Equations related to the Polchinski version of the exact renormalization group (RG) equations for scalar fields which extend the local potential approximation to first order in a derivative expansion, and which maintain reparameterization invariance, are postulated. Reparameterization invariance ensures that the equations determine the anomalous dimension η unambiguously and the equations are such that the result is exact to O (ε 2 ) in an ε-expansion for any multi-critical fixed point. It is also straightforward to determine η numerically. When the dimension d = 3 numerical results for a wide range of critical exponents are obtained in theories with O ( N ) symmetry, for various N and for ranges of η, are obtained within the local potential approximation. The associated η, which follows from the derivative approximation described here, is found for various N . The large N limit of the equations is also analysed. A corresponding discussion is also given in a perturbative RG framework and scaling dimensions for derivative operators are calculated to first order in ε.
Equations related to the Polchinski version of the exact renormalization group (RG) equations for scalar fields which extend the local potential approximation to first order in a derivative expansion, and which maintain reparameterization invariance, are postulated. Reparameterization invariance ensures that the equations determine the anomalous dimension η unambiguously and the equations are such that the result is exact to O (ε 2 ) in an ε-expansion for any multi-critical fixed point. It is also straightforward to determine η numerically. When the dimension d = 3 numerical results for a wide range of critical exponents are obtained in theories with O ( N ) symmetry, for various N and for ranges of η, are obtained within the local potential approximation. The associated η, which follows from the derivative approximation described here, is found for various N . The large N limit of the equations is also analysed. A corresponding discussion is also given in a perturbative RG framework and scaling dimensions for derivative operators are calculated to first order in ε.
The Polchinski version of the exact renormalisation group equations is applied to multicritical fixed points, which are present for dimensions between two and four, for scalar theories using both the local potential approximation and its extension, the derivative expansion. The results are compared with the epsilon expansion by showing that the non linear differential equations may be linearised at each multicritical point and the epsilon expansion treated as a perturbative expansion. The results for critical exponents are compared with corresponding epsilon expansion results from standard perturbation theory. The results provide a test for the validity of the local potential approximation and also the derivative expansion. An alternative truncation of the exact RG equation leads to equations which are similar to those found in the derivative expansion but which gives correct results for critical exponents to order ε and also for the field anomalous dimension to order ε 2. An exact marginal operator for the full RG equations is also constructed.
Continuous phase transitions provide a natural setting for understanding universality. Near criticality, the correlation length diverges, microscopic details become increasingly irrelevant, and physical observables obey scaling laws characterized by universal critical exponents. Within the Functional Renormalization Group (FRG), this scale dependence is described through the Effective Average Action, which progressively incorporates fluctuations as the momentum scale is lowered. A first part of this thesis is developed within the Local Potential Approximation (LPA), where we study multicritical fixed points of scalar field theories. Their universal properties are compared with results obtained at higher orders in the derivative expansion. Particular attention is devoted to the relation between global fixed-point potentials and their local representations through finite polynomial truncations. We show that the convergence of such expansions is constrained by the analytic structure of the global solution, and that expanding around a suitable nonzero stationary point significantly improves the reconstruction of multicritical branches. Beyond the structure of individual fixed points, a central part of this thesis concerns RG irreversibility. In two dimensions, Zamolodchikov’s c-theorem describes the flow from the ultraviolet to the infrared through a monotonically decreasing c-function. Within the FRG framework, we develop a variational procedure to reconstruct RG trajectories between fixed points and integrate the corresponding c-function along them. The resulting quantity displays monotonic behaviour and becomes increasingly stable as the truncation order is raised, providing a manifestation of RG irreversibility within the approximation considered. Finally, close to the upper critical dimensions, the linearized FRG flow around the Gaussian fixed point is naturally organized in terms of Hermite polynomials. Using the epsilon-expansion, we derive a curvature-based candidate c-function and obtain an analytic explanation for the alternating curvature of successive multicritical fixed points and for their progressive accumulation around the Gaussian fixed point.
