arXiv:cond-mat/0410118 · Physical Review Letters

Nonperturbative Functional Renormalization Group for Random-Field Models: The Way Out of Dimensional Reduction

Gilles Tarjus, Matthieu Tissier

Published Dec 23, 2004• 87 citations

Abstract

We develop a nonperturbative functional renormalization group approach for the random-field O(N) model that allows us to investigate the ordering transition in any dimension and for any value of N including the Ising case. We show that the failure of dimensional reduction and standard perturbation theory is due to the nonanalytic nature of the zero-temperature fixed point controlling the critical behavior, nonanalyticity, which is associated with the existence of many metastable states. We find that this nonanalyticity leads to critical exponents differing from the dimensional reduction prediction only below a critical dimension dc(N)<6, with dc(N=1)>3.

DOI: 10.1103/physrevlett.93.267008

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arXiv:cond-mat/0410118 · Physical Review Letters

Nonperturbative Functional Renormalization Group for Random-Field Models: The Way Out of Dimensional Reduction

Gilles Tarjus, Matthieu Tissier

Published Dec 23, 2004• 87 citations

arXiv HTML PDF

Abstract

DOI: 10.1103/physrevlett.93.267008

Related papers

Nonperturbative functional renormalization group for random field models and related disordered systems. II. Results for the random field O(N) model

Matthieu Tissier, Gilles Tarjus

We study the critical behavior and phase diagram of the $d$ -dimensional random field $O(N)$ 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 ${d}_{\text{DR}}(N)$ and we provide a description of criticality, ferromagnetic ordering, and quasi-long range order in the whole $(N,d)$ plane. Below ${d}_{\text{DR}}(N)$ , 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 the critical behavior and phase diagram of the $d$-dimensional random field $O(N)$ 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 ${d}_{\text{DR}}(N)$ and we provide a description of criticality, ferromagnetic ordering, and quasi-long range order in the whole $(N,d)$ plane. Below ${d}_{\text{DR}}(N)$, 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.

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Avalanches and perturbation theory in the random-field Ising model

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Abstract Perturbation theory for the random-field Ising model (RFIM) has the infamous attribute that it predicts at all orders a dimensional-reduction property for the critical behavior that turns out to be wrong in low dimensions. Guided by our previous work based on the nonperturbative functional renormalization group (NP-FRG), we show that one can still make some use of the perturbation theory for a finite range of dimensions below the upper critical dimension, d = 6. The new twist is to account for the influence of large-scale zero-temperature events known as avalanches. These avalanches induce nonanalyticities in the field dependence of the correlation functions and renormalized vertices, and we compute in a loop expansion the eigenvalue associated with the corresponding anomalous operator. The outcome confirms the NP-FRG prediction that the dimensional-reduction fixed point correctly describes the dominant critical scaling of the RFIM above some dimension close to 5 but not below.Abstract Perturbation theory for the random-field Ising model (RFIM) has the infamous attribute that it predicts at all orders a dimensional-reduction property for the critical behavior that turns out to be wrong in low dimensions. Guided by our previous work based on the nonperturbative functional renormalization group (NP-FRG), we show that one can still make some use of the perturbation theory for a finite range of dimensions below the upper critical dimension, d = 6. The new twist is to account for the influence of large-scale zero-temperature events known as avalanches. These avalanches induce nonanalyticities in the field dependence of the correlation functions and renormalized vertices, and we compute in a loop expansion the eigenvalue associated with the corresponding anomalous operator. The outcome confirms the NP-FRG prediction that the dimensional-reduction fixed point correctly describes the dominant critical scaling of the RFIM above some dimension close to 5 but not below.

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Dimensional reduction and its breakdown in the driven random-field O(N) model

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The critical behavior of a random-field $\mathbf{O}(N)$ model driven at a uniform velocity is investigated near three dimensions at zero temperature. From intuitive arguments, we predict that the large-scale behavior of the $D$ -dimensional driven random-field $\mathbf{O}(N)$ model is identical to that of the $(D\ensuremath{-}1)$-dimensional pure $\mathbf{O}(N)$ model. This is an analog of the dimensional reduction property of equilibrium cases, which states that the critical exponents of $D$ -dimensional random-field models are identical to those of $(D\ensuremath{-}2)$-dimensional pure models. However, the dimensional reduction property breaks down in low enough dimensions due to the presence of multiple metastable states. By employing the nonperturbative renormalization group approach, we calculate the critical exponents of the driven random-field $\mathbf{O}(N)$ model in the first order of $\ensuremath{\epsilon}=D\ensuremath{-}3$ and determine the range of $N$ in which the dimensional reduction breaks down.The critical behavior of a random-field $\mathbf{O}(N)$ model driven at a uniform velocity is investigated near three dimensions at zero temperature. From intuitive arguments, we predict that the large-scale behavior of the $D$-dimensional driven random-field $\mathbf{O}(N)$ model is identical to that of the $(D\ensuremath{-}1)$-dimensional pure $\mathbf{O}(N)$ model. This is an analog of the dimensional reduction property of equilibrium cases, which states that the critical exponents of $D$-dimensional random-field models are identical to those of $(D\ensuremath{-}2)$-dimensional pure models. However, the dimensional reduction property breaks down in low enough dimensions due to the presence of multiple metastable states. By employing the nonperturbative renormalization group approach, we calculate the critical exponents of the driven random-field $\mathbf{O}(N)$ model in the first order of $\ensuremath{\epsilon}=D\ensuremath{-}3$ and determine the range of $N$ in which the dimensional reduction breaks down.

