Physics 715:Conformal invariance at the critical point-I

In analytical studies of exactly solved lattice models of two-dimensional systems we seen that as a critical point is approached from above, the system becomes scale-invariant. This is also observed in experiments on three dimensional systems undergoing second order transitions.
This observation led to the scaling hypothesis. We developed three approaches to the problem of describing critical behavior:
$\bullet$ Analytic solution on the lattice, which is immensely challenging mathematically,
$\bullet$ renormalization group/scaling hypothesis to extract critical exponents,
$\bullet$ take the continuum limit and create a field theory (sums become integrals).

Continuum limits introduce many problems such as divergences. The largest $k$ -vector on a lattice is $\pi/a$ , but in the continuum it is infinite, and in some dimensions we encounter divergent integrals when the upper limit on $k$ is infinity.
In general one must use perturbation theory to do calculations in a field theory, and this is not apropos of strong interactions. Even if perturbation theory is appropriate, you need to handle the divergences inherited in the continuum limit.
If the lattice model was translationally and rotationally symmetric it will advance to a more complete translation/rotation symmetry in the continuum limit. Then at a critical point we gain scale invariance, and in two dimensions this leads to magic. Part of this magic is the bootstrap property

$\phi_i(z)\phi_j(w)=\sum_k C_{i,j,k}(z,w) \phi_k (w)$

that the (primary) fields form a fusion ring, an algebraically complete set, and that unitarity places very strict constraints on the fields themselves. If the fields obey this, we can work wonders.

What we learned from the renormalization group
The partition functions $Z= Tr e^{-\beta H}$ that we have studied should contain in $H$ a complete set of “operators” which we will call $\{S_a\}$ , together with a unique coupling constant $K_a$ for each

$-\beta H=\sum_a K_a \, S_a$

Decimation and rescaling of a lattice constant $a\rightarrow ba$ leads to a set of flow equations

$K_a'=\mathcal{R}_a(\{K\},b)$

We seek out the physical (unstable) fixed points

$K'_{a}^*=K_a^*=\mathcal{R}_a(\{K^*\},b)$

and linearize the equations around the fixed point

$(K'-K^*)_a=\sum_b T_{ab}(K-K^*)_b, \quad\mbox{or}\quad d\bm{K}'=T\cdot d\bm{K}, \quad T_{ab}={\partial\mathcal{R}_a\over \partial K_b}\Big{|}_{*}$

Next we made a variable change into the set of eigenvectors of $T_{ab}$ , and express the deviations of the coupling constants from the fixed point values

$T\cdot \bm{v}_i=\lambda_i \bm{v}_i\quad \implies\quad d\bm{K}=\sum_i u_i \bm{v}_i, \qquad u_j=d\bm{K}\cdot \bm{v}_j=\sum_a dK_a v_{i,a}$

The $u_i$ are called scaling variables. The scaling form of the free energy near the fixed point gotten from $Z$ then transforms under a renormalization as

$F_s(\{K\})=b^{-d}F_s(\{K'\})\rightarrow b^{-nd}F_s(\{K^{'n}\})$

and in terms of the new scaling variables $\{u\}$

$F_s(\{u\})\rightarrow b^{-nd}F_s(\{\lambda_i^n u_i\})$

and so we pick out the most relevent $u$ and call it $u_t$ , and one associated with an external field $h$ and call it $u_h$ , and further define $\lambda_i=b^{y_i}$ . Select $n$ such that $u_{t,0}=b^{n y_t}u_t$ :

$F_s(\{u\})\rightarrow \Big({u_t\over u_{t,0}}\Big)^{d/y_t}F_s(u_{t,0}, {u_h\over ({u_t\over u_{t,0}})^{{y_h\over y_t}}})$

and we have the fundamental form of the scaling hypothesis for the free energy.

RG works best when we begin with a complete set of operators $\{S_a\}$ and couplings $\{K_a\}$ in $\beta H$ , since we have seen that successive renormalizations can introduce new couplings if our set was incomplete, and some $K'$ rise and fall in relevence.
We assume that short-ranged operators and couplings such as $s(\bm{r})s(\bm{r}+\bm{a})$ are most important, and so our operators $S_a$ should be local ( $(\nabla(s))^2$ ).
What sort of advantage would be created if we were to transform from the set $\{S_a\}$ into a set $\{\phi_a\}$ such that

$-\beta H=\sum_a (K-K^*)_a S_a=\sum_i u_i \phi_i\quad ?$

The $\{\phi_i\}$ are called scaling fields. Clearly we will do this close to the fixed point.
In the continuum limit we would have in $-\beta H$

