arXiv:cond-mat/9611168 · Physical review. B, Condensed matter

Glass phase of two-dimensional triangular elastic lattices with disorder

David Carpentier, Pierre Le Doussal

Published May 1, 1997• 39 citations

Abstract

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 T${\mathrm{T}}_{\mathrm{g}}$, described by a plane of perturbative fixed points. The growth of displacements is found to be asymptotically isotropic with ${\mathrm{u}}_{\mathrm{T}}^{2}$\ensuremath{\sim}${\mathrm{u}}_{\mathrm{L}}^{2}$\ensuremath{\sim}${\mathrm{A}}_{1}$ln $^{2}\mathrm{r}$, with universal subdominant anisotropy ${\mathrm{u}}_{\mathrm{T}}^{2}$-${\mathrm{u}}_{\mathrm{L}}^{2}$\ensuremath{\sim}${\mathrm{A}}_{2}$ln r, where ${\mathrm{A}}_{1}$ and ${\mathrm{A}}_{2}$ 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 ${\mathrm{A}}_{1}$ 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.

DOI: 10.1103/physrevb.55.12128

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arXiv:cond-mat/9611168 · Physical review. B, Condensed matter

Glass phase of two-dimensional triangular elastic lattices with disorder

David Carpentier, Pierre Le Doussal

Published May 1, 1997• 39 citations

arXiv HTML PDF

Abstract

{\mathrm{T}}_{\mathrm{g}}

, described by a plane of perturbative fixed points. The growth of displacements is found to be asymptotically isotropic with

{\mathrm{u}}_{\mathrm{T}}^{2}

\ensuremath{\sim}

{\mathrm{u}}_{\mathrm{L}}^{2}

\ensuremath{\sim}

{\mathrm{A}}_{1}

^{2}\mathrm{r}

, with universal subdominant anisotropy

{\mathrm{u}}_{\mathrm{T}}^{2}

{\mathrm{u}}_{\mathrm{L}}^{2}

\ensuremath{\sim}

{\mathrm{A}}_{2}

ln r, where

{\mathrm{A}}_{1}

and

{\mathrm{A}}_{2}

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

{\mathrm{A}}_{1}

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.

DOI: 10.1103/physrevb.55.12128

Related papers

Application of the renormalization group to phase transitions in disordered systems

G. Grinstein, Alan H. Luther

The critical behavior of spin systems with quenched disorder is studied by renormalization-group methods. For the randomly dilute $m$ -vector model, the $n=0$ limit is used to construct a translationally invariant effective Hamiltonian which describes the original disordered system. This Hamiltonian is analyzed in the $\ensuremath{\epsilon}$ expansion to order ${\ensuremath{\epsilon}}^{2}$. Sharp second-order phase transitions with exponents which do not depend continuously on impurity concentration are predicted. For $m>{m}_{c}\ensuremath{\equiv}4\ensuremath{-}4\ensuremath{\epsilon}+O({\ensuremath{\epsilon}}^{2})$ the isotropic $m$ -component fixed point, which characterizes the critical behavior of the pure system, is stable. For $m<{m}_{c}$ , a new random fixed point becomes stable. The exponents corresponding to this fixed point are $\ensuremath{\eta}=[\frac{(5{m}^{2}\ensuremath{-}8m)}{256{(m\ensuremath{-}1)}^{2}}]{\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$, $\ensuremath{\nu}=\frac{1}{2}+[\frac{3m}{32(m\ensuremath{-}1)}]\ensuremath{\epsilon}+[\frac{m(127{m}^{2}\ensuremath{-}572m\ensuremath{-}32)}{4096{(m\ensuremath{-}1)}^{3}}]{\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$ for $m\ensuremath{\ne}1$, and $\ensuremath{\eta}=\ensuremath{-}\frac{\ensuremath{\epsilon}}{106}+O({\ensuremath{\epsilon}}^{\frac{3}{2}})$, $\ensuremath{\nu}=\frac{1}{2}+\frac{{(\frac{6\ensuremath{\epsilon}}{53})}^{\frac{1}{2}}}{4}+O(\ensuremath{\epsilon})$ for $m=1$ . More general random systems are qualitatively discussed from the effective-Hamiltonian viewpoint.The critical behavior of spin systems with quenched disorder is studied by renormalization-group methods. For the randomly dilute $m$-vector model, the $n=0$ limit is used to construct a translationally invariant effective Hamiltonian which describes the original disordered system. This Hamiltonian is analyzed in the $\ensuremath{\epsilon}$ expansion to order ${\ensuremath{\epsilon}}^{2}$. Sharp second-order phase transitions with exponents which do not depend continuously on impurity concentration are predicted. For $m>{m}_{c}\ensuremath{\equiv}4\ensuremath{-}4\ensuremath{\epsilon}+O({\ensuremath{\epsilon}}^{2})$ the isotropic $m$-component fixed point, which characterizes the critical behavior of the pure system, is stable. For $m<{m}_{c}$, a new random fixed point becomes stable. The exponents corresponding to this fixed point are $\ensuremath{\eta}=[\frac{(5{m}^{2}\ensuremath{-}8m)}{256{(m\ensuremath{-}1)}^{2}}]{\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$, $\ensuremath{\nu}=\frac{1}{2}+[\frac{3m}{32(m\ensuremath{-}1)}]\ensuremath{\epsilon}+[\frac{m(127{m}^{2}\ensuremath{-}572m\ensuremath{-}32)}{4096{(m\ensuremath{-}1)}^{3}}]{\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$ for $m\ensuremath{\ne}1$, and $\ensuremath{\eta}=\ensuremath{-}\frac{\ensuremath{\epsilon}}{106}+O({\ensuremath{\epsilon}}^{\frac{3}{2}})$, $\ensuremath{\nu}=\frac{1}{2}+\frac{{(\frac{6\ensuremath{\epsilon}}{53})}^{\frac{1}{2}}}{4}+O(\ensuremath{\epsilon})$ for $m=1$. More general random systems are qualitatively discussed from the effective-Hamiltonian viewpoint.

