Critical dynamics within the real-time fRG approach

The Schwinger-Keldysh functional renormalization group (fRG) developed in [1] is employed to investigate critical dynamics related to a second-order phase transition. The effective action of model A is expanded to the order of $O(\partial^2)$ in the derivative expansion for the $O(N)$ symmetry. By solving the fixed-point equations of effective potential and wave function, we obtain static and dynamic critical exponents for different values of the spatial dimension $d$ and the field component number $N$. It is found that one has $z \geq 2$ in the whole range of $2\leq d\leq 4$ for the case of $N=1$, while in the case of $N=4$ the dynamic critical exponent turns to $z<2$ when the dimension approach towards $d=2$.


I. INTRODUCTION
Non-equilibrium critical dynamics might play a significant role, when quark-gluon plasma (QGP) produced in relativistic heavy ion collisions evolves into the critical region of the critical end point (CEP) in the QCD phase diagram [2][3][4][5], though recently it has been found the critical region of QCD is extremely small [6].This is quite relevant for the search of CEP in experiments under way currently at, e.g., Relativistic Heavy Ion Collider (RHIC) [7,8] and other facilities.In the critical region, dynamics is dominated by the massless modes, and the significantly increased correlation length results in the well-known critical slowing down [9,10].On the other hand, the dynamics in the critical region is simplified, since it does not depend on the details of interactions of different systems, but is rather governed by some universal properties, which has been discussed in detail in the seminal paper [11].
In our former work [1], we have developed the formalism of fRG formulated on the Schwinger-Keldysh closed time path [33,34], see also, e.g., [35][36][37][38] for some relevant reviews about the Schwinger-Keldysh path integral.By the use of the Keldysh rotation, one is able to express the We begin with the dissipative relaxation model with no conservation laws, which is classified as model A in the seminal paper [11].The equation of motion for the scalar field of N components ϕ a with a = 0, 1, • • • N − 1 is described by the Langevin equation, viz., with a functional of fields for the free energy where Z ϕ (ρ) and V (ρ) are the field-dependent wave function and effective potential, respectively.The notations ∂ i = ∂/∂x i , ρ = ϕ 2 /2 with ϕ 2 = ϕ a ϕ a are used, and summation is assumed for repeated indices.Note that the O(N ) symmetry in Eq. ( 2) is explicitly broken by the last linear term in σ ≡ ϕ a=0 , with the breaking strength c.In Eq. ( 1) the diffusion constant Γ describes the relaxation rate and the last term denotes the Gaussian white noises with vanishing mean value, i.e., ⟨η a (x, t)⟩ = 0, and nonzero two-point correlations, as follows with the temperature T .
In this work, we would like to study the critical dynamics of model A within the functional renormalization group formulated on the Schwinger-Keldysh closed time path.The real-time fRG with the Schwinger-Keldysh path integral and the relevant techniques thereof have been discussed in detail in our former work [1], see also [41,42].Following the approach there, one is able to arrive at the renormalization group (RG) scale dependent effective action corresponding to Eq. ( 1) with Eq. ( 2), that is, with ∂ t = ∂/∂t, where ϕ c and ϕ q stand for the "classical" and "quantum" fields, respectively.Note that in Eq. ( 4) there are derivatives of the wave function and the effective potential, i.e., with ρ c = ϕ 2 c /4.The action here is expanded to the order of O(∂ 2 i ) in the derivative expansion.The kinetic coefficient Z t,k in Eq. ( 4) is related to the relaxation rate in Eq. (1) through Z t,k ∼ 1/Γ.Quantities with suffix k indicate their dependence on the RG scale.From the effective action in the Schwinger-Keldysh field theory in Eq. ( 4), one is able to obtain the retarded, advanced, and Keldysh propagators, three-and four-point vertices, etc., which are discussed in detail in Appendix A.

