Complex Bosonic ManyBody Models: Overview of the Small Field Parabolic Flow
Complex Bosonic ManyBody Models: Overview of the Small Field Parabolic Flow
Tadeusz Balaban
Joel Feldman
Horst Kno¨rrer
Eugene Trubowitz
This paper is a contribution to a program to see symmetry breaking in a weakly interacting many boson system on a threedimensional lattice at low temperature. It provides an overview of the analysis, given in Balaban et al. (The small field parabolic flow for bosonic manybody models: part 1main results and algebra, arXiv:1609.01745, 2016, The small field parabolic flow for bosonic manybody models: part 2fluctuation integral and renormalization, arXiv:1609.01746, 2016), of the 'small field' approximation to the 'parabolic flow' which exhibits the formation of a 'Mexican hat' potential well.

Research supported in part by the Natural Sciences and Engineering Research Council of
Canada and the Forschungsinstitut fu¨r Mathematik, ETH Zu¨rich.
The program was initiated in [3, 4], where we expressed the positive
temperature partition function and thermodynamic correlation functions in a
periodic box (a discrete threedimensional torus) as ‘temporal’ ultraviolet limits
of fourdimensional (coherent state) lattice functional integrals (see also [27]).
By a lattice functional integral, we mean an integral with one (in this case
complex) integration variable for each point of the lattice. By a ‘temporal’
ultraviolet limit, we mean a limit in which the lattice spacing in the inverse
temperature direction (imaginary time direction) is sent to zero while the
lattice spacing in the three spatial directions is held fixed.
In [7],1 by a complete large field/small field renormalization group
analysis, we expressed the temporal ultraviolet limit for the partition function,2
still in a periodic box, as a fourdimensional lattice functional integral with the
lattice spacing in all four directions being of the order one, preparing the way
for an infrared renormalization group analysis of the thermodynamic limit.
This overview concerns the next stage of the program, which is
contained in [13, 14] and the supporting papers [9–12, 15, 16]. There we initiate
the infrared analysis by tracking, in the small field region, the evolution of the
effective interaction generated by the iteration of a renormalization group map
that is taylored to a parabolic covariance3: in each renormalization group step
the spatial lattice directions expand by a factor4 L > 1, the inverse
temperature direction expands by a factor L2 and the running chemical potential grows
by a factor of L2, while the running coupling constant decreases by a factor
of L−1. Consequently, the effective potential, initially close to a paraboloid,
develops into a Mexican hat with a moderately large radius and a moderately
deep circular well of minima. [13, 14] ends after a finite number (of the order
of the magnitude of the logarithm of the coupling constant) of steps once the
chemical potential, which initially was of the order of the coupling constant,
has grown to a small ‘ ’ power of the coupling constant. Then we can no longer
base our analysis on expansions about zero field, because the renormalization
group iterations have moved the effective model away from the trivial
noninteracting fixed point.
In the next stage of the construction, we plan to continue the parabolic
evolution in the small field regime, but expanding around fields concentrated
at the bottom of the (Mexican hat shaped) potential well rather around zero
(much as is done in the Bogoliubov Ansatz) and track it through an
additional finite number of steps until the running chemical potential is sufficiently
larger than one. At that point, we will turn to a renormalization group map
with a scaling taylored to an elliptic covariance that expands both the
temporal (inverse temperature) and spatial lattice directions by the same factor L.
It is expected that the elliptic evolution can be controlled through infinitely
many steps, all the way to the symmetry broken fixed point. The system is
1 See also [8] for a more pedagogical introduction.
2 A similar analysis will yield the corresponding representations for the correlation functions.
3 Morally, the 1 + 3dimensional heat operator.
4 L is a fixed, sufficiently large, odd natural number.
superrenormalizable in the entire parabolic regime because the running
coupling constant is geometrically decreasing. However in the elliptic regime, the
system is only strictly renormalizable.
The final stage(s) of the program concern the control of the large field
contributions in both the parabolic and elliptic regimes.
The technical implementation of the parabolic renormalization group in
[13, 14] proceeds much as in [6, 7], except that we are restricting our attention
to the small field regime and
◦ we use 1 + 3dimensional block spin averages, as in [2, 24, 25]. In [7], we
had used decimation, which was suited to the effectively onedimensional
problem of evaluating the temporal ultraviolet limit.
◦ Otherwise, the stationary phase calculation that controls oscillations is
similar, but technically more elaborate.
◦ The essential complication is that the critical fields and background fields
are now solutions to (weakly) nonlinear systems of parabolic equations.
◦ The Stokes’ argument that allows us to shift the multidimensional
integration contour to the ‘reals’ and
◦ the evaluation of the fluctuation integrals is similar.
◦ However, there is an important new feature: The chemical potential has
to be renormalized.
To analyze the output of the block spin convolution (a single
renormalization group step), it is de rigueur for the small field/large field style of
renormalization group implementations to introduce local small field conditions on
the integrand and then decompose the integral into the sum over all partitions
of the discrete torus into small and large field regions on which the conditions
are satisfied and violated, respectively. Small field contributions are to be
controlled by powers of the coupling constant v0 (a suitable norm of the two
body interaction) uniformly in the volume of the small field region. Large field
contributions are to be controlled by a factor e−1/vε0 , ε > 0 , raised to the
volume of the large field region. Morally, in small field regions, perturbation
expansions in the coupling constant converge and exhibit all physical
phenomena. Large field regions give multiplicative corrections that are smaller than
any power of the coupling constant. So, in the leading terms, every point is
small field.
If the actions in our functional integrals were sums of positive terms (as in
a Euclidean O(n) model), it would be routine to extract an exponentially small
factor per point of a large field region. They are not. There are explicit purely
imaginary terms. In [13, 14], we analyze the parabolic flow of the leading term,
in which all points are small field, as long as it is possible to expand around
zero field. Nevertheless, we show (see, [15]) that our actions do have positivity
properties and consequently there is at least one factor e−1/vε0 whenever there
is a large field region. A stronger bound of a factor per point of a large field
region is reasonable and would be the main ingredient for controlling the full
parabolic renormalization group flow in this regime.
We now formally introduce the main objects of discussion and enough
machinery to allow technical (but simplified) statements of the main results of
[13, 14] and the methods used to establish them.
