SRQ seminar – October 19th, 2018
Notes from a talk in the SRQ series.
INI Seminar 20181019 Giuliani
Interacting dimers models (6/6)
Summary
\(\displaystyle \frac{Z_{\lambda}}{Z_0} = \int P_{\leqslant 0}
(\mathrm{d} \psi^{(\leqslant
0)}) e^{V^{(0)} (\psi^{(\leqslant
0)}_{\omega})} = e^{L^2 E^{(h)}} \int
P_{\leqslant h} (\mathrm{d}
\psi^{(\leqslant h)}) e^{V^{(h)} (\psi^{(\leqslant
h)})}\)
here \(\psi^{(\leqslant 0)} = \{ \psi^{(\leqslant 0), \pm}_{x, \omega}
\}_{x \in
\Lambda, \omega = \pm 1}\) with
\(\displaystyle V^{(0)} = \sum_{\underline{x} (P), \underline{\omega}
(P)}
v_{\underline{\omega} (P)} (\underline{x} (P))
\underbrace{\Psi_P}_{\prod_{f
\in P} \psi^{\varepsilon (f)}_{x (f),
\omega (f)}}\)
where \(P\) are sets of four indexes which represent the origina quartic
interaction. The propagator is given by
\(\displaystyle g_{\omega}^{(\leqslant 0)} (x, y) = \int
\frac{\mathrm{d} k}{(2 \pi)^2}
\frac{e^{i k \cdot (x - y)}}{\mu (k +
p^{\omega})} \chi (k)\)
with \(p^+ = (0, 0)\) and \(p^- = (\pi, \pi)\) the Fermi points. And we
decomposed it in a multiscale fashion as
\(\displaystyle g_{\omega}^{(\leqslant 0)} (x, y) = \sum_{h < h'
\leqslant 0}
g_{\omega}^{(h')} (x, y) + g_{\omega}^{(\leqslant h)} (x,
y)\)
and the effective potential at scale \(h \leqslant 0\) is represented as
\(\displaystyle V^{(h)} (\psi) = \sum_P \sum_{\underline{x} (P),
\underline{\omega} (P)} W_{|
P |, \underline{\omega} (P)}^{(h)}
(\underline{x} (P)) \Psi_P\)
with kernels \(W^{(h)}\) estimated by
\(\displaystyle \frac{1}{L^2} \| W^{(h)}_{\ell, \underline{\omega}}
\|_{L^1} \leqslant 2^{h (2
- \ell / 2)} \sum_{n \geqslant 1} C^n |
\alpha |^n \sum_{\tau \in J_{h, n}}
\sum_{\{ P_v \} : | P_{v_0} | =
\ell} \prod_{\text{$v$ branching point}}
2^{(h_v - h_{v'}) \overbrace{(2
- | P_v | / 2)}^{\text{scaling dimension}}}\)
\(\displaystyle
\raisebox{-0.5\height}{\includegraphics[width=14.8909550045914cm,height=8.92450478814115cm]{image-1.pdf}}\)
The sum restricted to \(P_v\)-s such that \(| P_v | \geqslant 6\) is
exponentially converging.
We have problems with vertices \(| P_v | = 2\) and \(| P_v | = 4\):
\(\displaystyle
\text{\raisebox{-0.5\height}{\includegraphics[width=14.8741637150728cm,height=8.92450478814115cm]{image-1.pdf}}}\)
Let us start with the two legs vertices.