III. FLOW EQUATIONS
The flow equation of effective potential can be obtained from the flow of one-point function as shown in Fig. 1.
1: Diagrammatic representation of the flow equation for the effective potential, obtained from the one-point correlation function of the effective action with an external leg of ϕ q .Here τ stands for the RG time τ = ln(k/Λ) with some reference scale Λ.The partial derivative ∂τ hits the k-dependence only through the regulator in propagators.See Appendix A for more details about the Feynman rule.
By employing the Feynman rules for the propagators and vertices in Appendix A, one is able to obtain the flow equation of effective potential (first-order derivative w.r.t. the field), to wit, where we have used ρ in replace of ρ c without ambiguity.The RG time is τ = ln(k/Λ) with Λ being some reference scale, e.g., an ultraviolet cutoff.The angular integral in d dimension gives rise to a constant where Γ(d/2) is the gamma function.In Eq. ( 6) one also has with Z ϕ,k = Z ϕ,k (ρ 0 ), that is field-independent, and ρ 0 is usually chosen to be the position of the minimum of potential, i.e., V ′ k (ρ 0 ) = 0.The anomalous dimension reads The dimensionless renormalized meson masses in Eq. ( 6) read where the bare masses are shown in Eq. (A4).In Appendix B it has been demonstrated that the flow of effective potential in the mesoscopic relaxation model in Eq. ( 6) corresponds to the high temperature limit of the flow of effective potential in the microscopic Klein-Gordon theory [1].
FIG. 2: Diagrammatic representation of the flow equation for the inverse retarded propagator, see Eq. (A1) or Fig. 8.
In fact in order to investigate scaling properties of Eq. ( 6), it is more convenient to adopt dimensionless renormalized variables, such as Then, one is left with The flows of the wave function renormalization Z ϕ,k (ρ c ) and the kinetic coefficient Z t,k in the effective action in Eq. ( 4) can be extracted from the flow equation of the two-point function, e.g., the inverse retarded propagator in Eq. (A1).The flow of the inverse retarded propagator is shown in Fig. 2. Inserting the different propagators, three-and four-point vertices in App.A into the flow equation, one is able to close the equations.The computation is straightforward, though a bit tedious.It is obvious from Eq. (A1) that the flow of the wave function renormalization can be obtained by performing the projection as follows Note that there is no summation for the index of field component a.In the same way, one finds for the kinetic coefficient In this work the projections in Eq. ( 12) and Eq. ( 16) are made on the pion field, i.e., the field component a ̸ = 0.Then, one arrives at with where the flat regulator, cf.Eq. (A7), is used, and Θ(x) denotes the Heaviside step function.
The flow of the kinetic coefficient reads The static anomalous dimension in Eq. ( 8) can be obtained by evaluating Eq. ( 14) at the minimum of potential ρ0 , and one arrives at with where the last equation follows from the requirement Note that since the field dependence of the kinetic coefficient is neglected, it is a natural choice to compute its flow in Eq. ( 16) at the physical point ρ0 .With the static anomalous dimension η in Eq. ( 8) or Eq. ( 18) and dynamic anomalous dimension η t in Eq. ( 17), one is able to calculate the dynamic critical exponent [11] It is interesting to find that the flows of the effective potential in Eq. ( 11) and the wave function in Eq. ( 14) do not receive contributions from the dynamical variable η t , indicating that the dynamics is decoupled from the static properties in the truncation as shown in the effective action in Eq. ( 4).In fact, in more sophisticated truncations, for instance, when the momentum or frequency dependence of the kinetic coefficient Z t,k is taken into account, there is no decoupling any more.Moreover, if the field dependence of the wave function is ignored, the truncation then is reduced to the modified local potential approximation, usually denoted by LPA ′ .Then the static anomalous dimension reads and the dynamic anomalous dimension is given by If the static anomalous dimension in Eq. ( 22) is assumed to be vanishing, i.e., η = 0, the truncation then returns to the local potential approximation (LPA).

IV. NUMERICAL RESULTS
In this section we solve fixed-point solutions of the flow equations of the effective potential and wave function numerically, that is ∂ τ u ′ (ρ) = 0 and ∂ τ z ϕ (ρ) = 0.In this work we focus on the Wilson-Fisher fixed point which is characterized by just one relevant eigenvalue of eigenperturbations around the fixed point [44,45], and this relevant eigenvalue is usually denoted by 1/ν, where ν is one of two static critical exponents besides the anomalous dimension η.
In this work we employ two different numerical methods to solve the fixed-point equations.One is the conventional grid method where the potential u ′ (ρ) and wave function z ϕ (ρ) are discretized on a grid of ρ.The other one is the high-precision direct integral of the fixed-point equation, that is recently proposed in [46], and more details can be found there.We find that these two different numerical methods produce identical results.
In Fig. 3 we show the fixed-point solutions of the global potential and wave function.It is found that with the decrease of dimension d, the zero crossing point ρ0 of u ′ (ρ), i.e., u ′ (ρ 0 ) = 0, moves right towards the direction of larger ρ.One can also find that with the decrease of dimension d, the dependence of the wave function z ϕ (ρ) on the field ρ becomes stronger, which indicates that when the dimension is small, say d ≲ 3, the field dependence of the wave function should be taken into account.In Fig. 4 we show the critical exponent ν as a function of the dimension d.Obviously, results obtained from the three truncations are convergent when the dimension is d ≳ 3. Deviations are observed in the region of small d, in particular in the vicinity of d = 2.In fact, the numerical calculations become more and more difficult as the dimension is approaching d = 2.This is already indicated in the results of the potential in the left panel of Fig. 3.The zero crossing point ρ0 is divergent when one has d = 2 and N ≥ 2. Therefore, the calculation of derivative expansion ceases at a value of d, where the computation is quite time-consuming.
The results of static anomalous dimension η are presented in Fig. 5. Since one has η = 0 in LPA, only two truncations are compared.One finds that the static anomalous dimension as a function of the dimension is monotonic for N = 1, whereas there is a non-monotonic dependence in the case of N = 4.This is closely related to the Mermin-Wagner-Hohenberg theorem [47][48][49], i.e., there is no phase transition in d = 2 dimension for the O(N ) symmetry with N ≥ 2. In Fig. 6 we show the dynamic anomalous dimension η t .Similar with the static one, the dependence of η t on the dimension is monotonic in N = 1 but non-monotonic for N ≥ 2.Moreover, one can see η t = 0 at d = 4, indicating there is no critical fluctuation or critical slowing down at the Gaussian fixed point.With the static and dynamic anomalous dimensions, one can obtain the dynamic critical exponent through Eq. ( 21).The relevant results are presented in Fig. 7.In the same way, we use three different truncations.Moreover, we also compare with the result of ϵ = 4 − d expansion in the order of three loops [12,50], which reads with the constant c c = 0.726(1 − 0.1885ϵ) , and the static anomalous dimension in the three-loop order [51] One can see that when the dimension is very close to d = 4, say d ≳ 3.5, where the ϵ expansion should work well, our results obtained with the LPA ′ or the derivative expansion are comparable with the ϵ expansion.However, the LPA computation overestimates the dynamic critical exponent apparently, since the static anomalous dimension in LPA is neglected.Furthermore, in the whole range of 2 ≤ d ≤ 4, our results prefer z ≥ 2 for the case of N = 1.The case of N = 4 is, however, more intricate, and one can see that the dynamic critical exponent calculated with the derivative expansion turns to z < 2 when the dimension is below d = 2.5.