One conclusion of our previous work in [7] is that the purely small field
contribution to the partition function for a gas of bosons hopping on a
threedimensional discrete torus X = Z3/LspZ3 (where Lsp, a power of L, is the
spatial infrared regulator which will ultimately be sent to infinity) takes the
form
S0 x ∈ X0
• X0 = Z/LtpZ × X is a 1 + 3dimensional discrete torus with points
x = (x0, x). Here, Ltp ≈ k1T , also a power of L, is the inverse temperature
infrared regulator, which can ultimately be sent to infinity to get the
temperature zero limit.
• ψ ∈ CX0 is a complex valued field on X0 , ψ∗ is the complex conjugate
field and, for each x ∈ X0, dψ(x)∗∧dψ(x) is the standard Lebesgue measure
2ı
on C.
ψ(x) ≤ v0−1/3+ , ∂ν ψ(x) ≤ v0−1/3+ , ν = 0, 1, 2, 3,
• S0 = ψ ∈ CX0
x ∈ X0
where the small ‘coupling constant’ v0 is an exponentially, tree length
weighted L1–L∞norm (see the discussion of norms at the end of this
overview or [13, Definition 1.9]) of an effective interaction V0 (see [13,
Proposition D.1]). Here, ∂ν , ν = 0, 1, 2, 3 , is the forward difference
operator in the xν direction.
• Let ψ∗ be another arbitrary element of CX0 . ( ψ∗ is not to be confused
with the complex conjugate ψ∗ of ψ .)
• A0(ψ∗, ψ) = −A0(ψ∗, ψ) + p0(ψ∗, ψ, ∇ψ∗, ∇ψ) . The action A0(ψ∗, ψ)
determining the partition function is the restriction
A0(ψ∗, ψ) = A0(ψ∗, ψ) ψ∗=ψ∗
of A0(ψ∗, ψ) to the ‘real’ subspace ψ∗ = ψ∗ of CX0 × CX0 . Here, ∇ is
the (fourdimensional) discrete gradient operator.
• ‘Morally,’ A0(ψ∗, ψ) = ψ∗, (−∂0 + h)ψ 0 + V0(ψ∗, ψ) − μ0 ψ∗, ψ 0,
where
f (x)g(x) is the natural real inner product on CX0
f , g 0 =
x ∈ X0
◦ h is a nonnegative, secondorder, elliptic (lattice) pseudodifferential
operator acting on X —for example, a constant times minus the
spatial discrete Laplacian
◦ V0(ψ∗, ψ) = 12 V0(x1, x2, x3, x4) ψ∗(x1)ψ(x2)ψ∗(x3)ψ(x4) is a
V0(0, x2, x3, x4) > 0
ψ∗, ψ, {ψ∗ν , ψν }3ν=0 ∈ C10X0
ψ(∗)(x), ψ(∗)ν (x) ≤ v0−1/3+ε,
where ‘(∗)’ means ‘either with ∗ or without ∗.’
See [13, Proposition D.1] for more details.
For convenience, set
With this notation, the partition function is
x ∈ X0
It is natural to study the partition function using a steepest descent
or stationary phase analysis. The exponential e ψ∗, ∂0ψ is purely oscillatory
because the quadratic form ψ∗, ∂0ψ is pure imaginary. Fortunately, our
partition function, Z , has the essential feature that there is an analytic function
A0(ψ∗, ψ) on a neighborhood of the origin in CX0 × CX0 whose restriction to
the real subspace is the ‘small field’ action. Our renormalization group
analysis of the oscillating integral defining Z is based on the critical points of
A0(ψ∗, ψ) = ψ∗, (−∂0 + h)ψ + V0(ψ∗, ψ) − μ0 ψ∗, ψ in CX0 × CX0 that
typically do not lie in the real subspace, and a multidimensional Stokes’
contour shifting construction that is only possible because p0(ψ∗, ψ) is analytic.
We now formally introduce the ‘block spin’ renormalization group
transformations that are used in this paper. Let X−1 be the subgroup L2Z/LtpZ ×
LZ3/LspZ3 of X0 . Observe that the distance between points of X−1 on
the inverse temperature axis is L2 and on the spatial axes is L , and that
X−1 = L−5X0 . Also, let Q(0) : CX0 → CX−1 be a linear operator that
commutes with complex conjugation. We will make a specific choice of Q(0)
later. It will be a ‘block spin averaging’ operator with, for each y ∈ X−1,
Q(0)ψ (y) being ‘morally’ the average value of ψ in the L2 × L × L × L block
centered on y. Insert into the integral of (2)
where f , g −1 = L5
y ∈ X−1
CX−1 and N (0) is a normalization constant. Then exchange the order of the
f (y)g(y) is the natural real inner product on
CX0 x∈X0
dψ(x)∗∧dψ(x) e− L12 θ∗ − Q(0)ψ∗ , θ−Q(0)ψ −1 F0(ψ∗, ψ).
2πı
X0(1) = Z/ LLt2p Z × Z3/ LLsp Z3
using the ‘parabolic’ scaling map x ∈ X0(1) → (L2x0, Lx) ∈ X−1 , which is
an isomorphism of Abelian groups. Abusing notation, we consciously use the
symbol ψ(x) as the name of a field on the unit torus X0(1) even though it
was used before as the name of a field on the unit torus X0 . By definition,
the block spin renormalization group transform of F0(ψ∗, ψ) associated with
Q(0) with external fields ψ and ψ∗ in CX0(1) is
F1(ψ∗, ψ) = B1 S−1ψ∗, S−1ψ
S−1ψ (y0, y) = L−3/2 ψ Ly02 , Ly (3)
CX0 x ∈ X0
the original small field part of the partition function.
Repeat the construction.
dψ(x)2∗π∧ıdψ(x) e− L12 θ∗−Q(1)ψ∗,θ−Q(1)ψ −1 F1(ψ∗, ψ)
f(y0+L2,y)−f(y0,y) . Similarly, for spatial difference operators.
L2
5 In θ∗, (∂0 + Δ)θ −1, ∂0 is the forward difference operator on X−1. That is, (∂0f )(y) =
and then rescale to obtain the block spin renormalization group transform
F2(ψ∗, ψ) = B2 S−1ψ∗, S−1ψ
where S−1ψ (y0, y) = L−3/2ψ( Ly02 , Ly ) for any ψ ∈ CX0(2) . Interchanging the
order of integration,
CX0 x ∈ X0
We keep repeating the construction to generate a sequence Fn(ψ∗, ψ) , n ≥ 1 ,
of functions defined on spaces CX0(n) × CX0(n) . Balaban et al. [13, 14] concern a
sequence Fn(SF )(ψ∗, ψ) of ‘small field’ approximations to the Fn’s. We expect
and provide some supporting motivation for, but do not prove, that Fn =
Fn(SF ) + O e−1/vε0 . For the precise definition, see [13, §1.2 and, in particular,
Definition 1.6]. For the supporting motivation see [15].
