- 1Laboratory of Semiconductor Physics, Institute of Technical Physics, Faculty of Materials Science and Applied Chemistry, Riga Technical University, Riga, Latvia
- 2Institute of Science and Innovative Technologies, University of Liepaja, Liepaja, Latvia
- 3The MS2 Discovery Interdisciplinary Research Institute, Wilfrid Laurier University, Waterloo, ON, Canada
- 4BCAM - Basque Center for Applied Mathematics, Bilbao, Spain
We consider the non-perturbative renormalization group (RG) equations, obtained as approximations of the exact Wetterich RG flow equation within the Blaizot–Mendez–Wschebor (BMW) truncation scheme. For the first time, we derive explicit RG flow equations for the scalar model at the arbitrary order of truncation. Moreover, we consider original, as well as modified, approximations, used to obtain a set of closed equations. We compare these equations at the s = 2 order of truncation with those recently derived in J. Phys. A: Math. Theor. 53, 415002 (2020) within a new truncation scheme and find a striking similarity. Namely, the first-order equations of the latter scheme, those of the original BMW scheme, and those of the modified BMW scheme (at s = 2) differ only in one term. We solved these equations by a recently proposed and tested method of semi-analytic approximations. Thus, the critical exponents η, ν, and ω were evaluated, recovering also the known results of the original BMW scheme. In addition, we estimated the subleading correction-to-scaling exponent ω2 for the three equations considered. To the best of our knowledge, this exponent has not yet been extracted from the Wetterich equation beyond the local potential (the zeroth order) approximation. Our current estimate for the 3D Ising model is ω2 = 2.02 (40), where the error bars include the expected truncation error in the BMW scheme.
1 Introduction
The renormalization group (RG) method is one of the most widely used approaches in the analysis of critical phenomena [1–5]. Here, we consider the non-perturbative RG approach [6–9], focusing on the Wetterich equation and its approximation schemes, developing further the research initiated in [10].
The functional renormalization group approach, which is at the heart of models such as the Wetterich and the Polchinski equations, provides an appropriate framework for the analysis of a range of central problems in many areas, including quantum systems, Yang–Mills theories, and statistical physics, see, e.g., [11], for a review. The Wetterich and Polchinski equations of the functional renormalization group belong to the so called non-perturbative RG equations. They have been successfully applied to many particular systems, including the Ginzburg–Landau model [10–18], quantum models and quantum gravity [19–30], and the tensorial group field theory [31]. It has been applied to equilibrium, as well as out-of-equilibrium systems and critical dynamics [32–34]. A recent review of all these applications is given in [35].
While the Wetterich equation itself is exact, it cannot be solved exactly. Approximate closed equations are obtained from it, applying certain truncation schemes for the effective action. The local potential approximation (LPA) is a widely known lowest-order approximation, which is traditionally improved step by step by the derivative expansion (DE) [9,14]. As alternative truncation schemes, the vertex expansion [14], the BMW scheme [17], and the recent scheme proposed in [10] can be mentioned. In distinction to the derivative expansion, which relies on the small momentum (wave vector magnitude) approximation, the latter two schemes preserve the full momentum dependence.
In this paper, we report the results of a significant further development of the BMW scheme for the scalar model, deriving explicit equations for calculations at the arbitrary order of truncation within this scheme, as well as proposing useful modifications of the original BMW scheme. Moreover, we established an intrinsic link between the equations of the BMW scheme and those of the recent scheme proposed in [10] at the first order of truncation. A recently developed method of semi-analytic approximations [36] was used to solve these equations.
The usefulness of our new developments is demonstrated in example calculations for the scalar model in three dimensions, where we evaluate the critical exponents, including the subleading correction-to-scaling exponent ω2, for which quite limited results are available in the literature. Apparently, such results for ω2 are obtained here for the first time from the Wetterich equation beyond the LPA, and we found only a relatively old estimate ω2 = 1.67(11) from the Polchinski equation at the
Our paper is organized as follows. The Wetterich equation is reviewed in Sec. 2, and the BMW scheme of its truncation is reviewed in Sec. 3. Our new developments of the BMW scheme, including the equations at the arbitrary order of truncation, are presented in Sec. 4 with details of the derivation being given in Supplementary Material SA1. The comparison of equations of the BMW scheme with those derived in [10] is made in Sec. 5, providing the dimensionless form of these equations in Sec. 6. The results of numerical calculations are collected in Sec. 7. The summary of results and outlook, pointing to potential further applications and developments, are provided in Sec. 8.
2 Wetterich equation
In statistical physics, equilibrium systems of interacting particles are routinely described by the action S = H/(kBT), where H is the Hamiltonian and kB is the Boltzmann constant. A basic example is the Ising model with S [σ] = −β∑⟨ij⟩σiσj − h∑iσi, where σi = ±1 is the spin variable at the ith lattice site, ⟨ij⟩ denotes the pairs of nearest neighbors, β is the coupling constant, and h is the normalized (divided by kBT) interaction energy between a spin with σi = ±1 and the external field. Another simple example is the φ4 model with
It is convenient to use the wave vector representation of the action S[φ] via the Fourier transformation φ(x) = Ω−1/2 ∑qφ(q)eiqx, where Ω is the volume of the system. The critical behavior is related to the long-wavelength (infrared) or small-|q| fluctuations [3], but one should also control the ultraviolet fluctuations. A point here is that S[φ] might be ill-defined without an appropriate upper or ultraviolet cut-off in the wave vector (momentum) space. In the φ4 model, one needs to introduce such a cut-off, e.g., by setting φ(q) ≡ 0 at q > Λ (where q = |q|) to avoid ultraviolet divergences in various calculations, e.g., in the calculation of ⟨φ2⟩. The cut-off parameter Λ refers to the shortest or “microscopic” length scale ∼ π/Λ of the model, like the lattice constant in the Ising model.
