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 ) 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 ) 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.
IntroductionLong-range spin models such as the Gaudin model [1] or the Haldane-Shastry model [2,3] have attracted the attention of physicists and mathematicians for a long period of time. In its original formulation, the Haldane-Shastry model describes the dynamics of SU(2) spins on a circle with inverse distance square interactions. It received a lot of attention due to its exact solvability and due to the form of its groundstate which is closely related to a bosonic Laughlin wavefunction at filling fraction ν = 1/2. The Haldane-Shastry model can be viewed as realizing a 1D analogue of a chiral spin liquid, with spinon excitations satisfying a generalized Pauli exclusion principle and obeying fractional statistics [4]. Many of the remarkable properties of the Haldane-Shastry model have their origin in the existence of an infinite-dimensional Yangian symmetry [5]. The latter also allowed to identify its thermodynamic limit as the SU(2) WZW conformal field theory at level k = 1 [6] (see also [7]). The Haldane-Shastry model admits obvious generalizations to symmetry groups such as the unitary series SU(N ) [5] or its supersymmetric analog SU(M |N ) [8,9].For our current work, there are two aspects of the SU(N ) Haldane-Shastry model that will be particularly important. First of all, it provides an efficient discretization of the SU(N ) WZW conformal field theory at level k = 1, where the scaling laws are not affected by logarithmic corrections. Secondly, wavefunctions of the groundstate as well as the excited states exhibit an intimate relation to the physics of fractional quantum Hall (FQH) systems, also for general values of N [10]. There are of course also differences: While the constituents of FQH systems are particles which are moving on a 2D surface, the degrees of freedom in the spin model are pinned to fixed discrete locations on a circle.The study of fractional quantum Hall systems is frequently based on the following intriguing dichotomy: One single chiral 2D conformal field theory (CFT) describes the two complementary aspects of the physical sample -its bulk and its boundary. It is known since the work of Mo...