Critical points of higher order can exist in complex magnetic and fluid systems. By definition, at a critical point of order $\mathcal{O}$ , $\mathcal{O}$ phases become identical simultaneously. Here the Wilson renormalization-group method is generalized from ordinary critical points ( $\mathcal{O}=2$ ) and Gaussian tricritical points ( $\mathcal{O}=3$ ) to critical points of arbitrary order $\mathcal{O}$ . An expansion scheme in ${\ensuremath{\epsilon}}_{\mathcal{O}}\ensuremath{\equiv}\frac{2\mathcal{O}}{(\mathcal{O}\ensuremath{-}1)\ensuremath{-}d}$ is proposed. The nontrivial fixed points and the critical exponents for $\mathcal{O}=3,4$ are calculated to order ${\ensuremath{\epsilon}}_{\mathcal{O}}$. We present the relevant scaling fields and densities for $\mathcal{O}=3$ , and, in an appendix, justify the validity of the approximate recursion relation to order ${\ensuremath{\epsilon}}_{\mathcal{O}}$.Critical points of higher order can exist in complex magnetic and fluid systems. By definition, at a critical point of order $\mathcal{O}$, $\mathcal{O}$ phases become identical simultaneously. Here the Wilson renormalization-group method is generalized from ordinary critical points ($\mathcal{O}=2$) and Gaussian tricritical points ($\mathcal{O}=3$) to critical points of arbitrary order $\mathcal{O}$. An expansion scheme in ${\ensuremath{\epsilon}}_{\mathcal{O}}\ensuremath{\equiv}\frac{2\mathcal{O}}{(\mathcal{O}\ensuremath{-}1)\ensuremath{-}d}$ is proposed. The nontrivial fixed points and the critical exponents for $\mathcal{O}=3,4$ are calculated to order ${\ensuremath{\epsilon}}_{\mathcal{O}}$. We present the relevant scaling fields and densities for $\mathcal{O}=3$, and, in an appendix, justify the validity of the approximate recursion relation to order ${\ensuremath{\epsilon}}_{\mathcal{O}}$.

