symmetry-protected topological phases in the SU($N>2$) spin chains for which a gapped system has been predicted. In this letter, I show how to realize this phase in a two-species spinor Bose gas. The proposed system consists of two intertwined species-dependent zigzag optical lattices with the two species labeling the quark and antiquark states of SU(3) symmetry. The Haldane phase is found connected to a position at which both the string order and entanglement spectrum degeneracy are absent, signaling the appearance of a critical point. I show how to understand this absence by a ground-state ansatz.Comment: 5 pages, 5 figure

Mixed global anomalies and boundary conformal field theories

We consider the relation of mixed global gauge gravitational anomalies and boundary conformal field theory in WZW models for simple Lie groups. The discrete symmetries of consideration are the centers of the simple Lie groups. These mixed anomalies prevent to gauge them i.e, take the orbifold by the center. The absence of anomalies impose conditions on the levels of WZW models. Next, we study the conformal boundary conditions for the original theories. We consider the existence of a conformal boundary state invariant under the action of the center. This also gives conditions on the levels of WZW models. By considering the combined action of the center and charge conjugation on boundary states, we reproduce the condition obtained in the orbifold analysis.Comment: 24pages, 1 figure, references adde