Scaling Solutions in the Derivative Expansion
PHYSICAL REVIEW D 98, 016013 (2018)
Scaling solutions in the derivative expansion
† N. Defenu1,* and A. Codello2,3, 1Institut für Theoretische Physik, Universität Heidelberg, D-69120 Heidelberg, Germany 2CP3-Origins, University of Southern Denmark, Campusvej 55, 5230 Odense M, Denmark 3INFN-Sezione di Bologna, via Irnerio 46, 40126 Bologna, Italy
(Received 24 February 2018; published 23 July 2018)
Scalar field theories with Z2 symmetry are the traditional playground of critical phenomena. In this work, these models are studied using functional renormalization group (FRG) equations at order ∂2 of the derivative expansion and, differently from previous approaches, the spike plot technique is employed to find the relative scaling solutions in two and three dimensions. The anomalous dimension of the first few universality classes in d ¼ 2 is given, and the phase structure predicted by conformal field theory is recovered (without the imposition of conformal invariance), while in d ¼ 3 a refined view of the standard Wilson-Fisher fixed point is found. Our study enlightens the strength of shooting techniques in studying FRG equations, suggesting them as candidates to investigate strongly nonperturbative theories even in more complex cases.
DOI: 10.1103/PhysRevD.98.016013
I. INTRODUCTION a new kind of nonperturbative expansion and to overcome the traditional limitations of diagrammatic techniques. Since the discovery of the phenomenon of universality In this paper, we use the flow equations for the EAA at and its explanation in terms of the renormalization group order Oð∂2Þ of the derivative expansion to investigate (RG) [1], one of the main goals of statistical and quantum scaling solutions (i.e., functional RG fixed points) of field theory has been the classification of all universality single-component scalar theories with Z2 symmetry in classes, i.e., the determination of fixed points of the RG two and three dimensions. Our approach generalizes the flow. Even though the phase diagram of scalar field theories spike plot technique already employed to solve the local has been subject to intense investigations over the decades, potential approximation (LPA) in [7–10] and refines the we still lack a complete picture of theory space even in the O ∂2 study [11] where a power like regulator and simplest cases, such as that of single-component scalar ð Þ numerical relaxation methods were employed. For related theories in two or three dimensions. This is not a surprise studies, but from a Polchinski equation perspective, see since the problem is inherently nonperturbative, but what is [12] and reference therein; for a proper time RG study also missing is an easy way to render and visualize the see [13]. complex landscape of theory space, since this is generally Our technique provides an extension of the shooting an infinite functional space. method previously employed in the literature to describe In recent years, the functional renormalization group single isolate fixed points in d>2 [14] to the investigation (FRG) [2,3] has shown its versatility and strength as a of the whole phase diagram of the d 2 case, where nonperturbative RG technique in many applications [4–6]. ¼ infinitely many fixed points exist. Our analysis paves the In this approach, the traditional RG is extended to work in way for the systematic application of shooting techniques the functional space of the effective average action (EAA), to high-order truncation in derivative expansions for more i.e., the scale-dependent generator of the one-particle complex field theories. In particular, we foresee possible irreducible vertexes of the theory, allowing us to pursue application of the present technique to OðNÞ symmetric field theories, where strongly nonperturbative fixed point exist [15,16], which cannot be investigated by means of *[email protected] other techniques. Indeed, standard solution procedures for † [email protected] flow equations rely on relaxation methods (which need fine-tuned initial conditions in presence of multiple fixed Published by the American Physical Society under the terms of points), polynomial expansions (which assume analytic the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to form for the effective potential) or full scale dependent the author(s) and the published article’s title, journal citation, solution (which need approximate knowledge of the final and DOI. Funded by SCOAP3. form for the effective potential). In contrast our numerical
2470-0010=2018=98(1)=016013(13) 016013-1 Published by the American Physical Society N. DEFENU and A. CODELLO PHYS. REV. D 98, 016013 (2018) procedure is unbiased and does not assume any further defined as t ¼ logðk=k0Þ, where k0 is an arbitrary refer- knowledge of the theory under consideration apart for its ence scale. global symmetry, neither it introduces further approxima- After inserting the ansatz (1) into the flow equation (2) tion rather than the EEA ansatz necessary to derive the flow and performing the appropriate projection one can derive equations. the beta functionals βV and βZ for the running effective potential and field-dependent wave-function renormaliza- II. DERIVATIVE EXPANSION tion function The derivative expansion is an approximation scheme ∂ V βd V00;Z where the EAA is expanded in powers of the field spatial t k ¼ Vð k kÞ d 00 000 0 00 derivatives [2,3,17,18]. This scheme is usually employed ∂tZk ¼ βZðVk;Vk ;Zk;Zk;ZkÞ: ð3Þ for matter field theories on flat space, where it becomes a series expansion in powers of the momentum. The deriva- The explicit form of the beta functionals can be obtained in tive expansion has been very successful in drawing phase arbitrary dimension and for arbitrary cutoff functions. The diagrams and computing accurate universal quantities, beta functional for the effective potential follows directly especially critical exponents. by evaluating (2) at a constant field configuration and reads If one considers a one-component scalar field, in d Z -dimensional flat space, with a 2 symmetry, the deriva- d 1 1 2 βV ¼ d Qd½Gk∂tRk : ð4Þ tive expansion for the EAA to order O ∂ reads 2 ð Þ ð4πÞ2 2 Z d 1 2 4 Γk½ϕ ¼ d x VkðϕÞþ ZkðϕÞð∂ϕÞ þ Oð∂ Þ: ð1Þ In (4) the regularized propagator at the constant field 2 configuration ϕ is defined as
