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arXiv:2112.10883 · Journal of Statistical Mechanics Theory and Experiment

Exact renormalization group equation for lattice Ginzburg–Landau models adapted to the solution in the local potential approximation

V. I. Tokar

Published Dec 1, 2022• 0 citations
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Abstract

Abstract The Wilson Green’s function approach and, alternatively, Feynman’s diffusion equation and the Hori representation have been used to derive an exact functional RG equation (EFRGE) that in the course of the RG flow interpolates between the interaction part of the lattice Ginzburg–Landau Hamiltonian and the logarithm of the generating functional of the S-matrix. Because the S-matrix vertices are the amputated correlation functions of the fluctuating field, it has been suggested that in the critical region the amputation of the long-range tails makes the S-matrix functional more localized and thus more amenable to the local potential approximation (LPA) than the renormalized free energy functional used in Wilson’s EFRGE. By means of a functional Legendre transform the S-matrix EFRGE has been converted into an EFRGE for the effective action (EA). It has been found that the field-dependent part of EA predicted by the equation is the same as calculated within the known EA EFRGE approaches but in addition it is accurately accounts for the field-independent terms. These are indispensable in calculation of such important quantities as the specific heat, the latent heat, etc. With the use of the derived EFRGE a closed expression for the renormalization counterterm has been obtained which when subtracted from the divergent solution of the Wetterich equation would lead to a finite exact expression for the EA thus making two approaches formally equivalent. The S-matrix equation has been found to be simply connected with a generalized functional Burgers’ equation which establishes a direct correspondence between the first order phase transitions and the shock wave solutions of the RG equation. The transparent semi-group structure of the S-matrix RG equation makes possible the use of different RG techniques at different stages of the RG flow in order to improve the LPA solution.

DOI
10.1088/1742-5468/aca0e6

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Abstract The Wilson Green’s function approach and, alternatively, Feynman’s diffusion equation and the Hori representation have been used to derive an exact functional RG equation (EFRGE) that in the course of the RG flow interpolates between the interaction part of the lattice Ginzburg–Landau Hamiltonian and the logarithm of the generating functional of the S-matrix. Because the S-matrix vertices are the amputated correlation functions of the fluctuating field, it has been suggested that in the critical region the amputation of the long-range tails makes the S-matrix functional more localized and thus more amenable to the local potential approximation (LPA) than the renormalized free energy functional used in Wilson’s EFRGE. By means of a functional Legendre transform the S-matrix EFRGE has been converted into an EFRGE for the effective action (EA). It has been found that the field-dependent part of EA predicted by the equation is the same as calculated within the known EA EFRGE approaches but in addition it is accurately accounts for the field-independent terms. These are indispensable in calculation of such important quantities as the specific heat, the latent heat, etc. With the use of the derived EFRGE a closed expression for the renormalization counterterm has been obtained which when subtracted from the divergent solution of the Wetterich equation would lead to a finite exact expression for the EA thus making two approaches formally equivalent. The S-matrix equation has been found to be simply connected with a generalized functional Burgers’ equation which establishes a direct correspondence between the first order phase transitions and the shock wave solutions of the RG equation. The transparent semi-group structure of the S-matrix RG equation makes possible the use of different RG techniques at different stages of the RG flow in order to improve the LPA solution.
The renormalization group (RG) equation in the self-consistent local potential approximation (SC-LPA) suggested earlier for the description of continuous phase transitions in lattice models of the Landau-Ginzburg type has been applied to the solution of the spin-1 Blume-Capel model on the simple cubic lattice. The calculated transition temperatures of both continuous and the first-order phase transitions (FOPTs) in zero external field have been found to be in excellent agreement with the best available estimates. It has been argued that the SC-LPA RG equation may give more accurate and complete description of the FOPTs than those reported in alternative approaches. It has been shown that the SC-LPA RG equation can be cast in the form of the generalized Burgers' equation (GBE). In this formulation of the RG the FOPTs have been shown to assume the form of the shock-wave solutions of GBE in the inviscid limit. Universality of the RG flow in the vicinity of the fixed point describing FOPTs has been discussed.
Abstract The known approximation schemes for the solution of the Wetterich exact renormalization group (RG) equation are critically reconsidered, and a new truncation scheme is proposed. In particular, the equations of the derivative expansion up to the ∂ 2 order for a scalar model are derived in a suitable form, clarifying the role of the off-diagonal terms in the matrix of functional derivatives. The natural domain of validity of the derivative expansion appears to be limited to small values of q / k in the calculation of the critical two-point correlation function, depending on the wave-vector magnitude q and the infrared cut-off scale k . The new approximation scheme has the advantage to be valid for any q / k , and, therefore, it can be auspicious in many current and potential applications of the celebrated Wetterich equation and similar models. Contrary to the derivative expansion, derivatives are not truncated at a finite order in the new scheme. The RG flow equations in the first approximation of this new scheme are derived and approximately solved as an example. It is shown that the derivative expansion up to the ∂ 2 order is just the small- q approximation of our new equations at the first order of truncation.
By simply applying the Local Potential Approximation (LPA) on the Polchinski's Exact Renormalization Group (ERG) flow equation for single Bosonic and spinless Fermionic fields, and initially considering only the coarse-graining (blocking) aspect of Wilson's Renormalization Group program. Within the LPA limit the Polchinski's ERG flow equation simplifies into a heat differential equation for both Bosonic and Fermionic fields. Solving the differential equations leads to logarithmic interactions (logarithmic vertex function) in both Bosonic and Fermionic fields at their fixed points.
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.
For the n-vector model of the general type a renormalization-group (RG) scheme in momentum space is developed. Using this scheme we obtain a generalized exact equation of the renormalization group. The main idea of this approach is not to allow the zero-order critical Hamiltonian to move away from the critical surface under the influence of the RG transformation. The equation found differs from the known ones in that all redundant operators are excluded. On the basis of the developed approach we argue that critical exponents are functionals of some function introducing the cutoff into the phase-transition theory.