Infinite Dimensional Matrix Product States for Long-Range Quantum Spin Models

Abstract

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...

Lattice Laughlin states on the torus from conformal field theory

“…One of them is to engineer a band structure that is reminiscent of a Landau level, ensure that the band is partially filled, and add interactions between the particles [21][22][23][24]. The other strategy is to start from an analytical FQH wave function in the continuum, modify it to a lattice wave function, and then analyze its properties and use analytical tools to derive a Hamiltonian for which the state is the ground state [19,[25][26][27][28][29][30][31][32][33][34][35][36]. Both approaches have been shown numerically to be successful.…”

“…This observation is also helpful for finding analytical expressions of FQH states with periodic boundary conditions [9]. More recently, the CFT formulation has turned out to be very fruitful for constructing FQH models in lattice systems with analytical wave functions and corresponding parent Hamiltonians [19,28,29,[31][32][33][34][35][36][37][38].…”

“…A natural way to transform a wave function in the continuum into a lattice wave function is to keep the analytical expression for the state, but to let the state be a sum over all possible distributions of the particles on a lattice rather than an integral over all possible positions of the particles in the plane. Part of the trick used to derive the parent Hamiltonians in [28,29,[31][32][33][34][35]38], however, is not just to do this, but to make an additional change. The Gaussian factor appearing in the continuum FQH wave functions is obtained by including a uniform background charge in the CFT correlator, and the additional change is to restrict the background charge to the same lattice as the particles.…”

Infinite matrix product states, boundary conformal field theory, and the open Haldane-Shastry model

“…This construction shares conceptual similarity to Moore and Read's approach [13] of writing 2D trial fractional quantum Hall states in terms of conformal blocks. For a variety of examples [12,[14][15][16][17][18][19][20], the infinite MPS (as well as their parent Hamiltonians) have been shown to describe critical chains with periodic boundary conditions (PBC) and, furthermore, their critical behaviors are often related to the CFT whose fields are used for constructing the wave functions [21]. In this sense, the infinite MPS introduced in Ref.…”

Chiral conformal field theory for topological states and the anyon eigenbasis on the torus

“…It is valid for many chiral topological orders and allows for a systematic construction of trial wave functions for such systems. This method has been generalized from the continuum to the lattice, both on the plane [13][14][15][16][17][18][19][20][21][22][23][24][25][26] and on the torus [27][28][29][30]. One instructive progress is to express the resulting wave functions as infinite-dimensional matrix product states [31].…”