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- High-Accuracy Calculations of the Critical Exponents of Dyson's Hierarchical Model
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- Finite-size scaling exponents in the interacting boson model
- The Local Potential Approximation for the Brueckner G-matrix

CERN-PH-TH-2007-006, SHEP-0702

High-accuracy scaling exponents in the local potential approximation

Claude Bervilliera , Andreas J¨ ttnerb , and Daniel F. Litimc,d u

a

Laboratoire de Math?matiques et Physique Th?orique, Universit? de Tours, Parc de Grandmont, F-37200 Tours e e e b School of Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, U.K. c Department of Physics and Astronomy, University of Sussex, Brighton, East Sussex, BN1 9QH, U.K. d Theory Group, Physics Division, CERN, CH-1211 Geneva 23. We test equivalences between di?erent realisations of Wilson’s renormalisation group by computing the leading, subleading, and anti-symmetric corrections-to-scaling exponents, and the full ?xed point potential for the Ising universality class to leading order in a derivative expansion. We discuss our methods with a special emphasis on accuracy and reliability. We establish numerical equivalence of Wilson-Polchinski ?ows and optimised renormalisation group ?ows with an unprecedented accuracy in the scaling exponents. Our results are contrasted with high-accuracy ?ndings from Dyson’s hierarchical model, where a tiny but systematic di?erence in all scaling exponents is established. Further applications for our numerical methods are brie?y indicated.

arXiv:hep-th/0701172v2 23 Jan 2007

I.

INTRODUCTION

Renormalisation group (RG) methods provide powerful tools in the computation of universal scaling exponents or anomalous dimensions in quantum ?eld theories [1]. Modern functional RG approaches are based on Wilson’s idea to integrateout momentum modes within a path-integral representation of the theory [2]. A particular strength of these methods is their ?exibility when it comes to approximations. Wilsonian ?ows can be implemented in many di?erent ways, which helps adapting the technique to the problem at hand [3]. A useful systematic expansion scheme is the derivative expansion, which, to leading order, is known as the local potential approximation (LPA). For the study of critical phenomena, the derivative expansion is expected to provide good results as long as quantum corrections to propagators and anomalous dimensions remain small [4]. Important results in LPA have been accumulated over the years based on di?erent implementations of the RG including the Wilson-Polchinski RG [5, 6], the functional RG [2] for the e?ective average action [7] and optimised versions thereof [8], and Dyson’s hierarchical RG [9, 10]. Interestingly, and despite qualitative di?erences in the respective formalisms, these approaches are closely related in LPA. In fact, it has been established by Felder [11] that the Wilson-Polchinski RG and the hierarchical RG are equivalent. More recently, following a conjecture ?rst stated in [12, 13], the equivalence between the Wilson-Polchinski ?ow and an optimised functional RG has been proven by Morris [14]. Consequently, universal scaling exponents from either of the three approaches should be identical. Quantitatively, these equivalences have been put to test for 3d critical scalar theories [13, 15]. While

scaling exponents from either approach agree at the order 10?4 , an unexpected disagreement starting at the order 10?5 was found. Presently, however, this last observation solely relies on results from the hierarchical RG and the optimised RG, where exponents have been obtained with su?ciently high accuracy. Therefore it becomes mandatory to obtain results from the Wilson-Polchinski RG with a higher precision, in order to con?rm or refute the above-mentioned discrepancy. In this paper, we close this gap in the literature and study both the Wilson-Polchinski RG and the optimised RG for the 3d Ising universality class with an unprecedented accuracy. To that end, we introduce several methods to solve non-linear eigenvalue problems with special emphasis on the numerical reliability. We expect that these techniques are equally useful for other non-linear problems in mathematical physics. The outline of the paper is as follows. We introduce the relevant di?erential equations (Sec. II) and discuss our numerical methods (Sec. III) and the error control (Sec. IV). Results for the ?xed point solution and scaling exponents are discussed (Sec. V) and compared with the hierarchical model (Sec. VI). We close with a discussion and conclusions (Sec. VII).

II.

