Physical review. B, Condensed matter

Scale-invariant quenched disorder and its stability criterion at random critical points

David Andelman, A. Nihat Berker

Published Mar 1, 1984• 104 citations

Abstract

The critical properties of systems with quenched bond disorder are determined from a fixed distribution, under renormalization group, of the random bonds. Full fixed distributions with all moments are obtained numerically by histograms and, to a good approximation, in terms of $\ensuremath{\Gamma}$ distributions. For such systems, the specific-heat exponent $\ensuremath{\alpha}$ does not equal the crossover exponent $\ensuremath{\varphi}$ at random criticality. We derive a new relation between $\ensuremath{\alpha}$ and $\ensuremath{\varphi}$, which invokes characteristics of the fixed distribution. The difference between $\ensuremath{\alpha}$ and $\ensuremath{\varphi}$ is noted for $n$-vector models in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions and for Potts models on hierarchical lattices solved exactly. In general, stable random critical behavior with positive $\ensuremath{\alpha}$ appears to be possible. We develop a general treatment of quenched disorder and illustrate it by calculating specific-heat curves. It is suggested that the critical exponents of the three- and four-state random-bond Potts models in two dimensions are $\ensuremath{\nu}\ensuremath{\simeq}1.06 \mathrm{and} 1.19$.

DOI: 10.1103/physrevb.29.2630

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Critical behavior with axially correlated random bonds

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Quenched bond dilution in two-dimensional Potts models

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Lack of self-averaging in critical disordered systems

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Critical properties of random Potts models

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Critical behavior of the random Potts chain

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Physical review. B, Condensed matter

Scale-invariant quenched disorder and its stability criterion at random critical points

David Andelman, A. Nihat Berker

Published Mar 1, 1984• 104 citations

Publisher

Abstract

\ensuremath{\Gamma}

distributions. For such systems, the specific-heat exponent

\ensuremath{\alpha}

does not equal the crossover exponent

\ensuremath{\varphi}

at random criticality. We derive a new relation between

\ensuremath{\alpha}

and

\ensuremath{\varphi}

, which invokes characteristics of the fixed distribution. The difference between

\ensuremath{\alpha}

and

\ensuremath{\varphi}

is noted for

n

-vector models in

4\ensuremath{-}\ensuremath{\epsilon}

dimensions and for Potts models on hierarchical lattices solved exactly. In general, stable random critical behavior with positive

\ensuremath{\alpha}

appears to be possible. We develop a general treatment of quenched disorder and illustrate it by calculating specific-heat curves. It is suggested that the critical exponents of the three- and four-state random-bond Potts models in two dimensions are

\ensuremath{\nu}\ensuremath{\simeq}1.06 \mathrm{and} 1.19

DOI: 10.1103/physrevb.29.2630

Related papers

Critical behavior with axially correlated random bonds

David Andelman, Amnon Aharony A

Critical properties are studied in systems with quenched bond disorder that is correlated along ${d}_{1}$ of d dimensions. A renormalization-group scheme (based on the Migdal-Kadanoff method) which follows the full distribution of the random bonds and which gives correctly the modified Harris criterion \ensuremath{\varphi}=\ensuremath{\alpha}+ ${d}_{1}$ \ensuremath{\nu} is used. For ${d}_{1}$ , we find fixed distributions at finite temperatures, yielding new ``random'' exponents for various q-state Potts models. For ${d}_{1}$ =d-1 there is no long-range order if there is a finite weight to zero coupling. Otherwise, we find a novel zero-temperature fixed distribution, for which all the moments diverge to infinity with finite ratios among them. This fixed distribution has a magnetic eigenvalue equal to d, indicating a first-order transition in the magnetization and possible related essential singularities. Thus, by analogy, the possibility of a magnetization jump is raised for the McCoy-Wu transition on a square lattice. The results for ${d}_{1}$ =1 are relevant to random quantum systems.Critical properties are studied in systems with quenched bond disorder that is correlated along ${d}_{1}$ of d dimensions. A renormalization-group scheme (based on the Migdal-Kadanoff method) which follows the full distribution of the random bonds and which gives correctly the modified Harris criterion \ensuremath{\varphi}=\ensuremath{\alpha}+${d}_{1}$\ensuremath{\nu} is used. For ${d}_{1}$, we find fixed distributions at finite temperatures, yielding new ``random'' exponents for various q-state Potts models. For ${d}_{1}$=d-1 there is no long-range order if there is a finite weight to zero coupling. Otherwise, we find a novel zero-temperature fixed distribution, for which all the moments diverge to infinity with finite ratios among them. This fixed distribution has a magnetic eigenvalue equal to d, indicating a first-order transition in the magnetization and possible related essential singularities. Thus, by analogy, the possibility of a magnetization jump is raised for the McCoy-Wu transition on a square lattice. The results for ${d}_{1}$=1 are relevant to random quantum systems.

