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Physical Review Letters

Large- n Limit of SU ( n ) Quantum "Spin" Chains

Ian K. Affleck

Published Mar 11, 1985• 128 citations
Publisher

Abstract

A class of $\mathrm{SU}(n)$ antiferromagnetic quantum "spin" chains is solved in the large-$n$ limit. While the low-lying spectrum of the normal spin chain ($n=2$) consists of massless solitons, in the large-$n$ limit the soliton is massive. Furthermore, the models exhibit many features of the $C{P}^{n}$ field theories at nonzero topological charge, which, it was recently argued, they become in the limit of a large representation.

DOI
10.1103/physrevlett.54.966

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A class of SU(n)\mathrm{SU}(n)SU(n) antiferromagnetic quantum "spin" chains is solved in the large-nnn limit. While the low-lying spectrum of the normal spin chain (n=2n=2n=2) consists of massless solitons, in the large-nnn limit the soliton is massive. Furthermore, the models exhibit many features of the CPnC{P}^{n}CPn field theories at nonzero topological charge, which, it was recently argued, they become in the limit of a large representation.
The structure of edge states in general finite antiferromagnetic quantum spin chains with arbitrary spin value S is discussed within the framework of the nonlinear-sigma model (NL\ensuremath{\sigma}M) plus a topological \ensuremath{\theta} term. Based on a large-N theory of SU(N) quantum antiferromagnets and strong-coupling expansion, we argue that edge states with fractionalized spin quantum number S' exist in all spin chains with S\ensuremath{\ge}1, with S'=S/2 for integer spin chains and S'=(S-1/2)/2 for half-integer spin chains.
We review the correspondence between strongly coupled lattice multiflavor Schwinger models and SU(N) antiferromagnetic chains. We show that finding the low lying states of the gauge models is equivalent to solving an SU(N) Heisenberg antiferromagnetic chain. For the two-flavor lattice Schwinger model the massless excitations correspond to gapless states of the Heisenberg chain, while the massive states are created by fermion transport in the ground state of the spin chain. Our analysis shows explicitly how spinons may arise in lattice gauge theories.
We study the spectrum of an integrable antiferromagnetic Hamiltonian of the gl(M|N) spin chain of alternating fundamental and dual representations. After extensive numerical analysis, we identify the vacuum and low lying excitations and with this knowledge perform the continuum limit, while keeping a finite gap. All antiferromagnetic gl(n + N|N)spin chains with n, N > 0 are shown to possess in the continuum limit 2n −2 multiplets of massive particles which scatter with gl(n)Gross-Neveu like S-matrices, meaning that their eigenvalues do not depend on N. We argue that the continuum theory is the gl(M|N) Gross-Neveu model, that is the massive deformation of the $ {{s\hat{l}}}{\left( {M\left| N \right.} \right)_1} $ Wess-Zumino-Witten model. As we can see in the example of gl(2 m|1) spin chains, the full particle spectrum is much richer. Our analysis suggests that for a complete characterization of the latter it is not enough to restrict to large volume calculations, as we do in this work.