LOCAL POTENTIAL APPROXIMATION

We restrict ourselves to a real scalar ?eld theory in three dimensions – the Ising universality class – and to the leading order in the derivative expansion, meaning that the scale-dependent e?ective action is approximated by Γk = d3 x[ 1 ?? φ?? φ + Vk (φ)]. 2 Here, k denotes the scale of the Wilsonian momentum cuto?. We provide the ?ows in terms of the dimensionless potentials u(ρ) = Vk (φ)/k 3 with

2 √ ρ = 1 φ2 /k, and v(?) = Vk (φ)/k 3 with ? = φ/ k. 2 Within an e?ective average action approach [7], the ?ow equation for u(ρ) depends on the infrared momentum cuto?. For an optimised choice of the latter [8], and neglecting an irrelevant (?eld-independent) term, it is given by ?t u = ?3u + ρu′ + 1+ u′ 1 . + 2ρ u′′ (1)

III. NUMERICAL METHODS

Here, t = ln k denotes the logarithmic scale parameter. At a ?xed point ?t u′ = 0, the potential obeys du′′ = dρ u′ u′′ ? 2 ρ (1 + u′ + 2ρ u′′ )2 ? 3u′′ . (2) 2ρ

In this section, we summarise the numerical techniques used to solve (1) – (8) and similar to high accuracy at a ?xed point, and to deduce the leading and subleading scaling exponents. These include tools for local polynomial expansions with various degrees of sophistication (a) – (c), tools for initial-value problems (d), and tools for solving (two-point) boundary-value problems (e). Analytical methods for solving (1) can be found in [16, 17], and are not further elaborated here.

There exists a unique non-trivial and well-de?ned (?nite, no poles) solution to (2) which extends over all ?elds. In terms of the potential v(?), the corresponding equations are 1 1 ?t v = ?3v + ? v ′ + , 2 1 + v ′′ dv ′′ 1 = (? v ′′ ? 5v ′ ) (1 + v ′′ )2 . d? 2 (3) (4)

A.

Local behaviour

For the Wilson-Polchinski RG, we follow an analogous ansatz for the e?ective action. In terms of the potential u(ρ) and with t = ln k we ?nd ?t u = ?3u + ρu′ ? u′ ? 2ρ u′′ + 2ρ (u′ )2 (5) du′′ u′′ u′ (u′ )2 3u′′ = ? + + 2u′ u′′ ? . (6) dρ 2 ρ ρ 2ρ The corresponding equations for v(?) are 1 ?t v = ?3v + ? v ′ ? v ′′ + (v ′ )2 , 2 1 dv ′′ = (? v ′′ ? 5v ′ ) + 2v ′ v ′′ . d? 2 (7) (8)

For small ?elds, we use polynomial expansions of the e?ective potential [18]. Their convergence has been addressed in [16, 19, 20]. We implement the expansion in di?erent ways (a) – (c). At a ?xed point, the potential is Z2 -symmetric under ? → ??. Therefore, we expand the e?ective potential in ρ rather than in ?. (a) We Taylor-expand the potential u(ρ) in ?eld monomials about vanishing ?eld ? ρn up to the maximum order m in the truncation, u(ρ) = 1 λn ρn . n! n=1

m

(12)

Numerical solutions for the ?xed point potentials and their derivatives (2), (4) and (8) are displayed in Figs. 1, 2, 3, 6 and 7 below. The potential v(?) in (3) is related to the WilsonPolchinski potential vWP (?WP ) in (7) by a Legendre transform [14], 1 2 (?WP ? ?) , 2 ′ ? = ?WP ? vWP , v = vWP + (9) (10)

up to an irrelevant constant (the absolute value of the potentials at vanishing ?eld). An immediate consequence of (10) is that the two potentials take their absolute minimum at the same ?eld value ?WP,0 = ?0 . It follows that the barrier heights v(0) ? v(?0 ) = vWP (0) ? vWP (?0 ) (11) are identical. Further similarities and di?erences between (1) – (8) have been studied in [13, 14].

