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/0002402 · Physical review. B, Condensed matter

Randomly dilute spin models: A six-loop field-theoretic study

Andrea Pelissetto C, Ettore Vicari

Published Sep 1, 2000• 105 citations
arXivHTMLPDF

Abstract

We consider the Ginzburg-Landau $\mathrm{MN}$ model that describes $MN$-vector cubic models with $O(M)$-symmetric couplings. We compute the renormalization-group functions to six-loop order in $d=3.$ We focus on the limit $\stackrel{\ensuremath{\rightarrow}}{N}0$ which describes the critical behavior of an M-vector model in the presence of weak quenched disorder. We perform a detailed analysis of the perturbative series for the random Ising model $(M=1).$ We obtain for the critical exponents: $\ensuremath{\gamma}=1.330(17),\ensuremath{\nu}=0.678(10),\ensuremath{\eta}=0.030(3),\ensuremath{\alpha}=\ensuremath{-}0.034(30),\ensuremath{\beta}=0.349(5),\ensuremath{\omega}=0.25(10).$ For $M>~2$ we show that the $O(M)$ fixed point is stable, in agreement with general nonperturbative arguments, and that no random fixed point exists.

DOI
10.1103/physrevb.62.6393

Related papers

Randomly dilute spin models with cubic symmetry

Pasquale Calabrese, Andrea Pelissetto C, Ettore Vicari

We study the combined effect of cubic anisotropy and quenched uncorrelated impurities on multicomponent spin models. For this purpose, we consider the field-theoretical approach based on the Ginzburg-Landau-Wilson ${\ensuremath{\varphi}}^{4}$ Hamiltonian with cubic-symmetric quartic interactions and quenched randomness coupled to the local energy density. We compute the renormalization-group functions to six loops in the fixed-dimension $(d=3)$ perturbative scheme. The analysis of such high-order series provides an accurate description of the renormalization-group flow. The results are also used to determine the critical behavior of three-dimensional antiferromagnetic three- and four-state Potts models in the presence of quenched impurities.

arXiv:cond-mat/0202292•Physical review. B, Condensed matter•Jan 22, 2003•27 citations

Critical behavior of the weakly disordered Ising model: Six-loop ɛ expansion study

Mikhail Vladimirovich Kompaniets, Andrey Kudlis, Aleksandr Ivanovich Sokolov

The critical behavior of three-dimensional weakly diluted quenched Ising model is examined on the base of six-loop renormalization group expansions obtained within the minimal subtraction scheme in 4-ɛ space dimensions. For this purpose the ϕ^{4} field theory with cubic symmetry was analyzed in the replica limit n→0. Along with renormalization group expansions in terms of renormalized couplings the sqrt[ɛ] expansions of critical exponents are presented. Corresponding numerical estimates for the physical, three-dimensional system are obtained by means of different resummation procedures applied both to the sqrt[ɛ] series and to initial renormalization group expansions. The results given by the latter approach are in a good agreement with their counterparts obtained experimentally and within the Monte Carlo simulations, while resumming of sqrt[ɛ] series themselves turned out to be disappointing.

arXiv:2011.10782•Physical review. E•Feb 19, 2021•9 citations

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.