To make a specific choice for the, to this point arbitrary, sequence Q(0),
. . . , Q(n), . . . of block averaging operators, let q(x) be a nonnegative,
compactly supported, even function on Z × Z3 and Q the associated convolution
operator6
x∈Z×Z3
q(x) ψ y + [x] , ψ ∈ CX0(n) , y ∈ X −(n1+1)
where [x] is the point in the quotient X0(n) = Z/ LL2tpn Z × Z3/ LLsnp Z3
represented by x ∈ Z × Z3 . By construction, Qψ ∈ CX −(n1+1) . We fix q(x)
to be the convolution of the indicator function of the (discrete) rectangle
[− L22−1 , L22−1 ] × [− L−21 , L−21 ]3 in Z × Z3 convolved with itself four times
and normalized so that its sum over Z × Z3 is one. In [13, 14], the basic objects
are the ‘small field’ block spin renormalization iterates Fn(SF )(ψ∗, ψ) , where
at each step Q is chosen to be convolution with the fixed kernel q .
If we had defined Q by convolving just with the indicator function of the
rectangle itself, properly normalized, then (Qψ)(y) would be the usual average
of ψ(x) over the rectangular box in X0(n) centered at y with sides L2 and
L . We work with the smoothed averaging kernel rather than the sharp one
for technical reasons: Commutators [∂ν , Q] are routinely generated and are
small enough when Q is smooth enough. For the rest of this overview, we will
pretend that q is just the indicator function of the rectangle and formulate
our results as if this were the case. We will also pretend that the operator h
on X appearing in the action A0(ψ∗, ψ) is (minus) the lattice Laplacian. Full,
technically complete, statements are in [13, §1.6].
6 By abuse of notation, we use the same symbol Q for the convolution operator acting on
(n)
all of the spaces CX0 .
− An ψ∗, ψ, φ∗n(ψ∗, ψ), φn(ψ∗, ψ)
An = an (ψ∗ − Qnφ∗n) , (ψ − Qnφn) 0 + φ∗n, (−∂0 − Δ)φn n
− μn φ∗n, φn n + Vn(φ∗n, φn)
on the domain
Sn =
ψ(∗)(x) ≤ κn , ∂ν ψ(∗)(x) ≤ κn,
and zero on its complement. Here,
• you can think of the radii κn and κn as being roughly L 43 nv0− 31 + and
L 83 nv− 31 + , respectively. Explicit expressions for κn and κn are given in
0
[13, Definition 1.11.a].
• φ∗n(ψ∗, ψ) and φn(ψ∗, ψ) are (nonlinear) maps from an open
neighborhood of the origin in CX0(n) × CX0(n) to CXn , where Xn is the discrete
torus, isomorphic to X0 , but scaled down to have lattice spacing L−2n
in the time direction and L−n in the spatial directions.9 We say more
about them in the last of this sequence of bullets. Given ‘external fields’
ψ∗, ψ , the functions φ∗n(ψ∗, ψ)(u) , φn(ψ∗, ψ)(u) on Xn are referred
to as the ‘background fields’ at scale n .
• f , g 0 = f (x)g(x) and f , g n = L−5n f (u)g(u) are the
Vn(u1, u2, u3, u4) f∗(u1)f (u2)f∗(u3)f (u4)
where Vn(u1, u2, u3, u4) is close to
Vn(u)(u1, u2, u3, u4) = L1n (L5n)3 V0(U1, U2, U3, U4) ,
Uj =
L2n uj0, Lnuj (4.)
7 An explicit formula for μ∗ is given in [13, (1.19)].
8 We are weakening some of the statements, for pedagogical reasons. In particular, the sets
of allowed μ0’s and n’s are a bit larger than the sets specified here.
9 Xn = L12n Z/ LL2tpn Z × L1n Z3/ LLsnp Z3 and the map u ∈ Xn → x = (L2nu0, Lnu) ∈ X0 is
an isomorphism of Abelian groups.
• The perturbative correction pn ψ∗, ψ, {ψ∗ν }3ν=0, {ψν }3ν=0 is a power
series in the ten variables ψ∗, ψ, {ψ∗ν , ψν }3ν=0 ∈ CX0(n) , with no
ψ∗(x)ψ(y) or constant terms, such that each nonzero term has as many
factors with asterisks as factors without asterisks. It converges10 when
ψ(∗)(x) ≤ κn and ψ(∗)ν (x) ≤ κn for all 0 ≤ ν ≤ 3 and x ∈ X0(n) .
• Zn is a normalization constant.11
• μn is the ‘renormalized’ chemical potential.12 It is close to L2nμ0 .
• For each pair in the polydisc
(ψ∗, ψ) ∈ CX0(n) × CX0(n) ψ(∗)(x) ≤ κn for all x ∈ X0(n)
the fields φ∗n(ψ∗, ψ)(u) , φn(ψ∗, ψ)(u) on Xn are critical points of the
functional
= an (ψ∗ − Qnφ∗), (ψ − Qnφ) 0 − φ∗, (∂0 + Δ + μn)φ) n
= X0 v20 z2 − μv00 2 − μv0220
V0(0, x2, x3, x4). The graph of the realvalued function
where, v0 =
v20 z2 − μv00 2 − μv0202
over the complex plane z = x1 + ıx2 is a surface of revolution around the
x3axis with the circular well of absolute minima z = μv00 . Our hypothesis
on μ0 implies that the radius and depth of the well are of order one and order
v0 , respectively. After n renormalization group steps, the effective potential
becomes
10 It is necessary to measure the size of pn by introducing an appropriate norm. See the
last paragraphs of this overview.
11 When we take logarithms and ultimately differentiate with respect to an external field to
obtain correlation functions, it will disappear.
12 We will describe the inductive construction of μn later on in this overview. The
dependence of pn on the derivatives of the fields arises because of the renormalization of the
chemical potential.
since, by [16, Remark 1.1], φn(ψ∗, ψ) ψ = z ≈ z and φ∗n(ψ∗, ψ) ψ = z ≈ z∗ .
The graph is again a surface of revolution with the circular well of absolute
minima z = v0μ/nLn , but now the radius and depth are of order L 23 n and
order L5nv0 , respectively; the well is developing. We stop the flow when the
well becomes so wide and so deep that we can no longer construct background
fields by expanding around ψ∗, ψ = 0 . This happens as μn approaches order
one.