Inserting (12) into (1) and solving for ?t u′ = 0 leads to coupled ordinary di?erential equations ?t λn = βn ({λi }) for the couplings λi , which are solved numerically. We note that the m di?erential equations ?t λn depend, in general, on m + 2 couplings up to order λm+2 . Therefore, we have to provide boundary conditions for the couplings λm+1 and λm+2 , as their values are undetermined by the truncated ?ow. For (12), we employ λm+1 = 0 as a boundary condition (in an expansion about vanishing ?eld, the dependence on λm+2 drops out), which shows good convergence with increasing m; see Tab. 1 for numerical results for the couplings at the ?xed point of (1). (b) The convergence is further enhanced by expanding u(ρ) about non-vanishing ?eld [16, 19]. In this case, we write u(ρ) = 1 λn (ρ ? λ1 )n , n! n=2

m

(13)

where the expansion point ρ = λ1 is de?ned implicitly by u′ (λ1 ) = c with c a free parameter. From (1) we conclude that c can take values between ?1

3

∞

8 4 2

∞

8 4 2 1 0

u(ρ)

u′ (ρ)

1

1 -2

0

-4 5

-4 -2 -1 0 1 2 4

-∞

ρ/ρ0

∞

-1 -∞

-4 -2 -1

0

1 2 4

ρ/ρ0

∞

Figure 1: Fixed point potential u(ρ) from (1), (2) of the Ising universality class for all ?elds ρ ∈ [?∞, ∞]. The absolute minimum is at ρ0 = 1.814 898 403 687 · · · , where the potential is normalised to u(ρ0 ) = 0. The axes x u are rescaled as x → 2+|x| with x = ρ/ρ0 , and u → 2+u .

Figure 2: First derivative of the ?xed point potential u′ (ρ) from (1), (2) in the Ising universality class. Numerically, it reads u′ (0) = ?0.186 064 249 470 · · · at vanishing argument. u′ is a monotonously increasing function of the ?eld. Same rescaling as in Fig. 1 (see text).

and ∞. This is con?rmed by the explicit result, see Fig. 2. A natural choice is c = 0, in which case λ1 denotes the unique potential minimum, u′ (λ1 ) = 0. The numerical convergence of the expansion is best for small c. In a given truncation to order m, the boundary condition λm+1 = 0 = λm+2 is used. The expansion displays very good convergence properties with increasing m [16]; see Tab. 1 for numerical results for the couplings at the ?xed point of (1). (c) The simple boundary conditions λm+1 = 0 or λm+1 = 0 = λm+2 used in (a) or (b), respectively, fail to provide convergent solutions for the Wilson-Polchinski ?ow (5). The reason behind this is that the Wilson-Polchinski ?ow at small ?elds is more strongly sensitive to the couplings at large ?elds [13]. Identifying a stable solution is then more demanding and boils down to providing better-adapted boundary conditions for those higher order couplings, which are undetermined in a given truncation. Equations which ?x the higher order couplings can be derived from the asymptotic behaviour of the scaling solution. An algebraic procedure which determines the highest order couplings from the asymptotic behaviour of (2), i.e. from u′′ → 0 for ρ → ∞ [and similarly for (6)] has been given in [21]. We have adopted this procedure for (6) (for technical details, see [22]). As

a result, we ?nd that the polynomial expansion (12) for the Wilson-Polchinski ?ow (6) with appropriate boundary condition converges nicely; see Tab. 2 for numerical results for the couplings at the ?xed point of (5). Although it is more demanding to stabilise the small ?eld expansion based on (5) as opposed to (1), we note that the Wilson-Polchinski potential for small ?elds di?ers only mildy from the optimised ?xed point potential (see Fig. 6 below). Polynomial expansions like (12) and (13) have, in general, a ?nite radius of convergence in ?eld space, restricted to ?eld values ρmin ≤ ρ ≤ ρmax . Here, we ?nd that ρmin < 0 and ρmax > ρ0 . In either of these cases (a) – (c), the ?ows (1) and (5) transform into a set of m coupled ordinary di?erential equations ?t λn = βn ({λi }) for the coe?cient functions λn . The ?xed point equations βn ({λi }) = 0 can be solved to very high accuracy with standard methods. The scaling exponent ν and subleading corrections-to-scaling exponents are deduced from the eigenvalues of the stability matrix M at criticality, Mij = ?λi βj |?t u′ =0 . Asymmetric corrections-to-scaling, i.e. eigenperturbations not symmetric under ? → ??, can be obtained as well [23].