Physical review. B, Solid state•Feb 1, 1976•529 citations

Cubic N -vector model and randomly dilute Ising model in general dimensions

Kathie E. Newman, Eberhard K. Riedel

Critical properties of models defined by continuous-spin Landau Hamiltonians of cubic symmetry are calculated as functions of spatial dimensionality, $2.8\ensuremath{\le}d\ensuremath{\le}4$, and number of spin components, $N$. The investigation employs the scaling-field method developed by Golner and Riedel for Wilson's exact momentum-space renormalization-group equation. Fixed points studied include the isotropic and decoupled Ising ($\ensuremath{-}2\ensuremath{\le}N<\ensuremath{\infty}$), the face-and corner-ordered cubic ($1\ensuremath{\le}N<\ensuremath{\infty}$), and, via the replica method for $N\ensuremath{\rightarrow}0$, the quenched random Ising fixed point. Variations of $N$ and $d$ are used to link the results to exact results or results from other calculational methods, such as $\ensuremath{\epsilon}$ expansions near two and four dimensions. This establishes the consistency of the calculation for three dimensions. Specifically, truncated sets involving seven (twelve) scaling-field equations are derived for the cubic $N$-vector model. A stable random Ising fixed point is found and shown to be distinct from the cubic fixed point and to connect, as a function of $d$, with the Khmelnitskii ${\ensuremath{\epsilon}}^{\frac{1}{2}}$ fixed point. At $d=3$, the short truncation yields $\ensuremath{\alpha}\ensuremath{\approx}0.11$ for the pure Ising and $\ensuremath{\alpha}\ensuremath{\approx}\ensuremath{-}0.09$ for the random Ising fixed point. A search for a random tricritical fixed point was inconclusive. For the $N$-component cubic model, the spin dimensionality ${N}_{c}$, at which the isotropic and cubic fixed points change stability, is determined as a function of $d$. The results support ${N}_{c}>3$ for three dimensions.

Physical review. B, Condensed matter•Jan 1, 1982•153 citations

Critical behavior of two-dimensional cubic and MN models in the five-loop renormalization group approximation

Pasquale Calabrese, E. V. Orlov, D. V. Pakhnin +1 more

The critical thermodynamics of the two-dimensional $N$-vector cubic and $MN$ models is studied within the field-theoretical renormalization group (RG) approach. The $\ensuremath{\beta}$ functions and critical exponents are calculated in the five-loop approximation and the RG series obtained are resummed using the Borel-Leroy transformation combined with the generalized Pad\'e approximant and conformal mapping techniques. For the cubic model, the RG flows for various $N$ are investigated. For $N=2$ it is found that the continuous line of fixed points running from the $XY$ fixed point to the Ising one is well reproduced by the resummed RG series and an account for the five-loop terms makes the lines of zeros of both $\ensuremath{\beta}$ functions closer to each another. For the cubic model with $N\ensuremath{\geqslant}3$, the five-loop contributions are shown to shift the cubic fixed point, given by the four-loop approximation, towards the Ising fixed point. This confirms the idea that the existence of the cubic fixed point in two dimensions under $N>2$ is an artifact of the perturbative analysis. For the quenched dilute $O(M)$ models ($MN$ models with $N=0$) the results are compatible with a stable pure fixed point for $M\ensuremath{\geqslant}1$. For the $MN$ model with $M,N\ensuremath{\geqslant}2$ all the nonperturbative results are reproduced. In addition a new stable fixed point is found for moderate values of $M$ and $N$.

arXiv:cond-mat/0405432•Physical Review B•Sep 29, 2004•34 citations

Effective and asymptotic critical exponents of a weakly diluted quenched Ising model: Three-dimensional approach versus ɛ expansion

R. Folk, Yurij V. Holovatch, Taras Yavors’kii

We present a field-theoretical treatment of the critical behavior of a three-dimensional weakly diluted quenched Ising model. To this end we analyze in the replica limit $\stackrel{\ensuremath{\rightarrow}}{n}0$ the five-loop renormalization-group functions of the ${\ensuremath{\varphi}}^{4}$ theory with $O(n)$-symmetric and cubic interactions [H. Kleinert and V. Schulte-Frohlinde, Phys. Lett. B 342, 284 (1995)]. The minimal subtraction scheme allows one to develop either the $\sqrt{\ensuremath{\varepsilon}}$-expansion series or to proceed within the three-dimensional approach, performing expansions in terms of renormalized couplings. Doing so, we compare both perturbation approaches and discuss their convergence and possible Borel summability. To study the crossover effect we calculate the effective critical exponents. We report resummed numerical values for the effective and asymptotic critical exponents. The results obtained within the three-dimensional approach agree pretty well with recent Monte Carlo simulations. $\sqrt{\ensuremath{\varepsilon}}$ expansion does not allow reliable estimates for $d=3.$

