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arXiv:1204.5685 · Physical Review Letters

Super-Rough Glassy Phase of the Random Field XY Model in Two Dimensions

Anthony Perret, Zoran Ristivojević, Pierre Le Doussal, Grégory Schehr, Kay Jörg Wiese

Published Oct 10, 2012• 14 citations
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Abstract

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 $XY$ model with quenched random symmetry-breaking fields and without vortices. In the super-rough glassy phase, i.e., below the critical temperature ${T}_{c}$, the disorder and thermally averaged correlation function $B(r)$ 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 $r=|\mathbf{r}|$ and $a$ 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 $\ensuremath{\tau}\ensuremath{\approx}0.5$.

DOI
10.1103/physrevlett.109.157205

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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 \ensuremathθ(x)\ensuremath{\theta}(\mathbf{x})\ensuremathθ(x), B(r)=⟨[\ensuremathθ(x)\ensuremath−\ensuremathθ(x+r)]2⟩‾B(r)=\overline{⟨[\ensuremath{\theta}(\mathbf{x})\ensuremath{-}\ensuremath{\theta}(\mathbf{x}+\mathbf{r}){]}^{2}⟩}B(r)=⟨[\ensuremathθ(x)\ensuremath−\ensuremathθ(x+r)]2⟩​ behaves, for r\ensuremath≫ar\ensuremath{\gg}ar\ensuremath≫a, as B(r)\ensuremath≃A(\ensuremathτ)ln2(r/a)B(r)\ensuremath{\simeq}A(\ensuremath{\tau}){ln}^{2}(r/a)B(r)\ensuremath≃A(\ensuremathτ)ln2(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 \ensuremathτ=(Tc\ensuremath−T)/Tc\ensuremath{\tau}=({T}_{c}\ensuremath{-}T)/{T}_{c}\ensuremathτ=(Tc​\ensuremath−T)/Tc​ and predict the universal amplitude A(\ensuremathτ)=2\ensuremathτ2\ensuremath−2\ensuremathτ3+O(\ensuremathτ4)A(\ensuremath{\tau})=2{\ensuremath{\tau}}^{2}\ensuremath{-}2{\ensuremath{\tau}}^{3}+\mathcal{O}({\ensuremath{\tau}}^{4})A(\ensuremathτ)=2\ensuremathτ2\ensuremath−2\ensuremathτ3+O(\ensuremathτ4). The universality of A(\ensuremathτ)A(\ensuremath{\tau})A(\ensuremathτ) 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(\ensuremathτ)A(\ensuremath{\tau})A(\ensuremathτ) numerically and obtain a remarkable agreement with our analytical prediction, up to \ensuremathτ\ensuremath≈0.5\ensuremath{\tau}\ensuremath{\approx}0.5\ensuremathτ\ensuremath≈0.5.
We consider the two-dimensional random-phase sine-Gordon and study the vicinity of its glass transition temperature Tc{T}_{c}Tc​, in an expansion in small $\ensuremath{\tau}=({T}_{c}\ensuremath{-}T)/{T}_{c}$, where TTT 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<TcT<{T}_{c}T<Tc​ 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 A\mathcal{A}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.
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The XY\mathrm{XY}XY model with quenched random disorder is studied by a zero-temperature domain wall renormalization group method in two dimensions (2D) and three dimensions (3D). Instead of the usual phase representation we use the charge (vortex) representation to compute the domain wall, or defect, energy. For the gauge glass corresponding to the maximum disorder we reconfirm earlier predictions that there is no ordered phase in 2D but an ordered phase can exist in 3D at low temperature. However, our simulations yield spin stiffness exponents ${\ensuremath{\theta}}_{s}\ensuremath{\approx}\ensuremath{-}0.36$ in 2D and ${\ensuremath{\theta}}_{s}\ensuremath{\approx}+0.31$ in 3D, which are considerably larger than previous estimates and strongly suggest that the lower critical dimension is less than three. For the $\ifmmode\pm\else\textpm\fi{}J \mathrm{XY}$ spin glass in 3D, we obtain a spin stiffness exponent ${\ensuremath{\theta}}_{s}\ensuremath{\approx}+0.10$ which supports the existence of spin glass order at finite temperature in contrast with previous estimates which obtain ${\ensuremath{\theta}}_{s}<0.$ Our method also allows us to study renormalization group flows of both the coupling constant and the disorder strength with a length scale L. Our results are consistent with recent analytic and numerical studies suggesting the absence of a reentrant transition in 2D at low temperature. Some possible consequences and connections with real vortex systems are discussed.
Using a functional renormalization group (RG) method, we reexamine the glass phase of the two-dimensional random-field sine Gordon model. It is described by a line of fixed points with a super-roughening amplitude $\overline{[u(0)\ensuremath{-}u(r){]}^{2}}\ensuremath{\sim}A(T){\mathrm{ln}}^{2}r$ as temperature TTT is varied. By speculating that this line is identical for all TTT to the one found in disordered free-fermion models via exact results from ``nearly conformal'' field theory, one would predict A(T=0)=0A(T=0)=0A(T=0)=0, contradicting numerics. We point out that this result may be related to failure of dimensional reduction, and that a functional RG method incorporating higher harmonics and nonanalytic operators predicts a nonzero A(T=0)A(T=0)A(T=0) which compares reasonably well with numerics.
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