\(\displaystyle \sum_{x, y, \omega} \psi^{(\leqslant h_{v'}), +}_{x,
\omega} W_{2,
\omega}^{(h_{v'})} (x, y) \psi^{(\leqslant h_{v'}), -}_{y,
\omega}\)
we decompose the kernel into a local and remainder parts:
\(\displaystyle W_{2, \omega}^{(h_{v'})} =\mathcal{L}W_{2,
\omega}^{(h_{v'})}
+\mathcal{R}W_{2, \omega}^{(h_{v'})}\)
where
\(\displaystyle \mathcal{L}W_{2, \omega}^{(h_{v'})} (x, y) :=
\underbrace{\sum_{y'} W_{2,
\omega}^{(h_{v'})} (x, y')}_{ \hat{W}_{2,
\omega}^{(h_{v'})} (k) |_{k = 0}}
\delta_{x, y}\)
Thanks to lattice symmetries (reflections and discrete rotations) we
have the cancellation
\(\displaystyle \sum_{y'} W_{2, \omega}^{(h_{v'})} (x, y') = 0.\)
So the local part of the two point kernel is zero and the rest is better
behaved, indeed it gives the following expression
\(\displaystyle \sum_{x, y, \omega} \psi^{(\leqslant h_{v'}), +}_{x,
\omega} \mathcal{R}W_{2,
\omega}^{(h_{v'})} (x, y) \psi^{(\leqslant
h_{v'}), -}_{y, \omega} = \sum_{x,
y, \omega} \psi^{(\leqslant h_{v'}),
+}_{x, \omega} W_{2, \omega}^{(h_{v'})}
(x, y) [\psi^{(\leqslant
h_{v'}), -}_{y, \omega} - \psi^{(\leqslant h_{v'}),
-}_{x, \omega}]\)
\(\displaystyle \approx \sum_{x, y, \omega} \psi^{(\leqslant h_{v'}),
+}_{x, \omega} W_{2,
\omega}^{(h_{v'})} (x, y) \underbrace{(y - x)
}_{2^{- h_v}}
\underbrace{\partial \psi^{(\leqslant h_{v'}), -}_{x,
\omega}}_{\propto
2^{h_{v'}} \psi^{(\leqslant h_{v'}), -}_{x, \omega}}\)
so we can estimate this as
\(\displaystyle \approx 2^{- (h_v - h_{v'})} \sum_{x, y, \omega}
\psi^{(\leqslant h_{v'}),
+}_{x, \omega} O (W_{2, \omega}^{(h_{v'})} (x,
y)) \psi^{(\leqslant h_{v'}),
-}_{x, \omega}\)
which has better behaviour of the original vertex function.
We refine the local part
\(\displaystyle \psi^-_y \rightarrow \psi^-_x + (y - x) \partial
\psi^-_x\)
\(\displaystyle \sum_{x, y, \omega} \psi^+_{x, \omega} (y - x) \cdot
\partial \psi^-_x W_{2,
\omega} (x, y) = \sum_{x, \omega} \psi^+_{x,
\omega} (\zeta_{\omega}^{(h)}
\cdot \partial) \psi^-_{x, \omega}\)
where \(\) in Fourier variables and due to lattice symmetries
\(\displaystyle - i \zeta_{\omega}^{(h)} \cdot k = -
\underbrace{\zeta_h}_{\in \mathbb{R}}
D_{\omega} (k)\)
where \(D_{\omega} (k) = (- i - \omega) k_1 + (- i - \omega) k_2\). So
this can be absorbed into the propagator
\(\displaystyle \mu (k) = D_w (k) + O (k^2),\)
Note that \(\zeta_h = O (\lambda^2)\). This will produce after a
multiplicative recursion to a critical exponents for the correlation
fuctions. We now remain only with a rest of the type
\(\displaystyle \sum_{x, y, \omega} \psi^{(\leqslant h_{v'}), +}_{x,
\omega} \mathcal{R}W_{2,
\omega}^{(h_{v'})} (x, y) \psi^{(\leqslant
h_{v'}), -}_{y, \omega} \approx
2^{- 2 (h_v - h_{v'})} \sum_{x, y,
\omega} \psi^{(\leqslant h_{v'}), +}_{x,
\omega} O (W_{2,
\omega}^{(h_{v'})} (x, y)) \psi^{(\leqslant h_{v'}), -}_{x,
\omega}\)
which is now decaying.
