Critical behaviour of SU(n) quantum chains and topological non-linear sigma-models

Ian K. Affleck

Published Dec 28, 1988• 0 citations

Abstract

The critical behaviour of SU(n) quantum ''spin'' chains, Wess-Zumino-Witten sigma-models and grassmanian sigma-models at topological angle theta = ..pi.. (of possible relevance to the quantum Hall effect) is reexamined. It is argued that an additional Z/sub n/ symmetry is generally necessary to stabilize the massless phase. This symmetry is not present for the sigma-models for n>2 and is only present for certain representations of ''spin'' chains.

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Abstract

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\mathrm{SU}(N)/\mathrm{SO}(N)

sigma models flow to the

\mathrm{SU}{(N)}_{1}

Wess-Zumino-Witten theory, while the $O(2N)/O(N)\ifmmode\times\else\texttimes\fi{}O(N)$ models flow to

{O(2N)}_{1}

(2N

-free Majorana fermions). These models are integrable, and the exact quasiparticle spectra and S matrices are found. One interesting feature is that charges fractionalize when $\ensuremath{\theta}=\ensuremath{\pi}.$ I compute the energy in a background field, and verify that the perturbative expansions for $\ensuremath{\theta}=0$ and \ensuremath{\pi} are the same as they must be. I discuss the flows between the two sequences of models, and also argue that the analogous sigma models with

\mathrm{Sp}(2N)

symmetry, the

\mathrm{Sp}(2N)/U(N)

models, flow to

\mathrm{Sp}{(2N)}_{1}.

Transition between critical antiferromagnetic phases in the

J_1

J_2

spin chain

The

J_1

J_2

spin chain is one of the canonical models of quantum magnetism, and has long been known to host a critical antiferromagnetic phase with power-law decay of spin correlations. We show in this Letter that there are, in fact, \textit{two} distinct critical antiferromagnetic phases, where the roles of the local dimer field and its dual field are interchanged: the `Affleck-Haldane' phase near the Heisenberg point

J_2 = 0

, where the dimer field that parametrises local singlet order is gapless and part of a joint

O(4)

Néel-singlet order parameter; and the `Zirnbauer' phase which appears at sufficiently large ferromagnetic

J_2

, where the dimer field is gapped out and its \textit{dual} field -- the instanton density of the

O(3)

Néel field -- is critical instead. The phases are so-named because each realises one of the competing pictures for how the

O(3)

non-linear sigma model with a topological theta term renormalises to the

\mathfrak{\hat{su}}(2)_1

Wess-Zumino-Witten model. We support these predictions with density matrix renormalisation group calculations.