Physical review. B, Solid stateJun 1, 197463 citations

THE RENORMALIZATION GROUP AND SCALING IN THE CRITICAL REGION

WORLD SCIENTIFIC eBooksJun 1, 20050 citations

Exact and approximate differential renormalization-group generators

Jeffrey F. Nicoll, T. S. Chang, Eugene Stanley

In dealing with critical phenomena of complex systems that simulate realistic materials, the full structure of the renormalization group is often unnecessarily cumbersome. For approximate calculations and for systems with special properties, specialized generators are simpler to apply. We derive several such exact and approximate differential generators and solve a number of interesting practical problems to illustrate this approach: (i) We derive a new approximate differential generator based on the Wilson incomplete-integration generator. Using this generator we calculate for an $n$ -component spin system the eigenvalues (critical-point exponents) associated with a critical point of arbitrary order $\mathcal{O}$ and "propagator exponent" $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\sigma}}$ to first order in the expansion parameter ${\ensuremath{\epsilon}}_{\mathcal{O}}(\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\sigma}})=d+\mathcal{O}(\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\sigma}}\ensuremath{-}d)$; this extends previous work for $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\sigma}}=2$, $\mathcal{O}$ arbitrary; $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\sigma}}\ensuremath{\le}2$, $\mathcal{O}=2$ (long-rang forces); and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\sigma}}=4$, $\mathcal{O}=2$ (the "Lifshitz point"). Our results agree with those obtainable using an approximate generator based on the Wegner-Houghton equation. The cases $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\sigma}}=2L$ ( $L$ a positive integer \ensuremath{\ge}2) describe the onset of helical ordering for which $|\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}|\ensuremath{\sim}{(\ensuremath{-}p)}^{{\ensuremath{\beta}}_{k}}$, where ${\ensuremath{\beta}}_{k}=\frac{1}{2(L\ensuremath{-}1)}+o({\ensuremath{\epsilon}}_{\mathcal{O}}^{2}(2L))$ and $p$ parametrizes the hypersurface of critical points. For $p>0$ , the ordered phase is uniform; for $p<0$ there is spiral order. The point $p=0$ , at which such nonuniform ordering commences, we term a generalized Lifshitz point of Lifschitz character $L$ . (ii) We consider the full Wilson and Wegner-Houghton generators in the paired spin-momenta limit and the $n\ensuremath{\rightarrow}\ensuremath{\infty}$ limit for even-order critical points. These limiting generators are identical for both full generators. This demonstrates that at least in these cases the Wilson and Wegner-Houghton generators agree exactly, without recourse to perturbation theory. These simple exact generators should provide "anchors" for calculations of exponents for higher-order critical points. (iii) We derive approximate generators which are suitable for compressible magnetic systems and more general systems with constraints for which the spin momenta are grouped in any arbitrary manner. We apply this to the case of a simple compressible magnet model and obtain the exact renormalization-group trajectories to order $\ensuremath{\epsilon}$ with $\ensuremath{\epsilon}\ensuremath{\equiv}{\ensuremath{\epsilon}}_{2}(2)=4\ensuremath{-}d$.In dealing with critical phenomena of complex systems that simulate realistic materials, the full structure of the renormalization group is often unnecessarily cumbersome. For approximate calculations and for systems with special properties, specialized generators are simpler to apply. We derive several such exact and approximate differential generators and solve a number of interesting practical problems to illustrate this approach: (i) We derive a new approximate differential generator based on the Wilson incomplete-integration generator. Using this generator we calculate for an $n$-component spin system the eigenvalues (critical-point exponents) associated with a critical point of arbitrary order $\mathcal{O}$ and "propagator exponent" $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\sigma}}$ to first order in the expansion parameter ${\ensuremath{\epsilon}}_{\mathcal{O}}(\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\sigma}})=d+\mathcal{O}(\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\sigma}}\ensuremath{-}d)$; this extends previous work for $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\sigma}}=2$, $\mathcal{O}$ arbitrary; $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\sigma}}\ensuremath{\le}2$, $\mathcal{O}=2$ (long-rang forces); and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\sigma}}=4$, $\mathcal{O}=2$ (the "Lifshitz point"). Our results agree with those obtainable using an approximate generator based on the Wegner-Houghton equation. The cases $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\sigma}}=2L$ ($L$ a positive integer \ensuremath{\ge}2) describe the onset of helical ordering for which $|\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}|\ensuremath{\sim}{(\ensuremath{-}p)}^{{\ensuremath{\beta}}_{k}}$, where ${\ensuremath{\beta}}_{k}=\frac{1}{2(L\ensuremath{-}1)}+o({\ensuremath{\epsilon}}_{\mathcal{O}}^{2}(2L))$ and $p$ parametrizes the hypersurface of critical points. For $p>0$, the ordered phase is uniform; for $p<0$ there is spiral order. The point $p=0$, at which such nonuniform ordering commences, we term a generalized Lifshitz point of Lifschitz character $L$. (ii) We consider the full Wilson and Wegner-Houghton generators in the paired spin-momenta limit and the $n\ensuremath{\rightarrow}\ensuremath{\infty}$ limit for even-order critical points. These limiting generators are identical for both full generators. This demonstrates that at least in these cases the Wilson and Wegner-Houghton generators agree exactly, without recourse to perturbation theory. These simple exact generators should provide "anchors" for calculations of exponents for higher-order critical points. (iii) We derive approximate generators which are suitable for compressible magnetic systems and more general systems with constraints for which the spin momenta are grouped in any arbitrary manner. We apply this to the case of a simple compressible magnet model and obtain the exact renormalization-group trajectories to order $\ensuremath{\epsilon}$ with $\ensuremath{\epsilon}\ensuremath{\equiv}{\ensuremath{\epsilon}}_{2}(2)=4\ensuremath{-}d$.

