Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/0606698 · Physical Review B

Two-loop functional renormalization group of the random field and random anisotropy O(N) models

Matthieu Tissier, Gilles Tarjus

Published Dec 18, 2006• 35 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.74.214419

Related papers

Two-loop Functional Renormalization Group of the Random Field and Random Anisotropy O(N) Models

Matthieu Tissier, Gilles Tarjus

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 2-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 2-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 non-perturbative FRG approach.

•Jul 20, 2006•0 citations

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.

arXiv:0712.3556•Physical Review B•Jul 9, 2008•66 citations

One-loop functional renormalization group study for the dimensional reduction and its breakdown in the long-range random field O( N ) spin model near lower critical dimension

Yoshinori Sakamoto

We consider the random-field $\mathrm{O}(N$) spin model with long-range exchange interactions which decay with distance $r$ between spins as ${r}^{\ensuremath{-}d\ensuremath{-}\ensuremath{\sigma}}$ and/or random fields which correlate with distance $r$ as ${r}^{\ensuremath{-}d+\ensuremath{\rho}}$, and reexamine the critical phenomena near the lower critical dimension by use of the perturbative functional renormalization group. We compute the analytic fixed points in the one-loop beta functions, and study their stability. We also calculate the critical exponents at the analytical fixed points. We show that the analytic fixed point which governs the phase transition in the system with the long-range correlations of random fields can be destabilized by the nonanalytic perturbation in both cases where the exchange interactions between spins are short ranged and long ranged. For the system with the long-range exchange interactions and uncorrelated random fields, we show that the $d\ensuremath{\rightarrow}d\ensuremath{-}\ensuremath{\sigma}$ dimensional reduction at the leading order of the $d\ensuremath{-}2\ensuremath{\sigma}$ expansion holds only for $N>2(4+3\sqrt{3})\ensuremath{\simeq}18.3923\dots{}$. Our investigation into the system with the long-range exchange interactions and uncorrelated random fields also gives the value of the boundary between critical behaviors in systems with long-range and short-range exchange interactions, which is identical to that predicted by Sak [Phys. Rev. B 8, 281 (1973)]. For the system with the long-range exchange interactions and the long-range correlated random fields, we show that the $d\ensuremath{\rightarrow}d\ensuremath{-}\ensuremath{\sigma}\ensuremath{-}\ensuremath{\rho}$ dimensional reduction does not hold within the present framework, as far as $N$ is finite.

arXiv:1902.10510•Physical review. B./Physical review. B•Jul 12, 2019•2 citations

Unified Picture of Ferromagnetism, Quasi-Long-Range Order, and Criticality in Random-Field Models

Matthieu Tissier, Gilles Tarjus

By applying the recently developed nonperturbative functional renormalization group (FRG) approach, we study the interplay between ferromagnetism, quasi-long-range order (QLRO), and criticality in the d-dimensional random-field O(N) model in the whole (N, d) diagram. Even though the "dimensional reduction" property breaks down below some critical line, the topology of the phase diagram is found similar to that of the pure O(N) model, with, however, no equivalent of the Kosterlitz-Thouless transition. In addition, we obtain that QLRO, namely, a topologically ordered "Bragg glass" phase, is absent in the 3-dimensional random-field XY model. The nonperturbative results are supplemented by a perturbative FRG analysis to two loops around d = 4.

arXiv:cond-mat/0511096•Physical Review Letters•Mar 1, 2006•81 citations

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

Gilles Tarjus, Matthieu Tissier

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.

arXiv:cond-mat/0410118•Physical Review Letters•Dec 23, 2004•87 citations

Avalanches and perturbation theory in the random-field Ising model

Gilles Tarjus, Matthieu Tissier

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.

arXiv:1510.01718•Journal of Statistical Mechanics Theory and Experiment•Feb 1, 2016•9 citations
We study by the perturbative functional renormalization group (FRG) the random field and random anisotropy O(N)O(N)O(N) models near d=4d=4d=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<4d<4d<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)O(N)O(N) models near d=4d=4d=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 2-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 2-loop level, most notably concerning the breakdown of dimensional reduction at the critical point and the stability of quasi-long range order in d<4d<4d<4, are shown to fit into the picture predicted by our recent non-perturbative FRG approach.
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 consider the random-field O(N\mathrm{O}(NO(N) spin model with long-range exchange interactions which decay with distance rrr between spins as ${r}^{\ensuremath{-}d\ensuremath{-}\ensuremath{\sigma}}$ and/or random fields which correlate with distance rrr as ${r}^{\ensuremath{-}d+\ensuremath{\rho}}$, and reexamine the critical phenomena near the lower critical dimension by use of the perturbative functional renormalization group. We compute the analytic fixed points in the one-loop beta functions, and study their stability. We also calculate the critical exponents at the analytical fixed points. We show that the analytic fixed point which governs the phase transition in the system with the long-range correlations of random fields can be destabilized by the nonanalytic perturbation in both cases where the exchange interactions between spins are short ranged and long ranged. For the system with the long-range exchange interactions and uncorrelated random fields, we show that the $d\ensuremath{\rightarrow}d\ensuremath{-}\ensuremath{\sigma}$ dimensional reduction at the leading order of the $d\ensuremath{-}2\ensuremath{\sigma}$ expansion holds only for $N>2(4+3\sqrt{3})\ensuremath{\simeq}18.3923\dots{}$. Our investigation into the system with the long-range exchange interactions and uncorrelated random fields also gives the value of the boundary between critical behaviors in systems with long-range and short-range exchange interactions, which is identical to that predicted by Sak [Phys. Rev. B 8, 281 (1973)]. For the system with the long-range exchange interactions and the long-range correlated random fields, we show that the $d\ensuremath{\rightarrow}d\ensuremath{-}\ensuremath{\sigma}\ensuremath{-}\ensuremath{\rho}$ dimensional reduction does not hold within the present framework, as far as NNN is finite.
By applying the recently developed nonperturbative functional renormalization group (FRG) approach, we study the interplay between ferromagnetism, quasi-long-range order (QLRO), and criticality in the d-dimensional random-field O(N) model in the whole (N, d) diagram. Even though the "dimensional reduction" property breaks down below some critical line, the topology of the phase diagram is found similar to that of the pure O(N) model, with, however, no equivalent of the Kosterlitz-Thouless transition. In addition, we obtain that QLRO, namely, a topologically ordered "Bragg glass" phase, is absent in the 3-dimensional random-field XY model. The nonperturbative results are supplemented by a perturbative FRG analysis to two loops around d = 4.
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.
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.