$\sum_i u_i\sum_r \phi_i(r)\rightarrow \sum_i u_i\int \phi_i(r)\, a^{-d} d^d r$

and in order for $-\beta H$ to be invariant under $a\rightarrow ba$ , when we already know that $u_i\rightarrow b^{y_i} u_i$ we would have to have

$\phi_i(r)\rightarrow b^{d-y_i}\phi(r')=b^{x_i}\phi_i(r')$

at the fixed point.
Add onto $-\beta H$ an interacttion $-\beta H\rightarrow -\beta H-\sum_r h(r) s(r)$ in a magnetic system, then

$\langle s(r_1)s(r_2)\rangle={\partial^2\over \partial h(r_1)\partial h(r_2)}\ln Z\Big{|}_{h=0}\sim {1\over |r_1-r_2|^{2x}}$

near the fixed point where $\xi\rightarrow \infty$ (recall the Orenstein-Zernicki formula from an earlier post on mean field theory.)

We know that in a renormalization $h'(r')=b^{y_h} h(r)$ , we also realize that the partition function is unchanged by all of these various operations, since decimation is nothing more than a partial evaluation of $Z$ , so that we can say that

$Z'(H',h')=Z(H,h)$

Suppose the the renormalization is a block spin transformation, so that $b^d$ spins $s_i$ are consolidated into a block $s'$ . The two blocks are at $r_1$ and $r_2$ which are far apart, so all of the spins in the block spin $s'(r_1)$ will have basically the same correlation with all of the spins in block spin $s'(r_2)$ . Then consider that

$\langle s'(r'_1)s'(r'_2)\rangle_{H'}={\partial^2\ln Z'(H',h')\over \partial h'(r'_1)\partial h'(r'_2)}={1\over b^{2y_h}}{\partial^2\ln Z'(H',h')\over \partial h(r_1)\partial h(r_2)}$

$\langle s'(r'_1)s'(r'_2)\rangle_{H'}={1\over b^{2y_h}}{\partial^2\ln Z'(H',h')\over \partial h(r_1)\partial h(r_2)}$

$=b^{-2y_h}\langle (\sum_{s\in s'(r'_1)}s)(\sum_{s\in s'(r'_2)}s)\rangle=b^{2d}b^{-2y_h}\langle s(r_1)s(r_2)\rangle_H$

Write this in terms of correlation functions

$G({|r_1-r_2|\over b}, H')=\langle s'(r'_1) s'(r'_2)\rangle=b^{2(d-y_h)}G(|r_1-r_2|, H)$

The solution to this is the scale invariant function

$G(|r_1-r_2|, H)={1\over |r_1-r_2|^{2x}}, \qquad x=d-y_h$

scale invariant, since $\xi\rightarrow\infty$ at the fixed point, compare with our Orenstein-Zernicke form).

But wait, there’s more! Full conformal invariance or scale invariance suggests that all of our $n$ -point functions for any scaling field must be homogeneous functions at the fixed point

$\langle \phi_1(r_1)\phi_2(r_2)\cdots\phi_n(r_n)\rangle=b^{-x_1-x_2-\cdots -x_n}\langle \phi_1(r_1/b)\phi_2(r_2/b)\cdots\phi_n(r_n/b)\rangle$

since we know that only homogeneous functions are scale invariant.

The variable change $-\beta H=\sum_a (K-K^*)_a S_a=\sum_i u_i \, \phi_i$
together with the fact that $\phi_i$ are scaling fields at the FP suggests that

$\langle \phi_i(r_1)\phi_i(r_2)\rangle={C_i\delta_{ij}\over |r_1-r_2|^{2x_i}}$

in which $x_i$ is the scaling dimension of $\phi_i$ .
At the critical point (fixed point), we have

$K'_*=K_*=\mathcal{R}(\{K\},b)$

the RG transformation, including a rescaling, does not change the FP coupling constant, so they must be dimensionless at the FP. If a quantity has dimension, it is not scale invariant. For example a length $\ell$ measured in meters may have a value of $10$ (meters), but change the scale to kilometers, and it now has a value of $0.01$ (km).

OPE and bootstrap hypothesis
The homogeneous nature of the correlation functions together with the fact that the $\{\phi_i\}$ form a sort of complete basis of scaling operators suggest the operator product expansion (OPE)

$\phi_i(r_1)\phi_i(r_2)=\sum_k C_{ijk}(r_1,r_2)\, \phi_k(r_2)$

(at least inside of $\langle\cdots\rangle$ ) with homogeneity requiring

$C_{ijk}(r_1,r_2)=C_{ijk}(r_1-r_2)={c_{ijk}\over |r_1-r_2|^{x_i+x_j-x_k}}$

This “bootstrap” hypothesis lets us intuitively and concisely perform the perturbative renormalization group program. When rewritten in terms of scaling fields, the perturbation series has a very simple form given entirely by the correlation functions.
Perturbative RG
Start out with some system with “Hamiltonian” $\beta H$ at the fixed point ( $\beta H_*$ ) , and add in some perturbations expressed in terms of the various scaling fields $\{\phi_i\}$ that emerged in the fixed point analysis of $K$