If the power series expansion of the perturbative correction pn had a
quadratic part K(x, y) ψ∗(x)ψ(y) the discussion of the evolving well in the
x,y∈X0(n)
last paragraph would be misleading, because the minimum of the total action
An − pn would not be close enough to the minimum of the dominant part An .
The requirement that pn must not contain quadratic terms is the
renormalization condition for the chemical potential. (See, Step 9 below.) Under the
scaling map (3), the local monomials
are relevant, and the local monomials
are marginal. The local monomials ψ∗, ∂ν ψ 0, 1 ≤ ν ≤ 3, do not appear,
because of reflection invariance. See [13, Definition B.1 and Lemma B.4]. So
pn does not contain any relevant monomials.
The parabolic renormalization group flow drives the system away from
the trivial (noninteracting) fixed point. To continue, we will have to construct
background fields by expanding about configurations supported near the
bottom of the developing well, analogously to the ‘Bogoliubov Ansatz.’ At present,
we expect to continue the parabolic flow, but expanding about configurations
supported near the bottom of the well, through a transition regime (which
overlaps with the regime of [13, 14]) until μn becomes large enough (but still
of order one), and then switch to a new ‘elliptic’ renormalization group flow
for the push to the symmetry broken, superfluid fixed point. In Appendix A,
below, we perform several model computations that contrast the parabolic
nature of the early renormalization group steps with the elliptical nature of
the late renormalization group steps.
The next part of this overview is an outline, in nine steps, of the
inductive construction that uses a steepest descent/stationary phase calculation to
build the desired form for Fn+1(ψ∗, ψ) = Bn+1 S−1ψ∗, S−1ψ , from that of
Fn(ψ∗, ψ) , n ≥ 0 , where
We are expecting that, by induction,
dψ(x)∗∧dψ(x) e− L12 θ∗−Qψ∗ , θ−Qψ −1 Fn(SF)(ψ∗, ψ)
2πı
Snx ∈ X0(n)
1 dψ(x)∗∧dψ(x) e− L12 θ∗−Qψ∗ , θ−Qψ −1 − An(ψ∗,ψ, φ∗n,φn) + pn
= N(n)Zn Snx ∈ X0(n) 2πı
= Dominant Part
Non Perturbative Correction.
We emphasize that Steps 1 and 6, which control the difference between
Fn+1(ψ∗, ψ) and its, dominant, ‘small field,’ part Fn(S+F1)(ψ∗, ψ), have not been
proven, though we do supply some motivation in [15].
Step 1 (Large field generates small factors). If Ψ ∈ CX0(n+1) is ‘large field,’ that
is Ψ ∈/ Sn+1, then we expect Bn+1(S−1Ψ∗, S−1Ψ) = O e−1/vε0 , since the real
part of the exponent appearing in the integrand of (6) is of order − v1ε0 . See
[15, Proposition 1, ‘Corollary’ 2 and the subsequent Steps 1 and 2].
Step 2 (Holomorphic form representation). We wish to analyze the integral
in (6) by a steepest descent/stationary phase argument. Recall that a critical
point of a function f (z) of one complex variable z = x + iy, that is not analytic
in z, is a point where both partial derivatives ∂∂fx and ∂∂fy , or equivalently, both
partial derivatives ∂∂fz = 12 ∂∂x − i ∂y f and ∂∂fz¯ = 12 ∂
∂ ∂x + i ∂∂y f vanish. We
prefer the latter formulation. So we rewrite the integral in (6) in a form that
allows us to treat ψ and its complex conjugate as independent fields. For each
fixed (θ∗, θ) , the ‘action’
− L12 θ∗ − Qψ∗ , θ − Qψ −1 − An ψ∗, ψ, φ∗n(ψ∗, ψ), φn(ψ∗, ψ)
= −An,eff (θ∗, θ, ψ∗, ψ) + pn(ψ∗, ψ, ∇ψ∗, ∇ψ)
is a holomorphic function of (ψ∗, ψ) on Sn × Sn . By design, the Dominant
Part of Bn+1(θ∗, θ) in (6) is expressed as (a constant times) the integral of
the holomorphic form
x ∈ X0(n)
of degree 2X0(n) over the real subspace in Sn × Sn given by ψ∗ = ψ∗ . We
shall see below that, typically, the critical point does not lie in the real subspace
and so is not in the domain of integration. This representation permits us to
use Stokes’ theorem,13 to shift the contour of integration to a nonreal contour
that does contain the critical point of (the principal terms of) the action. The
shift will be implemented in Step 6.
Step 3 (Critical Points). Our next task is to find critical points. In (7), above,
we wrote the exponent, An(θ∗, θ, ψ∗, ψ), as the sum of a very explicit, main,
part—An,eff and a not very explicit, smaller, part pn. We just find the
critical points of An,eff rather than the full An. Indeed, there is a unique pair of
holomorphic maps14 ψ∗cr(θ∗, θ) , ψcr(θ∗, θ) from (S−1Sn+1) × (S−1Sn+1)
to Sn such that the gradient ∇∇ψψ∗ of An,eff (θ∗, θ, ψ∗, ψ) vanishes when
ψ∗ = ψ∗cr(θ∗, θ) , ψ = ψcr(θ∗, θ) . This pair of ‘critical field maps’ can be
constructed by solving the critical point equations, a nonlinear parabolic
system of (discrete) partial difference equations, using the natural contraction
mapping argument to perturb off of the linearized equations.15 The analysis
of the linearized equations is based on a careful examination of some linear
operators given in [10]. Beware that, in general, ψ∗cr(θ∗, θ) = ψcr(θ∗, θ)∗.
To start the stationary phase calculation, we factor the integral of the
holomorphic form (8) over the real subspace (ψ∗, ψ) ∈ Sn ×Sn ψ∗ = ψ∗
as the product of
and the ‘fluctuation integral’
x ∈ X0(n)
Step 4 (The Value of the Action at the Critical Point). We would expect that
the biggest contribution to the integral would come from simply evaluating the
exponent at the critical point, and that the biggest contribution to the value
of the exponent An at the critical point would come from evaluating −An,eff
13 The argument is similar to the use of Cauchy’s theorem in stationary phase arguments
for functions of one variable.