arXiv:cond-mat/9909121•Physical review. B, Condensed matter•Jun 1, 2000•74 citations
We consider the Ginzburg-Landau MN\mathrm{MN}MN model that describes MNMNMN-vector cubic models with O(M)O(M)O(M)-symmetric couplings. We compute the renormalization-group functions to six-loop order in d=3.d=3.d=3. We focus on the limit N\ensuremath→0\stackrel{\ensuremath{\rightarrow}}{N}0N\ensuremath→​0 which describes the critical behavior of an M-vector model in the presence of weak quenched disorder. We perform a detailed analysis of the perturbative series for the random Ising model (M=1).(M=1).(M=1). We obtain for the critical exponents: \ensuremathγ=1.330(17),\ensuremathν=0.678(10),\ensuremathη=0.030(3),\ensuremathα=\ensuremath−0.034(30),\ensuremathβ=0.349(5),\ensuremathω=0.25(10).\ensuremath{\gamma}=1.330(17),\ensuremath{\nu}=0.678(10),\ensuremath{\eta}=0.030(3),\ensuremath{\alpha}=\ensuremath{-}0.034(30),\ensuremath{\beta}=0.349(5),\ensuremath{\omega}=0.25(10).\ensuremathγ=1.330(17),\ensuremathν=0.678(10),\ensuremathη=0.030(3),\ensuremathα=\ensuremath−0.034(30),\ensuremathβ=0.349(5),\ensuremathω=0.25(10). For M> 2M>~2M> 2 we show that the O(M)O(M)O(M) fixed point is stable, in agreement with general nonperturbative arguments, and that no random fixed point exists.
We study the combined effect of cubic anisotropy and quenched uncorrelated impurities on multicomponent spin models. For this purpose, we consider the field-theoretical approach based on the Ginzburg-Landau-Wilson ${\ensuremath{\varphi}}^{4}$ Hamiltonian with cubic-symmetric quartic interactions and quenched randomness coupled to the local energy density. We compute the renormalization-group functions to six loops in the fixed-dimension (d=3)(d=3)(d=3) perturbative scheme. The analysis of such high-order series provides an accurate description of the renormalization-group flow. The results are also used to determine the critical behavior of three-dimensional antiferromagnetic three- and four-state Potts models in the presence of quenched impurities.
The critical behavior of three-dimensional weakly diluted quenched Ising model is examined on the base of six-loop renormalization group expansions obtained within the minimal subtraction scheme in 4-ɛ space dimensions. For this purpose the ϕ^{4} field theory with cubic symmetry was analyzed in the replica limit n→0. Along with renormalization group expansions in terms of renormalized couplings the sqrt[ɛ] expansions of critical exponents are presented. Corresponding numerical estimates for the physical, three-dimensional system are obtained by means of different resummation procedures applied both to the sqrt[ɛ] series and to initial renormalization group expansions. The results given by the latter approach are in a good agreement with their counterparts obtained experimentally and within the Monte Carlo simulations, while resumming of sqrt[ɛ] series themselves turned out to be disappointing.
The critical behavior of spin systems with quenched disorder is studied by renormalization-group methods. For the randomly dilute mmm-vector model, the n=0n=0n=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 mmm-component fixed point, which characterizes the critical behavior of the pure system, is stable. For m<mcm<{m}_{c}m<mc​, 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=1m=1m=1. More general random systems are qualitatively discussed from the effective-Hamiltonian viewpoint.