Physical review. B, Condensed matterApr 1, 198531 citations

Exact Results for Quenched Bond Randomness at Criticality

Gesualdo Delfino A

We introduce an exact replica method for the study of critical systems with quenched bond randomness in two dimensions. For the q-state Potts model, we show that a line of renormalization group fixed points interpolates from weak to strong randomness as q-2 grows from small to large values. This theory exhibits a q-independent sector, and allows at the same time for a correlation length exponent which keeps the Ising value and continuously varying magnetization exponent and effective central charge. These findings appear to solve long-standing numerical and theoretical puzzles, and to illustrate the peculiarities which may characterize the conformal field theories of random fixed points.We introduce an exact replica method for the study of critical systems with quenched bond randomness in two dimensions. For the q-state Potts model, we show that a line of renormalization group fixed points interpolates from weak to strong randomness as q-2 grows from small to large values. This theory exhibits a q-independent sector, and allows at the same time for a correlation length exponent which keeps the Ising value and continuously varying magnetization exponent and effective central charge. These findings appear to solve long-standing numerical and theoretical puzzles, and to illustrate the peculiarities which may characterize the conformal field theories of random fixed points.

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Quenched bond dilution in two-dimensional Potts models

Christophe Chatelain, Bertrand Berche, Щур Лев Николаевич

We report a numerical study of the bond-diluted two-dimensional Potts model using transfer-matrix calculations. For different numbers of states per spin, we show that the critical exponents at the random fixed point are the same as in self-dual random-bond cases. In addition, we determine the multifractal spectrum associated with the scaling dimensions of the moments of the spin-spin correlation function in the cylinder geometry. We show that the behaviour is fully compatible with the one observed in the random-bond case, confirming the general picture according to which a unique fixed point describes the critical properties of different classes of disorder: dilution, self-dual binary random bond, self-dual continuous random bond.We report a numerical study of the bond-diluted two-dimensional Potts model using transfer-matrix calculations. For different numbers of states per spin, we show that the critical exponents at the random fixed point are the same as in self-dual random-bond cases. In addition, we determine the multifractal spectrum associated with the scaling dimensions of the moments of the spin-spin correlation function in the cylinder geometry. We show that the behaviour is fully compatible with the one observed in the random-bond case, confirming the general picture according to which a unique fixed point describes the critical properties of different classes of disorder: dilution, self-dual binary random bond, self-dual continuous random bond.

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Lack of self-averaging in critical disordered systems