What about the quartic terms with \(| P_v | = 4\)? They are of the form
\(\displaystyle \sum_{\text{\scriptsize{$\begin{array}{c}
x_1, \ldots
x_4\\
\omega_1, \ldots \omega_4
\end{array}$}}} \psi^{(\leqslant
h_{v'}), +}_{x_1, \omega_1} \psi^{(\leqslant
h_{v'}), -}_{x_2, \omega_2}
\psi^{(\leqslant h_{v'}), +}_{x_3, \omega_3}
\psi^{(\leqslant h_{v'}),
-}_{x_4, \omega_4} W_{4,
\underline{\omega}}^{(h_{v'})} (x_1, \ldots,
x_4)\)
The local part is now
\(\displaystyle \psi^{(\leqslant h_{v'}), +}_{x_1, \omega_1}
\psi^{(\leqslant h_{v'}),
-}_{x_1, \omega_2} \psi^{(\leqslant h_{v'}),
+}_{x_1, \omega_3}
\psi^{(\leqslant h_{v'}), -}_{x_1, \omega_4}\)
and up to permutation we remain only with (by symmetries, of the
lattice)
\(\displaystyle \psi^{(\leqslant h_{v'}), +}_{x_1, +} \psi^{(\leqslant
h_{v'}), -}_{x_1, +}
\psi^{(\leqslant h_{v'}), +}_{x_1, -}
\psi^{(\leqslant h_{v'}), -}_{x_1, -}\)
The remainder comes with an additional factor
\(\displaystyle 2^{- (h_v - h_{v'})}\)
which makes the remainder irrelevant. We have to live with the
“dangerous” local terms
\(\displaystyle \lambda_{h_{v'}} \sum_{x_1} \psi^{(\leqslant h_{v'}),
+}_{x_1, +}
\psi^{(\leqslant h_{v'}), -}_{x_1, +} \psi^{(\leqslant
h_{v'}), +}_{x_1, -}
\psi^{(\leqslant h_{v'}), -}_{x_1, -}\)
which cannot be reabsorbed effectively here
\(\displaystyle \lambda_h := 4 \sum_{x_2, x_3, x_4} W_{4,
\underline{\omega}}^{(h)} (x_1,
\ldots, x_4) \in \mathbb{R}\)
which is real by lattice symmetries.
We have now to modify the multiscale expansion in order to take into
account the absorption of these relevant and dangerous contributions.
The outcome of this procedure can be rewritten in the following way. We
have a modified multiscale expansion.
\(\displaystyle Z_{\lambda} = e^{L^2 E^{(h)}} \int P_{\leqslant h, Z_h}
(\mathrm{d}
\psi^{(\leqslant h)}) e^{V^{(h)} (Z_h^{1 / 2}
\psi^{(\leqslant h)})}\)
where \(P_{\leqslant h, Z_h}\) has a propagator: with \(r_{\omega} (k) =
\mu (k + p^{\omega}) - D_{\omega} (k) = O (k^2)\)
\(\displaystyle \frac{g_{\omega}^{(\leqslant h)} (x, y)}{Z_h} = \int
\frac{\mathrm{d} k}{(2
\pi)^2} \frac{e^{i k \cdot (x - y)}}{Z_h
D_{\omega} (k) + r_{\varepsilon} (k)}
\chi (2^h k)\)
and the interaction has the structure
\(\displaystyle V^{(h)} (\psi) = \lambda_h \sum_{x_1} \psi^+_{x_1, +}
\psi^-_{x_1, +}
\psi^+_{x_1, -} \psi^-_{x_1, -} +\mathcal{R}V^{(h)}
(\psi)\)
where \(\mathcal{R}V^{(h)} (\psi)\) includes all the terms with \(\ell
\geqslant 6\) and the second order remainder for \(\ell = 2\) and the
first order remainder for \(\ell = 4\). By our analysis
\(\mathcal{R}V^{(h)} (\psi)\) can be written as a convergent series but
is not going to zero, this would happen only if the theory is
asymptotically free which is not our case.
