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arXiv:1110.5956 · Physical Review B

Family of spin- S chain representations of SU(2) k Wess-Zumino-Witten models

Ronny Thomale, Stephan Rachel, Peter Schmitteckert, Martin Greiter

Published May 30, 2012• 40 citations
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Abstract

We investigate a family of spin-$S$ chain Hamiltonians recently introduced by one of us [Greiter, Mapping of Parent Hamiltonians, Springer Tracts in Modern Physics, Vol. 244 (Springer, Berlin, 2011)]. For $S=1/2$, it corresponds to the Haldane-Shastry model. For general spin $S$, we find indication that the low-energy theory of these spin chains is described by the SU(2)${}_{k}$ Wess-Zumino-Witten model with coupling $k=2S$. In particular, we investigate the $S=1$ model whose ground state is given by a Pfaffian for even number of sites $N$. We reconcile aspects of the spectrum of the Hamiltonian for arbitrary $N$ with trial states obtained by Schwinger projection of two Haldane-Shastry chains.

DOI
10.1103/physrevb.85.195149

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We investigate a family of spin-SSS chain Hamiltonians recently introduced by one of us [Greiter, Mapping of Parent Hamiltonians, Springer Tracts in Modern Physics, Vol. 244 (Springer, Berlin, 2011)]. For S=1/2S=1/2S=1/2, it corresponds to the Haldane-Shastry model. For general spin SSS, we find indication that the low-energy theory of these spin chains is described by the SU(2)k{}_{k}k​ Wess-Zumino-Witten model with coupling k=2Sk=2Sk=2S. In particular, we investigate the S=1S=1S=1 model whose ground state is given by a Pfaffian for even number of sites NNN. We reconcile aspects of the spectrum of the Hamiltonian for arbitrary NNN with trial states obtained by Schwinger projection of two Haldane-Shastry chains.
We develop a method of constructing excited states in one-dimensional spin chains which are derived from the SU(2)1\text{SU}{(2)}_{1}SU(2)1​ Wess-Zumino-Witten conformal field theory (CFT) using a parent Hamiltonian approach. The resulting systems are equivalent to the Haldane-Shastry model. In our ansatz, correlation functions between primary fields correspond to the ground state of the spin system, whereas excited states are obtained by insertion of descendant fields. Our construction is based on the current algebra of the CFT and emphasizes the close relation between the spectrum of the spin system and the underlying CFT. This general structure might imply that the method could be applied to a wider range of model systems.
We propose 1D and 2D lattice wave functions constructed from the SU(n)1 Wess–Zumino–Witten (WZW) model and derive their parent Hamiltonians. When all spins in the lattice transform under SU(n) fundamental representations, we obtain a two-body Hamiltonian in 1D, including the SU(n) Haldane–Shastry model as a special case. In 2D, we show that the wave function converges to a class of Halperin's multilayer fractional quantum Hall states and belongs to chiral spin liquids. Our result reveals a hidden SU(n) symmetry for this class of Halperin states. When the spins sit on bipartite lattices with alternating fundamental and conjugate representations, we provide numerical evidence that the state in 1D exhibits quantum criticality deviating from the expected behaviors of the SU(n)1 WZW model, while in 2D they are chiral spin liquids being consistent with the prediction of the SU(n)1 WZW model.
Building on the generalization of the exactly dimerized Majumdar-Ghosh ground state to arbitrary spin SSS for the Heisenberg chain with a three-site term $({\mathbf{S}}_{i\ensuremath{-}1}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{S}}_{i})({\mathbf{S}}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{S}}_{i+1})+\text{H.c.}$, we use density-matrix renormalization group simulations and exact diagonalizations to determine the nature of the dimerization transition for S=1S=1S=1, 3/23/23/2, and 2. The resulting central charge and critical exponent are in good agreement with the SU(2)k=2S\text{SU}{(2)}_{k=2S}SU(2)k=2S​ Wess-Zumino-Witten values c=3k/(2+k)c=3k/(2+k)c=3k/(2+k) and $\ensuremath{\eta}=3/(2+k)$. Since the three-site term that induces dimerization appears naturally if exchange interactions are calculated beyond second order, these results suggest that SU(2)k>1\text{SU}{(2)}_{k>1}SU(2)k>1​ Wess-Zumino-Witten models might finally be realized in actual spin chains.
We study the long-wavelength limit of a spin SSS Heisenberg antiferromagnetic chain. The fermionic Lagrangian obtained corresponds to a perturbed level 2S2S2S SU(2)\mathrm{SU}(2)SU(2) Wess-Zumino-Witten (WZW) model. This effective theory is then mapped into a compact U(1)U(1)U(1) boson interacting with Z2S{Z}_{2S}Z2S​ parafermions. The analysis of this effective theory allows us to show that when SSS is an integer there is a mass gap to all excitations, whereas this gap vanishes in the half-odd-integer spin case and the SU(2)2S{\mathrm{SU}(2)}_{2S}SU(2)2S​ WZW model flows towards the SU(2)1{\mathrm{SU}(2)}_{1}SU(2)1​ stable fixed point. This gives a field theory treatment of the so-called Haldane's conjecture for arbitrary values of the spin SSS.
To begin with, we introduce several exact models for SU(3) spin chains: First is a translationally invariant parent Hamiltonian involving four-site interactions for the trimer chain, with a threefold degenerate ground state. We provide numerical evidence that the elementary excitations of this model transform under representation 3‾\overline{\mathbf{3}}3 of SU(3) if the original spins of the model transform under representation 3. Second is a family of parent Hamiltonians for valence bond solids of SU(3) chains with spin representations 6, 10, and 8 on each lattice site. We argue that of these three models, only the latter two exhibit spinon confinement and, hence, a Haldane gap in the excitation spectrum. We generalize some of our models to SU(n)\mathrm{SU}(n)SU(n). Finally, we use the emerging rules for the construction of valence bond solid states to argue that models of antiferromagnetic chains of SU(n)\mathrm{SU}(n)SU(n) spins, in general, possess a Haldane gap if the spins transform under a representation corresponding to a Young tableau consisting of a number of boxes $\ensuremath{\lambda}$ which is divisible by nnn. If $\ensuremath{\lambda}$ and nnn have no common divisor, the spin chain will support deconfined spinons and not exhibit a Haldane gap. If $\ensuremath{\lambda}$ and nnn have a common divisor different from nnn, it will depend on the specifics of the model including the range of the interaction.