The general idea of the RG method is to look at a system on different length scales. In the Wetterich functional RG approach [6,9], this idea is realized by introducing an infrared running cut-off scale k, in such a way that fluctuations with q ≲ k are suppressed, those with q ≫ k being practically unaffected. For this purpose, the action is modified as S[φ] → S[φ] + ΔSk[φ] [9], where
Here, φ is an N-component vector with components φi, whereas Rk(q) is the infrared cut-off function with properties Rk(q) → 0 at k → 0 and Rk(q) → ∞ at k → ∞ for fixed q. A simple choice, proposed in [6,9], is
For q2 ≪ k2, this cut-off behaves as Rk(q) ∼ k2. It means that the Fourier modes φi(q) with small momenta q < k acquire a weight factor or effective mass ∼ k included in the term (1). This additional mass acts as an effective infrared cut-off for these low-momentum modes. More precisely, this effect is essential in vicinity of the critical point, where Rk(q) smoothly cuts off the critical infrared fluctuations at a finite k. The cut-off function (2) tends (exponentially fast) to zero at large q/k values; therefore, the high-momentum modes with q/k → ∞ are not disturbed by ΔSk[φ].
Further on, an averaged order parameter ϕ(x) = ⟨φ(x)⟩ is introduced, where the averaging is performed over φ(x) in the presence of external sources J(x). The averaged field ϕ(x) depends on J(x), and this dependence is affected by k, i.e., ϕ = ϕk(J). There exists also the inverse relation in the sense of the Legendre transformation J = Jk(ϕ). The effective average action Γk[ϕ] is considered as a functional of ϕ according to the definition
where Ji are the components of the vector J(x) and
In (3), J(x) is the Legendre transform of ϕ(x), i.e., J = Jk(ϕ).
An exact RG flow equation, describing the variation of Γk[ϕ] with the infrared cut-off scale k, has been obtained in [6]. A detailed non-perturbative derivation has been later reported in [9]. This equation, called the Wetterich equation, reads
In this equation, k decreases toward k = 0, starting from some initial value k0. Thus, it generally allows describing the physics of various models with various choices of the order-parameter field by smoothly interpolating between different length scales.
Starting with k0 = Λ (as in our calculations), only the fluctuations at the microscopic length scale q ∼Λ are properly taken into account in the initial stage of the integration of RG flow Eq. 5. Fluctuations of the original model (where ΔSk = 0) with longer and longer wavelengths are gradually restored as k → 0. At k = 0, all fluctuation modes are included so that Γk=0 [ϕ] is the effective action of the original model, defined by (3) without the term ΔSk [ϕ]. Another possibility, discussed in [35], is to choose k0 ≫Λ. In this case, all fluctuations are frozen at the beginning (since the original model contains only the modes with q < Λ, and these are suppressed at k ≫Λ) and
For the O(N) model, ϕ(x) is a vector with components ϕj(x), where j = 1, … , N. It is convenient to consider (5) in the space of wave vectors q, where
Ω being the volume of the system, for which periodic boundary conditions are assumed. The wave vectors are restricted by the upper cut-off, i.e., q < Λ. In this representation, the quantity
For the discrete wave vectors in (6), the cut-off function Rk is represented as [10]
The choice of Rk(q) is not unique, and several possibilities have been considered in [11,18]. For simplicity, here we have limited our choice to
traditionally used in many investigations [9,14,17], where Zk is a renormalization constant and α is an optimization parameter. This cut-off function coincides with (2) and, thus, has the required properties discussed before. It allowed us to compare the results with those of [17], where the same cut-off function was used.
3 The BMW scheme of truncation
Here, we review the BMW truncation scheme for the scalar model, considered in detail in [17], noting that the N-component case has also been considered there in the Appendix. In the BMW truncation scheme, one considers the s-point functions
The basic idea of this truncation scheme is to consider first a hierarchy of exact equations for these s-point functions, obtained by applying the functional derivatives to (5). Approximate closed equations are then obtained, using certain approximations of the higher order functions (at orders s + 2 and s + 1) by the lower-order functions (up to the order s).
For constant ϕ, one has
where Vk(ϕ) is the potential. The exact RG flow equation for it is obtained by evaluating the Wetterich equation at ϕ = const. Using the notations t = ln k,
where the integration is performed over the region q < Λ, this equation reads
where
is the two-point correlation function with
In the zeroth-order approximation (s = 0) or LPA, one sets [17]
where the coefficient 1 at q2 corresponds to the usually assumed coefficient 1/2 at the gradient term (∇ϕ)2 in the coordinate representation of Γk [ϕ], i.e.,
Using (16), Eq. 14 becomes a closed approximate RG flow equation for the potential Vk(ϕ).
