arXiv:0704.1514v2  [hep-th]  21 Sep 2007
CERN-TH/2007-037
Towards Functional Flows for Hierarchical Models
Daniel F. Litim1
Department of Physcis and Astronomy
University of Sussex, Brighton, BN1 9QH, U.K.
Theory Group, Physics Division
CERN, CH-1211 Geneva 23.
Abstract
The recursion relations of hierarchical models are studied and contrasted with functional
renormalisation group equations in corresponding approximations. The formalisms are
compared quantitatively for the Ising universality class, where the spectrum of universal
eigenvalues at criticality is studied. A significant correlation amongst scaling exponents is
pointed out and analysed in view of an underlying optimisation. Continuous functional flows
are provided which match with high accuracy all known scaling exponents from Dyson’s hi-
erarchical model for discrete block-spin transformations. Implications of the results are
discussed.
1 D.Litim@sussex.ac.uk, Daniel.Litim@cern.ch
http://arxiv.org/abs/0704.1514v2
1. Introduction
Renormalisation group methods [1], and in particular Wilson’s renormalisation group
[2], play an important role in the study of physical systems at strong coupling and/or
large correlations lengths. Differential implementations of Wilson’s idea [3–10] rely on an
appropriately-introduced momentum cutoff leading to flow equations for running couplings
and N -point functions, which can be studied with a large variety of analytical and numerical
methods. Numerical stability and reliability in the results is ensured through powerful
control and optimisation techniques [11–14]. A different implementation of Wilson’s idea is
realised in hierarchical models of lattice scalar theories [15–19]. Hierarchical renormalisation
group transformations are often discrete rather than continuous. Here, sophisticated
numerical methods have been developed to extract the relevant physics, most notably for
high-accuracy studies of scaling exponents for scalar models at criticality [20–22] and related
theories (see [19] and references therein).
Given the close similarity of the underlying principles, it is natural to ask whether
Wilsonian (functional) flows can be linked explicitly, and on a fundamental level, to hier-
archical models. If so, this link would provide a number of benefits. It will make powerful
functional and numerical methods available to the study of hierarchical models. Vice versa,
the numerical tools for hierarchical models could be employed for functional flows in specific
approximations. Furthermore, an explicit link may lead to a path integral representation of
hierarchical models, allowing for systematic improvements beyond a standard kinetic term.
Finally, well-developed optimisation techniques for functional flows could be taken over for
hierarchical models as well.
In the limit of continuous hierarchical block-spin transformations, an explicit link between
Dyson’s hierarchical model [15, 17] and the Wilson-Polchinski flow [23] in the local potential
approximation has been established long ago by Felder [24]. Following a conjecture of [13],
this link has been extended [25, 26] to include optimised versions [11, 12] of Wetterich’s
flow for the effective average action [27]. These interrelations have recently been backed-up
by extensive numerical studies of critical potentials and scaling exponents to high accuracy
from either formalism [28].
In this paper, we evaluate the more general case and ask whether hierarchical models for
discrete block-spin transformations are linked to functional flows with continuous renormali-
sation group transformations. We first contrast the basic setups for functional flows (Sec. 2),
background field flows (Sec. 3), and hierarchical models (Sec. 4). At a Wilson-Fisher fixed
point, underlying similarities and differences are worked out and compared for the leading
scaling exponent (Sec. 5). An extensive numerical study of the eigenvalue spectrum of the
Ising universality class from functional flows is performed (Sec. 6). A strong correlation of
scaling exponents is established and analysed (Sec. 7). It is shown that specific functional
flows match the leading and subleading scaling exponent from Dyson’s hierarchical models
for discrete transformation parameter to high accuracy (Sec. 8). We close with a discussion
of the results and further implications (Sec. 9).
2. Functional Flows
Wilsonian (functional) flows integrate-out quantum fluctuations within a path integral
representation of quantum field theory. In their simplest form, they are generated through
a cutoff term quadratic in the field added to the Schwinger functional, where the (classical)
action is replaced by S → S+∆Sk and ∆Sk ∼
dqφ(q)Rk(q
2)φ(−q). The infrared momentum
cutoff Rk(q
2) ensures that the propagation of small momentum modes q2 ≪ k2 is suppressed,
while the large momentum modes q2 ≫ k2 remain unaffected. Under an infinitesimal change
in the Wilsonian (infrared) cutoff scale k, the effective action Γk changes according to its
functional flow, which reads (t = ln k)
∂tΓk =
∂tRk (1)
in the form put forward by Wetterich [27]. The trace denotes a momentum integration and a
summation over fields. The factor ∂tRk in the integrand is peaked in the vicinity of q
2 ≈ k2.
The cutoff function Rk obeys Rk(q
2) → 0 as k2/q2 → 0, Rk(q2) > 0 as q2/k2 → 0, and
2) → ∞ as k → Λ, and can be chosen freely elsewise, e.g. [11]. It ensures that the flow
is well-defined, thereby interpolating between an initial action S at k = Λ in the ultraviolet
(UV) and the full quantum effective action Γ ≡ Γk=0 in the infrared k → 0.
In addition to providing a momentum cutoff, the function Rk(q
2) also controls the
stability and convergence of subsequent expansions [11–13, 29]. Therefore, it is possible to
identify optimised momentum cutoffs – within given systematic expansions – which improve
the physical result [11, 12, 30]. The construction of optimised cutoffs [11–14] is central to
extract reliable results also in more complex theories including e.g. QCD [31], quantum
gravity [32], thermal physics [5, 6, 33] and critical phenomena [28, 30, 34].
Below, we are interested in 3d scalar theories at criticality, where we can sent the ultraviolet
scale Λ → ∞. To leading order in the derivative expansion, the effective action reads Γk =
d3x[1
∂µφ∂µφ+Uk(ρ̄)] and ρ̄ =
φ2. Introducing r(y) = Rk(q
2)/q2 with y = q2/k2, we find
∂tu = −3u+ ρu′ +
−y3/2 r′(y)
y(1 + r) + u′ + 2ρu′′
with u(ρ) = Uk(ρ̄)/k
3 and ρ = ρ̄/k. An irrelevant constant originating from the angular
integration has been rescaled into the potential and the fields. For the optimal cutoff Ropt =
(k2 − q2)θ(k2 − q2) with ropt = (1/y − 1)θ(1− y), the flow reads [12]
∂tu = −3u+ ρu′ +
1 + u′ + 2ρu′′
after an additional rescaling. This flow is integrated analytically in the limit of a large
number of scalar fields [35]. We note that the universal content of the flow (3) is equivalent
to the Wilson-Polchinski flow in the local potential approximation [13, 25, 26].
3. Background Field Flows
A different form of the flow (1) is obtained for momentum cutoffs which depend addi-
tionally on a background field φ̄. Background fields are most commonly used for the study
of gauge theories [36], see [32, 37, 38] for applications. They have also been employed for a
path integral derivation of (generalised) proper-time flows [39, 40].
In the presence of background fields, the functional Γk[φ] turns into a functional of both
fields, Γk[φ, φ̄]. In order to maintain the one-loop exactness of (1), the momentum cutoff
can only depend on the background field, but not on the propagating field. Following [39],
we introduce x = Γ(2,0)[φ, φ] and x̄ = x[φ = φ̄], where Γ
(n,m)
k [φ, φ̄] ≡ δ
nδmΓk/δφ
nδφ̄m. We
chose momentum cutoffs of the form Rk(q
2)→ x̄ r[x̄], which depend now on the background
field. Here, the regulator cuts off both large momentum modes q2 ≫ k2 and large field
amplitudes with Γ
(2,0)
≫ k2. The full advantage of background fields becomes visible once
they are identified with the physical mean, leading to the functional Γk[φ, φ̄ = φ] → Γk[φ].
The resulting flow is closed provided Γ
k [φ] = Γ
(2,0)
k [φ, φ]. For scalars, this relation becomes
exact in the infrared limit studied below (for gauge fields, see [38]). Using the momentum
cutoffs [39]
rPT,m[x] = exp
2F1[m,m;m+ 1;−mk
− 1 ,
we are lead to the background field flow
∂tΓk = Tr
k2 + x/m
rPT,m
x(1 + rPT,m)
k2 + x/m
∂tx . (4)
If the term ∼ ∂tx on the right hand side is dropped – meaning that additional flow terms
originating from the implicit scale dependence in the momentum cutoff are neglected over
the leading term – the flow (4) reduces to the proper-time flow of Liao [41]. A general
proper-time flow is a linear combination of the first term in (4) for various m [40]; see [42, 43]
for applications.
Next, we specialise to the proper-time approximation to leading order in the derivative
expansion. The flow equation for the effective potential takes the very simple form
∂tu = −3u+ ρu′ +
(m+ u′ + 2ρu′′)m−3/2
, (5)
where m parameterises the momentum cutoff, and an irrelevant constant factor has been
rescaled into the potential and the fields. For m ∈ [1, 5
], the flow (5) is mapped onto the
flow (2) [39]. At m = 5
, the flow (5) is equivalent to (3), modulo a trivial rescaling. As a
final remark, we note that this proper-time flow is also obtained from linear combinations
of higher scale-derivatives of Callan-Szymanzik flows, without relying on background fields
[40]. In this representation, the approximation leading to (5) consists in the neglection of
higher order flow terms ∼ ∂nt Γ
4. Hierarchical Models
Several hierarchical models for an effective potential v(ϕ) of a lattice scalar field have
been introduced in the literature [15–17] (see also [19]). The hierarchical transformation
laws relate the potential v(ϕ) at momentum scale k/ℓ with an average in field space over
v(ϕ) at momentum scale k, where ℓ ≥ 1 is the renormalisation group step parameter. We
restrict ourselves to the three-dimensional case; the generalisation to arbitrary dimensions is
straightforward.
