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Physical Review A

Exact and approximate differential renormalization-group generators

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

Published Mar 1, 1976• 78 citations
Publisher

Abstract

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 $\ensuremath{\epsilon}\ensuremath{\equiv}{\ensuremath{\epsilon}}_{2}(2)=4\ensuremath{-}d$.

DOI
10.1103/physreva.13.1251

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We apply an approximate form of the exact one-particle-irreducible renormalization-group generator to the calculation of the equation of state. Several approaches are explored. (i) Global nonlinear trajectories for the Hamiltonian parameters are exploited to give nonlinear crossover equations of state. Logarithms of the reduced temperature $t\ensuremath{\equiv}\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}$ are automatically exponentiated to give power-law behavior. (ii) The form and asymptotic properties of the equation of state are described for those systems whose complete nonlinear trajectories cannot be explicitly obtained. (iii) Nonlinear trajectories for the irreducible Green's functions are solved to give a fully exponentiated equation of state containing no logarithmic terms. (iv) Simple critical (noncrossover) equations of state are obtained by iteration of the generator with linear and quasilinear trajectories. (v) Operator nonlinear equations are solved to give the crossover equation of state for an arbitrary order $\ensuremath{\theta}$ critical system, both in an $\ensuremath{\epsilon}$ expansion and at the borderline dimension. (vi) Combination of the Green's functions are formed into Green's eigenfunctions or operators and their nonlinear trajectories used to calculate fully exponentiated equations of state for order $\ensuremath{\theta}$ Ising systems. Systems considered include the usual Wilson-Fisher Hamiltonian [methods (i)-(iv)], the Sak-model compressible ferromagnet [(i)-(ii)], the $\mathrm{nm}$-hypercubical model (of which the dilute quenched random ferromagnet forms the $n\ensuremath{\rightarrow}0$ limit) [(ii)], and the isotropic $n$-component order $\ensuremath{\theta}$ ferromagnet [(iv)-(vi)]. The tricritical ($\ensuremath{\theta}=3$) case is given explicitly as an example of the general formalism of methods (iv)-(vi). Throughout, we use a bare critical propagator which is an arbitrary generalized homogeneous function of the components of the wave vector. This allows us to simultaneously describe ordinary critical systems and anisotropic Lifschitz points as well as certain structural and spin-reorientation phase transitions.

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Q -state Potts model by Wilson's exact renormalization-group equation

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Critical properties of the $Q$-state Potts model for dimensions $3\ensuremath{\le}d\ensuremath{\le}6$ are calculated by means of Wilson's exact momentum-space renormalization-group equation. The scaling-field method of Golner and Riedel is used to approximate the functional differential equation by a set of 11 ordinary coupled differential equations. For $d=4\ensuremath{-}\ensuremath{\epsilon}$, lines of critical and tricritical Potts fixed points are found as functions of $Q$ that annihilate as $Q$ approaches a critical value ${Q}_{c}=\frac{2+{\ensuremath{\epsilon}}^{2}}{a+O({\ensuremath{\epsilon}}^{3})}$. For $Q>{Q}_{c}$, the Potts transition is first order. Along these fixed lines the critical and tricritical exponents (upper and lower sign, respectively) are to leading order: $\frac{1}{\ensuremath{\nu}}=2\ensuremath{-}\frac{1}{6}[\ensuremath{\epsilon}\ifmmode\pm\else\textpm\fi{}{({\ensuremath{\epsilon}}^{2}\ensuremath{-}a\ensuremath{\delta})}^{\frac{1}{2}}]$, $\frac{\ensuremath{\varphi}}{\ensuremath{\nu}}=\ensuremath{\mp}{({\ensuremath{\epsilon}}^{2}\ensuremath{-}a\ensuremath{\delta})}^{\frac{1}{2}}$, and $\ensuremath{\eta}=\frac{{[\ensuremath{\epsilon}\ifmmode\pm\else\textpm\fi{}{({\ensuremath{\epsilon}}^{2}\ensuremath{-}a\ensuremath{\delta})}^{\frac{1}{2}}]}^{2}}{216}+b\ensuremath{\delta}$, where $\ensuremath{\epsilon}=4\ensuremath{-}d$, $\ensuremath{\delta}=Q\ensuremath{-}2$, and $\ensuremath{\delta}\ensuremath{\le}{\ensuremath{\delta}}_{c}=\frac{{\ensuremath{\epsilon}}^{2}}{a}+O({\ensuremath{\epsilon}}^{3})$. While the form of the $\ensuremath{\epsilon}$ and $\ensuremath{\delta}$ dependences is exact, the coefficients $a$ and $b$ cannot be obtained systematically by $\ensuremath{\epsilon}$ expansion, since the upper critical dimensionality of the Potts model is six when $Q\ensuremath{\ne}2$. In our truncation, $a=6.52$ and $b=0.065$. The results have been extended to dimensions $3.4\ensuremath{\le}d\ensuremath{\lesssim}4$ by solving the renormalization-group equations numerically. The percolation limit of the Potts model, $Q=1$, is also investigated and the critical exponents ${\ensuremath{\nu}}^{P},{\ensuremath{\varphi}}^{P}$, and ${\ensuremath{\eta}}^{P}$ determined as functions of dimension for $3\ensuremath{\le}d\ensuremath{\le}6$.

