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arXiv:hep-th/9802014 · Physical review. B, Condensed matter

Non-Abelian bosonization and Haldane’s conjecture

D. C. Cabra, Pierre Pujol, Cecilia von Reichenbach

Published Jul 1, 1998• 24 citations
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Abstract

We study the long-wavelength limit of a spin $S$ Heisenberg antiferromagnetic chain. The fermionic Lagrangian obtained corresponds to a perturbed level $2S$ $\mathrm{SU}(2)$ Wess-Zumino-Witten (WZW) model. This effective theory is then mapped into a compact $U(1)$ boson interacting with ${Z}_{2S}$ parafermions. The analysis of this effective theory allows us to show that when $S$ is an integer there is a mass gap to all excitations, whereas this gap vanishes in the half-odd-integer spin case and the ${\mathrm{SU}(2)}_{2S}$ WZW model flows towards the ${\mathrm{SU}(2)}_{1}$ stable fixed point. This gives a field theory treatment of the so-called Haldane's conjecture for arbitrary values of the spin $S$.

DOI
10.1103/physrevb.58.65

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Critical theory of quantum spin chains

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The relation between quantum spin chains and conformal field theories is reexamined. Using a generalized Hubbard model representation it is argued that the critical theory for generic half-odd-integer spin antiferromagnets is the Wess-Zumino-Witten model (WZW model) with topological coupling, k=1, whereas generic integer spin antiferromagnets have an energy gap. The higher-k WZW models (which describe integrable higher spin models) are multicritical points in the space of all spin Hamiltonians. The k=1 WZW model represents a stable fixed point for many theories including WZW models of arbitrary odd k with relevant operators added, generalized Hubbard or Thirring models with an odd number of colors and the O(3) \ensuremath{\sigma} model at topological angle \ensuremath{\theta}=\ensuremath{\pi}.

Physical review. B, Condensed matter•Oct 1, 1987•619 citations
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.
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.
In this thesis we study various one-dimensional quantum spin systems with SU(2) and SU(N) symmetry. We investigate the short-distance behavior of the SU(2) Heisenberg model in the limit of large spin and show that there exists an extended regime where perturbation theory, in the form of spin-wave theory, can be successfully applied. The reason for this perturbative regime stems from the asymptotic freedom of the nonlinear sigma model onto which the Heisenberg model can be mapped. When considering observables which respect the rotational invariance of the model, we observe a cancellation of infrared divergences in the perturbative expansions, leading to a meaningful description of correlation functions. We then turn to the study of SU(N) models. Building on the representation theory of the SU(N) group and on the matrix product state (MPS) formalism, we introduce a generic method to construct Affleck-Kennedy-Lieb-Tasaki (AKLT) models having edge states described by any self-conjugate irreducible representation (irrep) of SU(N). A simple example is given by a spin-1 AKLT model having spin-1 edge states. The phase transition between this model and the original AKLT model is shown to be continuous and to correspond to a topological phase transition described by the SU(2) level-1 Wess-Zumino-Witten conformal field theory universality class. In addition we study an SU(3) AKLT model for the 3-box symmetric irrep at each site. We demonstrate that the edge states are adjoint edge irreps, we extract its correlation length and provide a useful construction as an optimal MPS. Finally, we develop a density matrix renormalization group algorithm based on standard Young tableaus and subduction coefficients to make full use of the non-abelian symmetry and to investigate the SU(3) Heisenberg model with 3-box symmetric irrep at each site. We show that the model has a finite gap above the singlet ground state, in agreement with an extension of the Haldane conjecture to SU(3) chains in the fully symmetric irreps. We also argue that there are five branches of elementary excitations living in four different irreps, each of which is gapped.
According to Haldane’s conjecture, there is a spin gap of the excitation spectrum in integer Heisenberg spin chains whereas half-integer spin chains have a gapless excitation spectrum. In an earlier publication (Hung and Gong 2005 Phys. Rev. B 71 054413), the Haldane gap and other properties of the spin-1 system are explained using a spin-1/2 alternating bond chain model. The ferromagnetic–ferromagnetic–antiferromagnetic Heisenberg spin-1/2 chain is expected to behave similarly to the uniform spin-3/2 Heisenberg chain for strong ferromagnetic coupling. And the general nonuniform spin-1/2 chain may be used to explain the properties of the integer and half-integer high spin- S chains, consistently. In this paper, we provide a detailed numerical verification for the above problems.
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.
The relation between quantum spin chains and conformal field theories is reexamined. Using a generalized Hubbard model representation it is argued that the critical theory for generic half-odd-integer spin antiferromagnets is the Wess-Zumino-Witten model (WZW model) with topological coupling, k=1, whereas generic integer spin antiferromagnets have an energy gap. The higher-k WZW models (which describe integrable higher spin models) are multicritical points in the space of all spin Hamiltonians. The k=1 WZW model represents a stable fixed point for many theories including WZW models of arbitrary odd k with relevant operators added, generalized Hubbard or Thirring models with an odd number of colors and the O(3) \ensuremath{\sigma} model at topological angle \ensuremath{\theta}=\ensuremath{\pi}.