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arXiv:cond-mat/0006360

Critical points in two-dimensional replica sigma models

Paul Fendley

Published Jun 23, 2000• 1 citation
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Abstract

I survey the kinds of critical behavior believed to be exhibited in two-dimensional disordered systems. I review the different replica sigma models used to describe the low-energy physics, and discuss how critical points appear because of WZW and theta terms.
DOI
10.48550/arxiv.cond-mat/0006360

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The low-energy limits of models with disorder are frequently described by $\ensuremath{\sigma}$ models. In two dimensions, most $\ensuremath{\sigma}$ models admit either a Wess-Zumino-Witten term or a $\ensuremath{\theta}$ term. When such a term is present the model can have a stable critical point with gapless excitations. We describe how such a critical point appears, in particular in two-dimensional superconductors with disorder. The presence of such terms is required by the underlying (anomalous) symmetries of the original electron model. This indicates that the usual symmetry classes of disordered systems in two dimensions can be further refined. Conversely, our results also indicate that models previously thought to be in different universality classes are in fact the same once the appropriate extra terms are included.
We suggest the possibility that the two-dimensional SU(2)k\mathrm{SU}{(2)}_{k}SU(2)k​ Wess-Zumino-Witten (WZW) theory, which has global SO(4)\mathrm{SO}(4)SO(4) symmetry, can be continued to $2+\ensuremath{\epsilon}$ dimensions by enlarging the symmetry to $\mathrm{SO}(4+\ensuremath{\epsilon})$. This is motivated by the three-dimensional sigma model with SO(5)\mathrm{SO}(5)SO(5) symmetry and a WZW term, which is relevant to deconfined criticality. If such a continuation exists, the structure of the renormalization group flows at small $\ensuremath{\epsilon}$ may be fixed by assuming analyticity in $\ensuremath{\epsilon}$. This leads to the conjecture that the WZW fixed point annihilates with a new, unstable fixed point at a critical dimensionality dc>2{d}_{c}>2dc​>2. We suggest that dc<3{d}_{c}<3dc​<3 for all kkk, and we compute dc{d}_{c}dc​ in the limit of large kkk. The flows support the conjecture that the deconfined phase transition in SU(2)\mathrm{SU}(2)SU(2) magnets is a ``pseudocritical'' point with approximate SO(5)\mathrm{SO}(5)SO(5), controlled by a fixed point slightly outside the physical parameter space.