Shai Wiseman, Eytan Domany

We consider the sample to sample fluctuations that occur in the value of a thermodynamic quantity $P$ in an ensemble of finite systems with quenched disorder, at equilibrium. The variance of $P$ , ${V}_{P}$ , which characterizes these fluctuations is calculated as a function of the systems' linear size $l$ , focusing on the behavior at the critical point. The specific model considered is the bond-disordered Ashkin-Teller model on a square lattice [Phys. Rev. 64, 178 (1943)]. Using extensive Monte Carlo simulations, several bond-disordered Ashkin-Teller models were examined, including the bond-disordered Ising model and the bond-disordered four-state Potts model. It was found that far from criticality all thermodynamic quantities which were examined (energy, magnetization, specific heat, susceptibility) are strongly self-averaging, that is ${V}_{P}\ensuremath{\sim}{l}^{\ensuremath{-}d}$ (where $d=2$ is the dimension). At criticality though, the results indicate that the magnetization $M$ and the susceptibility $\ensuremath{\chi}$ are non-self-averaging, i.e., $\frac{{V}_{\ensuremath{\chi}}}{{\ensuremath{\chi}}^{2}}, \frac{{V}_{M}}{{M}^{2}}\ensuremath{\nrightarrow}0$. The energy $E$ at criticality is clearly weakly self-averaging, that is ${V}_{E}\ensuremath{\sim}{l}^{\ensuremath{-}{y}_{v}}$ with $0<{y}_{v}<d$ . Less conclusively, and possibly only as a transient behavior, the specific heat too is found to be weakly self-averaging. A phenomenological theory of finite size scaling for disordered systems is developed, based on physical considerations similar to those leading to the Harris criterion. Its main prediction is that when the specific heat exponent $\ensuremath{\alpha}<0$ ($\ensuremath{\alpha}$ of the disordered model) then, for a quantity $P$ which scales as ${l}^{\ensuremath{\rho}}$ at criticality, its variance ${V}_{P}$ will scale asymptotically as ${l}^{2\ensuremath{\rho}+\frac{\ensuremath{\alpha}}{\ensuremath{\nu}}}$. The theory is not applicable in the asymptotic limit ($l\ensuremath{\rightarrow}\ensuremath{\infty}$) to the bond-disordered Ashkin-Teller model where $\frac{\ensuremath{\alpha}}{\ensuremath{\nu}}={0}_{+}$. Nonetheless in the accessible range of lattice sizes we found very good agreement between the theory and the data for ${V}_{\ensuremath{\chi}}$ and ${V}_{E}$ . The theory may also be compatible with the data for the variance of the magnetization ${V}_{M}$ and the variance of the specific heat ${V}_{C}$ , but evidence for this is less convincing.We consider the sample to sample fluctuations that occur in the value of a thermodynamic quantity $P$ in an ensemble of finite systems with quenched disorder, at equilibrium. The variance of $P$, ${V}_{P}$, which characterizes these fluctuations is calculated as a function of the systems' linear size $l$, focusing on the behavior at the critical point. The specific model considered is the bond-disordered Ashkin-Teller model on a square lattice [Phys. Rev. 64, 178 (1943)]. Using extensive Monte Carlo simulations, several bond-disordered Ashkin-Teller models were examined, including the bond-disordered Ising model and the bond-disordered four-state Potts model. It was found that far from criticality all thermodynamic quantities which were examined (energy, magnetization, specific heat, susceptibility) are strongly self-averaging, that is ${V}_{P}\ensuremath{\sim}{l}^{\ensuremath{-}d}$ (where $d=2$ is the dimension). At criticality though, the results indicate that the magnetization $M$ and the susceptibility $\ensuremath{\chi}$ are non-self-averaging, i.e., $\frac{{V}_{\ensuremath{\chi}}}{{\ensuremath{\chi}}^{2}}, \frac{{V}_{M}}{{M}^{2}}\ensuremath{\nrightarrow}0$. The energy $E$ at criticality is clearly weakly self-averaging, that is ${V}_{E}\ensuremath{\sim}{l}^{\ensuremath{-}{y}_{v}}$ with $0<{y}_{v}<d$. Less conclusively, and possibly only as a transient behavior, the specific heat too is found to be weakly self-averaging. A phenomenological theory of finite size scaling for disordered systems is developed, based on physical considerations similar to those leading to the Harris criterion. Its main prediction is that when the specific heat exponent $\ensuremath{\alpha}<0$ ($\ensuremath{\alpha}$ of the disordered model) then, for a quantity $P$ which scales as ${l}^{\ensuremath{\rho}}$ at criticality, its variance ${V}_{P}$ will scale asymptotically as ${l}^{2\ensuremath{\rho}+\frac{\ensuremath{\alpha}}{\ensuremath{\nu}}}$. The theory is not applicable in the asymptotic limit ($l\ensuremath{\rightarrow}\ensuremath{\infty}$) to the bond-disordered Ashkin-Teller model where $\frac{\ensuremath{\alpha}}{\ensuremath{\nu}}={0}_{+}$. Nonetheless in the accessible range of lattice sizes we found very good agreement between the theory and the data for ${V}_{\ensuremath{\chi}}$ and ${V}_{E}$. The theory may also be compatible with the data for the variance of the magnetization ${V}_{M}$ and the variance of the specific heat ${V}_{C}$, but evidence for this is less convincing.