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Grégory Schehr, Pierre Le Doussal

Using the renormalization group method we investigate the nonequilibrium relaxation of the (Cardy-Ostlund) 2D random sine-Gordon model, which describes pinned arrays of lines. Its statics exhibit a marginal (theta = 0) glass phase for T < Tg described by a line of fixed points. We obtain the universal scaling functions for two-time dynamical response and correlations near Tg for various initial conditions, as well as the autocorrelation exponent. The fluctuation dissipation ratio is found to be nontrivial and continuously dependent on T.Using the renormalization group method we investigate the nonequilibrium relaxation of the (Cardy-Ostlund) 2D random sine-Gordon model, which describes pinned arrays of lines. Its statics exhibit a marginal (theta = 0) glass phase for T < Tg described by a line of fixed points. We obtain the universal scaling functions for two-time dynamical response and correlations near Tg for various initial conditions, as well as the autocorrelation exponent. The fluctuation dissipation ratio is found to be nontrivial and continuously dependent on T.

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A FUNCTIONAL RENORMALIZATION GROUP APPROACH TO SYSTEMS WITH LONG-RANGE CORRELATED DISORDER

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We studied the statics and dynamics of elastic manifolds in disordered media with long-range correlated disorder using functional renormalization group (FRG). We identified different universality classes and computed the critical exponents and universal amplitudes describing geometric and velocity-force characteristics. In contrast to uncorrelated disorder, the statistical tilt symmetry is broken resulting in a nontrivial response to a transverse tilting force. For instance, the vortex lattice in disordered superconductors shows a new glass phase whose properties interpolate between those of the Bragg and Bose glasses formed by pointlike and columnar disorder, respectively. Whereas there is no response in the Bose glass phase (transverse Meissner effect), the standard linear response expected in the Bragg-glass gets modified to a power law response in the presence of disorder correlations. We also studied the long distance properties of the O(N) spin system with random fields and random anisotropies correlated as 1/x^{d-sigma}. Using FRG we obtained the phase diagram in (d,sigma,N)-parameter space and computed the corresponding critical exponents. We found that below the lower critical dimension 4+sigma, there can exist two different types of quasi-long-range-order with zero order-parameter but infinite correlation length.We studied the statics and dynamics of elastic manifolds in disordered media with long-range correlated disorder using functional renormalization group (FRG). We identified different universality classes and computed the critical exponents and universal amplitudes describing geometric and velocity-force characteristics. In contrast to uncorrelated disorder, the statistical tilt symmetry is broken resulting in a nontrivial response to a transverse tilting force. For instance, the vortex lattice in disordered superconductors shows a new glass phase whose properties interpolate between those of the Bragg and Bose glasses formed by pointlike and columnar disorder, respectively. Whereas there is no response in the Bose glass phase (transverse Meissner effect), the standard linear response expected in the Bragg-glass gets modified to a power law response in the presence of disorder correlations. We also studied the long distance properties of the O(N) spin system with random fields and random anisotropies correlated as 1/x^{d-sigma}. Using FRG we obtained the phase diagram in (d,sigma,N)-parameter space and computed the corresponding critical exponents. We found that below the lower critical dimension 4+sigma, there can exist two different types of quasi-long-range-order with zero order-parameter but infinite correlation length.