The effective potential VkðϕÞ and the wave-function 1 Z ϕ G x; ϕ ; renormalization function kð Þ are arbitrary functions of kð Þ¼ 00 ð5Þ ZkðϕÞx þ V ðϕÞþRkðxÞ the field ϕ. The derivative expansion has been carried to k higher order in d 3 [17], where a beta function study was ¼ while the Q functionals are defined as performed. The flow equations for these functions are Z derived from the exact RG equation satisfied by the EAA ∞ 1 n−1 [2,3], Qn½f ≡ dxx fðxÞ: ð6Þ ΓðnÞ 0 1 ∂ Γ ϕ Γð2Þ ϕ R −1∂ R : t k½ ¼2 Trð k ½ þ kÞ t k ð2Þ Deriving the flow equation for the wave-function renorm- alization function is more involved. Taking the second For translational invariant systems the trace in latter functional derivative of (2) with respect to the fields it is equation represents a momentum integral and the cutoff possible to write down the flow equation for the two-point function Rk is a momentum-dependent mass term intro- function of the EAA in momentum space. Introducing on duced into the effective action in order to freeze the low the rhs of this equation the vertices of the EAA (1) and energy excitations responsible for the appearance of infra- extracting the Oðp2Þ terms one obtains, after some algebra, red (IR) divergences. The effective action explicitly the following beta functional for the wave-function renorm- depends on a scale k and the renormalization “time” is alization function
000 2 0 2 d ðVk Þ 2 2 ðZkÞ 2d þ 1 3 ðd þ 2Þðd þ 4Þ β Qd G G0 ∂ R Qd G G00∂ R Qd G ∂ R Z ¼ d f ½ k k t k þ þ1½ k k t k g þ d þ1½ k t k þ ð4πÞ2 2 2 ð4πÞ2 2 2 4 000 0 2 2 Vk Zk 3 2 Qd G G0 ∂ R Qd G G00∂ R 2Qd G ∂ R d 2 Qd G G0 ∂ R × ð þ2½ k k t k þ þ3½ k k t k Þ þ d f ½ k t k þð þ Þð þ1½ k k t k 2 2 ð4πÞ2 2 2 00 2 Zk 1 2 Qd G G00∂ R − Qd G ∂ R : þ þ2½ k k t k Þg þ d ½ k t k ð7Þ 2 ð4πÞ2 2 2
Equations (4) and (7) represent the flow equations for been chosen, the integrals in (4) and (7) can be performed. 2 VkðϕÞ and ZkðϕÞ for general cutoff function at Oð∂ Þ of the In this way one obtains a system of partial differential derivative expansion. Once an appropriate cutoff shape has equations for VkðϕÞ and ZkðϕÞ in the variables k and ϕ.
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Finally the flow equations are obtained introducing the Conditions (11) and (14) are a direct consequence of the Z2 dimensionless variables. The dimensionless quantities are symmetry. The condition (14) is obtained absorbing a defined as factor Zkð0Þ into the field redefinition. Thus the only unspecified condition remaining is the v00ð0Þ value. In −1 d 2 ð −1Þ d σ η ϕ ¼ Zk k 2 φ;VkðϕÞ¼k vðφÞ; Eq. (12) and are real values to be determined by requiring the global existence of the scaling solutions, Z ϕ Z ζ φ ; kð Þ¼ k ð Þ condition that, as we will see, reduces to a finite set the allowed functional fixed points. t from which we drive the following relations by The main goal of our paper is to show that it is possible differentiation to find solutions of the system (10) extending the simple spike plot technique used in LPA analysis [7,10]. This d − 2 þ η β −dv φv0 k−dβ method has diverse advantages: first of all it does not v ¼ þ 2 þ V require the solution of the flow equations in function of the d − 2 η þ 0 −1 t βζ ¼ ηζ þ φζ þ Zk βZ; ð8Þ renormalization time ; second it solves the fixed point 2 equations in their full functional form, without relying on truncations [17] (this property is necessary to be consistent where βv ≡∂tv and βζ ≡∂tζ. The anomalous dimension with the Mermin-Wagner theorem [20]); third it does not of the scalar field in (8) is defined by η ¼ −∂t log Zkð0Þ [4]. necessitate any external input as in [3,11], where the To obtain explicit expressions for the beta functional, we relaxation method was used starting from the exact spheri- need to specify the cutoff function Rk. In terms of the linear cal model solution. For these reasons the present method is cutoff1 introduced in [19], the most suited to uncover actual nonperturbative univer- 2 2 sality classes, which can not be investigated by means of RkðxÞ¼Zkðk − xÞθðk − xÞ; ð9Þ standard approaches [15]. Let us describe the procedure in more details, we solve the Q functionals in (4) and (7) can all be computed Eq. (10) for different values of σ and η. For any arbitrary analytically and they can be reduced to a single threshold σ; η integral, a hypergeometric function (see the Appendix A). point in the ð Þ plane the solution will become singular at a finite value of the field [21], we call this finite value φ , In integer dimension, explicit expressions for βv and βζ can sing and global scaling solutions correspond to those points in be written, as reported in the Appendix C. φ “ ” which sing shows a spike behaviour. Thus physical fixed points in the ðσ; ηÞ plane can be determined by a numerical III. SCALING SOLUTIONS φ analysis of the function sing. In this formalism the scaling solutions for the effective For the purpose of extending the resulting scaling v ζ φ actions appear as FRG fixed points and they are determined solutions and beyond sing the analysis is comple- by solving the coupled system of ordinary differential mented by the asymptotic behaviours computed using the equations large field solutions of Eq. (10)
( 2d βv 0 βζ 0: 10 d−2 η ¼ ¼ ð Þ vðφÞ ∼ v0φ þ for φ ≫ 1: ð15Þ − 2η For parity reasons we expect the following boundary ζðφÞ ∼ ζ0φ d−2þη conditions for the effective potential and the wave-function renormalization functional evaluated at the origin, Once the location of a fixed point in the ðσ; ηÞ plane has been determined, the coefficients ðv0; ζ0Þ can be evaluated v0ð0Þ¼0 ð11Þ by imposing continuity of the numerical solutions for v and ζ with the large field expansion in Eq, (15) at some value 00 φ < φ v ð0Þ¼σ ð12Þ max sing. We will show in subsection III A that, in d ¼ 3, there is ζð0Þ¼1 ð13Þ only one scaling solution to the system (10), which corresponds to the so-called Wilson-Fisher fixed point ζ0ð0Þ¼0: ð14Þ and describes the Ising d ¼ 3 universality class. This fixed point was extensively studied in the literature, also
1 with shooting techniques similar to the one discussed in this The linear cutoff is believed to be the optimal choice at low paper [14,22,23].Itisind ¼ 2, where every perturbative order in derivative expansion [19]. Indeed the results for the critical exponents found in this case are more accurate than approach fails to describe correctly the various universality the ones obtained with power law cutoff [3,11], as discussed in classes, first constructed exactly using conformal field the following. theory (CFT) methods [24], that the system (10) reveals
016013-3 N. DEFENU and A. CODELLO PHYS. REV. D 98, 016013 (2018) its nonperturbative potentialities. It was shown in [3] that the power-law cutoff version of the system (10), solved with the relaxation method [25], has scaling solutions in one-to-one correspondence with the minimal models known from CFT. In subsection III B, we will see that the same picture emerges also when the linear cutoff and the spike plot method are employed.