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We study by the perturbative functional renormalization group (FRG) the random field and random anisotropy $O(N)$ models near $d=4$ , the lower critical dimension of ferromagnetism. The long-distance physics is controlled by zero-temperature fixed points at which the renormalized effective action is nonanalytic. We obtain the beta functions at two-loop order, showing that despite the nonanalytic character of the renormalized effective action, the theory is perturbatively renormalizable at this order. The physical results obtained at the two-loop level, most notably concerning the breakdown of dimensional reduction at the critical point and the stability of quasi-long-range order in $d<4$ , are shown to fit into the picture predicted by our recent nonperturbative FRG approach.We study by the perturbative functional renormalization group (FRG) the random field and random anisotropy $O(N)$ models near $d=4$, the lower critical dimension of ferromagnetism. The long-distance physics is controlled by zero-temperature fixed points at which the renormalized effective action is nonanalytic. We obtain the beta functions at two-loop order, showing that despite the nonanalytic character of the renormalized effective action, the theory is perturbatively renormalizable at this order. The physical results obtained at the two-loop level, most notably concerning the breakdown of dimensional reduction at the critical point and the stability of quasi-long-range order in $d<4$, are shown to fit into the picture predicted by our recent nonperturbative FRG approach.

arXiv:cond-mat/0606698Physical Review BDec 18, 200635 citations

Activated dynamic scaling in the random-field Ising model: A nonperturbative functional renormalization group approach

Ivan Balog, Gilles Tarjus

The random-field Ising model shows an extreme critical slowdown that has been described by activated dynamic scaling: The characteristic time of relaxation toward equilibrium diverges exponentially with the correlation length $ln\ensuremath{\tau}\ensuremath{\sim}{\ensuremath{\xi}}^{\ensuremath{\psi}}/T$, with $\ensuremath{\psi}$ an a priori unknown barrier exponent. Through a nonperturbative functional renormalization group, we show that for spatial dimensions $d$ less than a critical value ${d}_{\text{DR}}\ensuremath{\simeq}5.1$, also associated with dimensional-reduction breakdown, $\ensuremath{\psi}=\ensuremath{\theta}$ with $\ensuremath{\theta}$ the temperature exponent near the zero-temperature fixed point that controls the critical behavior. For $d>{d}_{\text{DR}}$ , on the other hand, $\ensuremath{\psi}=\ensuremath{\theta}\ensuremath{-}2\ensuremath{\lambda}$, where $\ensuremath{\theta}=2$ and $\ensuremath{\lambda}>0$ an additional exponent. At the upper critical dimension $d=6,\ensuremath{\lambda}=1$, so that $\ensuremath{\psi}=0$, and activated scaling gives way to conventional scaling. We give a physical interpretation of the results in terms of collective events in real space, avalanches, and droplets. We also propose a way to check the two regimes by computer simulations of long-range one-dimensional systems.The random-field Ising model shows an extreme critical slowdown that has been described by activated dynamic scaling: The characteristic time of relaxation toward equilibrium diverges exponentially with the correlation length $ln\ensuremath{\tau}\ensuremath{\sim}{\ensuremath{\xi}}^{\ensuremath{\psi}}/T$, with $\ensuremath{\psi}$ an a priori unknown barrier exponent. Through a nonperturbative functional renormalization group, we show that for spatial dimensions $d$ less than a critical value ${d}_{\text{DR}}\ensuremath{\simeq}5.1$, also associated with dimensional-reduction breakdown, $\ensuremath{\psi}=\ensuremath{\theta}$ with $\ensuremath{\theta}$ the temperature exponent near the zero-temperature fixed point that controls the critical behavior. For $d>{d}_{\text{DR}}$, on the other hand, $\ensuremath{\psi}=\ensuremath{\theta}\ensuremath{-}2\ensuremath{\lambda}$, where $\ensuremath{\theta}=2$ and $\ensuremath{\lambda}>0$ an additional exponent. At the upper critical dimension $d=6,\ensuremath{\lambda}=1$, so that $\ensuremath{\psi}=0$, and activated scaling gives way to conventional scaling. We give a physical interpretation of the results in terms of collective events in real space, avalanches, and droplets. We also propose a way to check the two regimes by computer simulations of long-range one-dimensional systems.