14 In [13,14] these maps are called ψn∗ , ψn.
15 In [13,14,16] we take another route to the critical field maps. The background fields
φ(∗)n(ψ∗, ψ) are constructed first, using the natural contraction mapping argument to
perturb off of the linearized background field equations. See [16, Proposition 2.1]. The critical
fields can then be expressed as functions of the background fields. See [13, Proposition 3.4].
at the critical point. By [13, Proposition 3.4.c]
An,eff (θ∗, θ, ψ∗, ψ) ψ∗=ψ∗cr(θ∗, θ) , ψ=ψcr(θ∗, θ)
= Aˇn+1 θ∗, θ, φˇ∗n+1(θ∗, θ), φˇn+1(θ∗, θ)
is the background field evaluated at the critical point. Consequently,
= e−Aˇn+1(θ∗, θ, φˇ∗n+1(θ∗, θ), φˇn+1(θ∗, θ)) + pn(ψ∗cr,ψcr,∇ψ∗cr,∇ψcr).
Remark. Bear in mind that the checked fields depend implicitly on μn . In the
next steps, we will build a new ‘renormalized’ chemical potential μn+1 that
will appear in An+1 . If, for the purposes of discussion, we ignored the effects
of renormalization, An+1 would just be a rescaled Aˇn+1 (see [13, Definition
2.3 and Lemma 2.4.c]) and the new background field φ(∗)n+1 would just be
a rescaled φˇ(∗)n+1 (see [13, Definition 3.2 and Proposition 3.4.b]). So, we are
not far off.
Step 5 (Diagonalization of the Quadratic Form in the Fluctuation Integral).
Next consider the fluctuation integral (9). Make the change variables
δψ∗ = ψ∗ − ψ∗cr(θ∗, θ) , δψ = ψ − ψcr(θ∗, θ)
to shift the critical point to δψ∗ = δψ = 0. Substitute ψ(∗) = ψ(∗)cr(θ∗, θ) +
δψ(∗) into the main part
An,eff (θ∗, θ, ψ∗, ψ) − An,eff θ∗, θ, ψ∗cr(θ∗, θ), ψcr(θ∗, θ)
of the exponent and expand in powers of δψ(∗). The constant and, by criticality,
linear parts vanish. The quadratic term has a dominant part (see [13, Lemma
4.1, (4.13)] and [14, Lemma 5.5]), that is independent of θ∗, θ . All of the
eigenvalues of the kernel of that dominant part are bounded away from the
negative real axis, uniformly in n.16 So, it is invertible and its inverse, C(n), has
a square root D(n) , all of whose eigenvalues have strictly positive real parts.
See [10, Corollary 4.5]. Now, the Taylor expansion of the above difference of
effective actions in the new variables δψ∗ = D(n)T ζ∗ , δψ = D(n)ζ becomes
ζ∗, ζ 0 + smaller terms of degree 2 in ζ∗, ζ
+ terms of degree at least 3 in ζ∗, ζ
16 A major part of [10] is devoted to proving this vital technical statement.
and the fluctuation integral (9) becomes
e− ζ∗,ζ 0+qn(θ∗, θ, ζ∗ , ζ) det(D(n))2
x ∈ X0(n)
where the domain of integration Ωn(θ∗, θ) consists of the set of all pairs
(ζ∗, ζ) ∈ CX0(n) × CX0(n) such that ψ∗cr(θ∗, θ) + D(n)Tζ∗ , ψcr(θ∗, θ) + D(n)ζ
is in the real subspace of Sn ×Sn . The term qn is holomorphic on the complex
domain of all quadruples (θ∗, θ, ζ∗ , ζ) with (θ∗, θ) ∈ (S−1Sn+1) × (S−1Sn+1)
and
Step 6 (Stokes’ Theorem). For each pair (θ∗, θ) ∈ (S−1Sn+1) × (S−1Sn+1) ,
we construct, in [15, following (22)], a 2X0(n)+1 (real) dimensional ‘cylinder,’
inside the (ζ∗, ζ) domain of analyticity of qn, whose boundary consists of
◦ the original domain of integration Ωn(θ∗, θ) (which typically does not
contain the critical point ζ = ζ∗ = 0),
◦ the desired new domain of integration
Dn =
(ζ∗, ζ) ζ∗ = ζ∗ , ζ(x) < 14 Lvn0+1 ε/2 for all x ∈ X0(n)
(which does contain the critical point ζ = ζ∗ = 0)
◦ and components on which e− ζ∗,ζ 0 + qn(θ∗, θ, ζ∗ , ζ) is O(e−1/vε0 ) .
See [15, (23)]. The holomorphic differential form in Step 5 has maximal rank
and is therefore closed. It follows from Stokes’ theorem that the fluctuation
integral (10) is equal to the small field contribution
Dn x ∈ X0(n)
plus corrections that are expected to be nonperturbatively small.
on (S−1Sn+1) × (S−1Sn+1). See [14, Proposition 5.6].
Step 8 (Rescaling). To this point, we have determined that the small field part
of Bn+1(θ∗, θ) is a constant times the exponential of the sum of
◦ the contribution which comes from simply evaluating An at the critical
point—in Step 4, we saw that this was
−Aˇn+1(θ∗, θ, φˇ∗n+1(θ∗, θ), φˇn+1(θ∗, θ)) + pn(ψ∗cr, ψcr, ∇ψ∗cr, ∇ψcr)
◦ and an analytic function that came, in Step 7, from the fluctuation
integral.
We are now ready to scale to get the small field part of
Fn+1(Ψ∗, Ψ) = Bn+1 S−1Ψ∗, S−1Ψ .
L12 S−1Ψ∗ , S−1Ψ −1 = Ψ∗ , Ψ 0
SQQnS−1 = Qn+1
S−1f∗ , S−1f n = L2 f∗ , f n+1
S−1f∗, (∂0 +Δ)S−1f n = f,, (∂0 +Δ)f n+1
Aˇn+1(θ∗, θ, φˇ∗n+1(θ∗, θ), φˇn+1(θ∗, θ))
= An+1(Ψ∗, Ψ , φ∗n+1(Ψ∗, Ψ) , φn+1(Ψ∗, Ψ))
(see [13, Remark 2.2.c and Lemma 2.4.a,b]) we have that
− f∗ , (∂0 +Δ+L2μn)f n+1 + Vn+1(f∗, f )
φ(∗)n+1(Ψ∗, Ψ) = S φˇ(∗)n+1(S−1Ψ∗, S−1Ψ)
and if the kernel Vn of Vn were exactly the Vn(u) of (4), then the kernel of Vn+1
would be exactly Vn(+u)1. See [13, Remark 2.2.h]. Renormalization is going to
tweak, for example, the value of the chemical potential. As a result, An+1 is
not quite An+1, and φ(∗)n+1 is not quite φ(∗)n+1. That’s the reason for putting
the primes on.