In particular if we write
\(\displaystyle \mathcal{R}V^{(h)} (\psi) = \sum_P \sum_{\underline{x}
(P), \underline{\omega}
(P)} \mathcal{R}W_{| P |, \underline{\omega}
(P)}^{(h)} (\underline{x} (P))
\Psi_P\)
\(\displaystyle \frac{1}{L^2} \| \mathcal{R}W^{(h)}_{\ell,
\underline{\omega}} \|_{L^1}
\leqslant 2^{h (2 - \ell / 2)} \sum_{n
\geqslant 1} C^n | \alpha |^n
\sum_{\tau \in J_{h, n}} \times\)
\(\displaystyle \times \sum_{\{ P_v \} : | P_{v_0} | = \ell}
\prod_{\text{$v$ branching
point}} 2^{(h_v - h_{v'}) \left( 2 - \frac{|
P_v |}{2} + c \lambda^2 | P_v | -
\beta (P_v) \right)} \times \prod_{v
\text{e.p.}} | \lambda_{h_{v'}} |\)
where now we consider all trees and not only those ending at scale
\((0)\) and for them consider the contribution of \(\lambda_{h'}\) from
the leaves ending in scale \(h\).
Where
\(\displaystyle \beta (P_v) = \left\{ \begin{array}{ll}
- 2 & \text{if
$| P_v | = 2$}\\
- 1 & \text{if $| P_v | = 4$}\\
0 &
\operatorname{otherwise}
\end{array} \right.\)
and the factors \(c \lambda^2 | P_v |\) are coming from the wave
function renormalization if we assume that the following bound is true
\(\displaystyle \frac{Z_{h - 1}}{Z_h} \leqslant 2^{c \lambda_0^2}\)
and the sum make sense if \(| \lambda_h |\) remains small at all scales
in such a way that \(| \lambda_h | \leqslant C | \lambda_0 |\).
This procedure can be implemented recursively, indeed the \(\lambda_h\)
is a sum of contributions of trees in whichat each
vertex we apply an \(\mathcal{R}\) operator and at the first vertex an
\(\mathcal{L}\) operator. So we have an expression of the form
\(\displaystyle \lambda_h = \lambda_{h + 1} + \underbrace{\sum
\text{[trees with $n \geqslant
2$ endpoints]}}_{\beta_h (\lambda_{h +
1}, \lambda_{h + 2}, \ldots)}\)
and
\(\displaystyle \frac{Z_{h - 1}}{Z_h} = 1 + \zeta_h\)
and \(\zeta_h\) admits a similar representation as a convergent series
of trees, namely \(\zeta_h = \beta^Z_h (\lambda_{h + 1}, \lambda_{h +
2}, \ldots)\) with a similar beta function as for \(\lambda\).
We have to compute the \(\beta\) functions. At lowest order in
\(\lambda_h\)
\(\displaystyle \beta_h
=
\raisebox{-0.5\height}{\includegraphics[width=14.8909550045914cm,height=8.92450478814115cm]{image-1.pdf}}
+
\cdots \approx \text{} c_h \lambda^2_h = [c_{- \infty} + O
(2^h)]
\lambda^2_h\)
and
\(\displaystyle \beta^Z_h
\approx
\text{$\raisebox{-0.5\height}{\includegraphics[width=14.8909550045914cm,height=8.92450478814115cm]{image-1.pdf}}$}
+
\cdots\)
Carefully performing the computation we see that the first diagram is
given by
\(\displaystyle
\text{$\raisebox{-0.5\height}{\includegraphics[width=14.8909550045914cm,height=8.92450478814115cm]{image-1.pdf}}$}
\approx
\sum_y g^{(h)}_+ (x, y) (g^{(h)}_- (x, y) + g^{(h)}_- (y, x))\)
but \(g^{(h)}_- (x, y)\) is (approximatively) odd so the two terms
cancels and give a lower order contribution
\(\displaystyle \approx \sum_y 2^h \cdot 2^h \cdot e^{- 2^h | x - y |}
\approx 2^{2 h} \cdot
2^{- 2 h} \approx 1\)
and
\(\displaystyle \left| \sum_{\text{ways of
pairing}}
\raisebox{-0.5\height}{\includegraphics[width=2.99170274170274cm,height=1.98327430145612cm]{image-1.pdf}}
\right|
\leqslant C 2^h | \lambda_h |^2\)
so we get a recursion of the form
\(\displaystyle \lambda_{h - 1} = \lambda_h + c \lambda_h^2 2^h
\Rightarrow \lambda_h =
\lambda_0 (1 + O (\lambda_0))\)
thanks to this cancellation. What about higher orders? How to see these
cancellations???