In order to obtain the equations at the s = 2 order, one considers the RG flow for
According to the original idea of the BMW truncation scheme, a closed approximate equation is obtained from this by setting the internal momenta equal to zero, i.e., q = 0, in the arguments of
these simplified expressions for
At s = 2, the equation for
where
It is important to note that
keeping the term
4 Equations of the BMW scheme at the arbitrary order of truncation
4.1 Exact RG flow equations for the n-point functions
Here, we derive exact RG flow equations for the n-point function at any n, from which approximate closed equations at arbitrary truncation order s are then obtained, as described in Sec. 4.3. In the following, we summarize these equations, the details of the derivation being provided in Supplementary Material SA1. For brevity, we omitted ϕ in the list of arguments of the n-point functions
where
This cut-off is included here to follow a formal rigor in treating such cases, where the original action S[φ] is ill-defined without it, see Sec. 2. It influences the RG flow in the Wetterich equation only at k ∼Λ for k ≤ Λ. Therefore, it can be further omitted, considering the vicinity of the fixed point at k → 0, as it is usually performed in the literature.
For any given m1, m2, … mM at the considered n, the wave vectors Qi are defined by
where the latter sum is defined as zero for i = M (so that QM = q) and
At i = 1, Eqs 26, 27 reduce to
noting that
The known Eq. 18 can be easily recovered from (24). Following notations introduced after Eq. 15 and noting that p1 + p2 = 0 holds in (24) at n = 2, we have
to (24), noting that
At M = 2, the second sum reduces to m1 = m2 = 1 for n = 2, whereas the following sum comprises two cases: 1) j1(1) = 1, j1(2) = 2 and 2) j1(1) = 2, j1(2) = 1 (we have i = 1, 2 and ℓ = 1). In the first case, we obtain (using Eqs 26–28, where p1 + p2 = 0) P1 = p1, P2 = p2,
where
Equation 18 is obtained by summing up both contributions (31) and (33), exchanging the last two arguments of
If
which are all analytic functions of ρ = ϕ2/2. Substituting these new functions in RG flow Eq. 24, we easily obtain
where
where
4.2 Some useful relations
In this section, we introduce some exact relations, which are useful for building up the approximation schemes in Sec. 4.3.
First, we consider the structure of arguments of the highest-order term
Similarly, we consider the arguments of the
As the next step, we consider the exact relations
following from (19), where
Using these relations,
In the following, we derive such relations for the n-point functions
where
and the quantities
4.3 Closing approximations
Based on the relations of Sec. 4.2, we modify Eq. 35 for 2 ≤ n ≤ s to the form appropriate for making certain closing approximations. In addition, we consider two variants of such modifications.
1. The
2. The substitutions of
In addition, the change of variable from
is performed, which means that
in accordance with (23). In this case, the RG flow equation for Δk reads
with terms on the right-hand side being given in (35). Such a change of variable is meaningful, as it preserves the term
With the above replacements made in (35), we obtain a modified set of exact equations up to arbitrary order s. However, this set of equations is not closed because it contains the functions
(a) We set
(b) These quantities are approximated by their values at zero external momenta
where {0}n is the string 0, … , 0 of n zeroes, from which we obtain
with the help of (19), (34), and the definitions of
The closing approximations are represented by (52)–(53) at n = s, as well as by (52) at n = s − 1 if s > 2.
In summary, the final closed system of equations at the arbitrary order of truncation s is represented by Eq. 14 (where we omit ϕ for brevity) with
completed by explicit Eq. 35 for 2 ≤ n ≤ s, in which certain precisely defined replacements are performed. In particular, (49) is used for n = 2. In this sense, we derived an explicit closed system of equations for arbitrary s. There exist approximations with s = 0, 2, 3, 4, 5, etc., and only the order s = 1 does not exist.
We proposed four different versions of the replacements in (35), which can be numbered as 1a, 1b, 2a, and 2b, corresponding to options 1 and 2 for the exact modification of (35) and options (a) and (b) for the closing approximations used. Thus, we have four different modifications within the BMW scheme.
In fact, the original BMW scheme corresponds to 1a, since the q-dependence of
Option 2, including modifications for all n-point functions with 2 < n ≤ s, is indeed quite meaningful. Namely, in analogy with keeping the zeroth-order term
In fact, choice (a) relies on the smallness of the internal momentum q when approximating
From some other aspects, the versions of the BMW scheme are essentially different from the scheme of [10]. A truncation is applied directly to Γk[ϕ] in the latter scheme. Hence, one has to think how good or how well justified is the proposed specific truncation performed over an infinite set of terms or vertices contained in the exact Γk[ϕ]. It refers to any scheme, which deals with a direct truncation of Γk[ϕ]. The BMW scheme, including its current modifications, is quite different. Namely, the hierarchy of exact Eq. 24 appears in an unambiguous and natural way, and one only needs to think about an appropriate closing approximation of the highest-order n-point functions by lower order ones. The fact that this hierarchy is now explicitly written down for arbitrary n (via Eq. 24 or Eq. 35) can also be seen as an essential advantage of this scheme. According to the actual results at s = 2, discussed in the following sections, the closing approximations considered here are reasonably good, at least, as regards the critical exponents η, ν, and ω. Following the intuitive argument proposed above, we expect that the modified approximation (b) is generally better than the original one (a) since it should be better to approximate
A quite different closing approximation for the equations at s = 2 has been proposed in Appendix B of [17]. However, our first result obtained by this approximation was unsatisfactory, i.e., η = 0 at the simplest choice p0 = ρ0 = 0 of the free parameters p0 and ρ0 introduced in [17]. Therefore, we have further skipped this version.
5 Comparison of equations at s = 2
Here, we consider in detail the truncation order s = 2. We revealed a striking similarity between the RG flow equations in the original (option (a)) and the modified (option (b)) BMW schemes at s = 2 on one side and the scheme of [10] at the first order of truncation on the other side. According to Sec. 4 in this paper and Eq. 43 (together with the related equations) in [10], we have
where
and
noting only that Δk and Vk correspond to 2Ψk and Uk in [10], and the primes denote the derivatives with respect to ρ. Equation 56 with the first choice in (58) coincides with Eq. 23 in [17]. Thus, the RG flow equations differ only in one term ak (p, q) in the three cases considered. This is a really striking similarity, especially if we note that the equations of [10] have been derived in a completely different way than those of the BMW scheme.