A. The three-dimensional case First of all we consider the d ¼ 3 case, which is the less involved since only the standard Wilson-Fisher fixed point exists. The equations for the β functions in the linear regulator case, (C3) and (C4) from the Appendix, are set to zero and solved numerically, with a fourth-order Runge- Kutta approach, for several points in the ðσ; ηÞ plane. The value at which the numerical solver interrupts integration φ sing is represented as a contourplot in the plane of the initial conditions ðσ; ηÞ in Fig. 1 panel (a). Apart for the maximum in the origin, which represents the free Gaussian universality class, we see a family of maxima arising along a curve which separates the dark blue region of the severely ill-defined solutions from the light green region of intermediate solutions. Only a small portion φ of the plane has a nontrivial support for the sing function, which is identically zero for all σ < −0.2. Moreover the equations in the linear cutoff case develop a divergence for σ ¼ −1 often called spinodal instability. It is not clear wether this divergence, which is IR attractive in all directions, is the physical IR fixed point or rather an artifact of the linear regulator scheme. However it has been demonstrated that it is possible to recover information over the universal quantities studying the scaling of the φ σ; η solutions close to this instability [26]. FIG. 1. The contour plot of sing in the ð Þ planes has two The landscape in Fig. 1 can seem surprising due to the spikes: the one located at (0, 0) represents the Gaussian theory. Ising presence of several maxima rather than an isolated one, The other one indicates the universality class, which occurs at a finite positive value of the anomalous dimension and indicating the Ising universality class. However a closer for a negative dimensionless mass. In panel (b), the spike-plot is inspection reveals that the high of the peaks is nonmonot- performed along the maxima chain of the contour plot of panel onous, leading to the appearance of a prominent maximum (a) with different choices for the fit parameters (gray dashed at finite η and σ values. This prominent maximum repre- lines). The optimal choice (red solid line) is the one which sents the signature of the Wilson-Fisher fixed point. The maximizes the spike height, see the main text. connection of the Ising and Gaussian universalities through an extended chain is easily justified if one con- siders that the spike-plot technique furnishes a solution analysis along several trajectory in the ðσ; ηÞ, retrieving the even in the lower truncation scheme, where only the expected scenario. As anticipated above, a convenient potential equation is considered. Thus it appears that for approach to overcome these difficulties and to extract each value of η it exists a maximum as a function of σ the anomalous dimension of the interacting universality which represents the best approximation for the vðφÞ and classes is to reduce the problem of finding the maxima of φ σ; η ζðφÞ functions at that particular value of the anomalous the two-dimensional surface singð Þ to the simpler one dimension. of locating the maximum in a single one-dimensional On the other hand, due to the finite numerical grid in the chain. This is possible using the results in panels (a) of ðσ; ηÞ plane, to the complex three-dimensional shape of Figs. 1 and 3 as a guide. the chain as well as to the finite numerical accuracy in the First of all, we fit the locations of the maxima of a single solution of the flow equations, the peaks chain appears as a chain in the (σ, η) plane with a simple function; depending sequence of spikes rather than a continuous ridge. This on the case, linear or parabolic fit functions are employed. picture is confirmed in the following, where we pursue the This operation produces an explicit expression for the
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(gray dashed lines) show the result for different values of the fit coefficients, the optimized curve, red solid line, gives η ¼ 0.0496 with σ ¼ −0.1468. The fixed point potential and wave-function renormalization are shown in Fig. 2 panel (a) and (b) respectively. In correspondence of the minimum of the fixed point potential, a maximum of the wave function is found. The large field behavior can not be obtained by numerical integration, since a very precise identification of the correct fixed point values ðσ; ηÞ is necessary to push the integration forward. In Fig. 2, the large field branch of the two functions has been obtained using large field expansions reported in Eq. (15) and imposing continuity with the numerical solutions. The numerical value for the critical exponent η ¼ 0.0496 is in better agreement with exact Monte Carlo (MC) and conformal bootstrap results [27,28] η ¼ 0.0363, with respect to the power law regulator results present in literature η ¼ 0.0539 [14], in agreement with the expectation for the linear regulator of being the optimal cutoff for derivative expansions.
B. The two-dimensional case
In the d ¼ 2 case, scalar Z2 theories have an infinite number of universality classes, leading to a far more complicated landscape in the ðσ; ηÞ plane. These univer- sality classes correspond with the minimal models of CFT and the critical exponents are known exactly. The impor- tance of reproducing such results in an approximated scheme is not only the methodological one, deriving from the necessity to test our technique in a more complex scenario, but it also lies in the different assumptions FIG. 2. In panels (a) and (b), respectively, the solutions for the necessary to compute such quantities. Indeed, flow equa- functions ζðφÞ and vðφÞ are shown for the Wilson-Fisher fixed tions (4) and (7) have been derived without any additional point in three dimension. The lines have been obtained imposing condition rather than Z2 symmetry. Even in this over- continuity between the numerical solution (solid line) and simplified computation scheme and without any additional analytic large field behaviour of the scaling solutions (15) (dotted imposition on the symmetry of the model, such as con- lines). formal symmetry, the FRG technique is able to recover all the information on the location and the shape of the maxima chain as a function σðηÞ along which we can solutions, with a good numerical accuracy on