Abstract In Chapter 14, the singular behavior of ferromagnetic systems with O(N) symmetry and short-range interactions, near a second order phase transition has been determined in the mean-field approximation, which is also a quasi-Gaussian approximation. The mean-field approximation predicts a set of universal properties, properties independent of the detailed structure of the microscopic Hamiltonian, the dimension of space, and, to a large extent, of the symmetry of systems. However, the leading corrections to the mean-field approximation, in dimensions smaller than or equal to four, diverge at the critical temperature, and the universal predictions of the mean-field approximation cannot be correct. Such a problem originates from the non-decoupling of scales and leads to the question of possible universality. In Chapter 9, the question has been answered in four dimensions using renormalization theory, and related renormalization group (RG) equations. Moreover, below four dimensions, in an expansion around the mean-field, the most singular terms near criticality can be also formally recovered from a continuum, low-mass φ4 field theory. More generally, following Wilson, to understand universality beyond the mean-field approximation, it is necessary to build a general renormalization group in the form of flow equations for effective Hamiltonians and to find fixed points of the flow equations. Near four dimensions, the flow equations can be approximated by the renormalization group of quantum field theory (QFT), and the fixed points and critical behaviours derived within the framework of the Wilson-Fisher ϵ expansion.
The Polchinski exact renormalization group equation for a scalar field theory in arbitrary dimensions is translated, by means of a covariant Hamiltonian formalism, into a partial differential equation for an effective Hamiltonian density that depends on an infinite tower of momenta fields with higher spin. A natural approximation scheme is then expanding the Hamiltonian in momenta with increasing rank. The first order of this expansion, one next to the local potential approximation, is regulator-independent and already includes infinitely many derivative interactions. Further truncating this down to a quadratic dependence on the momenta leads to an alternative to the first order of the derivative expansion, which is used to compute η=0.03616(1) for the critical exponent of the three dimensional Ising model.
We employ perturbative renormalization group and $\ensuremath{\epsilon}$-expansion to study multicritical single-scalar field theories with higher derivative kinetic terms of the form $\ensuremath{\phi}(\ensuremath{-}\ensuremath{\square}{)}^{k}\ensuremath{\phi}$. We focus on those with a Z2-symmetric critical point which are characterized by an upper critical dimension ${d}_{c}=2nk/(n\ensuremath{-}1)$ accumulating at even integers. We distinguish two types of theories depending on whether or not the numbers k and $n\ensuremath{-}1$ are relatively prime. When they are, the critical theory involves a marginal powerlike interaction ${\ensuremath{\phi}}^{2n}$ and the deformations admit a derivative expansion that at leading order involves only the potential. In this case we present the beta functional of the potential and use this to calculate some anomalous dimensions and operator product expansion coefficients. These confirm some conformal field theory data obtained using conformal-block techniques, while giving new results. In the second case where k and $n\ensuremath{-}1$ have a common divisor, the theories show a much richer structure induced by the presence of marginal derivative operators at criticality. We study the case k=2 with odd values of n, which fall in the second class, and calculate the functional flows and spectrum. These theories have a phase diagram characterized at leading order in $\ensuremath{\epsilon}$ by four fixed points which apart from the Gaussian UV fixed point include an IR fixed point with a purely derivative interaction.
In this central and essential chapter, we develop the dynamic renormalization group approach to time-dependent critical phenomena. Again, we base our exposition on the simple O ( n )-symmetric relaxational models A and B; the generalization to other dynamical systems is straightforward. We begin with an analysis of the infrared and ultraviolet singularities appearing in the dynamic perturbation expansion. Although we are ultimately interested in the infrared critical region, we first take care of the ultraviolet divergences. Below and at the critical dimension d c = 4, only a finite set of Feynman diagrams carries ultraviolet singularities, which we evaluate by means of the dimensional regularization prescription, and then eliminate via multiplicative as well as additive renormalization (the latter takes into account the fluctuation-induced shift of the critical temperature). The renormalization group equation then permits us to explore the ensuing scaling behavior of the correlation and vertex functions of the renormalized theory upon varying the arbitrary renormalization scale. Of fundamental importance is the identification of an infrared-stable renormalization group fixed point, which describes scale invariance and hence allows the derivation of the critical power laws in the infrared limit from the renormalization constants determined in the ultraviolet regime. This program is explicitly carried through for the relaxational models A and B. For model B, we derive a scaling relation connecting the dynamic exponent to the Fisher exponent η . The critical exponents ν , η , and z are computed to first non-trivial order in an ∊ expansion around the upper critical dimension d c = 4.