4

ρ=0 u′ (ρ = λ1 ) = 0 coupling λ1 ?0.186 064 249 470 1.814 898 403 687 λ2 0.082 177 310 824 0.126 164 421 218 λ3 0.018 980 341 099 0.029 814 964 767 0.005 252 082 509 0.006 262 816 384 λ4 λ5 0.001 103 954 106 ?0.000 275 905 516 Table 1: The ?rst ?ve couplings at the ?xed point of the optimised ?ow (1) at vanishing ?eld (?rst column), and at the potential minimum (second column). coupling ρ=0 u′ (ρ = λ1 ) = 0 λWP ?0.228 598 202 437 1.814 898 403 687 1 λWP 0.187 236 893 730 0.086 535 420 434 2 ?0.105 930 606 484 ?0.028 253 169 622 λWP 3 λWP 0.101 611 201 027 0.015 928 269 983 4 λWP ?0.135 786 295 049 ?0.012 666 298 430 5 Table 2: The ?rst ?ve couplings at the ?xed point of the Wilson-Polchinski ?ow (5) at vanishing ?eld (?rst column), and at the potential minimum (second column). B. Global behaviour

∞

8 4 2 1 0

1 -2

u′ (ρ) + 2ρ u′′ (ρ)

-4 5

-1 -∞ -4 -2 -1 0 1 2 4

ρ/ρ0

∞

Figure 3: Mass function u′ + 2ρ u′′ of the ?xed point potential (2) in three dimensions, Ising universality class. 1 Same rescaling as in Fig. 1 (see text). For ρ = 2 ?2 ≥ 0, ′′ the mass function is equivalent to v (?) from (4).

Next we discuss two methods (d) and (e) which integrate (2) or (6) directly, without using polynomial approximations. (d) Initial value problem. The di?erential equations (2) or (6) are studied as initial value problems with boundary conditions given at some starting point ρ1 . The necessary boundary conditions can be obtained within the polynomial expansion (a) and (b) in their domain of validity, see Tab. 1 for the couplings at vanishing ?eld and at the potential minimum. The boundary conditions are u′ (0) = λ1 , u′ (0) = λWP , 1 u′′ (0) ≡ λ2 = ? 2 λ1 (1 + λ1 )2 3

We note that u displays only one global minimum. Both u′ and u′ + 2ρ u′′ are monotonously increasing functions of the ?eld variable. The numerical solution ?ts the expected analytical behaviour for asymptotically large ?elds. (e) Finally, we discuss solvers for two-point boundary value problems. The shooting method [24] requires boundary conditions at two points ρ1 and ρ2 in ?eld space. Initial conditions at ρ1 are varied until the boundary condition at ρ2 is matched. The procedure is iterated until the desired accuracy in the solution is achieved [24]. If one is not constrained by the Z2 symmetry ? → ?? and because of the potentially singular behaviour of the righthand sides of (2) and (6) for ρ → 0, it is preferable to implement (4) and (8). Then the di?erential equations at the origin ? = 0 are better under control and one may shoot from the origin ?1 = 0 (with the required symmetry conditions imposed) to large ?elds ?2 ≈ ?bound , where the asymptotic large-?eld behaviour is imposed. Shooting in the reverse direction may provide a better accuracy. The large?eld behaviour of (2), (6) and (4), (8) has previously been determined in the literature, e.g. [13]. The RG eigenvalues are deduced from (1) or (5) in the vicinity of the ?xed point solution. The implementation of (3), (4) and (7), (8) in terms of ? allows a direct computation of asymmetric correction-to-scaling exponents.

2 u′′ (0) ≡ λWP = ? 3 λWP (1 + λWP ) 1 1 2

at ρ1 = 0 for the optimised RG ?ow [16] and the Wilson-Polchinski ?ow, respectively. Identifying the well-de?ned ?xed point solution u′ (ρ) which extends over all ?elds ρ requires a high degree of ?ne-tuning in the boundary condition λ1 [20]. Integrating (2) towards larger ?elds, starting at some non-vanishing ?eld |ρ| ? O(1) with boundary conditions from (a) or (b) is numerically more stable. Following this strategy, we have computed in Figs. 1, 2 and 3 the ?xed point potential u, its ?rst derivative u′ , and the ?eld-dependent mass term u′ + 2ρ u′′ , respectively, for all ρ ∈ [?∞, ∞].