the universal pursue the standard one-dimensional spike plot technique. quantities. Obviously the coefficients of the fit will contain errors due The result for the landscape of interacting fixed points in to the finiteness of the grid in the landscapes of panels (a) in two-dimensional scalar field theories is reported in Fig. 3, Figs. 1 and 3. However the best value for the fit is obtained in the linear cutoff scheme. Identifying correctly the exact optimizing the coefficient in order to maximize the height value of the anomalous dimension for any universality class of the spike; as it is shown in panel (b) of Fig. 1 for the requires some care. Indeed, similarly to the d ¼ 3 case in φ three-dimensional Ising universality. The optimization Fig. 1, the maxima of sing are disposed on special lines of procedure is straightforward since one should allow only the ðσ; ηÞ plane, forming peak chains. In panel (a) of Fig. 3, for very small variation of the fit coefficients and the one- the Gaussian universality class appears as a infinitely tall dimensional spike plot computation is extremely fast. spike located at σ ¼ η ¼ 0. The two peaks chains at the In the three-dimensional case, the optimization pro- extrema of the origin are the longest in the η direction and cedure is shown in Fig. 1 panel (b). The grey dashed they represent the Wilson-Fisher and tricritical universal- curves are the spike plots along different fit representations ities which have the largest anomalous dimension values. of the peak chain in panel (a), Fig. 1. The equation for the Both these peak chains, going from η ¼ 0 to η ≃ 1,havea peak chain has been obtained by linear fit around the maxima at a finite value of the anomalous dimension which position of the nontrivial maximum. The various curves roughly corresponds to the expected one. Moreover we
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FIG. 3. In panel (a), spike plot in two dimensions for the linear regulator case. The landscape of scalar quantum field theories at this approximation level is quite complicated. The maxima of FIG. 4. In panel (a), we show the anomalous dimensions as a i φsing form mountain chains located on some special curves in the function of the critical index for the linear (green line) and ðσ; ηÞ plane. power law (blue line) cutoff case, compared to the exact CFT results (red line). In panel (b), the solutions for the functions ζðφÞ and vðφÞ are shown for the first six multicritical universalities. have infinitely many other peak chains starting at η ¼ 0. Each line has been obtained imposing continuity between the These chains accumulate in the origin, becoming shorter numerical solution (solid line) and analytic large field behaviour and shorter, as expected since they represent high-order of the scaling solutions (15) (dotted lines). universalities. As for the three-dimensional case, the finite numerical obtain a one-dimensional spike plot for every universality grid in the ðσ; ηÞ plane does not allow for a precise location class from the standard Wilson-Fisher case i ¼ 2 to the of the tallest maximum along each chain. Increasing the esacritical universality i ¼ 6 as it is shown in Fig. 3 panel φ precision of the grid will need an exponential growing (b). The values, at which the singularities of sing occur, are computation time due to the necessity to increase accord- in agreement with the expected values for the anomalous ingly the precision of the numerical solution of the differ- dimensions. During the optimization procedure of the fit ential equations. Once again, in order to maintain a low parameters it occurs that two minima are found in the line, computational cost, we consider a numerical fit for each proving the complex three-dimensional structure of the peak chain along which we pursue a one-dimensional spike peaks in the (σ, η) plane. In any case, the highest divergence plot procedure analogue to the LPA case. Optimizing the fit is always obtained for a single peak, confirming that the coefficients to maximize the height of the peak in every two peaks structure occurs only when the fit line does not chain of the contour plot in d ¼ 2, Fig. 3 panel (a), we cross the spike at its center. The distance between the two
016013-6 SCALING SOLUTIONS IN THE DERIVATIVE EXPANSION PHYS. REV. D 98, 016013 (2018) peaks can be easily reduced until a precision ≤5% is obtained in the η values, see Fig. 3 panel (b). In Fig. 4, panel (a), the anomalous dimension values obtained using the spike plot technique are shown. The results of this work (green line) are compared to the ones found in [11] (blue line) and to the exact results of CFT solutions (red line). The curve obtained here is more precise than the results found in the power law cutoff scheme of [11], which confirms the expected better performance of the linear cutoff scheme. It should be also noted that, as expected, the precision of FRG truncation scheme increases lowering the anomalous dimension values, with the esa- critical value for the η exponent reproduced up to 99% even in this rather simple approximation. The fact that the precision of FRG truncation scheme increases lowering the anomalous dimension values suggests that a truncation based on the real fixed point dimension of the operators included in the EAA might be better than the derivative expansion from the quantitative point of view. While in d ¼ 2 for η ¼ 0, the expansion in terms of the operator dimensions and derivative expansion coincides as η → 0 and, thus, as η grows, they start diverging. In d ¼ 3, the situation is very different and may explain the less precise numerical success of the derivative expansion in this dimension. In panel (b) of Fig. 4, we show the potential vðφÞ and the field-dependent wave function renormalization ζðφÞ for the first six universality classes in d ¼ 2. The solutions are shown only for positive values of the field φ, since the other branch can be simply obtained using reflection symmetry. Each potential shows a number of minima i, as indicated by its criticality order, and the corresponding wave function renormalization has a relative maximum in correspondence of each minimum position. The solutions shown in Fig. 4 panel (b) have been obtained solving Eqs. (4) and (7) with the values of η and σ found using the spike plot technique described in Fig. 1 panel (b).