If we could prove that
\(\displaystyle | \beta^{\lambda}_h | \leqslant C | \lambda_h |^2 2^h\)
then we would be happy (asymptotic cancellations in the \(\beta\)
function).
In order to have this key result at all orders we are going to follow
another strategy.
The really dangerous part is the one associated to the homogeneous part
of the propagator.
At each scale let us decompose
\(\displaystyle \frac{g_{\omega}^{(h)} (x, y)}{Z_h} = \overbrace{\int
\frac{\mathrm{d} k}{(2
\pi)^2} \frac{e^{i k \cdot (x - y)}}{Z_h
D_{\omega} (k)} [\chi (2^h k) - \chi
(2^{h - 1} k)]} ^{= g^{(h) }_R (x,
y) = \text{relativistic/Dirac propagator}}
+ \frac{g^{(h)}_S (x,
y)}{Z_h}\)
with \(| g^{(h)}_s (x, y) | \lesssim 2^{2 h} e^{- 2^h | x - y |}\) the
subdominant part of the propagator. And decompose the \(\beta\) function
as
\(\displaystyle \beta^{\lambda}_h = \beta^{\lambda}_{h, R} +
\beta^{\lambda}_{h, S},\)
and \(| \beta^{\lambda}_{h, S} | \leqslant C | \lambda_h |^2 2^h\)
thanks to the better behaviour of the propagator (since there is a least
one of the subdominants propagator in the expression of the trees which
give the important \(2^h\) factor outside). The function
\(\beta^{\lambda}_{h, R}\) is the same \(\beta\) function which one
obtains with the multi scale procedure for the model
\(\displaystyle \int P_{Z, R}^{(\leqslant N)} (\mathcal{D} \psi) e^{V
(Z^{1 / 2} \psi)},\)
where \(P_{Z, R}^{(\leqslant N)}\) is the reference measure associated
to
\(\displaystyle \frac{g^{(\leqslant N)} (x, y)}{Z} = \frac{1}{Z}
\int_{\mathbb{R}^2}
\frac{\mathrm{d} k}{(2 \pi)^2} \frac{e^{i k \cdot (x
- y)}}{D_{\omega} (k)}
\chi (2^N k),\)
(eventually \(N \rightarrow \infty\)). Here we are in the continuum and
\(\displaystyle V (\psi) = \lambda_{\infty} \int \mathrm{d} x \mathrm{d}
y \psi^+_{x, +}
\psi^-_{x, +} v (x - y) \psi^+_{x, -} \psi^-_{x, -},\)
where \(v\) is \(C^{\infty}_0\) potential, rotationally invariant, and
\(\hat{v} (0) = 0\). (We need other cutoffs and then use the tree
expansion to control this theory and then remove them). The \(\beta\)
function of this theory coincides in the infrared regime to our
\(\beta_R^{\lambda}\).
This model is exactly solvable. (if the IR cutoff is chosen and removed
with a specific procedure).
To solve it we can use bosonization. Or purely in fermionic variables
(cf. Benfatto and Mastropietro) using Ward identities.
