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Renormalization Group Calculation of Dynamic Exponent in the Models E and F with Hydrodynamic Fluctuations M

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Renormalization Group Calculation of Dynamic Exponent in the Models E and F with Hydrodynamic Fluctuations M Vol. 131 (2017) ACTA PHYSICA POLONICA A No. 4 Proceedings of the 16th Czech and Slovak Conference on Magnetism, Košice, Slovakia, June 13–17, 2016 Renormalization Group Calculation of Dynamic Exponent in the Models E and F with Hydrodynamic Fluctuations M. Dančoa;b;∗, M. Hnatiča;c;d, M.V. Komarovae, T. Lučivjanskýc;d and M.Yu. Nalimove aInstitute of Experimental Physics SAS, Watsonova 47, 040 01 Košice, Slovakia bBLTP, Joint Institute for Nuclear Research, Dubna, Russia cInstitute of Physics, Faculty of Sciences, P.J. Safarik University, Park Angelinum 9, 041 54 Košice, Slovakia d‘Peoples’ Friendship University of Russia (RUDN University) 6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation eDepartment of Theoretical Physics, St. Petersburg University, Ulyanovskaya 1, St. Petersburg, Petrodvorets, 198504 Russia The renormalization group method is applied in order to analyze models E and F of critical dynamics in the presence of velocity fluctuations generated by the stochastic Navier–Stokes equation. Results are given to the one-loop approximation for the anomalous dimension γλ and fixed-points’ structure. The dynamic exponent z is calculated in the turbulent regime and stability of the fixed points for the standard model E is discussed. DOI: 10.12693/APhysPolA.131.651 PACS/topics: 64.60.ae, 64.60.Ht, 67.25.dg, 47.27.Jv 1. Introduction measurable index α [4]. The index α has been also de- The liquid-vapor critical point, λ transition in three- termined in the framework of the renormalization group dimensional superfluid helium 4He belong to famous ex- (RG) approach using a resummation procedure [5] up to amples of continuous phase transitions. It is known that the four-loop perturbation precision and was measured regarding large-scale behavior such model lies in the same in the Shuttle experiment [6]. A contemporary accepted universality class as the XY model [1, 2]. This fact is value is α = −0:0127. The negative sign of the index α ∗ related to the observation that in both cases we have ensures that g2 = 0 at the stable fixed point. This me- two-component order parameter. The divergent length ans that the stability of model E can be considered as a at criticality reveals itself not only in static but also dy- particular realization of model F. Our main aim here is namic properties of the system [1]. An analysis of in- the calculation of the dynamic exponent z and a stability frared (IR) divergences and quantitative predictions of analysis of different IR scaling regimes due to an inclusion universal quantities are therefore indispensable in under- of velocity fluctuations in three-dimensional universality standing the dynamic behavior of spin systems. When class of the XY model. a system approaches the critical temperature at which 2. Dynamics of model F a phase transition occurs, the relaxation time τ diver- with hydrodynamic modes activated ges with the correlation length ξ as ξz. The exponent Models E and F with an activated hydrodynamic mo- z defines so-called dynamical critical exponent [1]. It is des were proposed and investigated by the RG method well known [1, 2] that to a given static universality class in [3, 7]. Let us refer to model F with activated hydro- different dynamic classes can be assigned. In contrast dynamic modes as model Fh. Corresponding field theo- to the static models now the proper slow modes have retic model given in terms of the De Dominicis–Janssen to be identified and their governing equation of motions action [1] can be written as a sum S = Snc + Sc + Sv, specified. According to the general scheme [2] the univer- where 0 0 0 sal behavior in critical region of a superfluid is captured S = 2λ y + y − @ − @ (v ) by model F. Recently this has been confirmed from mi- nc 0 t i i croscopic principles [3] using the local interaction approx- 2 y +λ0(1 + ib0)(@ − g01( ) =3 + g02m ) imation. Model F reduces to model E in a limiting case y in which coupling constants g2 and b (in our notation) +iλ0 (g07 − g03m + g03h0) + H.c. (1) are set to zero. Nowadays, there is no general consen- describes dynamics of the nonconserved order parameter sus which dynamic model (E or F) is genuine from the fields ( ; y) and H.c. stands for a Hermitian conjugate. point of view of experimentally measurable quantities. Further, the action In the corresponding static model, one of the ! indices 0 2 0 0h coincides directly with the well-known, experimentally Sc = −λ0u0m @ m + m − @t − vi@im 2 y −λ0u0@ (−m + g06 + h0) y 2 2 y i ∗corresponding author; e-mail: [email protected] +iλ0g05( @ − @ ) (2) (651) 652 M. Dančo et al. 5 3 2 2 describes dynamics of the conserved field m, which can be A2 = −6b u + 2b u[u − 6u − 6] − 6bu(1 + u) ; interpreted as a linear combination of density and tem- 4 2 2 3 perature field [4]. The velocity fluctuations are governed A3 = b (1 + 3u) − b (1 + u)[u − 4u − 2] + (1 + u) ; by the following action: A = 2ub5 + b3u[4 + 2u − 3u2] 1 4 S = v0Dv v0 + v0{−@ v + ν @2v − v @ v g; (3) v 2 i ij j i t i 0 i j j i −bu(1 + u)2[u2 + 2u − 2]; v where the explicit form of Dij can be found, e.g., in [4] 4 2 2 3 or [7]. For simplicity integrals over spatial and time vari- A5 = −2b u + 3b u (1 + u) + u(1 + u) (2 + u); ables in (1)–(3) have been omitted. The intermode cou- 6 2 pling of fields and y with the field m in (1) and (2) A6 = −2ub + b u(1 + u)[u(u − 2)(4 + u) − 6] corresponds physically to the exact relation between the 4 3 phase of the complex order parameter and the chemical +ub (u − 3)(2 + 3u) − u(1 + u) (2 + u); potential [2]. The interactions with velocity field vi are 5 3 2 A7 = 2ub + b u(4 + 2u − 3u ) introduced [4] via standard replacement @t ! @t + vi@i and from this point of view the passive advection is intro- +bu(2 + 2u − 3u2 − 4u3 − u4); duced into the model. We consider the velocity field to 2 be incompressible which is tantamount to the condition 3u1(1 + u1) A8 = − : (7) @ivi = 0 [4]. 8 The field-theoretic formulation summarized in These relations are in agreement with [7, 9] in the spe- Eqs. (1)–(3) has an advantage to be amenable to the cial limit obtained for b = g2 = g6 = g7 = 0, which full machinery of field theory [4]. Near criticality large corresponds to model E . fluctuations on all scales dominate the behavior of the h system, which results into the infrared divergences in 3. Scaling regimes and fixed-point structure the Feynman diagrams. The RG technique allows us to deal with them and as a result provides us with Scaling regimes are associated with fixed points of the RG equations. The fixed points are defined as the points information about the scaling behavior of the system. ∗ ∗ ∗ ∗ ∗ Moreover, it also leads to a perturbative computation g = (g1 ; ··· ; g7 ; u ; u1) at which all β-functions simul- of critical exponent in a formal expansion around the taneously vanish, i.e. ∗ upper critical dimension. In contrast to the standard βe(g ) = 0; e 2 fg1; ··· ; g7; u; u1g: (8) '4-theory we have to deal with two-parameter expansion The type of the fixed point is determined by the eigen- ("; δ), where " is the deviation from the upper critical values of the matrix of its first derivatives ! = f!ij = dimension dc = 4, and δ describes non-local behavior @βi=@ejg, where βi is the full set of β functions and ej is of random noise for velocity fluctuations. It follows the the full set of charges. The IR-asymptotic behavior is go- approach suggested in original work [8]. The model verned by IR-stable fixed points, for which all real parts under consideration is augmented [4, 9] by the following of eigenvalues of the matrix ! are positive. In fact, there relations between the charges: are two physically possible and interested regimes. The first one is the regime with hydrodynamic fluctuations g05 = g03; g06 = g02; g07 = g02g03: (4) The introduction of the new coupling constants are near the thermodynamic equilibrium that corresponds to needed in order to restore the multiplicative renor- the values " = 1 and δ = −1. The second one is the Kol- malizability of the model. Details of the perturbative mogorov turbulent regime with " = 1 and δ = 4. renormalization group calculations can be found in [9]. 4. Model Fh Here, we concentrate on the calculation of the dynamic exponent z, which has not been done previously. To A majority of the fixed points can be found only in this end we need to determine the anomalous dimension a numerical fashion. A fraction of them can be imme- γλ [1, 4], because the latter determines z through the diately discarded due to unacceptable values of physical relation parameters. This is why we have attempted to investi- gate the system specifically in different regimes, rather z = 2 − γ∗; (5) λ then solving it directly. In the turbulent scaling regime where the asterisk denotes the fixed point value. The numerical analysis reveals an IR stable fixed point with one-loop expression for γλ can be written in the form coordinates 2 2 ∗ ∗ ∗ γλ = g2A1 + g2g3A2 + g3A3 + g2g5A4 + g3g5A5 g4 = 10:6; u = 1; u1 = 0:7675919; . 2 23 ∗ ∗ ∗ ∗ ∗ ∗ ∗ +g2g6A6 + g3g6A7 + g4A8 b + (1 + u) ; (6) b = g1 = g2 = g3 = g5 = g6 = g7 = 0; (9) where where the overline symbol stands for a repeating decimal.
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