Critical properties of models defined by continuous-spin Landau Hamiltonians of cubic symmetry are calculated as functions of spatial dimensionality, $2.8\ensuremath{\le}d\ensuremath{\le}4$, and number of spin components, NNN. The investigation employs the scaling-field method developed by Golner and Riedel for Wilson's exact momentum-space renormalization-group equation. Fixed points studied include the isotropic and decoupled Ising ($\ensuremath{-}2\ensuremath{\le}N<\ensuremath{\infty}$), the face-and corner-ordered cubic ($1\ensuremath{\le}N<\ensuremath{\infty}$), and, via the replica method for $N\ensuremath{\rightarrow}0$, the quenched random Ising fixed point. Variations of NNN and ddd are used to link the results to exact results or results from other calculational methods, such as $\ensuremath{\epsilon}$ expansions near two and four dimensions. This establishes the consistency of the calculation for three dimensions. Specifically, truncated sets involving seven (twelve) scaling-field equations are derived for the cubic NNN-vector model. A stable random Ising fixed point is found and shown to be distinct from the cubic fixed point and to connect, as a function of ddd, with the Khmelnitskii ${\ensuremath{\epsilon}}^{\frac{1}{2}}$ fixed point. At d=3d=3d=3, the short truncation yields $\ensuremath{\alpha}\ensuremath{\approx}0.11$ for the pure Ising and $\ensuremath{\alpha}\ensuremath{\approx}\ensuremath{-}0.09$ for the random Ising fixed point. A search for a random tricritical fixed point was inconclusive. For the NNN-component cubic model, the spin dimensionality Nc{N}_{c}Nc​, at which the isotropic and cubic fixed points change stability, is determined as a function of ddd. The results support Nc>3{N}_{c}>3Nc​>3 for three dimensions.
The critical thermodynamics of the two-dimensional NNN-vector cubic and MNMNMN models is studied within the field-theoretical renormalization group (RG) approach. The $\ensuremath{\beta}$ functions and critical exponents are calculated in the five-loop approximation and the RG series obtained are resummed using the Borel-Leroy transformation combined with the generalized Pad\'e approximant and conformal mapping techniques. For the cubic model, the RG flows for various NNN are investigated. For N=2N=2N=2 it is found that the continuous line of fixed points running from the XYXYXY fixed point to the Ising one is well reproduced by the resummed RG series and an account for the five-loop terms makes the lines of zeros of both $\ensuremath{\beta}$ functions closer to each another. For the cubic model with $N\ensuremath{\geqslant}3$, the five-loop contributions are shown to shift the cubic fixed point, given by the four-loop approximation, towards the Ising fixed point. This confirms the idea that the existence of the cubic fixed point in two dimensions under N>2N>2N>2 is an artifact of the perturbative analysis. For the quenched dilute O(M)O(M)O(M) models (MNMNMN models with N=0N=0N=0) the results are compatible with a stable pure fixed point for $M\ensuremath{\geqslant}1$. For the MNMNMN model with $M,N\ensuremath{\geqslant}2$ all the nonperturbative results are reproduced. In addition a new stable fixed point is found for moderate values of MMM and NNN.
We present a field-theoretical treatment of the critical behavior of a three-dimensional weakly diluted quenched Ising model. To this end we analyze in the replica limit $\stackrel{\ensuremath{\rightarrow}}{n}0$ the five-loop renormalization-group functions of the ${\ensuremath{\varphi}}^{4}$ theory with O(n)O(n)O(n)-symmetric and cubic interactions [H. Kleinert and V. Schulte-Frohlinde, Phys. Lett. B 342, 284 (1995)]. The minimal subtraction scheme allows one to develop either the $\sqrt{\ensuremath{\varepsilon}}$-expansion series or to proceed within the three-dimensional approach, performing expansions in terms of renormalized couplings. Doing so, we compare both perturbation approaches and discuss their convergence and possible Borel summability. To study the crossover effect we calculate the effective critical exponents. We report resummed numerical values for the effective and asymptotic critical exponents. The results obtained within the three-dimensional approach agree pretty well with recent Monte Carlo simulations. $\sqrt{\ensuremath{\varepsilon}}$ expansion does not allow reliable estimates for d=3.d=3.d=3.