It is interesting to mention that the expression for ak (p, q) in the modified BMW scheme agrees with that in the original BMW scheme at q = 0 (since Δk (0) ≡ 0), whereas it agrees with that in the scheme of [10] at p = 0.
6 Dimensionless equations at s = 2
For the application of critical phenomena, we write the RG flow equations in a scaled (dimensionless) form, using the transformations
where
corresponds to (10). In addition, the dimensionless equations contain the running exponent η(k) = −d ln Zk/dt. As in [10], the above transformations led to the following RG flow equations:
where
and
where
Here,
It should be noted that fk (0; 0) ≡ 1 holds according to the definition of Zk; therefore, η(k) is found from the condition ∂tfk (0; 0) ≡ 0. Hence, the equation for η(k) reads
The dimensionless equations of [17] are recovered with the first choice in (69), whereas those of [10] are recovered with the third choice. However,
7 Calculation results
7.1 The method of solution
We solved the dimensionless equations of Sec. 6 in three dimensions (d = 3) as an example of the application of our general truncated equations within the BMW scheme, considering here the order s = 2. Moreover, we compared the results with those of the surprisingly similar equations in [10].
We used the method of semi-analytic approximations or functional truncations developed in [36]. According to this method, the dimensionless potential
where
Here,
The coefficients um,k and fm,k(y) are calculated adapting the algorithm described in [36]. Namely,
The integration of the RG flow equations (from t = 0 to t = −17) was performed by the fourth-order Runge–Kutta method, calculating fm,k(y) on a non-uniform grid of y values yn, where y0 = 0 and yn+1 = yn + (1 + ɛ)(yn − yn−1) for n ≥ 1 and yn ≤ ymax with large enough ymax to include the region q > Λ in the original variables. Typically, we used y1 = 0.02 and ɛ = 0.4, which corresponds to the “rough” grid considered in [36]. A similar “rough” grid with y1 = 0.002 and with restricted maximal step size 0.8 was used for the calculation of the integrals by the Simpson method, cutting the integration region at ymax = 30 and integrating over the angle θ with the step size π/5 (i.e., π/10 step size for subintervals).
Further details about the numerical procedures are provided in [36], noting that the “standard” grid has been mainly used there with twice smaller step sizes for y and θ. Here, we mainly use the “rough” grid, verifying in a subset of cases that the discrepancy with the results of the “standard” grid is practically negligible, as already pointed out in [36].
7.2 Estimation of the critical exponents η, ν, and ω
We extracted the critical exponents η, ν, and ω from the RG flow by a direct integration of the RG flow equations. In this case, the fixed point is reached by adjusting the initial condition iteratively, and the critical exponents are determined from the RG flow at and near the critical surface [36]. The critical exponent η describes the ∝ k−2+η divergence of the critical two-point correlation function, as well as the ∝ k−η divergence of the renormalization constant Zk on the critical surface at k → 0. It is determined as the fixed-point value of the running exponent η(k). The critical exponent ν describes the ∝|τ|−ν divergence of the correlation length at τ → 0, where τ is the deviation from the critical temperature. It describes also an infinitesimal deviation of the RG flow from the fixed point. It is extracted from a small ∝ k−1/ν deviation of this flow at small k values. The critical exponent ω describes corrections to scaling, as well as the ∝ kω distance of the RG flow from the fixed point on the critical surface at k → 0.
Practically, we determined the exponents yT = 1/ν and ω from the RG flow of the coupling coefficient u1,k in (73). We determined yT from the
For simpler RG flow equations, the critical exponents can be easily extracted from the linearized RG analysis in the vicinity of the fixed point. It is a standard method in the cases, where the RG flow is described by a finite number N of discrete parameters. It is true, e.g., for the LPA and the derivative expansion [11,14,15,18]. In this case, all the critical exponents, including the subleading correction-to-scaling exponents, are obtained as eigenvalues of the sensitivity matrix with dimensions N × N. Such a method cannot be easily applied to the equations with full momentum dependence in [10,17,36] and here, as one deals also with RG flow equations for continuous functions, in particular, for the functions fm,k(y) in (74). In [17], the critical exponents have been evaluated by a direct integration of the RG flow equations, like in our current study. A linearization around the fixed point, found by a fast iterative method without such integration, has been applied to equations with full momentum dependence in our earlier work [10]. However, it was only a toy example for a very simple approximation.
The critical exponents, evaluated here from the approximate RG flow equations with the dimensionless cut-off function (62), depend on the parameter α in (62). The exactly solved Wetterich equation would ensure the α-independence of these exponents; therefore, often such a value of α is assumed as optimal, at which the specific critical exponent shows a minimal variation with α. It is known as the principle of minimal sensitivity (PMS) [14]. In practice, the PMS values of η, ν, and ω appear to be the extremum points of the corresponding exponent vs. α plots. We found it meaningful to consider such plots depending on ln α rather than α. In this case, the plots appear to be more symmetric, and they are better approximated by spline curves. Moreover, the modulus of the local slope of such a plot is proportional to the magnitude of the variation when α → α(1 + ɛ) at a small ɛ and, therefore, it serves as a good measure of the sensitivity.
