Critical exponents of the O(N)-symmetric $$\phi ^4$$ model from the $$\varepsilon ^7$$ hypergeometric-Meijer resummation

Abstract

We extract the $$\varepsilon $$ ε -expansion from the recently obtained seven-loop g-expansion for the renormalization group functions of the O(N)-symmetric model. The different series obtained for the critical exponents $$\nu ,\ \omega $$ ν , ω and $$\eta $$ η have been resummed using our recently introduced hypergeometric-Meijer resummation algorithm. In three dimensions, very precise results have been obtained for all the critical exponents for $$N=0,1,2,3$$ N = 0 , 1 , 2 , 3 and 4. To shed light on the obvious improvement of the predictions at this order, we obtained the divergence of the specific heat critical exponent $$\alpha $$ α for the XY model. We found the result $$-0.0123(11)$$ - 0.0123 ( 11 ) which is compatible with the famous experimental result of $$-0.0127(3)$$ - 0.0127 ( 3 ) from the specific heat of zero gravity liquid helium superfluid transition while the six-loop Borel with conformal mapping resummation result in literature gives the value $$-0.007(3)$$ - 0.007 ( 3 ) . For the challenging case of resummation of the $$\varepsilon $$ ε -expansion series in two dimensions, we showed that our resummation results reflect a significant improvement to the previous six-loop resummation predictions.

Self-similar sequence transformation for critical exponents

“…Our aim in the present paper is to demonstrate that the accuracy of self-similar factor approximants for the O(N) -symmetric ϕ 4 theory in three dimensions can be drastically improved by employing the available seven-loop ε-expansions that are known for N = 1 [25] and are derived, using the seven-loop coupling parameter expansions [26], in Ref. [27] for N = 0, 1, 2, 3, 4. These expansions are as follows.…”

“…The results of this calculation for the exponent ν are illustrated in Table 1. Following the same procedure for the exponents η and ω, we obtain the values shown in Table 2, where we compare the results obtained by means of factor approximants (FA) with those of other methods: Monte Carlo simulations (MC) [28][29][30][31][32][33], Conformal bootstrap (CB) [34][35][36][37][38], Hypergeometric Meijer summation (HGM) [27], Borel summation complimented by additional conjectures on the behavior of coefficients (BAC) [39], Borel summation with conformal mapping (BCM) [40], and Nonperturbative renormalization group (NPRG) [5,[41][42][43].…”

“…4, where we compare our results with the most accurate values obtained by other methods summarized in Refs. [5,[19][20][21]. We compare our results with Monte Carlo simulations, conformal bootstrap, hypergeometric Meijer summation, Borel summation, Borel summation with conformal mapping, and the method of nonperturbative renormalization group.…”

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

“…The evolution equation for u should be obtained by substituting (36) into (28). However, in general case the equation will be incompatible with the LPA ansatz so the task is to satisfy it approximately.…”

“…To this end in the present paper a multi-step renormalization technique will be suggested that makes possible the use of different renormalization methods at different stages of the RG flow. For example, this will allow one to use LPA within the transient region at an early stage of renormalization when the interactions are strong [1] and to switch to the perturbative treatment in the critical region where, for example, the DE expansion can be efficient [22,27,28].…”