In Dyson’s model [15, 17], the renormalisation group step k → k/ℓ for the potential is
expressed as
e−vk/ℓ(ϕ) =
dξ µℓ(ξ) e
−ℓ3 vk(ℓ
−1/2ϕ+ξ) (6)
The details of the averaging procedure are encoded in the measure factor µℓ(ξ), in the ξ-
dependence of the potential on the right-hand side of (6), and in the choice for the decimation
parameter ℓ. As is evident from (6), a decimation parameter ℓ = 21/3 – employed for most
numerical studies [19–22] – corresponds to a volume decimation of ℓ3 = 2 at each iteration.
For Dyson’s model, the measure is chosen as µℓ(ξ) = (π σ(ℓ))
−1/2 exp(−ξ2/σ(ℓ)) [19], where
we require σ(ℓ) > 0 for ℓ 6= 1, and σ(1) = 0 with σ′(1) 6= 0. A standard choice is σ(ℓ) = 2(ℓ−1)
[24]. By definition, (6) describes a flow towards the infrared for decimation parameters ℓ ≥ 1.
For ℓ → 1, the hierarchical transformation (6) becomes continuous and the measure factor
turns into a δ-function µℓ→1(ξ) → δ(ξ). Performing −ℓ∂ℓ(6), which is equivalent to k∂k(6),
we arrive at a differential flow equation for the effective potential [24]
∂tv = −3v + 12ϕv
′ − v′′ + (v′)2 , (7)
where an irrelevant factor is rescaled into the fields and the potential; t = ln k. The
interaction terms in (7) originate from the scale-derivative of the measure −ℓ∂ℓ µℓ(ξ), which
reads 1
σ′(1) δ′′(ξ) in the limit ℓ → 1. This highlights the relevance of the measure factor
in hierarchical models. Our normalisation corresponds to the choice σ′(1) = 4 to match
with [28]. The limit (7) is independent of σ, but at ℓ 6= 1, we expect that scaling solutions
and exponents from (6) depend on it. Eq. (7) is the well-known Wilson-Polchinski flow
[2, 23]. We therefore conclude that the potential in (7) is related to the potential in (3) by a
Legendre transformation [26, 28].
A different version of a hierarchical model has been introduced by Wilson [16]. Here, the
recursion relation is written as
e−vk/ℓ(ϕ) =
dξ µℓ(ξ) e
ℓ3 [vk(ℓ
−1/2ϕ+ξ)+vk(ℓ
−1/2ϕ−ξ)] . (8)
In Wilson’s original model, the ξ-dependence of the measure is µℓ(ξ) = Nℓ exp(−ξ2), where
the normalisation factor Nℓ is ξ-independent [19]. The measure factor is different from the
one in Dyson’s model, because the Gaussian width is ℓ-independent. If instead we employ
the measure of Dyson’s model, the limit ℓ→ 1 can be performed analytically.2 Up to a trivial
rescaling, we find
∂tv = −3v + 12ϕv
′ − v′′ . (9)
In contrast to the Wilson-Polchinski flow (7), the non-linear term (v′)2 is absent. This comes
about because the integrand of (8) – as opposed to the integrand of (6) – is manifestly
symmetric under ξ → −ξ. Numerical evaluations of (8) have been reported in [16, 44]. For
other representations of hierarchical models we refer to [19] and references therein.
5. Matching Hierarchical Models
In order to match hierarchical models by functional flows, we have to detail the scheme
dependences of physical observables in either formalism. In the functional RG framework,
the fully integrated flow is independent of the momentum cutoff Rk(q
2) chosen for the
integration. Scheme dependences, which enter as a consequence of truncations of Γk[φ],
have been discussed extensively in the literature [11–13, 29, 45]. Their origin is easily
understood. Since the momentum cutoff R in (1) couples to all operators in the theory,
the missing back-coupling of operators neglected in a given truncation can result in a
spurious dependence of physical observables evaluated either from Γ0[φ], or from a fixed
point solution Γ∗[φ]. The scheme dependence is reduced by identifying those momentum
cutoffs, which, in a given truncation, lead to an improved convergence and stability of the flow.
In Fig. 1, we discuss the scheme dependence quantitatively for the leading order scaling
exponent ν at a fixed point of the 3d Ising universality class [29]. Within exact flows (2),
the full Rk-dependence has been studied in [30] by evaluating the fixed points of (2) for
general cutoffs (Fig. 1, first column). The main result is that the range of achievable values
is bounded from above and from below. The upper bound is attained for Callan-Symanzik
type flows with Rk ∼ k2. The lower bound with ν = νopt is attained with the optimal flow
(3), and hence equivalent to the Wilson-Polchinski flow. The sharp cutoff result is indicated
for comparison.
The proper-time flow (5) rests on an intrinsically different truncation, because implicit
dependences on the background field have been neglected as well as higher order flow terms
proportional to the flow of Γ(2); see Sec. 3. Therefore scheme dependences are quantitatively
different. In the approximation (4), the m-dependence of scaling exponents from (5) has
been studied in [39, 42] (Fig. 1, second column). The range of values is again bounded from
above by a Callan-Symanzik flow. The lower bound is achieved for m → ∞. We note that
the range of values exceeds those achievable within (standard) exact flows. The lower bound
2 The variance of the Gaussian measure in (8) can be changed by an explicit rescaling of the fields as
ϕ → ϕ/
σ for finite σ, see [19]. Rescaling also the integration variable ξ → ξ/
σ, and denoting the
potential in terms of the rescaled fields again as v(ϕ), we obtain (8) with a rescaled measure µℓ(ξ) =
−1/2 exp(−ξ2/σ). It agrees with the measure of Dyson’s model for Nℓ = π
−1/2 and σ = σ(ℓ). I thank
Y. Meurice for e-mail correspondence on this point.
Callan-Symanzik
sharp cutoff
WP/opt
mean field
exact RG
proper-time RG
Wilson HM
Dyson HM
0.626
0.649562
0.649570
0.650163
0.653
0.6895
Figure 1: (colour online) Comparison of scaling exponent ν from different functional flows (RG) in
the local potential approximation, and hierarchical models (HM). The solid lines indicate the range of
values obtained in the literature. The dashed lines, if present, indicate that the underlying parameter
space has not been exhausted. The horizontal lines from top to bottom indicate the results for
Callan-Symanzik flows, the sharp cutoff flow, the Wilson-Polchinski (optimal) flow, and the mean
field result. A non-linear rescaling of the ν-axis is introduced for display purposes only (see main
text). Colour coding: exact RG (red), exact background field RG in the proper-time approximation
(violet), Wilson’s hierarchical model (blue) and Dyson’s hierarchical model (light blue).
may be overcome once the additional flow terms, neglected here, are taken into account [39].
This is indicated by the dashed line.
Next we consider scheme dependences of hierarchical models. Based on their construction,
we expect that physical observables depend on the averaging procedure, on the measure
factor µℓ, and on the decimation parameter ℓ. It has proven difficult to systematically
include wave function renormalisations and higher order operators in hierarchical models,
and it is therefore not known whether the scheme dependence vanishes upon higher order
corrections [18, 19]. Still, the scheme dependence should give a reasonable estimate for
the underlying error in the model assumptions, in particular in comparison with functional
methods.
The ℓ-dependence of Wilson’s model (8), originally constructed for ℓ ≈ 2, has been studied
in [44] in the range ℓ ∈ [21/3, 2] (Fig. 1, third column). The full line covers the range of values
obtained in the literature, while the dashed lines indicate that the underlying parameter space
has not been exhausted. Because of (9) being linear in the potential as opposed to (7), we
expect a strong ℓ-dependence, possibly a discontinuity, in the limit ℓ→ 1. The slope ℓ∂ℓ ν(ℓ)
along the data points with ℓ > 21/3 is negative, meaning that ν(ℓ) increases for smaller ℓ.
We stress that Wilson’s HM has an overlap both with exact flows and proper-time flows.
Therefore, it is possible to map ν(ℓ) of Wilson’s model for certain decimation parameters ℓ
onto ν(R) from functional flows with appropriately chosen R. On the other hand, for some
decimation parameter ℓ, Wilson’s model can only be mapped onto proper-time flows but not
on exact flows, while for some decimation parameter it cannot be mapped onto either of them.
The ℓ-dependence of Dyson’s model (6) is displayed in Fig. 1, fourth column. The full line
connects the known results at ℓ = 1 [28, 30], ℓ = 21/3 [20, 22] and ℓ = 2 [20]. The dashed line
towards larger values for ν indicates that the parameter space ℓ ≥ 1 has not been exhausted.
We note that the ℓ-dependence is very weak, with a tiny slope in the range of ℓ-values
covered. The important observation is that the slope ℓ∂ℓ ν(ℓ) is positive in the vicinity of
ℓ ≈ 1−2, implying that ν(ℓ) > ν(1) for ℓ > 1. Consequently, it is possible to map the scaling
exponent ν(ℓ) at discrete block-spin transformation ℓ > 1 onto ν(R) from functional flows for
specific momentum cutoff R, both within the standard exact flows and within proper-time
flows. This supports the conjecture that Dyson’s model can be mapped onto functional flows.
6. Spectrum of Eigenvalues
Whether the observations of the preceeding section can be promoted to a full map
between the formalisms crucially depends on further observables including the subleading
scaling exponents. Here and in the following section, we study the spectrum of universal
eigenvalues (scaling exponents) form functional flows (2) to high accuracy. A fixed point
solution u∗ 6= const. of (2) is characterised by the universal eigenvalues of eigenperturbations
in its vicinity. We denote the ordered set of eigenvalues as O(R) = {ωi(R), i = 0, · · · ,∞},
with ωi < ωj for i < j.
3 In addition to the leading exponent ν(R) ≡ −1/ω0, we study the
first three subleading scaling exponents ω(R) ≡ ω1(R), ω2(R) and ω3(R) within the exact
flow (2) for various cutoffs and coarse graining parameters.