The plots of the critical exponents η, ν, and ω vs. ln α, calculated at the truncation orders n = 13 and n′ = 10 in (73)–(74), are shown in Figure 1. In this figure, the results of the (Secs. 5–6) approximate RG equations of the original BMW scheme, the modified BMW scheme, and the scheme of [10] considered here are shown. The PMS values correspond to the extremum points, indicated by the vertical dashed lines.
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FIGURE 1. Critical exponents η (A), ω (B), and ν (C) vs. ln α provided by the approximate RG flow equations (Sec. 6) of the original BMW scheme (squares), the modified BMW scheme (solid circles), and the scheme of [10] (empty circles), calculated at n = 13, n′ = 10 in (73)–(74). The values obtained using the principle of minimal sensitivity correspond to the extremum points, indicated by the vertical dashed lines.
These PMS values are collected in Table 1. The values in the third row are taken from [36], where the indicated error bars have been estimated based on a detailed analysis of the convergence in (73)–(74). We set the error bars in the first two rows in Table 1 based on a comparative analysis. We performed calculations at n = 10, n′ = 7 and found that the deviations in the critical exponents relative to those for n = 13, n′ = 10 strongly correlate in all three schemes considered and are comparable in magnitude with the corresponding error bars in row 3. For η and ν, these deviations in the BMW schemes are slightly (by 18% for ν and even less for η) larger than those in the scheme of [10]. For ω, they are at least by 38% smaller. In absence of a more detailed convergence test, we rounded up the expected error bars in rows 1–2.
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TABLE 1. Values of the critical exponents η, ω, and ν, extracted from approximate RG flow equations (Sec. 6) of three schemes, using the principle of minimal sensitivity to find the optimal parameter α for each of the exponents, as shown in Figure 1.
The critical exponents obtained here for the original BMW scheme coincide with those reported in [17], i.e., η ≈ 0.039, ν ≈ 0.632, and ω ≈ 0.78. The estimates in Table 1 can be compared with those of the conformal field theory (CFT), i. e., η = 0.0362978(20), ν = 0.6299709 [38], and ω = 0.82951(61) [39], which are claimed to be very accurate. This comparison shows that the considered truncated RG equations give rather accurate values of ν, but the values of η are less accurate and somewhat overestimated.
Concerning ω, there are still some doubts about the acceptable value. Probably, the CFT value ωCFT = 0.82951 (61) is just the acceptable one. However, as discussed in [40], some numerical estimates, including the Monte Carlo renormalization group (MCRG) values ω ≈ 0.7 [42], ω = 0.75 (5) [42], the MCRG estimate from the reanalyzed data of [41] ω = 0.741 (21) [40], and the large mass expansion result ω ≈ 0.8002 [43] tend to give smaller ω values. Seeking for a compromise between these estimations, we set 0.76 as the lower bound for ω in our formal analysis. This choice is partly motivated by the fact that even smaller ω values would poorly fit in this analysis, where the refined estimates are expected to be at least slightly better than those of the LPA.
Thus, we consider ω = ωCFT as one option, allowing also a theoretical possibility that ω is smaller than ωCFT, but not smaller than 0.76. In this case, the actual RG estimates of η, ν, and ω in Table 1 are closer to the exact values than the corresponding PMS estimates of LPA, i. e., ηLPA = 0, νLPA = 0.650601 (10), and ωLPA = 0.654115 (30) [36] for the cut-off function (62). It can be seen from the values of the parameter
where λ = η, ν, ω is the actual RG estimate of the critical exponent, λex is its exact value, and λLPA is its LPA value. The parameters ζλ, calculated from the PMS values of λ and λLPA, are collected in Table 2. We evaluated ζω at ωex = ωCFT, ωex = 0.8, and ωex = 0.76 in (76) to see how this parameter changes if the exact ω value is 0.76 ≤ ω ≤ ωCFT. It should be noted that ζλ > 1/2 would mean that the estimate of λ in Table 1 is worse than that of the LPA. It would be true for λ = ω in the scheme of [10] if ωex ≤ 0.757.
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TABLE 2. Parameters ζλ with λ = η, ν, ω, calculated from the PMS values of the critical exponents (Table 1) of three truncation schemes. The last three rows contain ζω values for ωex = ωCFT (top), ωex = 0.8 (middle), and ωex = 0.76 (bottom) in (76).
We further used these parameters for a rough estimation of possible error bars for the subleading correction-to-scaling exponent ω2 in Sec. 7.3, assuming that
7.3 Estimation of the critical exponent ω2
The developed techniques allow us to extract from the RG flow not only the leading correction-to-scaling exponent ω but also the subleading one ω2. To the best of our knowledge, it has not yet been obtained from the Wetterich equation beyond the LPA, where ω2 ≈ 3.18 has been reported in [15] for the actually considered scalar 3D model.
The RG flow is described by the state vector X depending on t, where X includes all the independent variables contained in the RG flow equations. In particular, considering our semi-analytic approximations, the components of X are the coefficients um,k in (73) and the functions fm,k(y) in (74). The RG flow equations have the form
where the explicit t dependence of F (X, t) shows up only in the upper cut-off and is irrelevant in vicinity of the fixed point X = X*. A standard method is to linearize X = X* + δX with respect to an infinitesimal deviation δX from the fixed point. It leads to the solution of the linearized RG, which has the form
on the critical surface at large −t values (t < 0), where ω is the leading correction-to-scaling exponent, ω2 > ω is the subleading correction-to-scaling exponent, ω3 > ω2 is the next exponent in this hierarchy of the eigenvalues, and Xi is the eigenvector corresponding to ωi (where ω1 ≡ ω). For a unique representation, the eigenvectors are normalized appropriately. The constants Ci depend on the initial conditions.