Similarly, the contributions from pn(ψ∗cr, ψcr, ∇ψ∗cr, ∇ψcr) and from the
fluctuation integral get scaled to
= pn ψ∗cr(θ∗, θ), ψcr(θ∗, θ), ∇ψ∗cr(θ∗, θ), ∇ψcr(θ∗,θ)
Fn(S+F1)(ψ∗, ψ) = e−An+1(ψ∗,ψ , φ∗n+1(ψ∗,ψ) , φn+1(ψ∗,ψ)) + pn+1(ψ∗,ψ,∇ψ∗,∇ψ)
Step 9 (Renormalization of the Chemical Potential). At this point, we are close
to the end of the induction step, but not there yet because the power series
pn+1 contains (renormalization group) relevant contributions, in particular a
quadratic term ψ∗, Kψ 0 , where K is a translation and (spatial) reflection
invariant linear operator mapping CX0(n+1) to itself. If such a term were to be
left in pn+1 it would, by the third line of (12), grow by roughly a factor of L2 in
each future renormalization group step. So we need to move (at least the local
part of) this term out of pn+1 and into An+1. By the discrete fundamental
theorem of calculus, for any translation invariant K,
where K ∈ C and Kν , ν = 0, 1, 2, 3 , are linear operators on CX0(n+1) . See [14,
3
Corollary B.2]. By reflection invariance, K is real and
ν=1
be rewritten as a sum of marginal and irrelevant monomials. See [14, Lemma
B.3.c].
So we would like to move K ψ∗, ψ 0 out of pn+1 into An+1. There are
two factors that complicate (but not seriously) this move.
◦ The chemical potential term in An+1(ψ∗, ψ , φ∗n+1(ψ∗, ψ) , φn+1(ψ∗, ψ))
is
L2μn φ∗n+1(ψ∗, ψ) , φn+1(ψ∗, ψ) n+1 .
ψ(∗).
◦ The prime fields φ∗n+1(ψ∗, ψ), φn+1(ψ∗, ψ) are background fields with
chemical potential L2μn, not with the chemical potential μn+1 that we
are going to end up with (and which we do not yet know).
To deal with the first complication, we use that φ(∗)n+1(ψ∗, ψ) = B(∗)ψ(∗) plus
terms of degree at least three in (ψ∗, ψ) (see [16, Proposition 2.1.a]). Because
the linear operators B(∗) have left inverses (see [10, Lemma 5.7] and the
beginning of the proof of [14, Lemma 6.3]), one can show that17 K ψ∗, ψ 0 =
K φ∗n+1(ψ∗, ψ), φn+1(ψ∗, ψ) n+1 plus a power series in ψ∗, ψ, ∇ψ∗, ∇ψ
that converges on the desired domain of analyticity and that does not contain
any relevant contributions. See [14, Lemma 6.3]. Thus,
pn+1 = K
φ∗n+1, φn+1 n+1 from pn+1
+ pn+1 = −An+1 ψ∗, ψ, φ∗n+1, φn+1
An+1 ψ∗, ψ, f∗, f = an+1 ψ∗ − Qn+1f∗, ψ − Qn+1f 0
− f∗ , (∂0 + Δ + (L2μn + K )f n+1 + Vn+1(f∗, f ).
But we are still not done—we still have the second complication to deal
with. The prime fields φ∗n+1(ψ∗, ψ), φn+1(ψ∗, ψ) are background fields for
17 For reasons that will be explained shortly, we do not actually use this fact expressed in
this way.
chemical potential L2μn, and not for chemical potential L2μn + K . That
is, the prime fields are critical for f∗, f → An+1 ψ∗, ψ, f∗, f and not for
f∗, f → An+1 ψ∗, ψ, f∗, f , as they must be to have An+1 = An+1. The way
out of this is of course a (straightforward) fixed point argument that yields a
self consistent μn+1 ≈ L2μn . See [14, Lemmas 6.2 and 6.6].
So far we have skirted the issue of bounding the perturbative correction
pn in our main result. To measure the size of pn , we introduce a norm whose
finiteness implies that all the kernels in its power series representation are
small with v0 and decay exponentially as their arguments separate in X0(n) .
For pedagogical simplicity pretend that pn is a function of only two fields—ψ
and one derivative field ψν . It has a power series expansion
pn (n) =
rr,+ss∈>N00
with the notations, N0 = N ∪ {0}, and
Each pn r s(x, y) is separately invariant under permutations of the components
of x and under permutations of the components of y. The norm of pn is
For a translation invariant kernel with four arguments, like the interaction
kernel V0(x1, x2, x3, x4) , V0 m is the (mass m) exponentially weighted L1–
L∞ norm of V0:
V0 m =
where τ (x1, x2, x3, x4) is the minimal length of a tree graph in X0 that has
x1, x2, x3, x4 among its vertices, and m ≥ 0 is a fixed decay rate. (The small
‘coupling constant’ v0 = 2 V0 2m .) The norm w m of a kernel w with an
arbitrary number of arguments is defined in much the same way. For details
see [13, §1.4 and Definition A.3].
Ideally, pn (n) would be bounded (and in fact small) uniformly in n .
Unfortunately, such a bound is too naive to achieve the upper limit on n stated
in our main result. The reason is that while the coefficient of an irrelevant
monomial decreases as the scale n increases, the maximum allowed size of
fields in the domain Sn also increases, so the monomial as a whole can be
relatively large. So we have chosen
• to move all quartic ψ∗ψ)2 monomials out of pn into An, i.e., to also
renormalize the interaction Vn, and
• to split pn into two parts,
◦ one, called En(ψ∗, ψ), is an analytic function whose size is measured
in terms of a normlike · (n) and is small (and decreasing with
n) and
◦ the other, called Rn, is a polynomial of fixed degree, the size of
whose coefficient kernels are measured in terms of a normlike
The details are stated in our main result, [13, Theorem 1.17].
Appendix A. Seeing the Parabolic and Elliptic Regimes
In this appendix, we perform several model computations that contrast the
parabolic nature of the early renormalization group steps with the elliptical
nature of the late renormalization group steps. We imagine that after n (block
spin) renormalization group steps we have an action whose dominant part
(that we are simplifying a bit18) is An ψ∗, ψ, φ∗n(ψ∗, ψ), φn(ψ∗, ψ) where
− μn φ∗, φ n + v2n φ∗φ, φ∗φ n .
f , g 0 =
1. This box contains ε˜nε3n points of Yn.
f (u)g(u) are the
nat18 In particular, for pedagogical purposes, we have replaced an by 1 and replaced Vn by a
local interaction.