C. Regulator dependence FIG. 5. Spike plot in two dimensions for the power law cutoff case. The landscape of scalar quantum field theories at this As it should be understood from the above investiga- φ approximation level is quite complicated. The maxima of sing tions, the efficiency of our spike plot technique crucially form mountain chains located on some special curves in the depends on the structure of the three-dimensional surface ðσ; ηÞ plane. φ σ; η singð Þ, which has nontrivial shape only on a finite number of quasi-two-dimensional manifolds, i.e., the lines in Fig. 3. It is then necessary to test whether this simplified rather peculiar shape, completely different from the linear structure is just an artifact of the linear cutoff function or if cutoff employed above. Indeed, while the last one is compact it is a general result valid for Eqs. (4) and (7) independently with extremely localized derivatives the first one has infinite from the particular form assumed by the Q functionals. support and it has finite derivatives at all orders. In order to check the stability of our approach, we We apply the same procedure already considered for the consider another cutoff function to explicitly compute the flow equations in the linear regulator case. Both in d ¼ 2 Q functionals. The most effective choice in this perspective and d ¼ 3 we retrieve the expected phase structure, with is the power law regulator already adopted in [11] only one correlated fixed point in the first case and 4 RkðxÞ¼k =x. Indeed, such a regulator, even if not optimal infinitely many solutions in d ¼ 2, as shown in Fig. 5.It to compute numerical quantities, produces simple results is worth noting that in the power law regulator case the for the flow equations. Moreover the power law cutoff has a height of the peaks in the contour plot is much smaller than
016013-7 N. DEFENU and A. CODELLO PHYS. REV. D 98, 016013 (2018) in the previous case. However both in 2, Fig. 5 panel (a), linear regulator (9), which leads to substantial numerical and 3, panel (b), the peak chains are evident, thus difficulties, especially in the study of the stability exponents. φ σ; η demonstrating that the structure of the singð Þ function, Then, it would be more convenient to consider different even if influenced by the regulator form, maintains a very regularization schemes which deliver considerably simpler small nontrivial support. In both the regulator cases, every functional flow equations [33]. Another interesting perspec- universality class of the theory appears as a chain of peaks tive is the study of the nonunitary family of fixed points in the ðσ; ηÞ plane. generalizing the LPA analysis [34] or long range interacting field theories, both in the classical [35] and quantum case IV. CONCLUSION [36,37] or, finally in presence of global symmetries as OðNÞ S interactions [8,9,15] or Potts model nþ1-symmetries [38]. After deriving the explicit expressions for the flow We leave all these applications to future works. equations of the effective potential Vk and the wave- Z Since the spike plot method does not need any prerequi- function renormalization function k, in two and three sites and avoids further approximations apart from the dimensions, and with the use of the linear cutoff, we have “ ” derivative expansion itself, it is the most suited technique shown that it is possible to extend the spike plot to uncover strongly nonperturbative aspects of critical phe- ∂2 technique to the study of scaling solutions at order of nomena. To this aim the generalization of the present the derivative expansion. In this approximation, theory technique to larger sets of flow equations will be necessary, space is projected to the two-dimensional plane parame- leading to an enlarged parameter space for the initial value σ ≡ v00 0 η trized by ð Þ and by the anomalous dimension . problem. However, we expect the physical solutions to The spike plot becomes a two-dimensional surface that always lie on low dimensional subsets of the parameter represents the landscape of scalar theories with Z2 sym- “ ” space as it happens in the present case, see panels (a) in metry. These landscapes are characterized by peak chains Figs. 3 and 1. Once these relevant regions have been along which the spikes appear; the shape of these chains φ σ η identified, looking for the maxima of sing on a rough grid, can be fitted by a simple linear or quadratic curve ð Þ and the investigation can be refined by constraining the param- the problem of finding the position of the maxima of these eters to lie in the relevant regions. Afterwards, the constraints surfaces can be reduced to a one-dimensional spike plot shall be optimized by magnifying the singularity of φ ,asit similar to those found at the LPA level. sing isshowninFig.1 panel (b). The procedure in an higher- In d ¼ 3, the landscape is characterized by a single dimensional space will thus be analogue to the one described nontrivial spike that represents the Ising universality class, in the present paper and it could lead to the identification and while in two dimensions the landscape is much more description of novel universality classes, whose existence is, complex: a whole chain is present for each Multi- till