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Wilson–Polchinski exact renormalization group equation for O (N) systems: leading and next-to-leading orders in the derivative expansion

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With a view to study the convergence properties of the derivative expansion of the exact renormalization group (RG) equation, I explicitly study the leading and next-to-leading orders of this expansion applied to the Wilson–Polchinski equation in the case of the N -vector model with the symmetry O ( N ). As a test, the critical exponents η and ν as well as the subcritical exponent ω (and higher ones) are estimated in three dimensions for values of N ranging from 1 to 20. I compare the results with the corresponding estimates obtained in preceding studies or treatments of other O ( N ) exact RG equations at second order. The possibility of varying N allows the derivative expansion method to be better valued. The values obtained from the resummation of high orders of perturbative field theory are used as standards to illustrate the eventual convergence in each case. Particular attention is drawn to the preservation (or not) of the reparameterization invariance.

arXiv:hep-th/0501087•Journal of Physics Condensed Matter•May 6, 2005•15 citations
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 nnn-component spin system the eigenvalues (critical-point exponents) associated with a critical point of arbitrary order O\mathcal{O}O and "propagator exponent" \ensuremathσ\ifmmode~\else˜\fi\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\sigma}}\ensuremathσ\ifmmode~\else˜\fi​ to first order in the expansion parameter \ensuremathϵO(\ensuremathσ\ifmmode~\else˜\fi)=d+O(\ensuremathσ\ifmmode~\else˜\fi\ensuremath−d){\ensuremath{\epsilon}}_{\mathcal{O}}(\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\sigma}})=d+\mathcal{O}(\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\sigma}}\ensuremath{-}d)\ensuremathϵO​(\ensuremathσ\ifmmode~\else˜\fi​)=d+O(\ensuremathσ\ifmmode~\else˜\fi​\ensuremath−d); this extends previous work for \ensuremathσ\ifmmode~\else˜\fi=2\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\sigma}}=2\ensuremathσ\ifmmode~\else˜\fi​=2, O\mathcal{O}O arbitrary; \ensuremathσ\ifmmode~\else˜\fi\ensuremath≤2\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\sigma}}\ensuremath{\le}2\ensuremathσ\ifmmode~\else˜\fi​\ensuremath≤2, O=2\mathcal{O}=2O=2 (long-rang forces); and \ensuremathσ\ifmmode~\else˜\fi=4\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\sigma}}=4\ensuremathσ\ifmmode~\else˜\fi​=4, O=2\mathcal{O}=2O=2 (the "Lifshitz point"). Our results agree with those obtainable using an approximate generator based on the Wegner-Houghton equation. The cases \ensuremathσ\ifmmode~\else˜\fi=2L\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\sigma}}=2L\ensuremathσ\ifmmode~\else˜\fi​=2L (LLL a positive integer \ensuremath{\ge}2) describe the onset of helical ordering for which ∣k\ensuremath→∣\ensuremath∼(\ensuremath−p)\ensuremathβk|\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}|\ensuremath{\sim}{(\ensuremath{-}p)}^{{\ensuremath{\beta}}_{k}}∣k\ensuremath→​∣\ensuremath∼(\ensuremath−p)\ensuremathβk​, where \ensuremathβk=12(L\ensuremath−1)+o(\ensuremathϵO2(2L)){\ensuremath{\beta}}_{k}=\frac{1}{2(L\ensuremath{-}1)}+o({\ensuremath{\epsilon}}_{\mathcal{O}}^{2}(2L))\ensuremathβk​=2(L\ensuremath−1)1​+o(\ensuremathϵO2​(2L)) and ppp parametrizes the hypersurface of critical points. For p>0p>0p>0, the ordered phase is uniform; for p<0p<0p<0 there is spiral order. The point p=0p=0p=0, at which such nonuniform ordering commences, we term a generalized Lifshitz point of Lifschitz character LLL. (ii) We consider the full Wilson and Wegner-Houghton generators in the paired spin-momenta limit and the n\ensuremath→\ensuremath∞n\ensuremath{\rightarrow}\ensuremath{\infty}n\ensuremath→\ensuremath∞ 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 \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ with \ensuremathϵ\ensuremath≡\ensuremathϵ2(2)=4\ensuremath−d\ensuremath{\epsilon}\ensuremath{\equiv}{\ensuremath{\epsilon}}_{2}(2)=4\ensuremath{-}d\ensuremathϵ\ensuremath≡\ensuremathϵ2​(2)=4\ensuremath−d.