19 The fine lattice Yn is a rescaled version of the original lattice X0 of (1).
They obey the background field equations
= 0
A.1. Constant Field Background Fields
To start getting a feel for the background field equations (A.2), we consider the
case that ψ∗ and ψ are constant fields with ψ∗ = ψ∗. We’ll look for solutions
φ(∗) which are also constant fields with φ∗ = φ∗. Since both Qn and Q∗n map the
constant function 1 to the constant function 1, the constant field background
fields obey
This is of the form ‘real number times φ equals real number times ψ’ so the
phase of φ and ψ will be the same (modulo π). So it suffices to consider the
case that ψ and φ are both real and obey
ddφ φ + vnφ2 − μn φ = 1 − μn + 3vnφ2 ⎨⎪ > 0 if μn > 1, φ >
there is always exactly one solution when μn ≤ 1, but the solution can be
nonunique when μn > 1. For example, when μn > 1 and ψ = 0, the solutions
are φ = 0 and φ = ± μnvn−1 .
A.2. The Background Field in the Parabolic Regime
Imagine that we wish to solve the background field equations (A.2) for φ(∗) as
analytic functions of ψ(∗), in the parabolic regime, when μn is small, so that
the minimum of the effective potential is still near the origin—see (5). Then
Q∗nQn + Dn − μn φ = Q∗nψ − vnφ∗φ2
Q∗nQn + Dn∗ − μn φ∗ = Q∗nψ∗ − vnφ2∗φ
φ = Q∗nQn − μn − dn∂0 − Δ −1Q∗nψ + O ψ(∗)
3
φ∗ = Q∗nQn − μn − dn∂0∗ − Δ −1Q∗nψ∗ + O ψ(3∗) .
A.3. The Background Field in the Elliptic Regime
Imagine that we again wish to solve the background field equations (A.2), but
this time in the elliptic regime when μn is large, vn is small and the effective
potential has a deep well, whose minima form a circle in the complex plane of
radius rn =
μvnn . We are interested in ψ(∗) and φ(∗) near the minimum of the
effective potential. That is, with ψ(∗) , φ(∗) ≈ rn. We write
and look for solutions when R, Θ are small. Substitute into (A.2) and divide
by rn. This gives
Dn eX+iH + Q∗n QneX+iH − eR+iΘ + μn e2X − 1 eX+iH = 0
Dn∗ eX−iH + Q∗n QneX−iH − eR−iΘ + μn e2X − 1 eX−iH = 0.
Expand the exponentials, keeping only terms to first order in R, Θ, X, H ,
to get
Now simplify, by adding together the two equations of (A.5) and dividing by 2,
and then subtracting the second equation of (A.5) from the first and dividing
by 2i. Pretend that ∂0 is a continuum partial derivative rather than a discrete
forward derivative. Then
12 (Dn + Dn∗) = − d2n (∂0 + ∂0∗) − Δ = −Δ
21i (Dn − Dn∗) = 2i dn (∂0 − ∂0∗) = i dn∂0
and (A.5) gives
or, in matrix form,
2μn − Δ + Q∗nQn X − i dn∂0H = Q∗nR
i dn∂0X +
i dn∂0∗ + Q∗nQn.
The Q∗nQn provides a mass which makes boundedly invertible. But, the
presence of this mass is a consequence of our having rescaled the original unit
lattice down to the very fine lattice Yn. To invert , ignoring the Q∗nQn, we
have to divide, essentially, by
• In the parabolic regime, μn is small and dn is essentially one so that the
operator in the curly brackets is approximately ∂0∗∂0 + (−Δ)2, which is
parabolic.
• In the elliptic regime, μn and dn are both very large with μd2n > 0 being
n
essentially independent of n. So the operator in the curly brackets is
approximately ∂0∗∂0 + +2 μd2nn (−Δ), which is elliptic.
A.4. The Quadratic Approximation to the Action
For the remaining model computations, we study the quadratic approximation
to the action (A.1).
A.4.a. Expanding Around Zero Field. We first consider the parabolic regime
as studied in [13, 14]. Substitute the linear approximation to the background
fields φ(∗) (as functions on ψ(∗)) of (A.3) into the action (A.1), keeping only
terms that are of degree at most two in ψ(∗). Writing
Sn(μn) = Q∗nQn − μn + Dn −1
4
An = (ψ∗ − Qnφ∗) , (ψ − Qnφ) 0 + φ∗, (Dn − μn)φ n + O ψ(∗)
= ψ∗ , ψ 0 − ψ∗, Qnφ 0 − Qnφ∗, ψ 0 + φ∗, (Q∗nQn + Dn − μn) φ n
= ψ∗ , ψ 0 − ψ∗, QnSn(μn)Q∗nψ 0 − QnSn(μn)∗Q∗nψ∗, ψ 0
4
+ Sn(μn)∗Q∗nψ∗, Q∗nψ n + O ψ(∗)
4
= ψ∗, (1l − QnSn(μn)Q∗n) ψ 0 + O ψ(∗) .
We now analyze the operator 1l − QnSn(μn)Q∗n in momentum space, in the
special case that μn = 0, and see that it is basically a (discrete) parabolic
differential operator. Set
Substituting in the definitions and simplifying, we see that
By [10, Remark 2.1.e], with q = 1,
un(p) = ε˜1n sin 12 ε˜np0 ν=1 ε1n sin 12 εnpν
Here k runs over the dual lattice of Y0 and k + runs over the dual lattice of
Yn. We do not need to know much about these dual lattices, except that the
dual lattice of Y0 is a discretization of R/2πZ × R3/2πZ3 , the dual lattice
of Yn is a discretization of R/ 2ε˜nπ Z × R3/ 2εnπ Z3 , and runs over
un(k + )Dˆn−1(k + )un(k + )Δˆ (n)(k)
1 + Dˆn(k)un(k)−2 + O k3
0= ∈Bˆn
0= ∈Bˆn
0= ∈Bˆn
+ O k3 = Dˆn(k) + O Dˆn(k)2 + O k3
and so is a parabolic operator.
A.4.b. Expanding Around the Bottom of the Effective Potential. For all μn =
0, it is appropriate to expand the action about the bottom of the effective
potential, rather than about the origin. That is, rather than in powers of ψ(∗).