now, doubted [39,40]. critical universality class, and each CFT minimal model emerges as the most prominent peak of each of these chains. AKNOWLEDGMENTS At high multicriticality, the obtained values of the anomalous dimensions are in very good agreement with the exact CFT We would like to thank M. Safari for his help and results. At the quantitative level, the increasing precision collaboration in the early stages of this work. N. D. acknowl- with which the anomalous dimension of the multicritical edges support by the Deutsche Forschungsgemeinschaft fixed points is obtained, for growing multicriticality index, Collaborative Research Centre “SFB 1225 (ISOQUANT).” suggests that an expansion scheme based on the relevancy of the real operator dimension (operator dimension expansion) APPENDIX A: DERIVATION OF ∂tVk AND ∂tZk of the terms included in the truncation ansatz of the effective average action might be quantitatively precise. This is sug- The effective average action to the second order in the gested by the fact that the derivative expansion, in d ¼ 2 and derivative expansion is given by as η → 0, becomes exactly the operator dimension expansion Z 1 η μ and indeed we obtain better and better values for as the Γk ¼ ZðφÞ∂μφ∂ φ þ VðφÞ ; ðA1Þ multicriticality grows. x 2 While the present analysis shows how the spike plot where VðφÞ is the effective potential and ZðφÞ is the technique can produce a complete and systematic analysis of generalized wave function renormalization. The functional fixed point solutions of functional flow equations, even derivatives of the action (A1) in momentum space for a beyond the well-known LPA case, it would be interesting to constant field configuration read disclose the full picture of the critical exponents for scalar field theories, including the correlation length exponent ν, ð2Þ d ð2Þ Γq q ¼ð2πÞ ð−Zq1 · q2 þ V Þδðq1 þ q2Þ: ðA2Þ even for fractional dimension beyond two and three where it 1 2 will be interesting to compare with the recent ϵ-expansion Γð3Þ 2π d −Zð1Þ q q q q q q Vð3Þ q1q2q3 ¼ð Þ ½ ð 1 · 2 þ 1 · 3 þ 2 · 3Þþ analysis [29–32]. However the latter task is hindered by the δ q q q ; complexity of the flow equations, even in the case of the × ð 1 þ 2 þ 3Þ ðA3Þ
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Γð4Þ 2π d −Zð2Þ q q q q q q The calculation is straightforward but complex, we shall q1q2q3q4 ¼ð Þ ½ ð 1 · 2 þ 1 · 3 þ 1 · 4 then carry it out term by term. We then define the first term q q q q q q Vð4Þ þ 2 · 3 þ 2 · 4 þ 3 · 4Þþ T1 as the right hand side of (A11)
× δðq1 þ q2 þ q3 þ q4Þ: ðA4Þ Z ddq 1 d2 T ∂ R q G q 2Γð3Þ The flow equations for the potential is 1 ¼ d t tð Þ ð Þ q;p;−p−q 2 Z ð2πÞ 2 dp 1 ddq 3 ∂tVt ¼ GðqÞRkðqÞ; ðA5Þ G p q Γð Þ 2 2π d × ð þ Þ −q;−p;pþq ðA16Þ ð Þ p¼0 and for the two point function in the standard form is Z with the other terms defined as in lines (A12)–(A15) ð2Þ d ð3Þ ∂tΓt;p;−p ¼ d qGðqÞΓq;p;−q−pGðq þ pÞ Z ddq T 2 ∂ R q G q 2Γð3Þ ð3Þ 2 ¼ d t tð Þ ð Þ q;p;−p−q × Γq p;−p;−qGðqÞ∂tRkðqÞ ð2πÞ Zþ 2 1 4 1 d − ddqG q Γð Þ G q ∂ R q ; G p q Γð3Þ ð Þ q;p;−p;−q ð Þ t kð Þ ðA6Þ × ð þ Þ 2 −q;−p;pþq ðA17Þ 2 2 dp p¼0 where the short hands are obvious. We have and ð4Þ 2 2 2 4 Γq;p;−p;−q ¼ Zð Þðq þ p ÞþVð Þ ðA7Þ Z d ð3Þ 1 2 2 3 d q 3 d Γq;p;−q−p ¼ Zð Þðq þ q · p þ p ÞþVð Þ ðA8Þ T 2 ∂ R q G q 2Γð Þ 3 ¼ 2π d t tð Þ ð Þ q;p;−p−q dp ð3Þ ð1Þ 2 2 ð3Þ ð Þ Γq p;−p;−q ¼ Z ðq þ q · p þ p ÞþV : ðA9Þ þ d G p q Γð3Þ × ð þ Þ −q;−p;pþq In order to evaluate the flow of the field-dependent wave dp p¼0 function, we use the definition A18 1 d2 ð Þ ∂ Z φ ∂ Γð2Þ −p; p : t ð Þ¼ lim 2 t t ð Þ ðA10Þ 2 p→0 dp and When we apply the derivatives on the right end side of Z ddq d 2 Eq. (A6), they go under the integral sign and act in the only T ∂ R q G q 2G p q Γð3Þ p G p q 4 ¼ d t tð Þ ð Þ ð þ Þ −q;−p;pþq part of the integrand which depends on , i.e., ð þ Þ; ð2πÞ dp p¼0 thus, we get Z ðA19Þ 1 ddq d2 ∂ Z φ ∂ R q G q 2Γð3Þ t ð Þ¼2 2π d t tð Þ ð Þ q;p;−p−q dp2 and ð Þ 3 Z d 2 G p q Γð Þ 1 d q 1 d 4 × ð þ Þ −q;−p;pþq ðA11Þ T − G q Γð Þ G q ∂ R q : p¼0 5 ¼ 2 2π d ð Þ 2 dp2 q;p;−p;−q ð Þ t kð Þ Z ð Þ ddq d2 2 ð3Þ ðA20Þ þ ∂tRtðqÞGðqÞ Γq;p;−p−qGðp þ qÞ ð2πÞd dp2 The most complex term to evaluate is T1 we will then left it ð3Þ × Γ−q;−p;p q ðA12Þ aside for the moment. It is then convenient to pursue the þ p 0 ¼ computation in reverse order. After deriving the vertex term Z d d q 2 ð3Þ d in (A20), we obtain þ 4 ∂tRtðqÞGðqÞ Γq;p;−p−q ð2πÞd dp Z Zð2Þ ddq d T − G q 2∂ R q : G p q Γð3Þ 5 ¼ 2 d ð Þ t kð Þ ðA21Þ × ð þ Þ −q;−p;pþq ðA13Þ ð2πÞ dp p¼0 Z d d q 2 we should then pass to spherical coordinates, integrate over þ ∂tRtðqÞGðqÞ Gðp þ qÞ ð2πÞd the angular variables and finally transform the integration q2 x d 2 variable accordingly to ¼ , Γð3Þ × −q;−p;pþq ðA14Þ Z dp p¼0 s s ð2Þ d d−1 2 ð2Þ d 2 _ Z T5 ¼ −Z x2 GðxÞ ∂tRkðxÞdx ¼ −Z Qd½G R : 1 d2 4 4 2 − ddqG q Γð4Þ G q ∂ R q : ð Þ 2 q;p;−p;−q ð Þ t kð Þ 4 dp p¼0 ðA22Þ ðA15Þ where we introduced the Mellin transformation,