Our basic idea is to find such initial conditions, at which C1 = 0. In this case,
A direct numerical extraction of ω2 from the RG flow at C1 ≠ 0 is difficult since δX contains a full set of correction terms in this case, including all ∝ knω terms with integer n ≥ 1. The vanishing of all these ∝ knω terms at C1 = 0 is a great advantage. We strictly verified numerically that these terms really vanish at C1 = 0. Namely, we have always obtained by our method such ω2 value, which is not just an integer multiple of ω, despite the fact that nω < ω2 holds for some n > 1.
We integrated the RG flow with a certain initial condition at k = k0, where k0 = Λ = 1, choosing
Considering the running exponent ω(k) (calculated just as described in Sec. 7.2) for various
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FIGURE 2. Running exponent ω(k) in the modified BMW scheme at α = 4.5. From top to bottom
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TABLE 3. Subleading correction-to-scaling exponent ω2, obtained from approximate RG flow equations of three truncation schemes at the given values of parameter α in (62).
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FIGURE 3. ω2 vs. ln α plots, obtained from approximate RG flow equations of the modified BMW scheme (the upper curve) and the scheme of [10] (the lower curve). The interval α ∈ [1.99, 5.65] of the preferable α values is between the vertical dotted lines. Our final result ω2 = 2.02 (40), evaluated in the middle of the corresponding ln α interval at α = 3.353 is indicated by the dashed lines, showing the error bars by the horizontal dotted lines and the vertical arrows. The LPA value
Because of a poor scaling of ω(k) at
The critical exponents η, ν, and ω are routinely estimated at the PMS values of α. The problem with ω2 is that no such α values can be identified in Figure 3. Moreover, an extrapolation of the plots in Figure 3 suggests that the extremum points, which could be identified with these PMS values, probably, are located at very large α values of about 20 or even
The estimation can be performed at reasonable values of α, chosen according to some extra criteria. In particular, ω2 is determined from the RG flow on the critical subsurface, where C1 = 0 holds; therefore, it makes sense to consider such α values, which allow describing this flow as far as possible accurately. The critical exponents η and ω2 are relevant for this flow; therefore, we are looking for such α values, which are acceptable or preferable for the estimation of both these exponents simultaneously.
We focus mainly on the estimation of ω2 from the equations of the modified BMW scheme. This method is advantageous from the point of view that it appears to be much more accurate than the LPA if ω = ωCFT, as well as if ω has a significantly smaller value within [0.76, ωCFT]. It is evident from the coefficients ζλ in Table 2. We argue that preferable values of α for the estimation of ω2 from these equations are
The estimation of the total error bars, including the systematic deviation from the exact value due to the closing approximation used in the modified BMW truncation scheme, is based on the assumption that, at appropriate α values (including α = 3.353),
for the scalar model in three dimensions, which belongs to the 3D Ising universality class. Although the error bars in (79) are not rigorous, no much larger discrepancy with the exact value is expected according to the arguments provided. These arguments basically rely on the assumption that the convergence of ω2 values, estimated at the truncation orders s = 0, 2, 3, 4, etc., in the modified BMW scheme, would not be essentially slower than the convergence for all other exponents (η, ν, and ω) considered here. Since the actual results include only the cases s = 0 and s = 2, no strict validation of this assumption is possible at the moment; therefore, the error bars in (79) are preliminary. From this point of view, one could allow even larger error bars, but the current information is insufficient to state how much larger. It can be clarified by extended calculations at higher orders.
The error bars of this estimation and the LPA value are shown in Figure 3 for an analysis and comparison. The results extracted from the equations of [10] support (79) since the corresponding lower curve in Figure 3 well fits within these error bars. This estimate agrees also with the RG value ω2 = 1.67 (11) obtained earlier in [37] from the approximated Polchinski RG equation within the derivative expansion at the
8 Summary and outlook
In this section, we give a brief summary of the obtained results, as well as discuss some possible further applications and developments.
In the current paper, we reported our new developments in the BMW truncation scheme for the Wetterich non-perturbative RG equation. We derived explicit RG flow equations for the scalar model at the arbitrary order of truncation, proposing also new closing approximations for the hierarchy of equations for the n-point functions in the BMW scheme (Sec. 4). A surprising similarity between the original and the modified equations of the BMW scheme at s = 2 order of truncation and those recently reported in [10] was shown (Secs. 5–6). As an example, calculations of the critical exponents for the 3D scalar model were performed, solving these equations by the method of semi-analytic approximations recently proposed in [36] (Sec. 7).
Particularly, the subleading correction-to-scaling exponent ω2 has been evaluated from the Wetterich equation beyond the LPA. Our estimate ω2 = 2.02 (40) is significantly smaller than the LPA value 3.18 reported in [15] and compares well with the value ω2 = 1.67 (11), obtained in [37] from the Polchinski equation at the
The calculations of ω2 demonstrate the usefulness of our new developments. In particular, just the modified equations of the BMW scheme and those derived in [10] appeared to be practically most useful for these calculations, allowing obtaining the results at reasonable values of the optimization parameter α.