So we rewrite the action (A.1)
An = (ψ∗ − Qnφ∗) , (ψ − Qnφ) 0 + φ∗, (−dn∂0 − Δ)φ n
and then substitute the representations (A.4) of ψ(∗) and φ(∗) in terms of
radial and tangential fields. Note that when R = Θ = X = H = 0, the field
magnitudes ψ(∗) = φ(∗) = rn and ψ(∗) and φ(∗) are at the bottom of the
effective potential. Still pretending that ∂0 is a continuous derivative, and using
the notation O[3] = O(X3+R3+H3+Θ3), we get the following representation
of the action, which is reminiscent of (A.8).
r1n2An = ΘR , 1l − Qn −1Q∗n ΘR 0 − rn22vn 1,1 n + O[3].
Proof. The three main terms in An are
= rn2 eR−iΘ − QneX−iH , eR+iΘ − QneX+iH 0
= rn2 R − iΘ − Qn(X − iH), R + iΘ − Qn(X + iH) 0 + O[3]
So all together
= rn2 HX , 2μ0n 00 HX + O[3].
r1n2An = ΘR , ΘR 0 − 2 ΘR , Qn HX 0
− rn22vn 1,1 n + O[3]
= ΘR , ΘR 0 − 2 ΘR , Qn HX 0 + HX ,
X
Substituting in (A.6), we have
r1n2 An =
− 2
0
−1Q∗n ΘR ,
− rn22vn 1, 1 n + O[3]
− rn22vn 1, 1 n + O[3]
and, analogously to (A.9),
Dn =
D˜ n = 1l + QnDn−1Q∗n −1.
Qn −1Q∗n = QnDn−1Q∗nD˜ n.
Dn−1Q∗nD˜ n and simplifying yields Q∗n so
The Fourier transform of Qn −1Q∗n is
un(k + ) Dˆ n−1(k + )un(k + )D#˜n(k)
= un(k)2 Dˆ n−1(k)D#˜n(k) +
un(k + )2 Dˆ n−1(k + )D#˜n(k)
0= ∈Bˆn
where, pretending that we have continuum, rather than discrete, differential
operators,
Dˆ n(p) =
−dn p0
⎛
D#˜ n(k) = ⎝ 1l +
un(k + )2 Dˆ n−1(k + )⎠
ˆ
∈Bn
During the course of the upcoming computation, we shall use the following
facts.
• The parameter dn ≥ 1. For small n, it takes the value 1 and for large n,
it decays quickly approaching 0 as n → ∞.
• The parameter μn > 0. For small n, it is very small and for large n, it is
very large, with dn−2μn bounded uniformly in n. When dn > 1, dn−2μn is
bounded away from zero.
• By [10, Lemma 2.2.b,c].
◦ un(k) = 1 + O(k2) and
⎡
The dominant term in (A.13) is
⎧
un(k)2 Dˆn−1(k)D#˜n(k) = un(k)2Dˆn−1(k)⎨ 1l +
⎩
◦ if = 0, un(k + ) ≤ ⎣ 0≤ν*ν=≤03 kν⎦ ν*=0  ν24+π.
0= ∈Bˆn
dn−2μn + k dn−1k 1"−1
= !1l + Dˆn(k)un(k)−2 + O 0k2 dn−1μn + dnk k
(by Lemma A.2.b)
with q1(k) = O(k2). The determinant of the matrix to be inverted in the last
line of (A.14.a) is
det −1d+nk20μ+n(O1+dn−q11(μkn))k+2k+2+OOd(nkk3)3 d1n+k0k+2 +OOd(nkk33)
= 1 + 2μn 1 + q1(k)) + 2(1 + μn)k2 + d2nk02
= d2n dn−2 1 + 2μn 1 + q1(k) + k02 + 2dn−2(1 + μn)k2 + O k3
= d2n dn−2 + 2dn−2μn + q2(k) + O k3
The tail of (A.13) is, by Lemma A.2.d,
un(k + )2Dˆ n−1(k + )D#˜n(k) = O
dn−1k 1
Combining (A.13) and the three (A.14)’s, we have that the Fourier
transform of QnDn−1Q∗nD˜ n = Qn −1Q∗n is
−1
dn−2
−dn k0 + O dnk3
dn k0 + O dn−1μnk2 + O dnk3
dn−1k57
d−1k0
1 − q4(k) + O k3
0= ∈Bˆn
q3(k) =
q4(k) =
So the Fourier transform of 1l − Qn −1Q∗n is
d−1k0
d−1k0
Unraveling the definitions and simplifying gives
q4(k) =
5
+ k2 .
k02
2dn−2μn
We see an elliptic operator in the tangential direction and a mass in the radial
direction.
On the other hand, when n is small, that is, early in the ‘parabolic regime,’
the parameter dn = 1 and μn 1 and the Fourier transform of 1l − Qn −1Q∗n
is roughly, for small k
−k0
The eigenvalues of this matrix are
which are parabolic operators.
±ik0 + k02 + k2 ≈ ±ik0 + k2
A.4.c. Some Operators in Momentum Space. We here gather together some
momentum space properties of the operators Dn and D˜ n that are used in the
computations leading up to (A.15).
Lemma A.2. (a) If p is bounded away from zero, then
Dˆ n−1(p) = O
⎝ ⎣ d−1
n
dn−1⎤ ⎞
(b) If = 0, then
(d) If = 0, then
Dˆ n−1(k + )Dˆ n(k) = O
dn−1k⎤ ⎞
D#˜n(k) = O ⎝ ⎣
dn−1k
dn−1k⎤ ⎞
Dˆ n−1(k + )D#˜n(k) = O 0 dn−4μn + dn−2k
dn−3μn + dn−1k
Proof. (a) If p is bounded away from zero, then
Dˆ n−1(p) =
= O
dnp05−1
−dnp0
64dn−2
dn−157
= O
−dnk0
dn−1k 1
(c) Using line 4 of (A.14.a),
= 1 + O(k2
−dnk0
1 + un2μ(kn)2 + k2 + O d−2μnk2 + O(k3)
n
−dn k0 + O dn−1μnk2 + O dnk3
udnn(kk0)2 + O(dn−1k3) −1
Next using (A.14.det)
= 1 + O(k2
= O(1) dn−2
= O(1) dn−2 ⎣
dn−2
− udnn(kk0)2 + O dn−1k3
−dnk0
−dnk0
− udnn(kk0)2 + O dn−1k3
D#˜n(k) = O
dn−1k
dn−1k 1
= 0, then
Dˆ n−1(k + )D#˜n(k) = O
= O
64dn−2
dn−1k57
dn−1k
dn−3k + dn−1k257
dn−2k + k2
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Communicated by Vieri Mastropietro . Received: January 20 , 2017 . Accepted: March 30 , 2017 .