For the numerical analysis, we introduce several classes of momentum cutoffs defined
through rmexp = b/((b+1)
y−1), rexp = 1/(exp cyb−1); rmod = 1/(exp[c(y+(b−1)yb)/b]−1),
with c = ln 2; and ropt,n = b(1/y − 1)nθ(1 − y). These cutoffs include the sharp cutoff
(b → ∞) and asymptotically smooth Callan-Symanzik type cutoffs Rk ∼ k2 as limiting
cases. The larger the parameter b, for each class, the ‘sharper’ the corresponding momentum
cutoff. The cutoff ropt,n probes a two-dimensional parameter space in the vicinity of ropt to
which it reduces for b = 1 and n = 1. For integer n, ropt,n is a C
(n+1) function. In addition,
we consider the cutoffs rmix = exp[−b(
y − 1/√y)] and rmix,opt = exp[− 1b (yb − y−b)], which
obey rmix(1/y) = 1/rmix(y). Note that we have covered a large variety of qualitatively
different momentum cutoffs including exponential, algebraic, power-law, sharp cutoffs and
cutoffs with compact support. Except for ropt,n, all cutoffs are C
(∞)-functions. We employ
the numerical techniques developed in [28, 30].
3 In our conventions, the sole negative eigenvalue at the Wilson-Fisher fixed point is ω0.
a) ω vs. −ω0 d) ω vs. ω2
1.46 1.48 1.5 1.52 1.54
2.8 2.9 3 3.1 3.2
b) ω2 vs. −ω0 e) ω2 vs. ω3
1.48 1.5 1.52 1.54
5.3 5.4 5.5 5.6 5.7 5.8 5.9 6
c) −ω0 vs. ω3 f) ω vs. ω3
5.3 5.4 5.5 5.6 5.7 5.8 5.9 6
5.3 5.4 5.5 5.6 5.7 5.8 5.9 6
ropt,1
rmexp
(rPT)
rmix,opt
rpower
Figure 2: (colour online) Six two-dimensional projections of the four leading scaling exponents in the
Ising universality class from the functional flow (2) for various cutoffs and coarse graining (approx-
imately 103 data sets). Here, ν ≡ −1/ω0. The Wilson-Polchinski result from the optimal flow (3)
(large black dot) corresponds to a local extremum for all scaling exponents. Data sets based on rpower,
rmix,opt, rmix, rexp, rmexp, ropt,1 and rmod; data from rPT is included in Fig. 4.
Our results for the universal eigenvalues at criticality are displayed in Fig. 2 for the six
two-dimensional projections of the four-dimensional subspace {−ω0, ω, ω2, ω3} of observables.
The plot contains roughly 103 data points, the different classes of cutoffs are colour-coded.
We focus on the relevant 10%-vicinity of the Wilson-Polchinski result with scaling exponents
Oopt ≡ O(Ropt) from (3), indicated by a large black dot, see [28] for the high-accuracy
numerical values. The central result of Fig. 2 is that scaling exponents are very strongly
correlated. Despite having probed the space of observables by many qualitatively different
momentum cutoffs, we find that only a small subset of values can actually be achieved. The
correlations increase the closer the eigenvalues O move towards Oopt. In the immediate
vicinity of the Wilson-Polchinski result, we only find a very narrow “throat” connecting
observables O(R) with Oopt. For the sub-leading scaling exponents ωi, the throat remains
very narrow even further away from Oopt. This is seen most clearly in the correlation of ω2
with ω3 in Fig. 2e), as well as in the correlations of ω with both ω2 and ω3 in Fig. 2d) and
f). In turn, for the leading exponents ν, the throat opens up more rapidly once its value is
further away from νopt, see Fig. 2a), b) and c).
7. Correlations of Eigenvalues
The strong correlation of scaling exponents is a structural fingerprint of Wilsonian flows
(2). Since the Wilson-Polchinski result is distinguished in the space of scaling exponents, it
is natural to normalise the data of Fig. 2 with respect to it. We introduce the distance of
any pair of scaling exponents (x, y) from the optimal result (xopt, yopt) as
ρ(x, y) =
(xopt − x)2 + (yopt − y)2 ≡ 10−Nρ(x,y) (10)
We have chosen a standard metric in the space of observables (other choices can be applied
as well). In this representation, full agreement with the (optimal) Wilson-Polchinski result
is achieved for ρ→ 0 and Nρ →∞. We also introduce the angles
ϕ(x, y) = arctan(x/y) . (11)
The critical indices ν and ωi, i ≥ 1, are positive numbers. Therefore, they can cover the
range x/y ∈ [0,∞] and ϕ ∈ [0, π
], and ρ ≥ 0 for any pair of observables (x, y). In the
subspace (ν, ω), the extremal values (νopt, ωopt) have the polar coordinates (ρopt, ϕopt), where
the angle reads ϕopt = 0.78066 · · · which is close to π/4 = 0.785398 · · · , and ρopt = 0. The
radial distance from the origin is ρ(0, 0) = 0.923002 · · · .
In the representation (10), we can study the close vicinity of the Wilson-Polchinski result.
In Fig. 3, we display our data points as functions of the angles ϕ, and their distance from
ρopt in a semi-logarithmic basis. It is noteworthy that only a very narrow range of angles ϕ
is actually achieved by the data, despite the fact that large fractions of the underlying space
of momentum cutoffs is covered. Also, and in contrast to Fig. 2a), many data points are
degenerate in the representation (Nρ, ϕ). A priori, the Wilson-Polchinski value (ρopt, ϕopt)
could have been approached along many different paths. Instead, we find that only a narrow
range of (Nρ, ϕ)-values is achieved for arbitrary momentum cutoff.
↑ Wilson-Polchinski
(optimal flow)Nρ(ν, ω)
← hierarchical model
(Dyson, ℓ = 21/3)
sharp cutoff →
Callan-Symanzik →
0 π/8 π/4 3π/8 π/2
ϕ(ν, ω)
ropt,1
rmexp
(rPT)
rmix,opt
rpower
Figure 3: (colour online) Distance Nρ of the pair of scaling exponents (ν, ω) from the optimal Wilson-
Polchinski values (νopt, ωopt) in the representation (10), (11). Only a narrow range of angles ϕ(ν, ω)
in the vicinity of ϕ ≈ π/4 is achieved by the data. Many data points are nearly degenerate. Data
from functional flows in the local potential approximation (2); same data sets and colour coding as in
Fig. 2, plus further high resolution data points from ropt,1 in the close vicinity of (νopt, ωopt). Results
from the sharp cutoff limit, the Callan-Symanzik type flow (with Rk ∼ k2) and Dyson’s hierarchical
model (with ℓ = 21/3) are also indicated (black dots). The Wilson-Polchinski (optimal flow) result
corresponds to ϕ = ϕopt and Nρ →∞.
This pattern is further highlighted in Fig. 4, where we have magnified the non-trivial
range of data sets from functional flows (2). In addition, we have added data points from the
background field flow (5) using the cutoff rPT,m for m <
. It is remarkable that these data
sets display the same pattern as the data from (2). Our results from this and the preceeding
section are summarised as follows:
Extremum.— The Wilson-Polchinski (optimal flow) result in Fig. 2 corresponds to an
extremum in the space of physical observables with |ωi| ≤ |ωi,opt| for all obervables in the
vicinity of Oopt. The extremum is local, because the exponents approach ωi = 2i − 1, i ≥ 0,
for very soft (Callan-Symanzik-type) momentum cutoffs [30]. For the eigenvalue products
Πni=0(ωi/ωi,opt), the Wilson-Polchinski extremum is a global one.
Uniqueness.— Our result indicates that the correlations of eigenvalues at the Wilson-
Pochinski result are strongest, in the sense that any flow of the form (2) with the exponent
ν(R) = νopt automatically also agrees with the Wilson-Polchinski result in all other
observables O(R) = Oopt. In general, for ν(R) > νopt, this is clearly not the case.
Nρ(ν, ω)
↑ Wilson-Polchinski
(optimal flow)
← hierarchical model
(Dyson, ℓ = 21/3)
sharp cutoff
← Callan-Symanzik
1 1.05 1.1 1.15 1.2
ropt,1
rmexp
rmix,opt
rpower
Figure 4: (colour online) Magnification of Fig. 3 in the vicinity of ϕ ≈ π/4 where ν/ω ≈ 1. The data
points for the distance Nρ(ν, ω) as a function of ν/ω remain highly degenerate. Same data sets and
colour coding as in Fig. 3, plus additional high resolution data points from background field flows
(5) using rPT,m with m <
. Results from the sharp cutoff limit, the Callan-Symanzik type flow
(with Rk ∼ k2) and Dyson’s hierarchical model (with ℓ = 21/3) are also indicated (black dots). The
Wilson-Polchinski (optimal flow) result corresponds to νopt/ωopt = 0.9905692 · · · and Nρ →∞.
Redundancy.— The eigenvalue correlations are so strong that the first two scaling
exponents ν(R) and ω(R), for a given R, contain enough information to fix the remaining
observables on the percent level or below. These ‘dynamical’ constraints point at a major
redundancy of (2) with respect to the underlying momentum cutoffs R. A relevant parameter
has been identified previously. The gap miny≥0 y(1 + r) for normalised cutoffs [11], when
maximised, leads towards the Wilson-Polchinski result [12, 13, 30].