5

15

12

10

10

Nacc

8

Nν

6

5

4

2

0 0

5

10

15

m

20

25

30

0 0

5

10

15

m

20

25

30

Figure 4: Accuracy of the ?xed point, as de?ned in (14), and its dependence on the order of the expansion m; from (1) using method (b) with c = 0. IV. ERROR CONTROL

Figure 5: Accuracy of the scaling exponent ν, as de?ned by (15), and its dependence on the order of the expansion m; from (1) using method (b) with c = 0.

Within the numerical approaches (a) – (e), there are several sources for numerical errors, the control of which is discussed in this section. (i) Within the polynomial expansions (a) – (c), the ?xed point is determined by seeking simultaneous zeros of all β-functions. Solutions for ?t λn = 0 are found to very high precision. To ensure that the ?xed point is numerically a good approximation to the full ?xed point solution, we have computed |?t u′ (ρ)| for all ρ within the domain of validity of the polynomial approximation. This serves as a measure for the quality of a polynomial expansion to order m. We de?ne the accuracy Nacc of the ?xed point solution to order m as 10?Nacc =

ρ∈[0,ρmax ]

place. This serves as an indicator for the reliability in the scaling exponents [see (ii)]. (ii) Within the polynomial expansions (a) – (c), we study the numerical convergence of both the ?xed point values and the scaling exponents with increasing degree of truncation m [16]. In analogy to (14), we de?ne the number of signi?cant ?gures NX in a ?xed point coupling or a critical index Xm at order m in the expansion as 10?NX = 1 ? Xm , X (15)

max

|?t u′ (ρ)| .

(14)

Here, [0, ρmax ] denotes a compact neighbourhood of the expansion point λ1 , which needs not to coincide with the potential minimum ρ0 . We take ρmax > ρ0 from u′ (ρmax ) + 2ρmax u′′ (ρmax ) = 1. This notion of accuracy is a good measure for how well a local Taylor expansion to order m approximates the full ?xed point potential on [0, ρmax ]. Using (b) with c = 0, we ?nd that (14) achieves its extremum typically at ρ = 0. In Fig. 4, we display Nacc as a function of m. Full red (open blue) dots indicate that max ?t u′ is positive (negative). The slope is approximately 1/2, indicating that an increase in the truncation by ?m ≈ 2 increases the accuracy in the ?xed point solution roughly by a decimal

where X denotes the full (asymptotic) result. For the leading scaling exponent X = ν, we display Nν as a function of m in Fig. 5, based on the expansion (13) with c = 0. An open blue (full red) dot indicates that ν(m) is smaller (larger) than the asymptotic value. The expansion converges roughly in the pattern + + ??. Note that the slope, in comparison with Fig. 4, slightly decreases towards larger m. This part of the analysis is conveniently performed with Mathematica. The accuracy of the matrix inversion (leading to the scaling exponents) has been checked independently with standard routines from Matlab. (iii) Within the polynomial expansions (a) and (b), we study the numerical stability of the result by varying the expansion point. The radius of convergence of polynomial expansions depends on the latter. This check serves as an indicator for a possible break-down of the expansion at the highest orders. We have con?rmed that only the rate of

6

0.6 0.5

∞

8

? ?6

? ?2

0.4

4

0.3 0.2 0.1 0 -0.1 -0.2 0 0.5

v(?)

2 1

v(?)

v ′ (?)

?/?0

1

1.5

2

0 0

1

2

4

?/?0

∞

Figure 6: Comparison of the ?xed point potentials v(?) and their ?rst derivatives v ′ (?) from the optimised RG (full lines) and the Wilson-Polchinski RG (dashed lines). Both potentials are normalised to v(?0 ) = 0, where√ they take their absolute minimum. In either case, ?0 = 2ρ0 = 1.905 202 563 344 · · · .

