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arXiv:cond-mat/0607657 · Physical Review B

Disordered free fermions and the Cardy-Ostlund fixed line at low temperature

Pierre Le Doussal, Grégory Schehr

Published May 1, 2007• 10 citations
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Abstract

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 $T$ is varied. By speculating that this line is identical for all $T$ to the one found in disordered free-fermion models via exact results from ``nearly conformal'' field theory, one would predict $A(T=0)=0$, 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 $A(T=0)$ which compares reasonably well with numerics.

DOI
10.1103/physrevb.75.184401

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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 [u(0)\ensuremath−u(r)]2‾\ensuremath∼A(T)ln2r\overline{[u(0)\ensuremath{-}u(r){]}^{2}}\ensuremath{\sim}A(T){\mathrm{ln}}^{2}r[u(0)\ensuremath−u(r)]2​\ensuremath∼A(T)ln2r 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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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 $\ensuremath{\tau}\ensuremath{\approx}0.5$.
We study the critical behavior and phase diagram of the ddd-dimensional random field O(N)O(N)O(N) model by means of the nonperturbative functional renormalization group approach presented in the preceding paper. We show that the dimensional-reduction predictions, which are obtained from conventional perturbation theory, break down below a critical dimension dDR(N){d}_{\text{DR}}(N)dDR​(N) and we provide a description of criticality, ferromagnetic ordering, and quasi-long range order in the whole (N,d)(N,d)(N,d) plane. Below dDR(N){d}_{\text{DR}}(N)dDR​(N), our formalism gives access to both the typical behavior of the system, which is controlled by zero-temperature fixed points at which the effective action has a nonanalytic field dependence, and to the physics of rare low-energy excitations (``droplets''), which is described at nonzero temperature by the rounding of the nonanalyticity in a thermal boundary layer.
We study two-dimensional triangular elastic lattices in a background of point disorder, excluding dislocations (tethered network). Using both (replica symmetric) static and (equilibrium) dynamic renormalization group (RG) for the corresponding N=2 component model, we find a transition to a glass phase for TTg{\mathrm{T}}_{\mathrm{g}}Tg​, described by a plane of perturbative fixed points. The growth of displacements is found to be asymptotically isotropic with uT2{\mathrm{u}}_{\mathrm{T}}^{2}uT2​\ensuremath{\sim}uL2{\mathrm{u}}_{\mathrm{L}}^{2}uL2​\ensuremath{\sim}A1{\mathrm{A}}_{1}A1​ln 2r^{2}\mathrm{r}2r, with universal subdominant anisotropy uT2{\mathrm{u}}_{\mathrm{T}}^{2}uT2​-uL2{\mathrm{u}}_{\mathrm{L}}^{2}uL2​\ensuremath{\sim}A2{\mathrm{A}}_{2}A2​ln r, where A1{\mathrm{A}}_{1}A1​ and A2{\mathrm{A}}_{2}A2​ depend continuously on temperature and the Poisson ratio \ensuremath{\sigma}. We also obtain the continuously varying dynamical exponent z. For the Cardy-Ostlund N=1 model, a particular case of the above model, we point out a discrepancy in the value of A1{\mathrm{A}}_{1}A1​ with other published results in the literature. We find that our result reconciles the order of magnitude of the RG predictions with the most recent numerical simulations.
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