016013-9 N. DEFENU and A. CODELLO PHYS. REV. D 98, 016013 (2018) Z 2 2 2 2 1 m−1 1 d 1 d x d dx d Qm½f ¼ x fðxÞdx ðA23Þ ¼ þ ðA30Þ ΓðmÞ 2 dp2 2 dp2 dx dp dx2 _ and the short hand notation A ¼ ∂tA. Applying the same The derivatives are notation to (A19) we obtain, Z dx s 1 2 d d 3 ¼ 2q cos θ: ðA31Þ T Zð Þ x2∂ R x G x dx dp 4 ¼ð Þ 2d t tð Þ ð Þ 0 sd ð1Þ 2 3 _ 2 ¼ðZ Þ Qd 1½G R ðA24Þ d x 2d 2þ 2: 2 ¼ ðA32Þ dp 0 while for (A18) Z We can also explicitly compute the derivatives of the 1 2 2sd d 1 2 T Zð Þ x2þ ∂ R x G x G x dx propagator with respect to x, 3 ¼ð Þ d t tð Þ ð Þ xð Þ Z 2s d ð1Þ ð3Þ d d 2 2 Z V x2∂tRt x G x Gx x dx A25 G q p G −G x Z φ R x ; þ d ð Þ ð Þ ð Þ ð Þ dx ð þ Þ¼ x ¼ ð Þ ð ð Þþ t;xð ÞÞ ðA33Þ
2s 2s 2 ð1Þ 2 d 2 _ ð1Þ ð3Þ d d ¼ðZ Þ Qd 2½G GxR þZ V 3 2 d 2þ d G q p G 2G x Z φ R x dx2 ð þ Þ¼ xx ¼ ð Þ ð ð Þþ t;xð ÞÞ 2 _ × Qd 1½G GxR ðA26Þ 2 2þ − GðxÞ Rt;xxðxÞ: ðA34Þ and for (A17) The same must be done for the vertexes, Z ð1Þ 2 d 3 ð1Þ ð3Þ T2 ¼ðZ Þ sd x2∂tRtðxÞGðxÞ dx þ Z V sd 1 d2 Γð3Þ Zð1Þ φ ; Z 2 −q;−p;pþq ¼ ð Þ ðA35Þ 2 dp p¼0 d−1 3 × x2 ∂tRtðxÞGðxÞ dx 2 1 d 4 ð1Þ 2 3 _ ð1Þ ð3Þ 3 _ Γð Þ Zð2Þ φ : ¼ðZ Þ sdQd 1½G R þZ V sdQd½G R ðA27Þ 2 −q;−p;pþq ¼ ð Þ ðA36Þ 2þ 2 2 dp p¼0
The computation of T1 needs some more efforts, we should We can then rewrite the T1 coefficient in the following firstly write the Green function form explicitly way, 1 Z G p q ; d ð þ Þ¼ 2 d q 2 ð Þ 2 ð1Þ ð3Þ 2 ZðφÞjp þ qj þ Uk ðφÞþRtðp þ qÞ T1 ¼ ∂tRtðqÞGðqÞ ðZ x þ V Þ ð2πÞd ðA28Þ G 2q2 2 θG : × ð x þ cos xxÞjp¼0 ðA37Þ We can transform the derivative using the definitions When evaluated in p ¼ 0 we obtain x ¼ q2, we then use x ¼ðp2 þ q2 þ 2pq cos θÞ; ðA29Þ latter relation and we rewrite the integrals using only the x variable, we also integrate over the angular variables We use the equivalence obtaining
Z sd d−1 2 1 3 2 2 T x2 dx∂ R x G x Zð Þx Vð Þ G x xG x 1 ¼ 2 t tð Þ ð Þ ð þ Þ xð Þþd xxð Þ s s ð1Þ 2 d 2 _ ð1Þ 2 d 2 _ ð1Þ ð3Þ 2 _ ¼ðZ Þ Qd 2½G GxR þðZ Þ Qd 3½G GxxR þZ V sdQd 1½G GxR 2 2þ d 2þ 2þ 2s s s ð1Þ ð3Þ d 2 _ ð3Þ 2 d 2 _ ð3Þ 2 d 2 _ þ Z V Qd 2½G GxxR þðV Þ Qd½G GxR þðV Þ Qd 1½G GxxR : ðA38Þ d 2þ 2 2 d 2þ We finally obtain for the flow equations, _ 1 Qd GR 2½ ∂tVkðφÞ¼ ðA39Þ 2 ð4πÞd=2
016013-10 SCALING SOLUTIONS IN THE DERIVATIVE EXPANSION PHYS. REV. D 98, 016013 (2018) Zð1Þ φ 2 1 ð Þ 2 _ 2 _ 3 _ ∂tZkðφÞ¼ Qd 1½G R þðd þ 2ÞQd 2½G GxR þdQd 1½G R ð4πÞd=2 2 2þ 2þ 2þ d d d d Zð2Þ φ 2 _ 2 _ ð Þ 2 _ þ þ 1 Qd 2½G GxR þ þ 2 þ 1 Qd 3½G GxxR − Qd½G R 2 2 2þ 2 2 2þ 2ð4πÞd=2 2 Zð1Þ φ Vð3Þ φ 2 ð Þ ð Þ 2 _ 3 _ 2 _ 2 _ þ 2Qd 1½G GxR þ Qd½G R þdQd 1½G GxR þðd þ 2ÞQd 2½G GxxR ð4πÞd=2 2þ d 2 2þ 2þ Vð3Þ φ 2 ð Þ 2 _ 2 _ þ ðQd½G GxR þQd 1½G GxxR Þ: ðA40Þ ð4πÞd=2 2 2þ
The derivatives of the propagator are n _ Qm½G GxR Z d k2 2Z k2 Z_ k2 − x Z φ − Z G q p G −G x 2 Z φ R x ; − xm−1dx ð k þ kð ÞÞð kð Þ kÞ ; ð þ Þ¼ x ¼ ð Þ ð ð Þþ t;xð ÞÞ ðA41Þ ¼ 2 dx 0 2 ð Þ nþ2 ððZkðφÞ − ZkÞx þ Zkk þ Vk ðφÞÞ d2 G q p G 2G x 3 Z φ R x 2 ðA48Þ dx2 ð þ Þ¼ xx ¼ ð Þ ð ð Þþ t;xð ÞÞ 2 Q GnG R_ − G x Rt;xx x : A42 m½ xx ð Þ ð Þ ð Þ Z k2 2 2Z k2 Z_ k2 − x Z φ − Z 2 xm−1dx ð k þ kð ÞÞð kð Þ kÞ To further proceed it is necessary to specify the cutoff ¼ 2 0 Z φ − Z x Z k2 Vð Þ φ nþ3 functions. The regulator employed in our calculations is the ðð kð Þ kÞ þ k þ k ð ÞÞ linear or optimized cutoff ðA49Þ
2 2 2 2 RkðxÞ¼Zkðk − xÞθðk − xÞðA43Þ 2Z δðk − xÞ − k : ðA50Þ 2 ð2Þ nþ2 2 ððZkðφÞ − ZkÞx þ Zkk þ Vk ðφÞÞ Rk;xðxÞ¼−Zkθðk − xÞðA44Þ The only ill-defined expression in the latter equations is on R x Z δ k2 − x k;xxð Þ¼ k ð ÞðA45Þ line (A50), where the δ function is evaluated at the boundary of the integration measure. In our convention, _ 2 2 _ 2 2 RkðxÞ¼2Zkk θðk − xÞ − Zkðk − xÞθðk − xÞ; this expression has to be evaluated as A46 Z ð Þ k2 1 fðxÞδðx − k2Þ¼ fðk2Þ: ðA51Þ where the terms proportional to ðk2 − xÞδðk2 − xÞ and their 0 2 successive derivatives have been neglected since they are zero in the distribution sense. The same result can be Latter definition is consistent with the heat kernel repre- δ formally obtained regularizing the δ-function via its heat sentation of the function as well as with other evaluation kernel representation and taking the singular limit at the schemes appeared in literature [41]. The integrated expres- end of the calculation. sions for the Q functionals are given in the next section. In order to give the expressions for the flow equations, one should consider the explicit form of the Q functionals APPENDIX B: Q FUNCTIONALS in the optimized cutoff case The Q functionals appearing in the beta functionals βV n _ and βZ of Sec. II can be evaluated analytically when the Qm½G R Z linear, or Litim, cutoff k2 2Z k2 Z_ k2 − x xm−1dx k þ kð Þ ; ¼ 2 2 2 0 2 ð Þ n Rk x Zk k − x θ k − x ððZkðφÞ − ZkÞx þ Zkk þ Vk ðφÞÞ ð Þ¼ ð Þ ð Þ ðA47Þ is employed. The explicit expressions are
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2ðn−mþ1Þ 1−m m k Zk Qn½Gk ∂tRk ¼ fð2 − ηÞqn;mðζ; ωÞþηqn 1;mðζ; ωÞg ΓðnÞ þ k2ðn−m−1ÞZ−m m 0 k Qn½Gk Gk∂tRk ¼− ðζ − 1Þfð2 − ηÞqn;m 2ðζ; ωÞþηqn 1;m 2ðζ; ωÞg ΓðnÞ þ þ þ k2ðn−m−2ÞZ−m k2ðn−m−2ÞZ−m 1 m 00 k 2 k Qn½Gk Gk∂tRk ¼2 ðζ − 1Þ fð2 − ηÞqn;m 3ðζ; ωÞþηqn 1;m 3ðζ; ωÞg − ; ðB1Þ ΓðnÞ þ þ þ ΓðnÞ ðζ þ ωÞmþ2 where we defined the threshold integral 1 1 − ζ qn;mðζ; ωÞ ≡ 2F1 m; n; n þ 1; ; ðB2Þ nð1 þ ωÞm 1 þ ω V00 ϕ k ð Þ ZkðϕÞ with ω ¼ 2 and ζ ¼ .Ind ¼ 2, 3, one obtains the expressions given in the following section. k Zk Zk
APPENDIX C: BETA FUNCTIONALS Here, we report the explicit linear cutoff expressions, in two and three dimensions, for the dimensionless beta functionals βv and βζ defined in Sec. II.