We apply an approximate form of the exact one-particle-irreducible renormalization-group generator to the calculation of the equation of state. Several approaches are explored. (i) Global nonlinear trajectories for the Hamiltonian parameters are exploited to give nonlinear crossover equations of state. Logarithms of the reduced temperature $t\ensuremath{\equiv}\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}$ are automatically exponentiated to give power-law behavior. (ii) The form and asymptotic properties of the equation of state are described for those systems whose complete nonlinear trajectories cannot be explicitly obtained. (iii) Nonlinear trajectories for the irreducible Green's functions are solved to give a fully exponentiated equation of state containing no logarithmic terms. (iv) Simple critical (noncrossover) equations of state are obtained by iteration of the generator with linear and quasilinear trajectories. (v) Operator nonlinear equations are solved to give the crossover equation of state for an arbitrary order $\ensuremath{\theta}$ critical system, both in an $\ensuremath{\epsilon}$ expansion and at the borderline dimension. (vi) Combination of the Green's functions are formed into Green's eigenfunctions or operators and their nonlinear trajectories used to calculate fully exponentiated equations of state for order $\ensuremath{\theta}$ Ising systems. Systems considered include the usual Wilson-Fisher Hamiltonian [methods (i)-(iv)], the Sak-model compressible ferromagnet [(i)-(ii)], the nm\mathrm{nm}nm-hypercubical model (of which the dilute quenched random ferromagnet forms the $n\ensuremath{\rightarrow}0$ limit) [(ii)], and the isotropic nnn-component order $\ensuremath{\theta}$ ferromagnet [(iv)-(vi)]. The tricritical ($\ensuremath{\theta}=3$) case is given explicitly as an example of the general formalism of methods (iv)-(vi). Throughout, we use a bare critical propagator which is an arbitrary generalized homogeneous function of the components of the wave vector. This allows us to simultaneously describe ordinary critical systems and anisotropic Lifschitz points as well as certain structural and spin-reorientation phase transitions.
We give an approximate renormalization-group formulation which parallels that of Wilson. The group generator represents the momentum-independent limit of the differential generator of Wegner and Houghton. The eigenfunctions near the Gaussian point are computed for all spin dimensions nnn and lattice dimensions ddd, including d=2d=2d=2. The nontrivial fixed-point Hamiltonian in dimensions near $d=\frac{2\mathcal{O}}{(\mathcal{O}\ensuremath{-}1)}$, together with the eigenvalues near that nontrivial fixed point, are found explicitly to first order in ${\ensuremath{\epsilon}}_{\mathcal{O}}\ensuremath{\equiv}\mathcal{O}(2\ensuremath{-}d)+d$ for all values of nnn and the order O\mathcal{O}O. Odd-dominated Ising systems and corresponding expansions in ${\ensuremath{\epsilon}}_{\mathcal{O}\ensuremath{-}\frac{1}{2}}$ are also treated.
Critical points of higher order can exist in complex magnetic and fluid systems. By definition, at a critical point of order O\mathcal{O}O, O\mathcal{O}O phases become identical simultaneously. Here the Wilson renormalization-group method is generalized from ordinary critical points (O=2\mathcal{O}=2O=2) and Gaussian tricritical points (O=3\mathcal{O}=3O=3) to critical points of arbitrary order O\mathcal{O}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 O=3,4\mathcal{O}=3,4O=3,4 are calculated to order ${\ensuremath{\epsilon}}_{\mathcal{O}}$. We present the relevant scaling fields and densities for O=3\mathcal{O}=3O=3, and, in an appendix, justify the validity of the approximate recursion relation to order ${\ensuremath{\epsilon}}_{\mathcal{O}}$.