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Nonperturbative functional renormalization group for random field models and related disordered systems. IV. Supersymmetry and its spontaneous breaking

Matthieu Tissier, Gilles Tarjus

We apply the nonperturbative functional renormalization group (NP-FRG) in the superfield formalism that we have developed in the preceding paper to study long-standing issues concerning the critical behavior of the random field Ising model. Through the introduction of an appropriate regulator and a supersymmetry-compatible nonperturbative approximation, we are able to follow the supersymmetry, more specifically, the superrotational invariance first unveiled by Parisi and Sourlas [Phys. Rev. Lett. 43, 744 (1979)] and its spontaneous breaking along the RG flow. Breaking occurs below a critical dimension ${d}_{\text{DR}}\ensuremath{\simeq}5.1$, and the supersymmetry-broken fixed point that controls the critical behavior then leads to a breakdown of the ``dimensional-reduction'' property. We solve the NP-FRG flow equations numerically and determine the critical exponents as a function of dimension down to $d\ensuremath{\lesssim}3$, with a good agreement in $d=3$ and 4 with the existing numerical estimates.We apply the nonperturbative functional renormalization group (NP-FRG) in the superfield formalism that we have developed in the preceding paper to study long-standing issues concerning the critical behavior of the random field Ising model. Through the introduction of an appropriate regulator and a supersymmetry-compatible nonperturbative approximation, we are able to follow the supersymmetry, more specifically, the superrotational invariance first unveiled by Parisi and Sourlas [Phys. Rev. Lett. 43, 744 (1979)] and its spontaneous breaking along the RG flow. Breaking occurs below a critical dimension ${d}_{\text{DR}}\ensuremath{\simeq}5.1$, and the supersymmetry-broken fixed point that controls the critical behavior then leads to a breakdown of the ``dimensional-reduction'' property. We solve the NP-FRG flow equations numerically and determine the critical exponents as a function of dimension down to $d\ensuremath{\lesssim}3$, with a good agreement in $d=3$ and 4 with the existing numerical estimates.

arXiv:1110.5500Physical Review BMar 2, 201264 citations

We study the critical behavior and phase diagram of the

d

-dimensional random field

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

{d}_{\text{DR}}(N)

and we provide a description of criticality, ferromagnetic ordering, and quasi-long range order in the whole

(N,d)

plane. Below

{d}_{\text{DR}}(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.

The critical behavior of a random-field

\mathbf{O}(N)

model driven at a uniform velocity is investigated near three dimensions at zero temperature. From intuitive arguments, we predict that the large-scale behavior of the

D

-dimensional driven random-field

\mathbf{O}(N)

model is identical to that of the $(D\ensuremath{-}1)$-dimensional pure

\mathbf{O}(N)

model. This is an analog of the dimensional reduction property of equilibrium cases, which states that the critical exponents of

D

-dimensional random-field models are identical to those of $(D\ensuremath{-}2)$-dimensional pure models. However, the dimensional reduction property breaks down in low enough dimensions due to the presence of multiple metastable states. By employing the nonperturbative renormalization group approach, we calculate the critical exponents of the driven random-field

\mathbf{O}(N)

model in the first order of $\ensuremath{\epsilon}=D\ensuremath{-}3$ and determine the range of

N

in which the dimensional reduction breaks down.

We study by the perturbative functional renormalization group (FRG) the random field and random anisotropy

O(N)

models near

d=4

, the lower critical dimension of ferromagnetism. The long-distance physics is controlled by zero-temperature fixed points at which the renormalized effective action is nonanalytic. We obtain the beta functions at two-loop order, showing that despite the nonanalytic character of the renormalized effective action, the theory is perturbatively renormalizable at this order. The physical results obtained at the two-loop level, most notably concerning the breakdown of dimensional reduction at the critical point and the stability of quasi-long-range order in

d<4

, are shown to fit into the picture predicted by our recent nonperturbative FRG approach.

d

less than a critical value ${d}_{\text{DR}}\ensuremath{\simeq}5.1$, also associated with dimensional-reduction breakdown, $\ensuremath{\psi}=\ensuremath{\theta}$ with $\ensuremath{\theta}$ the temperature exponent near the zero-temperature fixed point that controls the critical behavior. For

d>{d}_{\text{DR}}

, on the other hand, $\ensuremath{\psi}=\ensuremath{\theta}\ensuremath{-}2\ensuremath{\lambda}$, where $\ensuremath{\theta}=2$ and $\ensuremath{\lambda}>0$ an additional exponent. At the upper critical dimension $d=6,\ensuremath{\lambda}=1$, so that $\ensuremath{\psi}=0$, and activated scaling gives way to conventional scaling. We give a physical interpretation of the results in terms of collective events in real space, avalanches, and droplets. We also propose a way to check the two regimes by computer simulations of long-range one-dimensional systems.

d=3

and 4 with the existing numerical estimates.