1. d =2 η 1 η 2 ηω − 2 − η ζ ζ ω 0 þ ð Þ þ βv ¼ −2v þ φv þ − log ðC1Þ 2 8π ζ − 1 ðζ − 1Þ2 1 þ ω η ζ00 2 ηω − 2 − η ζ η ζ ω v000 2 2 − η 2 − η 2ζ 0 þ ð Þ þ ð Þ βζ ¼ ηζ þ φζ þ − log þ − − 2 8π ðζ − 1Þð1 þ ωÞðζ þ ωÞ ðζ − 1Þ2 1 þ ω 8π 3ðζ þ ωÞ3 3ð1 þ ωÞ3 ðζ þ ωÞ4 v000ζ0 4 − η ζð18 − 7ηÞ − 2η þ 12 2ζ2ð8 − 3ηÞþ2ζð2 − ηÞþ4 − η þ ω3 þ ω2 þ ω 4π ð1 þ ωÞ2ðζ þ ωÞ4 3ð1 þ ωÞ2ðζ þ ωÞ4 3ð1 þ ωÞ2ðζ þ ωÞ4 2ζ3ð2 − ηÞþ4ζ2 − ζð2 þ ηÞ ðζ0Þ2 3η 3ηð7ζ þ 1Þ þ þ − ω4 − ω3 3ð1 þ ωÞ2ðζ þ ωÞ4 8π ðζ − 1Þ2ð1 þ ωÞðζ þ ωÞ4 2ðζ − 1Þ2ð1 þ ωÞðζ þ ωÞ4 ζ2ð30 − 83ηÞ − ζð23η þ 60Þ − 2η þ 30 ζ3ð36 − 47ηÞ − ζ2ð25η þ 60Þþ2ζð6 þ ηÞþ12 − 2η þ ω2 þ ω 6ðζ − 1Þ2ð1 þ ωÞðζ þ ωÞ4 6ðζ − 1Þ2ð1 þ ωÞðζ þ ωÞ4 9ζ4ð2 − ηÞ − ζ3ð11η þ 36Þþζ2ð18 þ 4ηÞ − 2ζη 3η ζ þ ω þ þ log : ðC2Þ 6ðζ − 1Þ2ð1 þ ωÞðζ þ ωÞ4 ðζ − 1Þ3 1 þ ω
2. d =3 rffiffiffiffiffiffiffiffiffiffiffiffi 1 η 1 6 1 3ω η − 2 3 − η ζ 2 ηω − 2 − η ζ pffiffiffiffiffiffiffiffiffiffiffiffi 1 − ζ þ 0 þð þ Þ ð Þ þ ð Þ βv ¼ −3v þ φv þ − þ 1 þ ω arctanh ðC3Þ 2 4π2 3ðζ − 1Þ2 ð1 − ζÞ5=2 1 þ ω
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rffiffiffiffiffiffiffiffiffiffiffiffi 1 η ζ00 ζ 2 − 3η − 2 − 3ηω 2 η 2 3ω − ζ 2 − η 1 − ζ β ηζ þ φζ0 ð Þ þ ð þ Þ ffiffiffiffiffiffiffiffiffiffiffiffið Þ ζ ¼ þ þ 2 2 þ p arctanh 2 8π ðζ − 1Þ ðζ þ ωÞ ð1 − ζÞ5=2 ω þ 1 1 þ ω rffiffiffiffiffiffiffiffiffiffiffiffi ðv000Þ2 −3ζðη − 2Þþηðω þ 4Þ − 6 1 − ζ 1 η − 1 1 η − 1 þ arctanh − þ ð4πÞ2 24ð1 − ζÞ3=2ðω þ 1Þ5=2 1 þ ω 12 ðζ − 1Þ2ðω þ 1Þ 12 ðζ − 1Þ2ðζ þ ωÞ 1 η − 2 2 ðη − 3Þ 4 ζ 1 1 þ − − þ 18 ðζ − 1Þðω þ 1Þ2 9 ðζ þ ωÞ3 3 ðζ þ ωÞ4 12 ðζ − 1Þðζ þ ωÞ2 rffiffiffiffiffiffiffiffiffiffiffiffi v000ζ0 7 ζðη − 2Þ − ηð3ω þ 4Þþ2 1 − ζ 7 η − 2 4 η − 5 8 ζ þ − arctanh − − − 8π2 12 ð1 − ζÞ5=2ðω þ 1Þ3=2 1 þ ω 12 ðζ − 1Þ2ðω þ 1Þ 9 ðζ þ ωÞ3 3 ðζ þ ωÞ4 rffiffiffiffiffiffiffiffiffiffiffiffi 7 η 1 7 1 ζ0 2 49 ζ η − 2 η 5ω 4 2 1 − ζ − þ − ð Þ ð Þþ ð ffiffiffiffiffiffiffiffiffiffiffiffiþ Þþ 2 2 þ 2 p arctanh 6 ðζ − 1Þ ðζ þ ωÞ 3 ðζ − 1Þðζ þ ωÞ 8π 24 ð1 − ζÞ7=2 ω þ 1 1 þ ω 2 η − 7 245η 4 ζ 49 η þ 3 49 1 − þ − − − : ðC4Þ 9 ðζ þ ωÞ3 24ðζ − 1Þ3 3 ðζ þ ωÞ4 32 ðζ − 1Þ2ðζ þ ωÞ 18 ðζ − 1Þðζ þ ωÞ2
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