From the solution we have exact formula for correlations which gives in
particular the vanishing of the \(\beta\) function and density-density
correlations which give fine asymptotics for the dimer-dimer
correlations (which were our final goal).
\(\displaystyle \langle \mathbb{I}_e ; \mathbb{I}_{e'} \rangle = A
\sum_{\omega} \langle
\psi^+_{x, \omega} \psi^-_{x, \omega} ; \psi^+_{y,
\omega} \psi^-_{y, \omega}
\rangle_{\operatorname{Lutt}} + B (- 1)^{x -
y} \sum_{\omega} \langle
\psi^+_{x, \omega} \psi^-_{x, - \omega} ;
\psi^+_{y, \omega} \psi^-_{y, -
\omega} \rangle_{\operatorname{Lutt}} +
O \left( \frac{1}{| x - y |^3} \right)\)
All the infrared behaviour is the same modulo lower order contributions.
To prove the Haldane relation \(A = \nu\) then we have to recognize that
the lattice theory has a Ward identity (which is that at each site you
have an exiting dimer) which can be used to put in relation prefactors
in correlations.
On the Luttinger model:
\(\displaystyle P_{Z, R}^{(\leqslant N)} (\mathrm{D} \psi) \propto
\mathrm{D} \psi e^{-
\sum_{x, \omega} (\hat{\psi}^+_{k, \omega}, \chi^{-
1} (2^N k) D_{\omega} (k)
\hat{\psi}^-_{k, \omega})}\)
the free action take the form (neglecting the cutoff)
\(\displaystyle \sum_{\omega} \int \mathrm{d} x (\psi^+_{k, \omega}
\underbrace{[(1 - i
\omega) \partial_1 + (1 + i \omega) \partial_2]}_{=:
\partial_{\omega}}
\hat{\psi}^-_{k, \omega})\)
and the model is covariant under local chiral transformation
\(\displaystyle \psi^{\pm}_{x, \omega} \rightarrow e^{\pm i
\alpha_{\omega} (x)}
\psi^{\pm}_{x. \omega}\)
Indeed the interaction is locally gauge invariant, the measure also and
the action is covariant and the transformation generates some
correction:
\(\displaystyle \sum_{\omega} \int \mathrm{d} x (\psi^+_{k, \omega}
\partial_{\omega}
\hat{\psi}^-_{k, \omega}) \longrightarrow
\sum_{\omega} \int \mathrm{d} x
(\psi^+_{k, \omega} \partial_{\omega}
\hat{\psi}^-_{k, \omega}) +
\sum_{\omega} \int \mathrm{d} x ((- i
\partial_{\omega} \alpha_{\omega} (x))
\psi^+_{k, \omega}
\hat{\psi}^-_{k, \omega})\)
and we can introduce an external field and differentiate wrt.
\(\alpha_{\omega} (x)\) and set it to zero to get identities among
correlation functions.
We get the following kind of identities:
\(\displaystyle ZD_{\omega}
(p)
\raisebox{-0.5\height}{\includegraphics[width=14.8909550045914cm,height=8.92450478814115cm]{image-1.pdf}}
=
\raisebox{-0.5\height}{\includegraphics[width=14.8909550045914cm,height=8.92450478814115cm]{image-2.pdf}}\)
Where the diagram on the left corresponds to the expression:
\(\displaystyle
\raisebox{-0.5\height}{\includegraphics[width=14.8909550045914cm,height=8.92450478814115cm]{image-1.pdf}}
=
\int \mathrm{d} k' \langle \hat{\psi}^+_{k' + p, \omega}
\hat{\psi}^-_{k',
\omega} ; \hat{\psi}^-_{k, \omega} \hat{\psi}^+_{k +
p, \omega} \rangle\)
And then also
\(\displaystyle Z D_{\omega} (p) \int \mathrm{d} k' \langle
\hat{\psi}^+_{k' + p, - \omega}
\hat{\psi}^-_{k', - \omega} ;
\hat{\psi}^-_{k, \omega} \hat{\psi}^+_{k + p,
\omega} \rangle = 0\)
On the other hand Dyson–Schwinger equations gives
\(\displaystyle \langle \hat{\psi}^+_{k, \omega} \hat{\psi}^-_{k,
\omega} \rangle = (Z
D_{\omega} (k))^{- 1} + (Z D_{\omega} (k))^{- 1} (Z
D_{\omega} (k + p))^{- 1}
\hat{v} (p) \int \mathrm{d} k' \langle
\hat{\psi}^+_{k' + p, \omega}
\hat{\psi}^-_{k', \omega} ;
\hat{\psi}^-_{k, \omega} \hat{\psi}^+_{k + p,
\omega} \rangle\)
However there are corrections due to \(\chi^{- 1}\) to the local gauge
invariance and we will have corrections to the vanishing of the r.h.s.