The estimation of the critical exponent ω2 is crucial for testing the consistency between the RG exponents and those of the CFT. While the RG critical exponents η, ν, and ω appear to be reasonably close to the corresponding CFT values (see Sec. 7.2), there is clearly a problem with ω2. As already pointed out in Sec. 1, the known estimation in [37] gives a much smaller ω2 (i. e., ω2 = 1.67 (11)) than the CFT value 3.8956 (43) [39]. Our current estimation ω2 = 2.02 (40) allows for significantly larger than 1.67 values; however, these are still very small to speak about a consistency with CFT. In fact, the current estimations urge us to think that the RG exponent ω2, probably, is unrelated to the conformal symmetry since the RG values appear to be incompatibly smaller than 3.8956 (43).
The proposed scheme has the potential for various applications in calculations at higher than s = 2 orders. In particular, it would help in a further clarification of the question about the relation between the RG exponents and the CFT exponents. Development of improved solution techniques would be quite important for such calculations. Following [44,45], an interesting option is to use the expansion in Chebyshev polynomials for semi-analytic approximations. It could be an advantageous method owing to the guaranteed fast convergence properties of such expansions [45]. These can be used as an alternative to the expansions in powers of z in (73)–(74). One can also think about appropriate semi-analytic approximations for the momentum dependence of the n-point functions to make the calculations at higher than s = 2 orders more feasible.
Apart from very extensive and systematic applications of the non-perturbative RG approach to equilibrium statistical physics, it has also been applied to the out-of-equilibrium systems and critical dynamics [32–34]. The employed truncation schemes include the LPA and its refined modification [34]. There is still room for potential applications of other truncation schemes, including the currently introduced modified BMW scheme, which could be adjusted to this problem. It would be a quite interesting application, eventually, allowing obtaining refined RG estimates for the dynamical exponent z. Indeed, the actual estimates of this exponent are not very accurate, and there is a continued discussion of z values obtained from Monte Carlo simulations, experiments, and RG equations [34, 46].
Data availability statement
The raw data supporting the conclusion of this article will be made available by the authors, without undue reservation.
Author contributions
All authors listed made a substantial, direct, and intellectual contribution to the work and approved it for publication.
Acknowledgments
The authors acknowledge the Latvian Grid Infrastructure and High Performance Computing Centre of Riga Technical University for providing resources. JK acknowledges the support from the Science Support Fund of Riga Technical University. RM acknowledges the support from the NSERC and CRC programs.
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s note
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Supplementary material
The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fphy.2023.1182056/full#supplementary-material
References
1. Amit DJ. Field theory, the renormalization group, and critical phenomena. Singapore: World Scientific (1984).
3. Shang–Keng Ma. Modern theory of critical phenomena. New York, NY, USA: W.A. Benjamin, Inc. (1976).
4. Zinn–Justin J. Quantum field theory and critical phenomena. Oxford, England: Clarendon Press (1996).
5. Kleinert H, Schulte–Frohlinde V. Critical properties of ϕ4 theories. Singapore: World Scientific (2001).
6. Wetterich C. Exact evolution equation for the effective potential. Phys Lett B (1993) 301:90. doi:10.1016/0370-2693(93)90726-X
7. Polchinski J. Renormalization and effective Lagrangians. Nucl Phys B (1984) 231:269. doi:10.1016/0550-3213(84)90287-6
8. Bagnuls C, Bervillier C. Exact renormalization group equations. An introductory review. Phys Rep (2001) 348:91. doi:10.1016/S0370-1573%2800%2900137-X
9. Berges J, Tetradis N, Wetterich C. Non-perturbative renormalization flow in quantum field theory and statistical physics. Phys Rep (2002) 363:223. doi:10.1016/S0370-1573%2801%2900098-9
10. Kaupužs J, Melnik RVN. A new method of solution of the Wetterich equation and its applications. J Phys A: Math Theor (2020) 53:415002. doi:10.1088/1751-8121/abac96
11. Balog I, Chate H, Delamotte B, Marohnic M, Wschebor N. Convergence of nonperturbative approximations to the renormalization group. Phys Rev Lett (2019) 123:240604. doi:10.1103/PhysRevLett.123.240604
12. Papenbrock T, Wetterich C. Two-loop results from improved one loop computations. Z Phys C (1995) 65:519. doi:10.1007/BF01556140
13. Litim D. Optimized renormalization group flows. Phys Rev D (2001) 64:105007. doi:10.1103/PhysRevD.64.105007
14. Canet L, Delamotte B, Mouhanna D, Vidal J. Optimization of the derivative expansion in the nonperturbative renormalization group. Phys Rev D (2003) 67:065004. doi:10.1103/PhysRevD.67.065004
15. Litim DF. Critical exponents from optimised renormalisation group flows. Nucl.Phys B (2002) 631:128. doi:10.1016/S0550-3213(02)00186-4
16. Bender CM, Sarkar S. Asymptotic analysis of the local potential approximation to the Wetterich equation. J Phys A: Math Theor (2018) 51:225202. doi:10.1088/1751-8121/aabf63
17. Benitez F, Blaizot J-P, Chate H, Delamotte B, Mendez-Galain R, Wschebor N. Non-perturbative renormalization group preserving full-momentum dependence: implementation and quantitative evaluation. Phys Rev E (2012) 85:026707. doi:10.1103/PhysRevE.85.026707
18. De Polsi G, Balog I, Tissier M, Wschebor N. Precision calculation of critical exponents in the O(N) universality classes with the nonperturbative renormalization group. Phys Rev E (2020) 101:042113. doi:10.1103/PhysRevE.101.042113
19. Berges J, Jungnickel D-U, Wetterich C. Two flavor chiral phase transition from nonperturbative flow equations. Phys Rev D (1999) 59:034010. doi:10.1103/PhysRevD.59.034010
20. Schütz F, Kopietz P. Functional renormalization group with vacuum expectation values and spontaneous symmetry breaking. J Phys A: Math Gen (2006) 39:8205. doi:10.1088/0305-4470/39/25/S28
21. Benedetti D, Groh K, Machado PF, Saueressig F. The universal RG machine. JHEP (2011) 06:079. doi:10.1007/JHEP06(2011)079
22. Demmel M, Saueressig F, Zanusso O. RG flows of Quantum Einstein Gravity on maximally symmetric spaces. JHEP (2014) 06:026. doi:10.1007/JHEP06(2014)026
23. Wetterich C, Yamada M. Variable Planck mass from the gauge invariant flow equation. Phys Rev D (2019) 100:066017. doi:10.1103/PhysRevD.100.066017
24. Platania AB. Asymptotically safe gravity: from spacetime foliation to cosmology. Berlin, Germany: Springer (2018). p. 29–46.
