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}}}$ $\displaystyle \raisebox{-0.5\height}{\includegraphics[width=14.8825429620884cm,height=8.92447199265381cm]{image-1.pdf}}$

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 $| P_v | = 4$:

$\displaystyle \text{\raisebox{-0.5\height}{\includegraphics[width=14.8741473173291cm,height=8.92447199265381cm]{image-1.pdf}}}$

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

$\displaystyle \frac{Z_{h - 1}}{Z_h} \leqslant 2^{c \lambda_0^2}$

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 $\lambda_h$

$\displaystyle \beta_h = \raisebox{-0.5\height}{\includegraphics[width=14.8825429620884cm,height=8.92447199265381cm]{image-1.pdf}} + \cdots \approx \text{} c_h \lambda^2_h = [c_{- \infty} + O (2^h)] \lambda^2_h$

and

$\displaystyle \beta^Z_h \approx \text{$\raisebox{-0.5\height}{\includegraphics[width=14.8825429620884cm,height=8.92447199265381cm]{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=14.8825429620884cm,height=8.92447199265381cm]{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

$\displaystyle \left| \sum_{\text{ways of pairing}} \raisebox{-0.5\height}{\includegraphics[width=2.98329889807163cm,height=1.98327430145612cm]{image-1.pdf}} \right| \leqslant C 2^h | \lambda_h |^2$

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:

$\displaystyle ZD_{\omega} (p) \raisebox{-0.5\height}{\includegraphics[width=14.8825429620884cm,height=8.92447199265381cm]{image-1.pdf}} = \raisebox{-0.5\height}{\includegraphics[width=14.8825429620884cm,height=8.92447199265381cm]{image-2.pdf}}$

Where the diagram on the left corresponds to the expression:

$\displaystyle \raisebox{-0.5\height}{\includegraphics[width=14.8825429620884cm,height=8.92447199265381cm]{image-1.pdf}} = \int \mathrm{d} k' \langle \hat{\psi}^+_{k' + p, \omega} \hat{\psi}^-_{k', \omega} ; \hat{\psi}^-_{k, \omega} \hat{\psi}^+_{k + p, \omega} \rangle$

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

$\displaystyle Z \left[ D_{\omega} (p) \raisebox{-0.5\height}{\includegraphics[width=3.0824642529188cm,height=2.47909287682015cm]{image-1.pdf}} + \frac{\lambda_{\infty}}{8 \pi} \hat{v} (p) D_{- \omega} (p) \raisebox{-0.5\height}{\includegraphics[width=3.37996031746032cm,height=1.88414994096812cm]{image-2.pdf}} \right] =$ $\displaystyle = \raisebox{-0.5\height}{\includegraphics[width=14.8825429620884cm,height=8.92447199265381cm]{image-1.pdf}}$

and a second equation of the form

$\displaystyle Z \left[ D_{- \omega} (p) \raisebox{-0.5\height}{\includegraphics[width=3.0824642529188cm,height=2.47909287682015cm]{image-1.pdf}} + \frac{\lambda_{\infty}}{8 \pi} \hat{v} (p) D_{\omega} (p) \raisebox{-0.500002431823819\height}{\includegraphics[width=3.08249704840614cm,height=1.68574544142726cm]{image-2.pdf}} \right] = 0$

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.