arXiv:cond-mat/9506101Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topicsOct 1, 1995132 citations

Critical properties of random Potts models

Wolfgang Kinzel, Eytan Domany

The critical properties of Potts models with random bonds are considered in two dimensions. A position-space renormalization-group procedure, based on the Migdal-Kadanoff method, is developed. While all previous position-space calculations satisfied the Harris criterion and the resulting scaling relation only approximately, we found conditions under which these relations are exactly satisfied, and constructed our renormalization-group procedure accordingly. Numerical results for phase diagrams and thermodynamic functions for various random-bond Potts models are presented. In addition, some exact results obtained using a duality transformation, as well as an heuristic derivation of scaling properties that correspond to the percolation problem are given.The critical properties of Potts models with random bonds are considered in two dimensions. A position-space renormalization-group procedure, based on the Migdal-Kadanoff method, is developed. While all previous position-space calculations satisfied the Harris criterion and the resulting scaling relation only approximately, we found conditions under which these relations are exactly satisfied, and constructed our renormalization-group procedure accordingly. Numerical results for phase diagrams and thermodynamic functions for various random-bond Potts models are presented. In addition, some exact results obtained using a duality transformation, as well as an heuristic derivation of scaling properties that correspond to the percolation problem are given.

Physical review. B, Condensed matterApr 1, 1981131 citations

Critical behavior of the random Potts chain

Enrico Carlon, Christophe Chatelain, Bertrand Berche

We study the critical behavior of the random q-state Potts quantum chain by density-matrix renormalization techniques. Critical exponents are calculated by scaling analysis of finite lattice data of short chains $(L<~16)$ averaging over all possible realizations of disorder configurations chosen according to a binary distribution. Our numerical results show that the critical properties of the model are independent of q in agreement with a renormalization group analysis of Senthil and Majumdar [Phys. Rev. Lett. 76, 3001 (1996)]. We show how an accurate analysis of moments of the distribution of magnetizations allows a precise determination of critical exponents, circumventing some problems related to binary disorder. Multiscaling properties of the model and dynamical correlation functions are also investigated.We study the critical behavior of the random q-state Potts quantum chain by density-matrix renormalization techniques. Critical exponents are calculated by scaling analysis of finite lattice data of short chains $(L<~16)$ averaging over all possible realizations of disorder configurations chosen according to a binary distribution. Our numerical results show that the critical properties of the model are independent of q in agreement with a renormalization group analysis of Senthil and Majumdar [Phys. Rev. Lett. 76, 3001 (1996)]. We show how an accurate analysis of moments of the distribution of magnetizations allows a precise determination of critical exponents, circumventing some problems related to binary disorder. Multiscaling properties of the model and dynamical correlation functions are also investigated.

arXiv:cond-mat/9906059Physical review. B, Condensed matterNov 1, 199910 citations

\ensuremath{\Gamma}

distributions. For such systems, the specific-heat exponent

\ensuremath{\alpha}

does not equal the crossover exponent

\ensuremath{\varphi}

at random criticality. We derive a new relation between

\ensuremath{\alpha}

and

\ensuremath{\varphi}

, which invokes characteristics of the fixed distribution. The difference between

\ensuremath{\alpha}

and

\ensuremath{\varphi}

is noted for

n

-vector models in

4\ensuremath{-}\ensuremath{\epsilon}

dimensions and for Potts models on hierarchical lattices solved exactly. In general, stable random critical behavior with positive

\ensuremath{\alpha}

\ensuremath{\nu}\ensuremath{\simeq}1.06 \mathrm{and} 1.19

Critical properties are studied in systems with quenched bond disorder that is correlated along