in the Dyson–Schwinger equation.
We have the scale of the interaction, set by \(v\) and the scale of the
UV regularization which in \(N\). There will be an error term in the
Ward identity which once produced tend to stay there and that we have to
control. The correction proportional to \((\chi^{- 1} - 1)\) is
marginal.
We can work out the multiscale construction of this theory, including
the effect of the cutoff. What happens is that one can single out the
contributions to the flow of the cutoff coupling which are not
irrelevant in the UV and compute them explicitly to obtain a remarkable
identity of the form
\(\displaystyle Z \left[ D_{\omega}
(p)
\raisebox{-0.5\height}{\includegraphics[width=3.09085989767808cm,height=2.47907647907648cm]{image-1.pdf}}
+
\frac{\lambda_{\infty}}{8 \pi} \hat{v} (p) D_{- \omega}
(p)
\raisebox{-0.5\height}{\includegraphics[width=3.38836416109143cm,height=1.88413354322445cm]{image-2.pdf}}
\right]
=\)
\(\displaystyle
=
\raisebox{-0.5\height}{\includegraphics[width=14.8909550045914cm,height=8.92450478814115cm]{image-1.pdf}}\)
and a second equation of the form
\(\displaystyle Z \left[ D_{- \omega}
(p)
\raisebox{-0.5\height}{\includegraphics[width=3.09085989767808cm,height=2.47907647907648cm]{image-1.pdf}}
+
\frac{\lambda_{\infty}}{8 \pi} \hat{v} (p) D_{\omega}
(p)
\raisebox{-0.5\height}{\includegraphics[width=3.09089269316542cm,height=1.68575364029909cm]{image-2.pdf}}
\right]
= 0\)
Then [I didnt' followed the discussion] we get that the asymptotics of
the propagator is given by
\(\displaystyle G_{\omega, R} (x) = \langle \psi^-_{x, \omega}
\psi^+_{y, \omega}
\rangle_{\text{Lutt,$\lambda_{\infty}$}} = \langle
\psi^-_{x, \omega}
\psi^+_{y, \omega} \rangle_{\text{Lutt,0}} \times
e^{- \lambda_{\infty} \Delta
(x)}\)
and
\(\displaystyle \Delta (x) = \int \frac{\mathrm{d} p}{(2 \pi)^2}
\frac{e^{- i px}}{D_{\omega}
(p)} \frac{(\hat{v} (p))^2 \left( -
\frac{\lambda_{\infty}}{8 \pi}
\right)}{D_{\omega} (p) \left[ 1 - \left(
\frac{\lambda_{\infty}}{8 \pi}
\hat{v} (p) \right)^2 \right]} \approx
\frac{\lambda_{\infty}}{8 \pi}
\frac{1}{1 - \left(
\frac{\lambda_{\infty}}{8 \pi} \right)^2} \log | x |\)
and also other asymptotics of the correlations follows from the
symmetry.
In order to compare to the lattice theory we have to tune the bare
parameters of this model in order to match the behaviours in the IR and
then the exact relations in this models for the critical exponents give
the same relations of the lattice model.