Optimised observables.— Previous reasonings in favour of an optimisation only invoked
properties of the underlying flow (1), e.g. its convergence, locality, stability and boundedness,
allowing for improved physical predictions. This has been exemplified quantitatively for
the observable ν(R) which obeys 1 ≥ ν(R) ≥ νopt [30], where the lower bound νopt is
closest to the physical result [25]. Fig. 2 now shows that this pattern extends to subleading
eigenvalues. This equally extends to asymmetric corrections-to-scaling [46]. Therefore, one
may turn the original reasoning around and argue that – because of the extremum property
of the observables O(R) – an extremisation of the functional flow along the lines discussed
in [11, 12, 14], or similar, should naturally lead towards the values Oopt. Stated differently,
Figs. 2-4 show that observables derived from (1) admit an optimisation.
method cutoff parameter ν ω
hierarchical model Dyson (ℓ = 1) 0.649 561 773 880a 0.655 745 939 193a
ropt,n (n = 1, b = 1) 0.649 561 773 880
a 0.655 745 939 193a
functional RG rcompact (b→ 0) 0.649 561 773 880 0.655 745 939 193
rint (b→ 1) 0.649 561 773 880 0.655 745 939 193
rPT,m (m = 5/2) 0.649 561 773 880
a 0.655 745 939 193a
Table 1: Matching scaling exponents ν and ω from continuous hierarchical transformations with
functional flows. Results agree at least to the order 10−12. Data from this work, and from a) [28].
Finally, we note that the data point from Dyson’s hierarchical model with ℓ = 21/3 –
as plotted in Figs. 3 and 4 – nicely fits into the set of data points covered by functional
flows, extending the link observed in Sec. 5 beyond the leading exponent. This observation
is addressed quantitatively in the following section.
8. Matching Beyond the Leading Exponent
To further substantiate our conjecture that hierarchical models could be mapped onto
functional flows, we have to show quantitatively that results from hierarchical model are
reproduced by specific functional flows. Here, we study the close vicinity of the Wilson-
Polchinski result Oopt, where the correlations are strongest, see Figs. 2 and 3. We have to
restrict our search to Dyson’s hierarchical model, where high-accuracy data for the first
sub-leading scaling exponent ω is available. No subleading exponents have been computed
for Wilson’s model.
For the numerical analysis, we introduce additional classes of momentum cutoffs R which
contain the optimal flow (3) in some limit. In addition to the two-parameter family of cutoffs
ropt,n, we also study the cutoff rcompact = y
−1 exp[−e−1/y/(b − y)]θ(b − y) for b > 0 which
is C(∞), and the cutoff rint = exp(−y)θ(1 − y)θ(y − b) with b ∈ [0, 1], which is effective for
a finite interval of momenta q2 ∈ [bk2, k2]. In the limit b → 0 (b → 1), the corresponding
flows are equivalent to (3). Hence, ropt,n, rcompact and rint parametrise substantially different
classes of cutoffs. More generally, there are infinitely many cutoffs Rk leading to scaling
exponents identical with Oopt, and the examples provided above serve to illustrate this.
At ℓ = 1, Dyson’s hierarchical transformation is continuous, and the scaling exponents
are equivalent to those from the optimal flow (3) and the Wilson-Polchinski flow (7). In
Tab. 1, we compare exponents from different functional flows. We confirm numerically, and
with high accuracy, that the cutoffs ropt,n, rcompact and rint lead to the Wilson-Polchinski
result for specific parameter values.
method cutoff parameter ν ω
hierarchical model Dyson (ℓ = 21/3) 0.649 570b 0.655 736b
ropt,n (n = 1, b = 1.048) 0.649 570(9) 0.655 736(6)
ropt,n (n = 1, b = 0.9545) 0.649 570(9) 0.655 736(9)
ropt,n (n = 1.135, b = 1) 0.649 570(6) 0.655 736(8)
ropt,n (n = 1.1, b = 1.028) 0.649 570(6) 0.655 736(8)
functional RG rcompact (b = 0.04775) 0.649 570(9) 0.655 736(9)
rint (b = 0.944) 0.649 570(9) 0.655 736(8)
rint (b = 0.9444) 0.649 570(7) 0.655 736(9)
rPT,m (m = 2.499785) 0.649 564(9) 0.655 736(1)
rPT,m (m = 2.49944) 0.649 570(1) 0.655 720(6)
Table 2: Matching scaling exponents ν and ω from discrete hierarchical transformations with functional
flows. Results agree to the order 10−6 for all cutoffs except the proper-time flow, which matches up
to the order 10−5. Data from this work, and from b) [22].
At ℓ = 21/3, Dyson’s hierarchical transformation is discrete. The reference data reads
= 0.649570 and ω
= 0.655736 [22].4 These values differ only at the order 10−5 from
the optimal (Wilson-Polchinski) result, and are therefore sufficiently close to Oopt to confirm
or refute the correlations observed in the previous section. Figs. 3 and 4 indicate that the
result from Dyson’s hierarchical model is fully matched by functional flows. Our numerical
results are given in Tab. 2; brackets indicate that a digit is possibly affected by numerical
errors. We have found several sets of parameter values, such that the scaling exponents
agree with Dyson’s model to order 10−6. More importantly, the momentum cutoffs are quite
different. Hence, our analysis also confirms the strong correlation of scaling exponents in
the immediate vicinity of the Wilson-Polchinski result. Based on the eigenvalue correlations
within functional flows, we conjecture that the subleading eigenvalues ωi with i ≥ 2 of
Dyson’s model at ℓ = 21/3 also agree to the corresponding accuracy with the values implied
through the functional flows in Tab. 2.
Our results based on the proper-time flow (5) with rPT has also been given in Tab. 2. Full
agreement is achieved either with the exponent ν
or the subleading exponent ω
, but
not with both of them. Once one of them is matched, the deviation in the other observable
is of the order 10−5. The relevant parameters are m < 5
, the regime where (5) is mapped
onto (2) [39]. Therefore, the values in Tab. 2 reflects well the range covered by standard
Wilsonian flows (2). We expect that full agreement is achieved for proper-time flows which
are linear combinations of (5) for different m, but we did not attempt to do so here.
4 In [22], high-accuracy results at ℓ = 21/3 have been given for γ = 2 ν and ∆ = ν ω (and η = 0) with 13
significant digits. They imply νDHM = 0.649570365 · · · and ωDHM = 0.655736286 · · · . For the present study,
only the first six figures are required.
In summary, we have provided numerical maps from several functional flows onto
Dyson’s model at a non-trivial ℓ 6= 1, with an accuracy of the order 10−6. The set of
achievable values for scaling exponents from functional flows in the close vicinity of the
optimal result is just wide enough to accommodate for the data from Dyson’s model.
This is a non-trivial result, also showing that the ℓ-dependence of Dyson’s model and
the Rk-dependence of functional flows are very intimately related. Based on our results
for ν and ω at ℓ 6= 1, and on continuity in ℓ, we expect that this map extends to other
universal quantities in the same approximation, analogous to the full map which is known
for ℓ = 1. Data for further symmetric and asymmetric corrections-to-scaling exponents,
once available, will allow for additional checks of this picture. Full equivalence is guaranteed
as soon as an explicit link in the form ℓ = ℓ(Rk) or Rk(q
2) = Rk(q
2, ℓ) is furnished. For
the local potential approximation, our results indicate that this map, if it exists, is not unique.
9. Discussion and Conclusions
Establishing equivalences between implementations of Wilson’s renormalisation group as
different as discrete hierarchical models of lattice scalar fields on one side and continuous
functional flows on the other, allows for new views and insights on the respective formalisms
and on the underlying physics. Previously, equivalences were known only in the limit where
the hierarchical transformation becomes continuous. In this paper, based on similarities
in the dependences related to the underlying coarse-graining, we have extended this link
towards discrete hierarchical transformations. This correspondence shows that continuous
RG flows (1) are sensitive to implicit discretisation effects via the momentum cutoff.
Specifically, for the 3d Ising universality class, we have compared the formalisms on the
level of scaling exponents. Their dependence on the step-size parameter ℓ within Dyson’s
hierarchical model (6) is qualitatively and quantitatively similar to their dependence on
the momentum cutoff R within functional flows (2). In either case, scaling exponents
are bounded by the Wilson-Polchinski values obtained for ℓ → 1 and R → Ropt. Once
the hierarchical transformations are discrete, ℓ 6= 1, slight variations in all known scaling
exponents from Dyson’s model are matched by functional flows with non-optimal momentum
cutoffs R 6= Ropt. This is quite remarkable, particularily in view of the strong eigenvalue
correlations found amongst functional flows. In this light, the optimisation of functional
flows with R→ Ropt can now, alternatively, be viewed as the removal of discretisation effects,
at least to leading order in a derivative expansion as studied here. It will be interesting to
contrast these findings with the construction of improved or perfect actions on the lattice.
More generally, it is conceivable that Dyson’s model for arbitrary ℓ is mapped by functional
flows on a fundamental level beyond the numerical map provided for ℓ = 21/3. An explicit
map would be very welcome, also in view of linking hierarchical models to a path integral
representation of the theory. In Wilson’s hierarchical model (8), the range covered by the
leading scaling exponent indicates that a partial map onto functional flows exists, though
only for a restricted domain of ℓ-values. Interestingly, the overlap with background field
flows is even larger. Whether these maps extend beyond the leading exponent cannot be
settled presently due to a lack of data for subleading exponents from Wilson’s model.
In addition, we found a distinct correlation of scaling exponents from functional flows
(2). The eigenvalue spectrum, a fingerprint of the physics in a local potential approxima-
tion, is severely constrained and achieves the Wilson-Polchinski values as an extremum.
Furthermore, the full space of physical observables is described by very few parameters only,
instead of the infinitely many moments of the momentum cutoff. Resolving this redundancy
should prove useful for studies of e.g. non-trivial momentum structures and higher orders in
the derivative expansion. Finally this pattern amongst physical observables highlights the
extremum property of cutoffs leading to the optimal Wilson-Polchinski result [11–14]. We
expect that the intimate link between optimised flows on one side, and extremum points in
the space of observables on the other, persists in more complex theories. This observation
will prove useful for studies in QCD and quantum gravity, where an appropriate optimisation
is even more important to extract the relevant physics.