Figure 7: Comparison of the ?xed point potentials v(?) from the optimised RG (full line) and the WilsonPolchinski RG (dashed line) for all ?elds ? ∈ [0, ∞]. As in Fig. 6, the potentials are normalised to v(?0 ) = 0. x For display purposes, the axes are rescaled as x → 2+|x| v with x = ?/?0 , and v → 2+v .

convergence depends on the expansion point, but not the asymptotic result. (iv) The numerical approach (d) is checked in several ways. First, starting at intermediate ?elds, the numerical accuracy in the integration can be made large. Second, the domain of validity for (d) and (a) – (c) overlap. This allows for a quantitative cross-check. Third, for large ?elds |ρ| → ∞, the results from (d) match the expected asymptotic behaviour. Fourth, we have cross-checked the result with a two-point boundary value routine. (v) The shooting method (e) is controlled in several ways. First, the numerical precision for the solution on [0, ρbound] is only limited by the machine precision. We use standard routines under Fortran f77 with double precision. The boundary condition at large ?elds has not to be known to very high accuracy to achieve a reliable result. Stability in the result is con?rmed by varying ρbound as well as the boundary condition at ρbound. These procedures are applied for both (2) and (6). The shooting method is checked independently using a di?erent solver under Maple, based on a di?erent boundary value integration algorithm.

V.

RESULTS

Our results for the scaling exponents from (1) and (7) are given in Tab. 3 and 4. All digits are signi?cant, except for possible rounding e?ects. Earlier ?ndings with a lower level of accuracy [16, 25, 26] are fully consistent with Tab. 3. Specifically, in Tab. 3, the exponents from the optimised RG ?ows have been obtained using (a), (b), and (e), whereas those for the Wilson-Polchinski ?ow follow from (c) and (e). The method (b) has been used up to the order m = 32, leading to roughly 11 signi?cant digits in the result for ν, and roughly one signi?cant digit less with increasing order for the subleading exponents, see [16, 23]. Method (e) has been pushed to an accuracy of 14 digits for all scaling exponents. The asymmetric corrections-to-scaling exponents in Tab. 4 have been computed using (e), again to an accuracy of 14 digits. Again, all earlier results to lower order in the accuracy [23, 26] are consistent with Tab. 4. In Figs. 1 – 3, we have plotted our results for the ?xed point potential u(ρ) from (2) and its derivatives for all ρ ∈ [?∞, ∞]. It is noteworthy that the solution extends to all negative ρ, which in terms of the ?eld ? corresponds to the analytical

7

optimised RG exponent ν 0.649 561 773 880 65 ω 0.655 745 939 193 3 ω2 3.180 006 512 059 2 5.912 230 612 747 7 ω3 ω4 8.796 092 825 414 ω5 11.798 087 658 337 14.896 053 175 688 ω6

Wilson?Polchinski RG 0.649 561 773 880 65 0.655 745 939 193 35 3.180 006 512 059 2 5.912 230 612 747 7 8.796 092 825 414 11.798 087 658 336 9 14.896 053 175 688

exponent optimised RG Wilson?Polchinski ω ? 1.886 703 838 091 4 1.886 703 838 091 4 ω2 ? 4.524 390 733 670 8 4.524 390 733 670 8 ω3 ? 7.337 650 643 354 7.337 650 643 354 4 10.283 900 724 026 10.283 900 724 026 ω4 ? Table 4: The ?rst four asymmetric correction-to-scaling exponents for the Ising universality class in three dimensions.

Table 3: The leading scaling exponent ν and the ?rst six subleading eigenvalues for the Ising universality class in three dimensions.