$Z\rightarrow Tr \, e^{-\beta H_*-\sum_i g_i\sum_r \phi_i(r)}= Tr \, e^{-\beta H_*-\sum_i g_i\int {d^d r\over a^d} \phi_i(r)}$

(1) $\begin{eqnarray*} \mbox{Expand...}\qquad Z&=&Z_*\, \Big(1-\sum_i g_ia^{-d}\int d^dr \, \langle \phi_i(r)\rangle_*\nonumber\\ &+&\tfrac{1}{2}\sum_{ij}g_ig_j a^{-2d}\int d^dr_1 d^d r_2 \, \langle \phi_i(r_1)\phi_j(r_2)\rangle_*+\cdots\Big)\nonumber\end{eqnarray*}$

If we were working on a lattice, the smallest $\Delta r$ would be $a$ , and we would encounter no divergences. To avoid such divergences in the continuum limit we must cut off $|r_1-r_2|>a$ in the multiple integrals appearing in the expansion.

Perform a simple rescaling $a\rightarrow ba=(1+\delta \ell)a$ . How must the coupling constants change if $Z$ is scale invariant? $\int d^d r\langle \phi_i(r)\rangle$ knows nothing about $a$ . The first term in the expansion is invariant if

$g_i\int {d^dr\over a^d}\phi_i(r)=g'_i \int {d^dr/b\over (ba)^d}b^{x_i}\phi_i(r/b)=g'_i b^{x_i-d}\int {d^dr'\over a^d}\phi_i(r')$

$g_i\rightarrow g'_i=b^{d-x_i}g_i\approx (1+(d-x_i)\, d\ell)\, g_i$

You can also see that we will pick up contributions $\propto g_j g_k$ because of the shift in the lower cutoff on the double integral.

$\int d^dr_1d^d r_2=\int d^dR\int_{|r|>a} d^d r\rightarrow \int d^dR\int_{|r|>ba} d^d r, \quad r=r_1-r_2$

$\mbox{Write this as}\quad \int_{|r|>ba} d^d r=\int_{|r|>a}d^dr-\int_{ba>|r|>a}d^dr$

and within this very small shell we can use the OPE (operator product expansion)

$-\int_{ba>|r|>a}{d^dr\over a^d} \langle \phi_i(r_1)\phi_j(r_2)\rangle\approx -\sum_k \int_{ba>|r|>a}{d^dr\over a^d} \, c_{ijk}\langle \phi_k(r)\rangle$

$\approx -\sum_k {S_d\over d} (b^d-1){a^d\over a^d}\, c_{ijk}\langle \phi_k(r)\rangle$

This contributes to the coefficient of $\int d^dr \langle \phi_k(r)\rangle$ :

$-g_k\rightarrow -g'_k b^{x_k-d}+\tfrac{1}{2}\sum_{ij}\Big(-{S_d\over d} (b^d-1)\, c_{ijk}g'_i g'_j b^{x_i+x_j-d}\Big)$

with $b=1+d\ell$

$g_k\rightarrow g'_k(1+(x_k-d) d\ell+\cdots)-\tfrac{1}{2}\sum_{ij}S_d \, d\ell\, c_{ijk}g'_i g'_j (1+(x_i+x_j-d)\, d\ell+\cdots)$

Invert to lowest order

$g_k\approx g'_k(1+(x_k-d) d\ell), \qquad g'_k\approx {g_k\over 1+(x_k-d) d\ell}\approx g_k \, (1-(x_k-d) d\ell)$

and then correct to lowest order in $d\ell$

$g'_k-g_k=dg_k=\Big((d-x_k)g_k-\tfrac{1}{2}\sum_{ij}S_d c_{ijk}g_i g_j \Big)\, d\ell$

$\mbox{or}\qquad {dg_k\over d\ell}=(d-x_k)g_k -\tfrac{1}{2}S_d\sum_{ij} g_i g_j c_{ijk}=y_k g_k -\tfrac{1}{2}S_d\sum_{ij} g_i g_j c_{ijk}$

give us the full perturbative system of RG flow equations correct to second order in terms of the exponents $y_k$ and fusion coefficients $c_{ijk}$ !

Up next: The Gaussian and Fischer-Wilson fixed points.

References: “Scaling and Renormalization in Statistical Physics” by John Cardy.

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Physics 715:Conformal invariance at the critical point-I

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