25. Wetterich C. Quantum correlations for the metric. Phys Rev D (2017) 95:123525. doi:10.1103/PhysRevD.95.123525
26. Eichhorn A. An asymptotically safe guide to quantum gravity and matter. Front Astron Space Sci (2019) 5:47. doi:10.3389/fspas.2018.00047
27. Alwis SP. Exact RG flow equations and quantum gravity. JHEP (2018) 03:118. doi:10.1007/JHEP03(2018)118
28. Litim DF, Trott MJ. Asymptotic safety of scalar field theories. Phys Rev D (2018) 98:125006. doi:10.1103/PhysRevD.98.125006
29. Bond AD, Litim DF. Price of asymptotic safety. Phys Rev Lett (2019) 122:211601. doi:10.1103/PhysRevLett.122.211601
30. Falls KG, Litim DF, Schroder J. Aspects of asymptotic safety for quantum gravity. Phys Rev D (2019) 99:126015. doi:10.1103/PhysRevD.99.126015
31. Lahoche V, Samary DO. Progress in solving the nonperturbative renormalization group for tensorial group field theory. Universe (2019) 5:86. doi:10.3390/universe5030086
32. Canet L, Chate H. General framework of the non-perturbative renormalization group for non-equilibrium steady states. J Phys A (2007) 40:1937. doi:10.1088/1751-8113/40/9/002
33. Canet L, Chate H, Delamotte B. General framework of the non-perturbative renormalization group for non-equilibrium steady states. J Phys A: Math Theor (2011) 44:495001. doi:10.1088/1751-8113/44/49/495001
34. Roth JV, Smekal L. Critical dynamics in a real-time formulation of the functional renormalization group (2023). https://arxiv.org/pdf/2303.11817.pdf.
35. Dupuis N, Canet L, Eichhorn A, Metzner W, Pawlowski JM, Tissier M, The nonperturbative functional renormalization group and its applications. Phys Rep (2021) 910:1. doi:10.1016/j.physrep.2021.01.001
36. Kaupužs J, Melnik RVN. Functional truncations for the solution of the nonperturbative RG equations. J Phys A: Math Theor (2022) 55:465002. doi:10.1088/1751-8121/ac9f8c
37. Newman KE, Riedel EK. Critical exponents by the scaling-field method: the isotropic N-vector model in three dimensions. Phys Rev B (1984) 30:6615. doi:10.1103/PhysRevB.30.6615
38. Poland D, Simmons-Duffin D. The conformal bootstrap. Nat Phys (2016) 12:535. doi:10.1038/nphys3761
39. Reehorst M. Rigorous bounds on irrelevant operators in the 3d Ising model CFT. JHEP (2022) 09:177. doi:10.1007/JHEP09(2022)177
40. Kaupužs J, Melnik RVN. Corrections to scaling in the 3D Ising model: A comparison between MC and MCRG results. Int J Mod Phys C (2023):2350079. doi:10.1142/S0129183123500791
41. Gupta R, Tamayo P. Critical exponents of the 3D Ising model. Int J Mod Phys C (1996) 7:305–19. doi:10.1142/S0129183196000247
42. Ron D, Brandt A, Swendsen RH. Surprising convergence of the Monte Carlo renormalization group for the three-dimensional Ising model. Phys Rev E (2017) 95:053305. doi:10.1103/PhysRevE.95.053305
43. Yamada H. Critical exponents from large mass expansion (2014). https://arxiv.org/pdf/1408.4584.pdf.
44. Borchardt J, Knorr B. Global solutions of functional fixed point equations via pseudospectral methods. Phys Rev D (2015) 91:10. doi:10.1103/PhysRevD.91.105011
45. Borchardt J, Knorr B. Solving functional flow equations with pseudospectral methods. Phys Rev D (2016) 94:025027. doi:10.1103/PhysRevD.94.025027
Keywords: functional renormalization, Wetterich equation, truncation schemes, exact renormalization group equations, non-perturbative approaches, quantum and statistical field theories, strongly interacting systems
Citation: Kaupužs J and Melnik RVN (2024) Original and modified non-perturbative renormalization group equations of the BMW scheme at the arbitrary order of truncation. Front. Phys. 11:1182056. doi: 10.3389/fphy.2023.1182056
Received: 08 March 2023; Accepted: 01 December 2023;
Published: 15 January 2024.
Edited by:
Manuel Asorey, University of Zaragoza, SpainReviewed by:
Frank Simon Saueressig, Radboud University, NetherlandsGerd Roepke, University of Rostock, Germany
Copyright © 2024 Kaupužs and Melnik. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: J. Kaupužs, kaupuzs@latnet.lv