{d}_{1}

of d dimensions. A renormalization-group scheme (based on the Migdal-Kadanoff method) which follows the full distribution of the random bonds and which gives correctly the modified Harris criterion \ensuremath{\varphi}=\ensuremath{\alpha}+

{d}_{1}

\ensuremath{\nu} is used. For

{d}_{1}

, we find fixed distributions at finite temperatures, yielding new ``random'' exponents for various q-state Potts models. For

{d}_{1}

=d-1 there is no long-range order if there is a finite weight to zero coupling. Otherwise, we find a novel zero-temperature fixed distribution, for which all the moments diverge to infinity with finite ratios among them. This fixed distribution has a magnetic eigenvalue equal to d, indicating a first-order transition in the magnetization and possible related essential singularities. Thus, by analogy, the possibility of a magnetization jump is raised for the McCoy-Wu transition on a square lattice. The results for

{d}_{1}

=1 are relevant to random quantum systems.

We consider the sample to sample fluctuations that occur in the value of a thermodynamic quantity

P

in an ensemble of finite systems with quenched disorder, at equilibrium. The variance of

P

{V}_{P}

, which characterizes these fluctuations is calculated as a function of the systems' linear size

l

, focusing on the behavior at the critical point. The specific model considered is the bond-disordered Ashkin-Teller model on a square lattice [Phys. Rev. 64, 178 (1943)]. Using extensive Monte Carlo simulations, several bond-disordered Ashkin-Teller models were examined, including the bond-disordered Ising model and the bond-disordered four-state Potts model. It was found that far from criticality all thermodynamic quantities which were examined (energy, magnetization, specific heat, susceptibility) are strongly self-averaging, that is ${V}_{P}\ensuremath{\sim}{l}^{\ensuremath{-}d}$ (where

d=2

is the dimension). At criticality though, the results indicate that the magnetization

M

and the susceptibility $\ensuremath{\chi}$ are non-self-averaging, i.e., $\frac{{V}_{\ensuremath{\chi}}}{{\ensuremath{\chi}}^{2}}, \frac{{V}_{M}}{{M}^{2}}\ensuremath{\nrightarrow}0$. The energy

E

at criticality is clearly weakly self-averaging, that is ${V}_{E}\ensuremath{\sim}{l}^{\ensuremath{-}{y}_{v}}$ with

0<{y}_{v}<d

. Less conclusively, and possibly only as a transient behavior, the specific heat too is found to be weakly self-averaging. A phenomenological theory of finite size scaling for disordered systems is developed, based on physical considerations similar to those leading to the Harris criterion. Its main prediction is that when the specific heat exponent $\ensuremath{\alpha}<0$ ($\ensuremath{\alpha}$ of the disordered model) then, for a quantity

P

which scales as ${l}^{\ensuremath{\rho}}$ at criticality, its variance

{V}_{P}

will scale asymptotically as ${l}^{2\ensuremath{\rho}+\frac{\ensuremath{\alpha}}{\ensuremath{\nu}}}$. The theory is not applicable in the asymptotic limit ($l\ensuremath{\rightarrow}\ensuremath{\infty}$) to the bond-disordered Ashkin-Teller model where $\frac{\ensuremath{\alpha}}{\ensuremath{\nu}}={0}_{+}$. Nonetheless in the accessible range of lattice sizes we found very good agreement between the theory and the data for ${V}_{\ensuremath{\chi}}$ and

{V}_{E}

. The theory may also be compatible with the data for the variance of the magnetization

{V}_{M}

and the variance of the specific heat

{V}_{C}

, but evidence for this is less convincing.

(L<~16)

averaging over all possible realizations of disorder configurations chosen according to a binary distribution. Our numerical results show that the critical properties of the model are independent of q in agreement with a renormalization group analysis of Senthil and Majumdar [Phys. Rev. Lett. 76, 3001 (1996)]. We show how an accurate analysis of moments of the distribution of magnetizations allows a precise determination of critical exponents, circumventing some problems related to binary disorder. Multiscaling properties of the model and dynamical correlation functions are also investigated.