continuation to purely imaginary ?eld values. For ρ → ?∞, the solution approaches the convexity bound where u′ + 2ρu′′ → ?1. In the ?ow equation (1), this limit is potentially singular as it corresponds to a pole. However, the explicit solution for ?t u′ = 0 shows that the pole is supressed, leading to a well-behaved solution even in this limit. In the physical domain where ρ ≥ 0, the ?xed point solution stays far away from the pole, since minρ≥0 (u′ + 2ρu′′ ) = u′ (0) > ?1. General (non-?xed point) solutions to (1) approach the pole only in a phase with spontaneous symmetry breaking where Vk′′ (φ) is negative. Then the pole ensures convexity of the physical potential Vk=0 (φ) ′′ in the infrared limit where Vk=0 (φ) ≥ 0 [27]. In Figs. 6 and 7, we compare the ?xed point potentials v(?) from the optimised RG and the Wilson-Polchinski RG for small and large ?elds. We have checked that u(ρ) in Fig. 1 matches v(?) in Fig. 7, as it should. Both potentials have their absolute minimum at the same ?eld value ρWP = ρ0 = 1.814 898 403 687; see Tab. 1 and 2. 0 It is also con?rmed that the potentials display the same barrier height v(0) ? v(?0 ) as expected from (11). For large ?elds, the potentials scale di?erently with the ?elds [13]. We have checked numerically that the ?xed point solutions to (2) and (6) are related by a Legendre transform, see (9), (10). Consequently, at vanishing ?eld, the couplings from Tab. 1 are related to the Wilson-Polchinski couplings in Tab. 2 by λ1 = λWP 1 λWP 1 , 1 ? λWP 1 λ1 , = 1 + λ1 λ2 = λWP 2 λWP 2 , (1 ? λWP )4 1 λ2 = . (1 + λ1 )4 (16) (17)

In summary, we have con?rmed the equivalence of an optimised RG and the Wilson-Polchinski RG to the order 10?14 in the universal indices, and in the Legendre-transformed ?xed point solutions.

Our value λWP = ?0.228 598 202 437 02 for the 1 dimensionless mass term squared at vanishing ?eld deviates at the order 10?5 from the corresponding value ?0.228 601 293 102 given in [28].

VI.

COMPARISON

Next we compare our results with those based on Dyson’s hierarchical model [9]. To leading order in the derivative expansion, it has been proven by Felder [11] that the hierarchical RG is equivalent to the Wilson-Polchinski RG, and hence to the optimised RG. Then scaling exponents should come out identical. With high-accuracy numerical data at hand, we can test this assertion quantitatively. In hierarchical models, the block-spin transformations are characterised by a decimation parameter ? ≥ 1, where ? → 1 refers to Felder’s limit of continuous transformations. Numerical studies with ? = 21/3 ≈ 1.26 have been performed in [15, 30, 31, 32, 33, 34, 35]. In Tab. 5, we compare our results with previous computations of ν from the Wilson-Polchinski RG [28, 29], the optimised RG [12, 16], and the hierarchical RG [15, 30, 31, 32, 33, 34, 35]. The most advanced hierarchical RG (HRG) and functional RG (FRG) studies agree amongst themselves at least to the order 10?6 and 10?12 , respectively. We emphasize that the functional RG results clearly deviate from the hierarchical RG results, although the di?erence (νFRG ? νHRG )/νFRG ≈ ?1.3 × 10?5 is quantitatively small. In Tab. 6 we compare the best results from the functional RG (i.e. both the WilsonPolchinski RG and the optimised RG) with those from the hierarchical RG for various other indices, some of which have been obtained

Similar expressions are found for the higher order couplings. The numerical results in Tab. 1 and 2 con?rm these relations within the present accuracy.

8

Wilson ? Polchinski optimised RG 0.649c 0.64956g d 0.6496 0.649562h 0.649561773881 0.649561773881

hierarchical RG 0.6496a 0.64957b 0.649570e,f,i

Table 5: Comparison of the scaling exponent ν for the Ising universality class in three dimensions (see text). Data from this work, and from a) [30], b) [31], c) [28], d) [29], e) [32], f ) [33], g) [12], h) [16], i) [15].

exponent ν ω γ ? α

functional RG 0.649 561 773 880 65 0.655 745 939 193 3 1.299 123 547 761 3 0.425 947 495 477 4 0.051 314 678 358 05

hierarchical RG 0.649 570 e,f,i 0.655 736 i 1.299 140 730 159e 0.425 946 859 881e 0.051 289 i

Table 6: Comparison of universal indices ν, ω, γ = 2 ν, ? = ω ν and α = 2 ? 3ν (for η = 0) from the functional RG (this work) and the hierarchical RG (same referencing as in Tab. 5).