Acknowledgements.— This work is supported by an EPSRC Advanced Fellowship. I thank
Y. Meurice for e-mail correspondence, the Galileo Galilei Institute for Theoretical Physics
for hospitality, and the INFN for partial support.
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  The recursion relations of hierarchical models are studied and contrasted
with functional renormalisation group equations in corresponding
approximations. The formalisms are compared quantitatively for the Ising
universality class, where the spectrum of universal eigenvalues at criticality
is studied. A significant correlation amongst scaling exponents is pointed out
and analysed in view of an underlying optimisation. Functional flows are
provided which match with high accuracy all known scaling exponents from
Dyson's hierarchical model for discrete block-spin transformations.
Implications of the results are discussed.

Introduction
Renormalisation group methods [1], and in particular Wilson’s renormalisation group
[2], play an important role in the study of physical systems at strong coupling and/or
large correlations lengths. Differential implementations of Wilson’s idea [3–10] rely on an
appropriately-introduced momentum cutoff leading to flow equations for running couplings
and N -point functions, which can be studied with a large variety of analytical and numerical
methods. Numerical stability and reliability in the results is ensured through powerful
control and optimisation techniques [11–14]. A different implementation of Wilson’s idea is
realised in hierarchical models of lattice scalar theories [15–19]. Hierarchical renormalisation
group transformations are often discrete rather than continuous. Here, sophisticated
numerical methods have been developed to extract the relevant physics, most notably for
high-accuracy studies of scaling exponents for scalar models at criticality [20–22] and related
theories (see [19] and references therein).
Given the close similarity of the underlying principles, it is natural to ask whether
Wilsonian (functional) flows can be linked explicitly, and on a fundamental level, to hier-
archical models. If so, this link would provide a number of benefits. It will make powerful
functional and numerical methods available to the study of hierarchical models. Vice versa,
the numerical tools for hierarchical models could be employed for functional flows in specific
approximations. Furthermore, an explicit link may lead to a path integral representation of
hierarchical models, allowing for systematic improvements beyond a standard kinetic term.
Finally, well-developed optimisation techniques for functional flows could be taken over for
hierarchical models as well.
In the limit of continuous hierarchical block-spin transformations, an explicit link between
Dyson’s hierarchical model [15, 17] and the Wilson-Polchinski flow [23] in the local potential
approximation has been established long ago by Felder [24]. Following a conjecture of [13],
this link has been extended [25, 26] to include optimised versions [11, 12] of Wetterich’s
flow for the effective average action [27]. These interrelations have recently been backed-up
by extensive numerical studies of critical potentials and scaling exponents to high accuracy
from either formalism [28].
In this paper, we evaluate the more general case and ask whether hierarchical models for
discrete block-spin transformations are linked to functional flows with continuous renormali-
sation group transformations. We first contrast the basic setups for functional flows (Sec. 2),
background field flows (Sec. 3), and hierarchical models (Sec. 4). At a Wilson-Fisher fixed
point, underlying similarities and differences are worked out and compared for the leading
scaling exponent (Sec. 5). An extensive numerical study of the eigenvalue spectrum of the
Ising universality class from functional flows is performed (Sec. 6). A strong correlation of
scaling exponents is established and analysed (Sec. 7). It is shown that specific functional
flows match the leading and subleading scaling exponent from Dyson’s hierarchical models
for discrete transformation parameter to high accuracy (Sec. 8). We close with a discussion
of the results and further implications (Sec. 9).
2. Functional Flows
Wilsonian (functional) flows integrate-out quantum fluctuations within a path integral
representation of quantum field theory. In their simplest form, they are generated through
a cutoff term quadratic in the field added to the Schwinger functional, where the (classical)
action is replaced by S → S+∆Sk and ∆Sk ∼
dqφ(q)Rk(q
2)φ(−q). The infrared momentum
cutoff Rk(q
2) ensures that the propagation of small momentum modes q2 ≪ k2 is suppressed,
while the large momentum modes q2 ≫ k2 remain unaffected. Under an infinitesimal change
in the Wilsonian (infrared) cutoff scale k, the effective action Γk changes according to its
functional flow, which reads (t = ln k)
∂tΓk =
∂tRk (1)
in the form put forward by Wetterich [27]. The trace denotes a momentum integration and a
summation over fields. The factor ∂tRk in the integrand is peaked in the vicinity of q
2 ≈ k2.
The cutoff function Rk obeys Rk(q
2) → 0 as k2/q2 → 0, Rk(q2) > 0 as q2/k2 → 0, and
2) → ∞ as k → Λ, and can be chosen freely elsewise, e.g. [11]. It ensures that the flow
is well-defined, thereby interpolating between an initial action S at k = Λ in the ultraviolet
(UV) and the full quantum effective action Γ ≡ Γk=0 in the infrared k → 0.
In addition to providing a momentum cutoff, the function Rk(q
2) also controls the
stability and convergence of subsequent expansions [11–13, 29]. Therefore, it is possible to
identify optimised momentum cutoffs – within given systematic expansions – which improve
the physical result [11, 12, 30]. The construction of optimised cutoffs [11–14] is central to
extract reliable results also in more complex theories including e.g. QCD [31], quantum
gravity [32], thermal physics [5, 6, 33] and critical phenomena [28, 30, 34].
Below, we are interested in 3d scalar theories at criticality, where we can sent the ultraviolet
scale Λ → ∞. To leading order in the derivative expansion, the effective action reads Γk =
d3x[1
∂µφ∂µφ+Uk(ρ̄)] and ρ̄ =
φ2. Introducing r(y) = Rk(q
2)/q2 with y = q2/k2, we find
∂tu = −3u+ ρu′ +
−y3/2 r′(y)
y(1 + r) + u′ + 2ρu′′
with u(ρ) = Uk(ρ̄)/k
3 and ρ = ρ̄/k. An irrelevant constant originating from the angular
integration has been rescaled into the potential and the fields. For the optimal cutoff Ropt =
(k2 − q2)θ(k2 − q2) with ropt = (1/y − 1)θ(1− y), the flow reads [12]
∂tu = −3u+ ρu′ +
1 + u′ + 2ρu′′
after an additional rescaling. This flow is integrated analytically in the limit of a large
number of scalar fields [35]. We note that the universal content of the flow (3) is equivalent
to the Wilson-Polchinski flow in the local potential approximation [13, 25, 26].
3. Background Field Flows
A different form of the flow (1) is obtained for momentum cutoffs which depend addi-
tionally on a background field φ̄. Background fields are most commonly used for the study
of gauge theories [36], see [32, 37, 38] for applications. They have also been employed for a
path integral derivation of (generalised) proper-time flows [39, 40].
In the presence of background fields, the functional Γk[φ] turns into a functional of both
fields, Γk[φ, φ̄]. In order to maintain the one-loop exactness of (1), the momentum cutoff
can only depend on the background field, but not on the propagating field. Following [39],
we introduce x = Γ(2,0)[φ, φ] and x̄ = x[φ = φ̄], where Γ
(n,m)
k [φ, φ̄] ≡ δ
nδmΓk/δφ
nδφ̄m. We
chose momentum cutoffs of the form Rk(q
2)→ x̄ r[x̄], which depend now on the background
field. Here, the regulator cuts off both large momentum modes q2 ≫ k2 and large field
amplitudes with Γ
(2,0)
≫ k2. The full advantage of background fields becomes visible once
they are identified with the physical mean, leading to the functional Γk[φ, φ̄ = φ] → Γk[φ].
The resulting flow is closed provided Γ
k [φ] = Γ
(2,0)
k [φ, φ]. For scalars, this relation becomes
exact in the infrared limit studied below (for gauge fields, see [38]). Using the momentum
cutoffs [39]
rPT,m[x] = exp
2F1[m,m;m+ 1;−mk
− 1 ,
we are lead to the background field flow
∂tΓk = Tr
k2 + x/m
rPT,m
x(1 + rPT,m)
k2 + x/m
∂tx . (4)
If the term ∼ ∂tx on the right hand side is dropped – meaning that additional flow terms
originating from the implicit scale dependence in the momentum cutoff are neglected over
the leading term – the flow (4) reduces to the proper-time flow of Liao [41]. A general
proper-time flow is a linear combination of the first term in (4) for various m [40]; see [42, 43]
for applications.
Next, we specialise to the proper-time approximation to leading order in the derivative
expansion. The flow equation for the effective potential takes the very simple form
∂tu = −3u+ ρu′ +
(m+ u′ + 2ρu′′)m−3/2
, (5)
where m parameterises the momentum cutoff, and an irrelevant constant factor has been
rescaled into the potential and the fields. For m ∈ [1, 5
], the flow (5) is mapped onto the
flow (2) [39]. At m = 5
, the flow (5) is equivalent to (3), modulo a trivial rescaling. As a
final remark, we note that this proper-time flow is also obtained from linear combinations
of higher scale-derivatives of Callan-Szymanzik flows, without relying on background fields
[40]. In this representation, the approximation leading to (5) consists in the neglection of
higher order flow terms ∼ ∂nt Γ
4. Hierarchical Models
Several hierarchical models for an effective potential v(ϕ) of a lattice scalar field have
been introduced in the literature [15–17] (see also [19]). The hierarchical transformation
laws relate the potential v(ϕ) at momentum scale k/ℓ with an average in field space over
v(ϕ) at momentum scale k, where ℓ ≥ 1 is the renormalisation group step parameter. We
restrict ourselves to the three-dimensional case; the generalisation to arbitrary dimensions is
straightforward.