Physical Review AMar 1, 197678 citations

Critical exponents for higher order phase transitions: Landau theory and RG flow

Joydeep Chakravarty, Diksha Jain

Abstract In this work, we define and calculate critical exponents associated with higher order thermodynamic phase transitions. Such phase transitions can be classified into two classes: with or without a local order parameter. For phase transitions involving a local order parameter, we write down the Landau theory and calculate critical exponents using the saddle point approximation. Further, we investigate fluctuations about the saddle point and demarcate when such fluctuations dominate over saddle point calculations by introducing the generalized Ginzburg criteria. We use Wilsonian renormalization group (RG) to derive scaling forms for observables near criticality and obtain scaling relations between the critical exponents. Afterwards, we find out fixed points of the RG flow using the one-loop beta function and calculate critical exponents about the fixed points for third and fourth order phase transitions.Abstract In this work, we define and calculate critical exponents associated with higher order thermodynamic phase transitions. Such phase transitions can be classified into two classes: with or without a local order parameter. For phase transitions involving a local order parameter, we write down the Landau theory and calculate critical exponents using the saddle point approximation. Further, we investigate fluctuations about the saddle point and demarcate when such fluctuations dominate over saddle point calculations by introducing the generalized Ginzburg criteria. We use Wilsonian renormalization group (RG) to derive scaling forms for observables near criticality and obtain scaling relations between the critical exponents. Afterwards, we find out fixed points of the RG flow using the one-loop beta function and calculate critical exponents about the fixed points for third and fourth order phase transitions.

arXiv:2102.08398Journal of Statistical Mechanics Theory and ExperimentSep 1, 20218 citations

THE RENORMALIZATION GROUP AND SCALING IN THE CRITICAL REGION

WORLD SCIENTIFIC eBooksApr 1, 19930 citations

Critical points of higher order can exist in complex magnetic and fluid systems. By definition, at a critical point of order

\mathcal{O}

\mathcal{O}

phases become identical simultaneously. Here the Wilson renormalization-group method is generalized from ordinary critical points (

\mathcal{O}=2

) and Gaussian tricritical points (

\mathcal{O}=3

) to critical points of arbitrary order

\mathcal{O}

. An expansion scheme in ${\ensuremath{\epsilon}}_{\mathcal{O}}\ensuremath{\equiv}\frac{2\mathcal{O}}{(\mathcal{O}\ensuremath{-}1)\ensuremath{-}d}$ is proposed. The nontrivial fixed points and the critical exponents for

\mathcal{O}=3,4

are calculated to order ${\ensuremath{\epsilon}}_{\mathcal{O}}$. We present the relevant scaling fields and densities for

\mathcal{O}=3

, and, in an appendix, justify the validity of the approximate recursion relation to order ${\ensuremath{\epsilon}}_{\mathcal{O}}$.

n

-component spin system the eigenvalues (critical-point exponents) associated with a critical point of arbitrary order

\mathcal{O}

and "propagator exponent" $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\sigma}}$ to first order in the expansion parameter ${\ensuremath{\epsilon}}_{\mathcal{O}}(\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\sigma}})=d+\mathcal{O}(\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\sigma}}\ensuremath{-}d)$; this extends previous work for $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\sigma}}=2$,

\mathcal{O}

arbitrary; $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\sigma}}\ensuremath{\le}2$,

\mathcal{O}=2

(long-rang forces); and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\sigma}}=4$,

\mathcal{O}=2

(the "Lifshitz point"). Our results agree with those obtainable using an approximate generator based on the Wegner-Houghton equation. The cases $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\sigma}}=2L$ (

L

a positive integer \ensuremath{\ge}2) describe the onset of helical ordering for which $|\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}|\ensuremath{\sim}{(\ensuremath{-}p)}^{{\ensuremath{\beta}}_{k}}$, where ${\ensuremath{\beta}}_{k}=\frac{1}{2(L\ensuremath{-}1)}+o({\ensuremath{\epsilon}}_{\mathcal{O}}^{2}(2L))$ and

p

parametrizes the hypersurface of critical points. For

p>0

, the ordered phase is uniform; for

p<0

there is spiral order. The point

p=0

, at which such nonuniform ordering commences, we term a generalized Lifshitz point of Lifschitz character

L

. (ii) We consider the full Wilson and Wegner-Houghton generators in the paired spin-momenta limit and the $n\ensuremath{\rightarrow}\ensuremath{\infty}$ limit for even-order critical points. These limiting generators are identical for both full generators. This demonstrates that at least in these cases the Wilson and Wegner-Houghton generators agree exactly, without recourse to perturbation theory. These simple exact generators should provide "anchors" for calculations of exponents for higher-order critical points. (iii) We derive approximate generators which are suitable for compressible magnetic systems and more general systems with constraints for which the spin momenta are grouped in any arbitrary manner. We apply this to the case of a simple compressible magnet model and obtain the exact renormalization-group trajectories to order $\ensuremath{\epsilon}$ with $\ensuremath{\epsilon}\ensuremath{\equiv}{\ensuremath{\epsilon}}_{2}(2)=4\ensuremath{-}d$.