with up to 12 signi?cant ?gures [32] (see also [15]). The variations (XFRG ? XHRG )/XFRG in the exponents X = ν, ω, γ, ?, α read (?1.3, 1.5, ?1.3, 0.15, 5.0) × 10?5 , respectively. Hence, we con?rm that the ?ndings disagree, beginning at the order 10?5 . Given Felder’s proof of equivalence for the limit ? → 1 [11], we conclude that the hierarchical model diplays a dependence on the decimation parameter ?, with a tiny slope of the order of 10?6 for variations in the exponents.

VII.

DISCUSSION AND CONCLUSIONS

literature by computing scaling exponents for the Ising universality class from the Wilson-Polchinski RG and the optimised RG with an unprecedented accuracy of the order 10?14 . We con?rm their equivalence, ?rst conjectured in [12], based on the leading, subleading and asymmetric correction-toscaling exponents. Equally important, our central numerical results are obtained in several ways, and furthermore independently from both the Wilson-Polchinski and the optimised RG ?ow. We conclude that their equivalence is rock solid for all technical purposes, and in full agreement with the explicit proof given by Morris [14]. In contrast to this, we now have clear indications for a non-equivalence of our results with those from Dyson’s hierarchical model for discrete block-spin transformations. In the 3d Ising universality class, all scaling exponents from the functional RG, i.e. from both the Wilson-Polchinski and the optimised RG, di?er systematically at the order 10?5 from high accuracy studies based on Dyson’s hierarchical RG. Given the high degree of accuracy in the critical indices from the present and earlier studies, and the fact that the scaling exponents in all three approaches have been obtained from several independent numerical implementations and collaborations, it is unlikely that the discrepancy originates from numerical insu?ciencies. Furthermore, these di?erences are absent in the limit of continuous block-spin transformations [11]. We conclude that the tacit assumption of ?-independence in the scaling exponents of the hierarchical model [15] cannot be maintained. Rather, the hierarchical model carries an inherent ?-dependence, similar to a dependence previously observed in [36] (see also [37]). In functional RG approaches, scheme-dependences related to the underlying Wilsonian momentum cuto? can arise in truncations [2]. Here, powerful control and optimisation mechanisms are available which decrease truncational artefacts and increase convergence

Equivalences between non-linear di?erential or di?erence equations, and more generally functional relationships between di?erent implementations of the renormalisation group, often allow for a deeper understanding of both the underlying physics and the adequacy and e?ciency of the methods at hand. It has been proven previously that three different implementations of Wilson’s renormalisation group – the Wilson-Polchinski RG, an optimised version of the e?ective average action RG, and Dyson’s hierarchical RG in the limit of continuous block-spin transformations – are equivalent in the local potential approximation. This implies that the corresponding non-linear RG equations carry identical universal content, e.g. identical scaling exponents. Quantitatively, these equivalences have been con?rmed up to the order 10?4 in the literature, but a discrepancy at the order 10?5 has recently been pointed out. This last observation relies on studies within an optimised RG and Dyson’s hierarchical RG with decimation parameter ? = 21/3 , where results with a su?ciently high accuracy are available, while previous results from the Wilson-Polchinski RG were in agreement with either of them. Here, we have closed this gap in the

9 towards the physical theory [3, 8, 38]. It will be interesting to see if the ?-dependence of hierarchical models or its extensions can be understood along similar lines. For our numerical work, we have developed methods to solve non-linear eigenvalue problems with high precision. The combined use of local polynomial expansions techniques with global integration methods allows for an e?cient determination of the full ?xed point solutions with reasonable numerical e?orts. We have also put some emphasis on a reliable error control, in view of the high accuracy aimed for in the universal eigenvalues. We expect that this combination of techniques will be equally useful for other non-linear problems in mathematical physics, including e.g. the derivative expansion to higher order. Acknowledgements. We thank Humboldt University for computer time. AJ is supported by PPARC grants PPA/G/S/2002/00467 and PPA/G/O/2002/0046. DFL is supported by an EPSRC Advanced Fellowship.

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