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arXiv:1204.6221 · Physical Review B

Super-rough phase of the random-phase sine-Gordon model: Two-loop results

Zoran Ristivojević, Pierre Le Doussal, Kay Jörg Wiese

Published Aug 3, 2012• 8 citations
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Abstract

We consider the two-dimensional random-phase sine-Gordon and study the vicinity of its glass transition temperature ${T}_{c}$, in an expansion in small $\ensuremath{\tau}=({T}_{c}\ensuremath{-}T)/{T}_{c}$, where $T$ denotes the temperature. We derive renormalization group equations in cubic order in the anharmonicity, and show that they contain two universal invariants. Using them we obtain that the correlation function in the super-rough phase for temperature $T<{T}_{c}$ behaves at large distances as $\overline{\ensuremath{\langle}{[\ensuremath{\theta}(x)\ensuremath{-}\ensuremath{\theta}(0)]}^{2}\ensuremath{\rangle}}=\mathcal{A}{\mathrm{ln}}^{2}(|x|/a)+\mathcal{O}[\mathrm{ln}(|x|/a)]$, where the amplitude $\mathcal{A}$ is a universal function of temperature $\mathcal{A}=2{\ensuremath{\tau}}^{2}\ensuremath{-}2{\ensuremath{\tau}}^{3}+\mathcal{O}({\ensuremath{\tau}}^{4})$. This result differs at two-loop order, i.e., $\mathcal{O}({\ensuremath{\tau}}^{3})$, from the prediction based on results from the ``nearly conformal'' field theory of a related fermion model. We also obtain the correction-to-scaling exponent.

DOI
10.1103/physrevb.86.054201

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We study both analytically, using the renormalization group (RG) to two loop order, and numerically, using an exact polynomial algorithm, the disorder-induced glass phase of the two-dimensional XYXYXY model with quenched random symmetry-breaking fields and without vortices. In the super-rough glassy phase, i.e., below the critical temperature Tc{T}_{c}Tc​, the disorder and thermally averaged correlation function B(r)B(r)B(r) of the phase field $\ensuremath{\theta}(\mathbf{x})$, $B(r)=\overline{⟨[\ensuremath{\theta}(\mathbf{x})\ensuremath{-}\ensuremath{\theta}(\mathbf{x}+\mathbf{r}){]}^{2}⟩}$ behaves, for $r\ensuremath{\gg}a$, as $B(r)\ensuremath{\simeq}A(\ensuremath{\tau}){ln}^{2}(r/a)$ where r=∣r∣r=|\mathbf{r}|r=∣r∣ and aaa is a microscopic length scale. We derive the RG equations up to cubic order in $\ensuremath{\tau}=({T}_{c}\ensuremath{-}T)/{T}_{c}$ and predict the universal amplitude $A(\ensuremath{\tau})=2{\ensuremath{\tau}}^{2}\ensuremath{-}2{\ensuremath{\tau}}^{3}+\mathcal{O}({\ensuremath{\tau}}^{4})$. The universality of $A(\ensuremath{\tau})$ results from nontrivial cancellations between nonuniversal constants of RG equations. Using an exact polynomial algorithm on an equivalent dimer version of the model we compute $A(\ensuremath{\tau})$ numerically and obtain a remarkable agreement with our analytical prediction, up to $\ensuremath{\tau}\ensuremath{\approx}0.5$.
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At large distances and in the low-temperature phase, the quenched correlation functions in the 2 d random phase sine-Gordon model have been argued to be of the form: , with = ( T − T c ). However, renormalization group computations predict B ≠ 0, while variational approaches (which are supposed to be exact for models with a large number of components) give B = 0. We introduce a large- N version of the random phase sine-Gordon model. Using non-Abelian bosonization and renormalization group techniques, we show that the correlation functions of our models have the above form but with a coefficient B suppressed by a factor 1/ N 3 compared to A .
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