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arXiv:1910.08196 · Nuclear Physics B

Generalization of the Haldane conjecture to SU(n) chains

Kyle Wamer, Miklós Lajkó, Frédéric Mila, Ian K. Affleck

Published Jan 21, 2020• 25 citations
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Abstract

Recently, SU(3) chains in the symmetric and self-conjugate representations have been studied using field theory techniques. For certain representations, namely rank-p symmetric ones with p not a multiple of 3, it was argued that the ground state exhibits gapless excitations. For the remaining representations considered, a finite energy gap exists above the ground state. In this paper, we extend these results to SU(n) chains in the symmetric representation. For a rank-p symmetric representation with n and p coprime, we predict gapless excitations above the ground state. If p is a multiple of n, we predict a unique ground state with a finite energy gap. Finally, if p and n have a greatest common divisor 1<q<n, we predict a ground state degeneracy of n/q, with a finite energy gap. To arrive at these results, we derive a non-Lorentz invariant flag manifold sigma model description of the SU(n) chains, and use the renormalization group to show that Lorentz invariance is restored at low energies. We then make use of recently developed anomaly matching conditions for these Lorentz-invariant models. We also review the Lieb-Schultz-Mattis-Affleck theorem, and extend it to SU(n) models with longer range interactions.

DOI
10.1016/j.nuclphysb.2020.114932

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Recently, SU(3) chains in the symmetric and self-conjugate representations have been studied using field theory techniques. For certain representations, namely rank-p symmetric ones with p not a multiple of 3, it was argued that the ground state exhibits gapless excitations. For the remaining representations considered, a finite energy gap exists above the ground state. In this paper, we extend these results to SU(n) chains in the symmetric representation. For a rank-p symmetric representation with n and p coprime, we predict gapless excitations above the ground state. If p is a multiple of n, we predict a unique ground state with a finite energy gap. Finally, if p and n have a greatest common divisor 1<q<n, we predict a ground state degeneracy of n/q, with a finite energy gap. To arrive at these results, we derive a non-Lorentz invariant flag manifold sigma model description of the SU(n) chains, and use the renormalization group to show that Lorentz invariance is restored at low energies. We then make use of recently developed anomaly matching conditions for these Lorentz-invariant models. We also review the Lieb-Schultz-Mattis-Affleck theorem, and extend it to SU(n) models with longer range interactions.
One dimensional SU(n) chains with the same irreducible representation R at each site are considered. We determine which R admit low-energy mappings to a SU(n)/[U(1)]n−1 flag manifold sigma model, and calculate the topological angles for such theories. Generically, these models will have fields with both linear and quadratic dispersion relations; for each R, we determine how many fields of each dispersion type there are. Finally, for purely linearly-dispersing theories, we list the irreps that also possess a Zn symmetry that acts transitively on the SU(n)/[U(1)]n−1 fields. Such SU(n) chains have an 't Hooft anomaly in certain cases, allowing for a generalisation of Haldane's conjecture to these novel representations. In particular, for even n and for representations whose Young tableaux have two rows, of lengths p1 and p2 satisfying p1≠p2, we predict a gapless ground state when p1+p2 is coprime with n. Otherwise, we predict a gapped ground state that necessarily has spontaneously broken symmetry if p1+p2 is not a multiple of n.
Recently, Haldane's conjecture about SU(2) chains has been generalized to SU(n)\mathrm{SU}(n)SU(n) chains in the symmetric representations. For a rank-ppp representation, a gapless phase is predicted when ppp and nnn are coprime; otherwise, a finite energy gap is present above the ground state. In this work we offer an intuitive explanation of this behavior based on fractional topological excitations, which are able to generate a mass gap except when ppp and nnn have no common divisor. This is a generalization of an older work in SU(2), which explains the generation of the Haldane gap in terms of merons in the O(3) nonlinear $\ensuremath{\sigma}$ model.
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.
Motivated by the recent generalization of the Haldane conjecture to SU(3) chains [Lajkó et al., Nucl. Phys. B924, 508 (2017)NUPBBO0550-321310.1016/j.nuclphysb.2017.09.015] according to which a Haldane gap should be present for symmetric representations if the number of boxes in the Young diagram is a multiple of three, we develop a density matrix renormalization group algorithm based on standard Young tableaus to study the model with three boxes directly in the representations of the global SU(3) symmetry. We show that there is a finite gap between the singlet and the symmetric [3 0 0] sector Δ_{[3 0 0]}/J=0.040±0.006 where J is the antiferromagnetic Heisenberg coupling, and we argue on the basis of the structure of the low energy states that this is sufficient to conclude that the spectrum is gapped.
It was recently argued that SU(3) chains in the ppp-box symmetric irreducible representation (irrep) exhibit a ``Haldane gap'' when ppp is a multiple of 3 and are otherwise gapless [Nucl. Phys. B 924, 508 (2017)]. We extend this argument to the self-conjugate irreps of SU(3) with ppp columns of length 2 and ppp columns of length 1 in the Young tableau (p=1p=1p=1 corresponding to the adjoint irrep), arguing that they are always gapped but have spontaneously broken parity symmetry for ppp odd but not even.