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arXiv:2001.09114 · Physical review. B./Physical review. B

Mass generation by fractional instantons in SU( n ) chains

Kyle Wamer, Ian K. Affleck

Published Jun 17, 2020• 10 citations
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Abstract

Recently, Haldane's conjecture about SU(2) chains has been generalized to $\mathrm{SU}(n)$ chains in the symmetric representations. For a rank-$p$ representation, a gapless phase is predicted when $p$ and $n$ 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 $p$ and $n$ 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.

DOI
10.1103/physrevb.101.245143

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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 \ensuremathσ\ensuremath{\sigma}\ensuremathσ model.
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.
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.
The nature of symmetry-protected topological phases of Heisenberg spin chains in totally symmetric representations of rank NNN of the SU(N\mathrm{SU}(NSU(N) group is investigated through a Majorana fermion study starting from an integrable point. The latter approach generalizes the one pioneered by Tsvelik [Phys. Rev. B 42, 10499 (1990)] to describe the low-energy properties of the Haldane phase of the spin-1 Heisenberg chain from three massive Majorana fermions. We find, for all NNN's, the emergence of a nondegenerate gapped phase with edge states whose topological protection depends on the parity of NNN. Whereas for NNN odd, there is no such protection, the phase with even NNN is shown to be topologically protected. We find that the phase belongs to the same topological class as the phase with edge states living in self-conjugate fully antisymmetric representation of the SU(N)\mathrm{SU}(N)SU(N) group.
There are many interesting parallels between systems of interacting non-Abelian anyons and quantum magnetism occurring in ordinary SU(2) quantum magnets. Here we consider theories of so-called su(2)k{}_{k}k​ anyons, well-known deformations of SU(2), in which only the first k+1k+1k+1 angular momenta of SU(2) occur. In this paper, we discuss in particular anyonic generalizations of ordinary SU(2) spin chains with an emphasis on anyonic spin S=1S=1S=1 chains. We find that the overall phase diagrams for these anyonic spin-1 chains closely mirror the phase diagram of the ordinary bilinear-biquadratic spin-1 chain including anyonic generalizations of the Haldane phase, the AKLT construction, and supersymmetric quantum critical points. A novel feature of the anyonic spin-1 chains is an additional topological symmetry that protects the gapless phases. Distinctions further arise in the form of an even/odd effect in the deformation parameter kkk when considering su(2)k{}_{k}k​ anyonic theories with $k\ensuremath{\ge}5$, as well as for the special case of the su(2)4{}_{4}4​ theory for which the spin-1 representation plays a special role. We also address anyonic generalizations of spin-12\frac{1}{2}21​ chains with a focus on the topological protection provided for their gapless ground states. Finally, we put our results into the context of earlier generalizations of SU(2) quantum spin chains, in particular so-called (fused) Temperley-Lieb chains.
Gapped quantum spin chains with symmetry PSU(N)=SU(N)/ZN\text{PSU}(N)=\text{SU}(N)/{\mathbb{Z}}_{N}PSU(N)=SU(N)/ZN​ are known to possess NNN distinct symmetry protected topological phases. Besides the trivial phase, there are $N\ensuremath{-}1$ Haldane phases which are distinguished by the occurrence of massless boundary spins. We give an expression for a universal AKLT Hamiltonian for an arbitrary symmetry group, including the case where inversion symmetry is broken. Using a diagrammatic method, we construct Hamiltonians for two of the topological classes in SU(N)\text{SU}(N)SU(N) that arise when the physical spin is in the adjoint representation. Since our results are valid for all NNN, we also discuss the large-NNN limit.