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Abstract We recently showed that the two-dimensional Ising spin glass allows for a line of renormalization group fixed points which explains properties observed in numerical studies. We observe that this exact result corresponds to enhancement to a one-generator continuous internal symmetry. This finally explains why no finite temperature transition to a spin glass phase is observed in two dimensions. In more than two dimensions, instead, the continuous symmetry can be broken spontaneously and yields a spin glass order parameter which, for fixed temperature and disorder strength, takes continuous values in an interval. Such a feature is shared by the order parameter of the known mean field solution of the model with infinite-range interactions, which corresponds to infinitely many dimensions.Abstract We recently showed that the two-dimensional Ising spin glass allows for a line of renormalization group fixed points which explains properties observed in numerical studies. We observe that this exact result corresponds to enhancement to a one-generator continuous internal symmetry. This finally explains why no finite temperature transition to a spin glass phase is observed in two dimensions. In more than two dimensions, instead, the continuous symmetry can be broken spontaneously and yields a spin glass order parameter which, for fixed temperature and disorder strength, takes continuous values in an interval. Such a feature is shared by the order parameter of the known mean field solution of the model with infinite-range interactions, which corresponds to infinitely many dimensions.

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Critical behavior of the three-dimensional random-anisotropy Heisenberg model

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Simulation of the zero-temperature behavior of a three-dimensional elastic medium

David McNamara, A. Alan Middleton, Chen Zeng

We have performed numerical simulation of a three-dimensional elastic medium, with scalar displacements, subject to quenched disorder. In the absence of topological defects this system is equivalent to a $(3+1)$ -dimensional interface subject to a periodic pinning potential. We have applied an efficient combinatorial optimization algorithm to generate exact ground states for this interface representation. Our results indicate that this Bragg glass is characterized by power law divergences in the structure factor $S(k)\ensuremath{\sim}{\mathrm{Ak}}^{\ensuremath{-}3}.$ We have found numerically consistent values of the coefficient A for two lattice discretizations of the medium, supporting universality for A in the isotropic systems considered here. We also examine the response of the ground state to the change in boundary conditions that corresponds to introducing a single dislocation loop encircling the system. The rearrangement of the ground state caused by this change is equivalent to the domain wall of elastic deformations which span the dislocation loop. Our results indicate that these domain walls are highly convoluted, with a fractal dimension ${d}_{f}=2.60(5).$ We also discuss the implications of the domain wall energetics for the stability of the Bragg glass phase. Elastic excitations similar to these domain walls arise when the pinning potential is slightly perturbed. As in other disordered systems, perturbations of relative strength \ensuremath{\delta} introduce a new length scale ${L}^{*}\ensuremath{\sim}{\ensuremath{\delta}}^{\ensuremath{-}1/\ensuremath{\zeta}}$ beyond which the perturbed ground state becomes uncorrelated with the reference (unperturbed) ground state. We have performed a scaling analysis of the response of the ground state to the perturbations and obtain $\ensuremath{\zeta}=0.385(40).$ This value is consistent with the scaling relation $\ensuremath{\zeta}{=d}_{f}/2\ensuremath{-}\ensuremath{\theta},$ where \ensuremath{\theta} characterizes the scaling of the energy fluctuations of low energy excitations.We have performed numerical simulation of a three-dimensional elastic medium, with scalar displacements, subject to quenched disorder. In the absence of topological defects this system is equivalent to a $(3+1)$-dimensional interface subject to a periodic pinning potential. We have applied an efficient combinatorial optimization algorithm to generate exact ground states for this interface representation. Our results indicate that this Bragg glass is characterized by power law divergences in the structure factor $S(k)\ensuremath{\sim}{\mathrm{Ak}}^{\ensuremath{-}3}.$ We have found numerically consistent values of the coefficient A for two lattice discretizations of the medium, supporting universality for A in the isotropic systems considered here. We also examine the response of the ground state to the change in boundary conditions that corresponds to introducing a single dislocation loop encircling the system. The rearrangement of the ground state caused by this change is equivalent to the domain wall of elastic deformations which span the dislocation loop. Our results indicate that these domain walls are highly convoluted, with a fractal dimension ${d}_{f}=2.60(5).$ We also discuss the implications of the domain wall energetics for the stability of the Bragg glass phase. Elastic excitations similar to these domain walls arise when the pinning potential is slightly perturbed. As in other disordered systems, perturbations of relative strength \ensuremath{\delta} introduce a new length scale ${L}^{*}\ensuremath{\sim}{\ensuremath{\delta}}^{\ensuremath{-}1/\ensuremath{\zeta}}$ beyond which the perturbed ground state becomes uncorrelated with the reference (unperturbed) ground state. We have performed a scaling analysis of the response of the ground state to the perturbations and obtain $\ensuremath{\zeta}=0.385(40).$ This value is consistent with the scaling relation $\ensuremath{\zeta}{=d}_{f}/2\ensuremath{-}\ensuremath{\theta},$ where \ensuremath{\theta} characterizes the scaling of the energy fluctuations of low energy excitations.

