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Physical review. B, Solid state

Invariance of critical exponents for renormalization groups generated by a flow vector

Melville S. Green

Published Jun 1, 1977• 5 citations
Publisher

Abstract

An invariance theorem for exponents of scaling variables is proved for a class of renormalization groups introduced by Wegner. (Renormalization groups generated by a flow vector.) A convenient form for the renormalization group linearized both around the fixed point and around the renormalization trajectory is derived. The latter is used to derive a coordinate system for perturbations around a critical Hamiltonian from that around a fixed point. It is shown that the classification scaling redundant applies also to the vectors of this critical-point coordinate system. The effect of an infinitesimal perturbation of the flow vector on the fixed point and on the renormalization group linearized around the fixed point is investigated. If the unperturbed linearized group has no marginal scaling operator, two cases arise. (i) The perturbed group has no fixed point near that of the unperturbed group. This case may obtain if there is a marginal redundant observable. (ii) The perturbed group has a fixed point which differs from that of the unperturbed group by a redundant perturbation. In the second case the scaling eigenvalues of the perturbed linearized group will be unchanged. Eigenvalues of redundant observables may change. The effect of a perturbation in a flow vector on a renormalized trajectory is considered and shown to suggest the concept of manifolds of equivalent Hamiltonians. Some difficulties involved in this concept are discussed.

DOI
10.1103/physrevb.15.5367

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Physical review. B, Solid state•Jul 1, 1975•34 citations
An invariance theorem for exponents of scaling variables is proved for a class of renormalization groups introduced by Wegner. (Renormalization groups generated by a flow vector.) A convenient form for the renormalization group linearized both around the fixed point and around the renormalization trajectory is derived. The latter is used to derive a coordinate system for perturbations around a critical Hamiltonian from that around a fixed point. It is shown that the classification scaling redundant applies also to the vectors of this critical-point coordinate system. The effect of an infinitesimal perturbation of the flow vector on the fixed point and on the renormalization group linearized around the fixed point is investigated. If the unperturbed linearized group has no marginal scaling operator, two cases arise. (i) The perturbed group has no fixed point near that of the unperturbed group. This case may obtain if there is a marginal redundant observable. (ii) The perturbed group has a fixed point which differs from that of the unperturbed group by a redundant perturbation. In the second case the scaling eigenvalues of the perturbed linearized group will be unchanged. Eigenvalues of redundant observables may change. The effect of a perturbation in a flow vector on a renormalized trajectory is considered and shown to suggest the concept of manifolds of equivalent Hamiltonians. Some difficulties involved in this concept are discussed.
A generalization of Wilson's renormalization group equation is discussed. A large class of equivalent fixed points exists for each fixed point. The eigen perturbations fall into two classes: (i) scaling operators whose scaling exponents are independent of the fixed point within the class of equivalent fixed points, and (ii) redundant operators whose exponents depend on the choice of the renormalization group equation. The exponents of the redundant operators are spurious since the free energy depends only on the scaling fields of the scaling operators but not on the scaling fields of the redundant operators.
Wegner's paradigm for constructing renormalization groups in statistical mechanics is applied to fluids. The coordinates of the fraction ($1\ensuremath{-}y$) of the molecules are integrated over and a system of interactions describing the remaining fraction yyy is constructed with the aid of the McMillan-Mayer solution theory. Renormalized Ursell functions are defined in terms of which the group operations are shown to be linear. The group operations commute with a scaling operation and it is found that a series of renormalization groups depending on a parameter xxx can be constructed by combining the operation of integrating over coordinates with the scaling operation. By comparing the renormalized cluster functions with the form of the critical cluster functions given by scaling it is shown that a fixed point of one of the series of groups corresponds to the critical point. The parameter xxx turns out to be the scaling exponent for the chemical potential. The question of the universality of critical phenomena is expressed as the question of the discreteness or continuity of the spectrum of values of the parameter xxx for which the linear fixed-point equations have a nontrivial solution. It is suggested that this question can be answered definitively only when a precise formulation is made of the function space in which the sequence of renormalized Ursell functions must lie.
Renormalization group (RG) flows for systems subject to an external field can be derived either by fixing the field or by fixing the conjugate order parameter. We show how to implement these boundary conditions within the functional renormalization group approach. We derive and compare the corresponding flow equations for ϕ4ϕ^4ϕ4 theory and show that some scaling variables and scaling exponents associated with the linearized flow of the relevant three-point vertex in the vicinity of the Ising fixed point depend on the boundary conditions. We also discuss the effect of elasticity on Ising criticality. In this case two of the fixed points (namely the renormalized Ising fixed point and the spherical fixed point) obtained for constant strain are shifted to physically inaccessible regimes of parameter space when we perform the RG at constant stress.
We saw in Chapter 3 that the hamiltonian for a System at a critical point flows under the renormalization group into a critical fixed point. Under a renormalization group transformation, the microscopic length scale is rescaled by a constant factor b , and so the coordinates of a given point, as measured in units of this length scale, transform according to r → b −1 r. This is called a scale transformation . Once the flows reach such a fixed point, the parameters of the hamiltonian no longer change, and it is said to be scale invariant . As well as being scale invariant, the fixed point hamiltonian usually possesses other spatial symmetries. For example, if the underlying model is defined on a lattice, so that its hamiltonian is invariant under lattice translations, the corresponding critical fixed point hamiltonian is generally invariant under arbitrary uniform translations. This is because terms which might be added to the hamiltonian which break the symmetry under continuous translations down to its subgroup of lattice translations are irrelevant at such a fixed point. Similarly, if the lattice model is invariant under a sufficiently large subgroup of the rotation group (for example, if the interactions in the x and y directions on a Square lattice are equal), then the fixed point hamiltonian enjoys full rotational invariance. As discussed on p.58, even if the interactions are anisotropic, rotational invariance may often be recovered by a suitable finite relative rescaling of the coordinates.
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.
The analysis of selected nonlinear problems in the renormalization group is found to show striking contrasts between the usual local linearized fixed-point analysis and the properties of global solutions of nonlinear equations derived from an approximation of the Wegner-Houghton differential formulation. The competition between various fixed points that is incorporated in general global solutions can upset the asymptotically valid critical behavior deduced from the local analysis. In general, the critical-point exponents of such a solution will not satisfy equalities, but rather the corresponding inequalities. However, these nonscaling solutions have extraneous singularities that are not related to the thermodynamic singularities of the system. If singularities of this type are excluded, then the global solution has the same critical-point exponents as the local solution derived by linearizing around the stablest fixed point. It is shown that in this case the critical surface in the Hamiltonian space is closely related to the surface of order-2 critical points in a thermodynamic field space. The boundaries of this surface are correspondingly related to the critical points of higher order in this thermodynamic space. The nonlinear global solution predicts multiple power scaling behavior from a single scaling equation deduced from the renormalization group. Previously such behavior was obtained by postulating the simultaneous validity of two of more "linear" scaling hypotheses.