In Dyson’s model [15, 17], the renormalisation group step k → k/ℓ for the potential is
expressed as
e−vk/ℓ(ϕ) =
dξ µℓ(ξ) e
−ℓ3 vk(ℓ
−1/2ϕ+ξ) (6)
The details of the averaging procedure are encoded in the measure factor µℓ(ξ), in the ξ-
dependence of the potential on the right-hand side of (6), and in the choice for the decimation
parameter ℓ. As is evident from (6), a decimation parameter ℓ = 21/3 – employed for most
numerical studies [19–22] – corresponds to a volume decimation of ℓ3 = 2 at each iteration.
For Dyson’s model, the measure is chosen as µℓ(ξ) = (π σ(ℓ))
−1/2 exp(−ξ2/σ(ℓ)) [19], where
we require σ(ℓ) > 0 for ℓ 6= 1, and σ(1) = 0 with σ′(1) 6= 0. A standard choice is σ(ℓ) = 2(ℓ−1)
[24]. By definition, (6) describes a flow towards the infrared for decimation parameters ℓ ≥ 1.
For ℓ → 1, the hierarchical transformation (6) becomes continuous and the measure factor
turns into a δ-function µℓ→1(ξ) → δ(ξ). Performing −ℓ∂ℓ(6), which is equivalent to k∂k(6),
we arrive at a differential flow equation for the effective potential [24]
∂tv = −3v + 12ϕv
′ − v′′ + (v′)2 , (7)
where an irrelevant factor is rescaled into the fields and the potential; t = ln k. The
interaction terms in (7) originate from the scale-derivative of the measure −ℓ∂ℓ µℓ(ξ), which
reads 1
σ′(1) δ′′(ξ) in the limit ℓ → 1. This highlights the relevance of the measure factor
in hierarchical models. Our normalisation corresponds to the choice σ′(1) = 4 to match
with [28]. The limit (7) is independent of σ, but at ℓ 6= 1, we expect that scaling solutions
and exponents from (6) depend on it. Eq. (7) is the well-known Wilson-Polchinski flow
[2, 23]. We therefore conclude that the potential in (7) is related to the potential in (3) by a
Legendre transformation [26, 28].
A different version of a hierarchical model has been introduced by Wilson [16]. Here, the
recursion relation is written as
e−vk/ℓ(ϕ) =
dξ µℓ(ξ) e
ℓ3 [vk(ℓ
−1/2ϕ+ξ)+vk(ℓ
−1/2ϕ−ξ)] . (8)
In Wilson’s original model, the ξ-dependence of the measure is µℓ(ξ) = Nℓ exp(−ξ2), where
the normalisation factor Nℓ is ξ-independent [19]. The measure factor is different from the
one in Dyson’s model, because the Gaussian width is ℓ-independent. If instead we employ
the measure of Dyson’s model, the limit ℓ→ 1 can be performed analytically.2 Up to a trivial
rescaling, we find
∂tv = −3v + 12ϕv
′ − v′′ . (9)
In contrast to the Wilson-Polchinski flow (7), the non-linear term (v′)2 is absent. This comes
about because the integrand of (8) – as opposed to the integrand of (6) – is manifestly
symmetric under ξ → −ξ. Numerical evaluations of (8) have been reported in [16, 44]. For
other representations of hierarchical models we refer to [19] and references therein.
5. Matching Hierarchical Models
In order to match hierarchical models by functional flows, we have to detail the scheme
dependences of physical observables in either formalism. In the functional RG framework,
the fully integrated flow is independent of the momentum cutoff Rk(q
2) chosen for the
integration. Scheme dependences, which enter as a consequence of truncations of Γk[φ],
have been discussed extensively in the literature [11–13, 29, 45]. Their origin is easily
understood. Since the momentum cutoff R in (1) couples to all operators in the theory,
the missing back-coupling of operators neglected in a given truncation can result in a
spurious dependence of physical observables evaluated either from Γ0[φ], or from a fixed
point solution Γ∗[φ]. The scheme dependence is reduced by identifying those momentum
cutoffs, which, in a given truncation, lead to an improved convergence and stability of the flow.
In Fig. 1, we discuss the scheme dependence quantitatively for the leading order scaling
exponent ν at a fixed point of the 3d Ising universality class [29]. Within exact flows (2),
the full Rk-dependence has been studied in [30] by evaluating the fixed points of (2) for
general cutoffs (Fig. 1, first column). The main result is that the range of achievable values
is bounded from above and from below. The upper bound is attained for Callan-Symanzik
type flows with Rk ∼ k2. The lower bound with ν = νopt is attained with the optimal flow
(3), and hence equivalent to the Wilson-Polchinski flow. The sharp cutoff result is indicated
for comparison.
The proper-time flow (5) rests on an intrinsically different truncation, because implicit
dependences on the background field have been neglected as well as higher order flow terms
proportional to the flow of Γ(2); see Sec. 3. Therefore scheme dependences are quantitatively
different. In the approximation (4), the m-dependence of scaling exponents from (5) has
been studied in [39, 42] (Fig. 1, second column). The range of values is again bounded from
above by a Callan-Symanzik flow. The lower bound is achieved for m → ∞. We note that
the range of values exceeds those achievable within (standard) exact flows. The lower bound
2 The variance of the Gaussian measure in (8) can be changed by an explicit rescaling of the fields as
ϕ → ϕ/
σ for finite σ, see [19]. Rescaling also the integration variable ξ → ξ/
σ, and denoting the
potential in terms of the rescaled fields again as v(ϕ), we obtain (8) with a rescaled measure µℓ(ξ) =
−1/2 exp(−ξ2/σ). It agrees with the measure of Dyson’s model for Nℓ = π
−1/2 and σ = σ(ℓ). I thank
Y. Meurice for e-mail correspondence on this point.
Callan-Symanzik
sharp cutoff
WP/opt
mean field
exact RG
proper-time RG
Wilson HM
Dyson HM
0.626
0.649562
0.649570
0.650163
0.653
0.6895
Figure 1: (colour online) Comparison of scaling exponent ν from different functional flows (RG) in
the local potential approximation, and hierarchical models (HM). The solid lines indicate the range of
values obtained in the literature. The dashed lines, if present, indicate that the underlying parameter
space has not been exhausted. The horizontal lines from top to bottom indicate the results for
Callan-Symanzik flows, the sharp cutoff flow, the Wilson-Polchinski (optimal) flow, and the mean
field result. A non-linear rescaling of the ν-axis is introduced for display purposes only (see main
text). Colour coding: exact RG (red), exact background field RG in the proper-time approximation
(violet), Wilson’s hierarchical model (blue) and Dyson’s hierarchical model (light blue).
may be overcome once the additional flow terms, neglected here, are taken into account [39].
This is indicated by the dashed line.
Next we consider scheme dependences of hierarchical models. Based on their construction,
we expect that physical observables depend on the averaging procedure, on the measure
factor µℓ, and on the decimation parameter ℓ. It has proven difficult to systematically
include wave function renormalisations and higher order operators in hierarchical models,
and it is therefore not known whether the scheme dependence vanishes upon higher order
corrections [18, 19]. Still, the scheme dependence should give a reasonable estimate for
the underlying error in the model assumptions, in particular in comparison with functional
methods.
The ℓ-dependence of Wilson’s model (8), originally constructed for ℓ ≈ 2, has been studied
in [44] in the range ℓ ∈ [21/3, 2] (Fig. 1, third column). The full line covers the range of values
obtained in the literature, while the dashed lines indicate that the underlying parameter space
has not been exhausted. Because of (9) being linear in the potential as opposed to (7), we
expect a strong ℓ-dependence, possibly a discontinuity, in the limit ℓ→ 1. The slope ℓ∂ℓ ν(ℓ)
along the data points with ℓ > 21/3 is negative, meaning that ν(ℓ) increases for smaller ℓ.
We stress that Wilson’s HM has an overlap both with exact flows and proper-time flows.
Therefore, it is possible to map ν(ℓ) of Wilson’s model for certain decimation parameters ℓ
onto ν(R) from functional flows with appropriately chosen R. On the other hand, for some
decimation parameter ℓ, Wilson’s model can only be mapped onto proper-time flows but not
on exact flows, while for some decimation parameter it cannot be mapped onto either of them.
The ℓ-dependence of Dyson’s model (6) is displayed in Fig. 1, fourth column. The full line
connects the known results at ℓ = 1 [28, 30], ℓ = 21/3 [20, 22] and ℓ = 2 [20]. The dashed line
towards larger values for ν indicates that the parameter space ℓ ≥ 1 has not been exhausted.
We note that the ℓ-dependence is very weak, with a tiny slope in the range of ℓ-values
covered. The important observation is that the slope ℓ∂ℓ ν(ℓ) is positive in the vicinity of
ℓ ≈ 1−2, implying that ν(ℓ) > ν(1) for ℓ > 1. Consequently, it is possible to map the scaling
exponent ν(ℓ) at discrete block-spin transformation ℓ > 1 onto ν(R) from functional flows for
specific momentum cutoff R, both within the standard exact flows and within proper-time
flows. This supports the conjecture that Dyson’s model can be mapped onto functional flows.
6. Spectrum of Eigenvalues
Whether the observations of the preceeding section can be promoted to a full map
between the formalisms crucially depends on further observables including the subleading
scaling exponents. Here and in the following section, we study the spectrum of universal
eigenvalues (scaling exponents) form functional flows (2) to high accuracy. A fixed point
solution u∗ 6= const. of (2) is characterised by the universal eigenvalues of eigenperturbations
in its vicinity. We denote the ordered set of eigenvalues as O(R) = {ωi(R), i = 0, · · · ,∞},
with ωi < ωj for i < j.
3 In addition to the leading exponent ν(R) ≡ −1/ω0, we study the
first three subleading scaling exponents ω(R) ≡ ω1(R), ω2(R) and ω3(R) within the exact
flow (2) for various cutoffs and coarse graining parameters.