Critical properties of the QQQ-state Potts model for dimensions $3\ensuremath{\le}d\ensuremath{\le}6$ are calculated by means of Wilson's exact momentum-space renormalization-group equation. The scaling-field method of Golner and Riedel is used to approximate the functional differential equation by a set of 11 ordinary coupled differential equations. For $d=4\ensuremath{-}\ensuremath{\epsilon}$, lines of critical and tricritical Potts fixed points are found as functions of QQQ that annihilate as QQQ approaches a critical value ${Q}_{c}=\frac{2+{\ensuremath{\epsilon}}^{2}}{a+O({\ensuremath{\epsilon}}^{3})}$. For Q>QcQ>{Q}_{c}Q>Qc​, the Potts transition is first order. Along these fixed lines the critical and tricritical exponents (upper and lower sign, respectively) are to leading order: $\frac{1}{\ensuremath{\nu}}=2\ensuremath{-}\frac{1}{6}[\ensuremath{\epsilon}\ifmmode\pm\else\textpm\fi{}{({\ensuremath{\epsilon}}^{2}\ensuremath{-}a\ensuremath{\delta})}^{\frac{1}{2}}]$, $\frac{\ensuremath{\varphi}}{\ensuremath{\nu}}=\ensuremath{\mp}{({\ensuremath{\epsilon}}^{2}\ensuremath{-}a\ensuremath{\delta})}^{\frac{1}{2}}$, and $\ensuremath{\eta}=\frac{{[\ensuremath{\epsilon}\ifmmode\pm\else\textpm\fi{}{({\ensuremath{\epsilon}}^{2}\ensuremath{-}a\ensuremath{\delta})}^{\frac{1}{2}}]}^{2}}{216}+b\ensuremath{\delta}$, where $\ensuremath{\epsilon}=4\ensuremath{-}d$, $\ensuremath{\delta}=Q\ensuremath{-}2$, and $\ensuremath{\delta}\ensuremath{\le}{\ensuremath{\delta}}_{c}=\frac{{\ensuremath{\epsilon}}^{2}}{a}+O({\ensuremath{\epsilon}}^{3})$. While the form of the $\ensuremath{\epsilon}$ and $\ensuremath{\delta}$ dependences is exact, the coefficients aaa and bbb cannot be obtained systematically by $\ensuremath{\epsilon}$ expansion, since the upper critical dimensionality of the Potts model is six when $Q\ensuremath{\ne}2$. In our truncation, a=6.52a=6.52a=6.52 and b=0.065b=0.065b=0.065. The results have been extended to dimensions $3.4\ensuremath{\le}d\ensuremath{\lesssim}4$ by solving the renormalization-group equations numerically. The percolation limit of the Potts model, Q=1Q=1Q=1, is also investigated and the critical exponents ${\ensuremath{\nu}}^{P},{\ensuremath{\varphi}}^{P}$, and ${\ensuremath{\eta}}^{P}$ determined as functions of dimension for $3\ensuremath{\le}d\ensuremath{\le}6$.
We give the first nonlinear solution of renormalization-group equations. This solution, based on the differential generator of Wegner and Houghton, exhibits an explicit mixing of (or crossover between) critical and mean-field behavior. The solution is given for all values of the spin dimension nnn and to first order in $\ensuremath{\epsilon}\ensuremath{\equiv}4\ensuremath{-}d$, where ddd is the lattice dimensionality.
With a view to study the convergence properties of the derivative expansion of the exact renormalization group (RG) equation, I explicitly study the leading and next-to-leading orders of this expansion applied to the Wilson–Polchinski equation in the case of the N -vector model with the symmetry O ( N ). As a test, the critical exponents η and ν as well as the subcritical exponent ω (and higher ones) are estimated in three dimensions for values of N ranging from 1 to 20. I compare the results with the corresponding estimates obtained in preceding studies or treatments of other O ( N ) exact RG equations at second order. The possibility of varying N allows the derivative expansion method to be better valued. The values obtained from the resummation of high orders of perturbative field theory are used as standards to illustrate the eventual convergence in each case. Particular attention is drawn to the preservation (or not) of the reparameterization invariance.