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 and symmetry broken phases of Z_N parafermions in one dimension

We classify the gapped phases of Z_N parafermions in one dimension and construct a representative of each phase. Even in the absence of additional symmetries besides parafermionic parity, parafermions may be realized in a variety of phases, one for each divisor n of N. The phases can be characterized by spontaneous symmetry breaking, topology, or a mixture of the two. Purely topological phases arise if n is a unitary divisor, i.e. if n and N/n are co-prime. Our analysis is based on the explicit realization of all symmetry broken gapped phases in the dual Z_N-invariant quantum spin chains.Comment: 16 pages; v2: improved exposition and additional reference

Classical topological paramagnetism

Topological phases of matter are one of the hallmarks of quantum condensed matter physics. One of their striking features is a bulk-boundary correspondence wherein the topological nature of the bulk manifests itself on boundaries via exotic massless phases. In classical wave phenomena analogous effects may arise; however, these cannot be viewed as equilibrium phases of matter. Here we identify a set of rules under which robust equilibrium classical topological phenomena exist. We write down simple and analytically tractable classical lattice models of spins and rotors in two and three dimensions which, at suitable parameter ranges, are paramagnetic in the bulk but nonetheless exhibit some unusual long-range or critical order on their boundaries. We point out the role of simplicial cohomology as a means of classifying, writing-down, and analyzing such models. This opens a new experimental route for studying strongly interacting topological phases of spins.Comment: 12 pages + 1 page Appendix; 5 figures; new version with Z_N cas

Rectangular amplitudes, conformal blocks, and applications to loop models

In this paper we continue the investigation of partition functions of critical systems on a rectangle initiated in [R. Bondesan et al, Nucl.Phys.B862:553-575,2012]. Here we develop a general formalism of rectangle boundary states using conformal field theory, adapted to describe geometries supporting different boundary conditions. We discuss the computation of rectangular amplitudes and their modular properties, presenting explicit results for the case of free theories. In a second part of the paper we focus on applications to loop models, discussing in details lattice discretizations using both numerical and analytical calculations. These results allow to interpret geometrically conformal blocks, and as an application we derive new probability formulas for self-avoiding walks.Comment: 46 page