For the numerical analysis, we introduce several classes of momentum cutoffs defined
through rmexp = b/((b+1)
y−1), rexp = 1/(exp cyb−1); rmod = 1/(exp[c(y+(b−1)yb)/b]−1),
with c = ln 2; and ropt,n = b(1/y − 1)nθ(1 − y). These cutoffs include the sharp cutoff
(b → ∞) and asymptotically smooth Callan-Symanzik type cutoffs Rk ∼ k2 as limiting
cases. The larger the parameter b, for each class, the ‘sharper’ the corresponding momentum
cutoff. The cutoff ropt,n probes a two-dimensional parameter space in the vicinity of ropt to
which it reduces for b = 1 and n = 1. For integer n, ropt,n is a C
(n+1) function. In addition,
we consider the cutoffs rmix = exp[−b(
y − 1/√y)] and rmix,opt = exp[− 1b (yb − y−b)], which
obey rmix(1/y) = 1/rmix(y). Note that we have covered a large variety of qualitatively
different momentum cutoffs including exponential, algebraic, power-law, sharp cutoffs and
cutoffs with compact support. Except for ropt,n, all cutoffs are C
(∞)-functions. We employ
the numerical techniques developed in [28, 30].
3 In our conventions, the sole negative eigenvalue at the Wilson-Fisher fixed point is ω0.
a) ω vs. −ω0 d) ω vs. ω2
1.46 1.48 1.5 1.52 1.54
2.8 2.9 3 3.1 3.2
b) ω2 vs. −ω0 e) ω2 vs. ω3
1.48 1.5 1.52 1.54
5.3 5.4 5.5 5.6 5.7 5.8 5.9 6
c) −ω0 vs. ω3 f) ω vs. ω3
5.3 5.4 5.5 5.6 5.7 5.8 5.9 6
5.3 5.4 5.5 5.6 5.7 5.8 5.9 6
ropt,1
rmexp
(rPT)
rmix,opt
rpower
Figure 2: (colour online) Six two-dimensional projections of the four leading scaling exponents in the
Ising universality class from the functional flow (2) for various cutoffs and coarse graining (approx-
imately 103 data sets). Here, ν ≡ −1/ω0. The Wilson-Polchinski result from the optimal flow (3)
(large black dot) corresponds to a local extremum for all scaling exponents. Data sets based on rpower,
rmix,opt, rmix, rexp, rmexp, ropt,1 and rmod; data from rPT is included in Fig. 4.
Our results for the universal eigenvalues at criticality are displayed in Fig. 2 for the six
two-dimensional projections of the four-dimensional subspace {−ω0, ω, ω2, ω3} of observables.
The plot contains roughly 103 data points, the different classes of cutoffs are colour-coded.
We focus on the relevant 10%-vicinity of the Wilson-Polchinski result with scaling exponents
Oopt ≡ O(Ropt) from (3), indicated by a large black dot, see [28] for the high-accuracy
numerical values. The central result of Fig. 2 is that scaling exponents are very strongly
correlated. Despite having probed the space of observables by many qualitatively different
momentum cutoffs, we find that only a small subset of values can actually be achieved. The
correlations increase the closer the eigenvalues O move towards Oopt. In the immediate
vicinity of the Wilson-Polchinski result, we only find a very narrow “throat” connecting
observables O(R) with Oopt. For the sub-leading scaling exponents ωi, the throat remains
very narrow even further away from Oopt. This is seen most clearly in the correlation of ω2
with ω3 in Fig. 2e), as well as in the correlations of ω with both ω2 and ω3 in Fig. 2d) and
f). In turn, for the leading exponents ν, the throat opens up more rapidly once its value is
further away from νopt, see Fig. 2a), b) and c).
7. Correlations of Eigenvalues
The strong correlation of scaling exponents is a structural fingerprint of Wilsonian flows
(2). Since the Wilson-Polchinski result is distinguished in the space of scaling exponents, it
is natural to normalise the data of Fig. 2 with respect to it. We introduce the distance of
any pair of scaling exponents (x, y) from the optimal result (xopt, yopt) as
ρ(x, y) =
(xopt − x)2 + (yopt − y)2 ≡ 10−Nρ(x,y) (10)
We have chosen a standard metric in the space of observables (other choices can be applied
as well). In this representation, full agreement with the (optimal) Wilson-Polchinski result
is achieved for ρ→ 0 and Nρ →∞. We also introduce the angles
ϕ(x, y) = arctan(x/y) . (11)
The critical indices ν and ωi, i ≥ 1, are positive numbers. Therefore, they can cover the
range x/y ∈ [0,∞] and ϕ ∈ [0, π
], and ρ ≥ 0 for any pair of observables (x, y). In the
subspace (ν, ω), the extremal values (νopt, ωopt) have the polar coordinates (ρopt, ϕopt), where
the angle reads ϕopt = 0.78066 · · · which is close to π/4 = 0.785398 · · · , and ρopt = 0. The
radial distance from the origin is ρ(0, 0) = 0.923002 · · · .
In the representation (10), we can study the close vicinity of the Wilson-Polchinski result.
In Fig. 3, we display our data points as functions of the angles ϕ, and their distance from
ρopt in a semi-logarithmic basis. It is noteworthy that only a very narrow range of angles ϕ
is actually achieved by the data, despite the fact that large fractions of the underlying space
of momentum cutoffs is covered. Also, and in contrast to Fig. 2a), many data points are
degenerate in the representation (Nρ, ϕ). A priori, the Wilson-Polchinski value (ρopt, ϕopt)
could have been approached along many different paths. Instead, we find that only a narrow
range of (Nρ, ϕ)-values is achieved for arbitrary momentum cutoff.
↑ Wilson-Polchinski
(optimal flow)Nρ(ν, ω)
← hierarchical model
(Dyson, ℓ = 21/3)
sharp cutoff →
Callan-Symanzik →
0 π/8 π/4 3π/8 π/2
ϕ(ν, ω)
ropt,1
rmexp
(rPT)
rmix,opt
rpower
Figure 3: (colour online) Distance Nρ of the pair of scaling exponents (ν, ω) from the optimal Wilson-
Polchinski values (νopt, ωopt) in the representation (10), (11). Only a narrow range of angles ϕ(ν, ω)
in the vicinity of ϕ ≈ π/4 is achieved by the data. Many data points are nearly degenerate. Data
from functional flows in the local potential approximation (2); same data sets and colour coding as in
Fig. 2, plus further high resolution data points from ropt,1 in the close vicinity of (νopt, ωopt). Results
from the sharp cutoff limit, the Callan-Symanzik type flow (with Rk ∼ k2) and Dyson’s hierarchical
model (with ℓ = 21/3) are also indicated (black dots). The Wilson-Polchinski (optimal flow) result
corresponds to ϕ = ϕopt and Nρ →∞.
This pattern is further highlighted in Fig. 4, where we have magnified the non-trivial
range of data sets from functional flows (2). In addition, we have added data points from the
background field flow (5) using the cutoff rPT,m for m <
. It is remarkable that these data
sets display the same pattern as the data from (2). Our results from this and the preceeding
section are summarised as follows:
Extremum.— The Wilson-Polchinski (optimal flow) result in Fig. 2 corresponds to an
extremum in the space of physical observables with |ωi| ≤ |ωi,opt| for all obervables in the
vicinity of Oopt. The extremum is local, because the exponents approach ωi = 2i − 1, i ≥ 0,
for very soft (Callan-Symanzik-type) momentum cutoffs [30]. For the eigenvalue products
Πni=0(ωi/ωi,opt), the Wilson-Polchinski extremum is a global one.
Uniqueness.— Our result indicates that the correlations of eigenvalues at the Wilson-
Pochinski result are strongest, in the sense that any flow of the form (2) with the exponent
ν(R) = νopt automatically also agrees with the Wilson-Polchinski result in all other
observables O(R) = Oopt. In general, for ν(R) > νopt, this is clearly not the case.
Nρ(ν, ω)
↑ Wilson-Polchinski
(optimal flow)
← hierarchical model
(Dyson, ℓ = 21/3)
sharp cutoff
← Callan-Symanzik
1 1.05 1.1 1.15 1.2
ropt,1
rmexp
rmix,opt
rpower
Figure 4: (colour online) Magnification of Fig. 3 in the vicinity of ϕ ≈ π/4 where ν/ω ≈ 1. The data
points for the distance Nρ(ν, ω) as a function of ν/ω remain highly degenerate. Same data sets and
colour coding as in Fig. 3, plus additional high resolution data points from background field flows
(5) using rPT,m with m <
. Results from the sharp cutoff limit, the Callan-Symanzik type flow
(with Rk ∼ k2) and Dyson’s hierarchical model (with ℓ = 21/3) are also indicated (black dots). The
Wilson-Polchinski (optimal flow) result corresponds to νopt/ωopt = 0.9905692 · · · and Nρ →∞.
Redundancy.— The eigenvalue correlations are so strong that the first two scaling
exponents ν(R) and ω(R), for a given R, contain enough information to fix the remaining
observables on the percent level or below. These ‘dynamical’ constraints point at a major
redundancy of (2) with respect to the underlying momentum cutoffs R. A relevant parameter
has been identified previously. The gap miny≥0 y(1 + r) for normalised cutoffs [11], when
maximised, leads towards the Wilson-Polchinski result [12, 13, 30].