arXiv:cond-mat/9905058Physical review. B, Condensed matterOct 1, 199918 citations

{\mathrm{T}}_{\mathrm{g}}

, described by a plane of perturbative fixed points. The growth of displacements is found to be asymptotically isotropic with

{\mathrm{u}}_{\mathrm{T}}^{2}

\ensuremath{\sim}

{\mathrm{u}}_{\mathrm{L}}^{2}

\ensuremath{\sim}

{\mathrm{A}}_{1}

^{2}\mathrm{r}

, with universal subdominant anisotropy

{\mathrm{u}}_{\mathrm{T}}^{2}

{\mathrm{u}}_{\mathrm{L}}^{2}

\ensuremath{\sim}

{\mathrm{A}}_{2}

ln r, where

{\mathrm{A}}_{1}

and

{\mathrm{A}}_{2}

{\mathrm{A}}_{1}

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.

The critical behavior of spin systems with quenched disorder is studied by renormalization-group methods. For the randomly dilute

m

-vector model, the

n=0

limit is used to construct a translationally invariant effective Hamiltonian which describes the original disordered system. This Hamiltonian is analyzed in the $\ensuremath{\epsilon}$ expansion to order ${\ensuremath{\epsilon}}^{2}$. Sharp second-order phase transitions with exponents which do not depend continuously on impurity concentration are predicted. For $m>{m}_{c}\ensuremath{\equiv}4\ensuremath{-}4\ensuremath{\epsilon}+O({\ensuremath{\epsilon}}^{2})$ the isotropic

m

-component fixed point, which characterizes the critical behavior of the pure system, is stable. For

m<{m}_{c}

, a new random fixed point becomes stable. The exponents corresponding to this fixed point are $\ensuremath{\eta}=[\frac{(5{m}^{2}\ensuremath{-}8m)}{256{(m\ensuremath{-}1)}^{2}}]{\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$, $\ensuremath{\nu}=\frac{1}{2}+[\frac{3m}{32(m\ensuremath{-}1)}]\ensuremath{\epsilon}+[\frac{m(127{m}^{2}\ensuremath{-}572m\ensuremath{-}32)}{4096{(m\ensuremath{-}1)}^{3}}]{\ensuremath{\epsilon}}^{2}+O({\ensuremath{\epsilon}}^{3})$ for $m\ensuremath{\ne}1$, and $\ensuremath{\eta}=\ensuremath{-}\frac{\ensuremath{\epsilon}}{106}+O({\ensuremath{\epsilon}}^{\frac{3}{2}})$, $\ensuremath{\nu}=\frac{1}{2}+\frac{{(\frac{6\ensuremath{\epsilon}}{53})}^{\frac{1}{2}}}{4}+O(\ensuremath{\epsilon})$ for

m=1

. More general random systems are qualitatively discussed from the effective-Hamiltonian viewpoint.

(3+1)

-dimensional interface subject to a periodic pinning potential. We have applied an efficient combinatorial optimization algorithm to generate exact ground states for this interface representation. Our results indicate that this Bragg glass is characterized by power law divergences in the structure factor $S(k)\ensuremath{\sim}{\mathrm{Ak}}^{\ensuremath{-}3}.$ We have found numerically consistent values of the coefficient A for two lattice discretizations of the medium, supporting universality for A in the isotropic systems considered here. We also examine the response of the ground state to the change in boundary conditions that corresponds to introducing a single dislocation loop encircling the system. The rearrangement of the ground state caused by this change is equivalent to the domain wall of elastic deformations which span the dislocation loop. Our results indicate that these domain walls are highly convoluted, with a fractal dimension

{d}_{f}=2.60(5).

We also discuss the implications of the domain wall energetics for the stability of the Bragg glass phase. Elastic excitations similar to these domain walls arise when the pinning potential is slightly perturbed. As in other disordered systems, perturbations of relative strength \ensuremath{\delta} introduce a new length scale ${L}^{*}\ensuremath{\sim}{\ensuremath{\delta}}^{\ensuremath{-}1/\ensuremath{\zeta}}$ beyond which the perturbed ground state becomes uncorrelated with the reference (unperturbed) ground state. We have performed a scaling analysis of the response of the ground state to the perturbations and obtain $\ensuremath{\zeta}=0.385(40).$ This value is consistent with the scaling relation $\ensuremath{\zeta}{=d}_{f}/2\ensuremath{-}\ensuremath{\theta},$ where \ensuremath{\theta} characterizes the scaling of the energy fluctuations of low energy excitations.