V. SUMMARY AND CONCLUSIONS
In this work, we have used the Schwinger-Keldysh fRG developed in [1] to study critical dynamics related to a second-order phase transition.As a concrete example, a dissipative relaxation model classified as model A is employed.In the formalism of Schwinger-Keldysh fRG, the RG scale dependent effective action is expressed in terms of two different kinds of fields.One is similar with the physical classical field, i.e. the "classical" field, and the other plays the role of fluctuations, called the "quantum" field.This formalism of double fields provides us with a very powerful approach to study real-time critical dynamics by employing systematic expansions for the truncation.For example, the derivative expansion can be applied to the sector of classical fields.For the sector of quantum fields, one is able to study the transition from a microscopic theory to a mesoscopic model, e.g., the semiclassical limit of a quantum action, see e.g.[38] for more relevant discussions.
We expand the effective action of O(N ) symmetry to the order of O(∂ 2 ) in the derivative expansion.The flow equations of the effective potential, wave function and the kinetic coefficient are obtained.By solving the fixedpoint equations of the dimensionless renormalized potential and wave function, one is able to find the solution of the Wilson-Fisher fixed point and the relevant static and dynamic critical exponents.It is found that both the static anomalous dimension η and the dynamic anomalous dimension η t behave as monotonic functions of the spatial dimension d in the range of 2 ≤ d ≤ 4 in the case of N = 1, whereas they are both non-monotonic when N ≥ 2.
The dynamic critical exponent z is obtained as a function of the spatial dimension d for different values of N .Our results are also compared with those of ϵ expansion in the order of three loops.It is found that results obtained from derivative expansion and LPA ′ are consistent with that from the ϵ expansion when the dimension is close to d = 4, while LPA overestimates the dynamic critical exponent.Furthermore, we find that z ≥ 2 in the whole range of 2 ≤ d ≤ 4 for the case of N = 1, while in the case of N = 4 the dynamic critical exponent turns to z < 2 when the dimension approach towards d = 2. scales are integrated in successively with the evolution of RG scale [52].It is a nonperturbative continuum field theory, see, e.g., [3,53] for reviews and [54][55][56][57][58][59] for recent progresses.

2 FIG. 3 :
FIG. 3: Derivative of the effective potential u ′ (ρ) (left panel) and the wave function z ϕ (ρ) (right panel) at the Wilson-Fisher fixed point as functions of ρ for the O(N ) symmetry with N = 4. Three different values of dimension d are chosen.

FIG. 4 :FIG. 5 :
FIG. 4: Critical exponent ν as a function of the dimension d for the O(N ) symmetry with N = 1 (left panel) and N = 4 (right panel).Three different truncations, LPA, LPA ′ and derivative expansion up to the order of O(∂ 2 ) are used, and their respective results are compared.

FIG. 6 :FIG. 7 :
FIG. 6: Dynamic anomalous dimension η t as a function of the dimension d for the O(N ) symmetry with N = 1 (left panel) and N = 4 (right panel).Results obtained in three different truncations are compared.