Optimised observables.— Previous reasonings in favour of an optimisation only invoked
properties of the underlying flow (1), e.g. its convergence, locality, stability and boundedness,
allowing for improved physical predictions. This has been exemplified quantitatively for
the observable ν(R) which obeys 1 ≥ ν(R) ≥ νopt [30], where the lower bound νopt is
closest to the physical result [25]. Fig. 2 now shows that this pattern extends to subleading
eigenvalues. This equally extends to asymmetric corrections-to-scaling [46]. Therefore, one
may turn the original reasoning around and argue that – because of the extremum property
of the observables O(R) – an extremisation of the functional flow along the lines discussed
in [11, 12, 14], or similar, should naturally lead towards the values Oopt. Stated differently,
Figs. 2-4 show that observables derived from (1) admit an optimisation.
method cutoff parameter ν ω
hierarchical model Dyson (ℓ = 1) 0.649 561 773 880a 0.655 745 939 193a
ropt,n (n = 1, b = 1) 0.649 561 773 880
a 0.655 745 939 193a
functional RG rcompact (b→ 0) 0.649 561 773 880 0.655 745 939 193
rint (b→ 1) 0.649 561 773 880 0.655 745 939 193
rPT,m (m = 5/2) 0.649 561 773 880
a 0.655 745 939 193a
Table 1: Matching scaling exponents ν and ω from continuous hierarchical transformations with
functional flows. Results agree at least to the order 10−12. Data from this work, and from a) [28].
Finally, we note that the data point from Dyson’s hierarchical model with ℓ = 21/3 –
as plotted in Figs. 3 and 4 – nicely fits into the set of data points covered by functional
flows, extending the link observed in Sec. 5 beyond the leading exponent. This observation
is addressed quantitatively in the following section.
8. Matching Beyond the Leading Exponent
To further substantiate our conjecture that hierarchical models could be mapped onto
functional flows, we have to show quantitatively that results from hierarchical model are
reproduced by specific functional flows. Here, we study the close vicinity of the Wilson-
Polchinski result Oopt, where the correlations are strongest, see Figs. 2 and 3. We have to
restrict our search to Dyson’s hierarchical model, where high-accuracy data for the first
sub-leading scaling exponent ω is available. No subleading exponents have been computed
for Wilson’s model.
For the numerical analysis, we introduce additional classes of momentum cutoffs R which
contain the optimal flow (3) in some limit. In addition to the two-parameter family of cutoffs
ropt,n, we also study the cutoff rcompact = y
−1 exp[−e−1/y/(b − y)]θ(b − y) for b > 0 which
is C(∞), and the cutoff rint = exp(−y)θ(1 − y)θ(y − b) with b ∈ [0, 1], which is effective for
a finite interval of momenta q2 ∈ [bk2, k2]. In the limit b → 0 (b → 1), the corresponding
flows are equivalent to (3). Hence, ropt,n, rcompact and rint parametrise substantially different
classes of cutoffs. More generally, there are infinitely many cutoffs Rk leading to scaling
exponents identical with Oopt, and the examples provided above serve to illustrate this.
At ℓ = 1, Dyson’s hierarchical transformation is continuous, and the scaling exponents
are equivalent to those from the optimal flow (3) and the Wilson-Polchinski flow (7). In
Tab. 1, we compare exponents from different functional flows. We confirm numerically, and
with high accuracy, that the cutoffs ropt,n, rcompact and rint lead to the Wilson-Polchinski
result for specific parameter values.
method cutoff parameter ν ω
hierarchical model Dyson (ℓ = 21/3) 0.649 570b 0.655 736b
ropt,n (n = 1, b = 1.048) 0.649 570(9) 0.655 736(6)
ropt,n (n = 1, b = 0.9545) 0.649 570(9) 0.655 736(9)
ropt,n (n = 1.135, b = 1) 0.649 570(6) 0.655 736(8)
ropt,n (n = 1.1, b = 1.028) 0.649 570(6) 0.655 736(8)
functional RG rcompact (b = 0.04775) 0.649 570(9) 0.655 736(9)
rint (b = 0.944) 0.649 570(9) 0.655 736(8)
rint (b = 0.9444) 0.649 570(7) 0.655 736(9)
rPT,m (m = 2.499785) 0.649 564(9) 0.655 736(1)
rPT,m (m = 2.49944) 0.649 570(1) 0.655 720(6)
Table 2: Matching scaling exponents ν and ω from discrete hierarchical transformations with functional
flows. Results agree to the order 10−6 for all cutoffs except the proper-time flow, which matches up
to the order 10−5. Data from this work, and from b) [22].
At ℓ = 21/3, Dyson’s hierarchical transformation is discrete. The reference data reads
= 0.649570 and ω
= 0.655736 [22].4 These values differ only at the order 10−5 from
the optimal (Wilson-Polchinski) result, and are therefore sufficiently close to Oopt to confirm
or refute the correlations observed in the previous section. Figs. 3 and 4 indicate that the
result from Dyson’s hierarchical model is fully matched by functional flows. Our numerical
results are given in Tab. 2; brackets indicate that a digit is possibly affected by numerical
errors. We have found several sets of parameter values, such that the scaling exponents
agree with Dyson’s model to order 10−6. More importantly, the momentum cutoffs are quite
different. Hence, our analysis also confirms the strong correlation of scaling exponents in
the immediate vicinity of the Wilson-Polchinski result. Based on the eigenvalue correlations
within functional flows, we conjecture that the subleading eigenvalues ωi with i ≥ 2 of
Dyson’s model at ℓ = 21/3 also agree to the corresponding accuracy with the values implied
through the functional flows in Tab. 2.
Our results based on the proper-time flow (5) with rPT has also been given in Tab. 2. Full
agreement is achieved either with the exponent ν
or the subleading exponent ω
, but
not with both of them. Once one of them is matched, the deviation in the other observable
is of the order 10−5. The relevant parameters are m < 5
, the regime where (5) is mapped
onto (2) [39]. Therefore, the values in Tab. 2 reflects well the range covered by standard
Wilsonian flows (2). We expect that full agreement is achieved for proper-time flows which
are linear combinations of (5) for different m, but we did not attempt to do so here.
4 In [22], high-accuracy results at ℓ = 21/3 have been given for γ = 2 ν and ∆ = ν ω (and η = 0) with 13
significant digits. They imply νDHM = 0.649570365 · · · and ωDHM = 0.655736286 · · · . For the present study,
only the first six figures are required.
In summary, we have provided numerical maps from several functional flows onto
Dyson’s model at a non-trivial ℓ 6= 1, with an accuracy of the order 10−6. The set of
achievable values for scaling exponents from functional flows in the close vicinity of the
optimal result is just wide enough to accommodate for the data from Dyson’s model.
This is a non-trivial result, also showing that the ℓ-dependence of Dyson’s model and
the Rk-dependence of functional flows are very intimately related. Based on our results
for ν and ω at ℓ 6= 1, and on continuity in ℓ, we expect that this map extends to other
universal quantities in the same approximation, analogous to the full map which is known
for ℓ = 1. Data for further symmetric and asymmetric corrections-to-scaling exponents,
once available, will allow for additional checks of this picture. Full equivalence is guaranteed
as soon as an explicit link in the form ℓ = ℓ(Rk) or Rk(q
2) = Rk(q
2, ℓ) is furnished. For
the local potential approximation, our results indicate that this map, if it exists, is not unique.
9. Discussion and Conclusions
Establishing equivalences between implementations of Wilson’s renormalisation group as
different as discrete hierarchical models of lattice scalar fields on one side and continuous
functional flows on the other, allows for new views and insights on the respective formalisms
and on the underlying physics. Previously, equivalences were known only in the limit where
the hierarchical transformation becomes continuous. In this paper, based on similarities
in the dependences related to the underlying coarse-graining, we have extended this link
towards discrete hierarchical transformations. This correspondence shows that continuous
RG flows (1) are sensitive to implicit discretisation effects via the momentum cutoff.
Specifically, for the 3d Ising universality class, we have compared the formalisms on the
level of scaling exponents. Their dependence on the step-size parameter ℓ within Dyson’s
hierarchical model (6) is qualitatively and quantitatively similar to their dependence on
the momentum cutoff R within functional flows (2). In either case, scaling exponents
are bounded by the Wilson-Polchinski values obtained for ℓ → 1 and R → Ropt. Once
the hierarchical transformations are discrete, ℓ 6= 1, slight variations in all known scaling
exponents from Dyson’s model are matched by functional flows with non-optimal momentum
cutoffs R 6= Ropt. This is quite remarkable, particularily in view of the strong eigenvalue
correlations found amongst functional flows. In this light, the optimisation of functional
flows with R→ Ropt can now, alternatively, be viewed as the removal of discretisation effects,
at least to leading order in a derivative expansion as studied here. It will be interesting to
contrast these findings with the construction of improved or perfect actions on the lattice.
More generally, it is conceivable that Dyson’s model for arbitrary ℓ is mapped by functional
flows on a fundamental level beyond the numerical map provided for ℓ = 21/3. An explicit
map would be very welcome, also in view of linking hierarchical models to a path integral
representation of the theory. In Wilson’s hierarchical model (8), the range covered by the
leading scaling exponent indicates that a partial map onto functional flows exists, though
only for a restricted domain of ℓ-values. Interestingly, the overlap with background field
flows is even larger. Whether these maps extend beyond the leading exponent cannot be
settled presently due to a lack of data for subleading exponents from Wilson’s model.
In addition, we found a distinct correlation of scaling exponents from functional flows
(2). The eigenvalue spectrum, a fingerprint of the physics in a local potential approxima-
tion, is severely constrained and achieves the Wilson-Polchinski values as an extremum.
Furthermore, the full space of physical observables is described by very few parameters only,
instead of the infinitely many moments of the momentum cutoff. Resolving this redundancy
should prove useful for studies of e.g. non-trivial momentum structures and higher orders in
the derivative expansion. Finally this pattern amongst physical observables highlights the
extremum property of cutoffs leading to the optimal Wilson-Polchinski result [11–14]. We
expect that the intimate link between optimised flows on one side, and extremum points in
the space of observables on the other, persists in more complex theories. This observation
will prove useful for studies in QCD and quantum gravity, where an appropriate optimisation
is even more important to extract the relevant physics.
Acknowledgements.— This work is supported by an EPSRC Advanced Fellowship. I thank
Y. Meurice for e-mail correspondence, the Galileo Galilei Institute for Theoretical Physics
for hospitality, and the INFN for partial support.
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