Linearization of RG and Critical Exponents

Hello! Welcome back to our exploration of the Renormalization Group.

In our last lesson, we developed a powerful qualitative picture of the RG. We saw how the RG transformation creates a flow in a space of Hamiltonian parameters, and how the fixed points of this flow correspond to scale-invariant states. We established that stable fixed points represent thermodynamic phases, while unstable fixed points correspond to critical points, with the critical surface acting as the phase boundary.

Today, we will make this picture quantitative. Our goal is to analyze the RG flow in the immediate vicinity of a fixed point by linearizing the transformation. This will allow us to precisely determine the stability of a fixed point and, most importantly, to derive the universal critical exponents that characterize the phase transition.

The Mathematics of Linearization

Let's represent the set of all coupling constants in our Hamiltonian as a point $S=\{K_1,K_2,\dots \}$ in a parameter space. An RG transformation with a rescaling factor $b$ maps this point to a new one:
$S^{\prime }=R_b(S)$
A fixed point, $S^{\ast }$, is a point that remains unchanged by this transformation, i.e., $S^{\ast }=R_b(S^{\ast })$.

To understand the flow near this fixed point, we consider a small deviation $\delta S=S-S^{\ast }$. The new point $S^{\prime }$ will be:
$S^{\prime }=S^{\ast }+\delta S^{\prime }=R_b(S^{\ast }+\delta S)$
Since $\delta S$ is small, we can perform a Taylor expansion of the function $R_b$ around $S^{\ast }$:
$S^{\ast }+\delta S^{\prime }\approx R_b(S^{\ast })+\frac{\partial R_b}{\partial S}{|}_{S^{\ast }}\cdot \delta S$
Using $R_b(S^{\ast })=S^{\ast }$, this simplifies to a linear matrix equation for the deviation:
$\delta S^{\prime }=M\cdot \delta S$
where $M$ is the Jacobian matrix of the RG transformation, evaluated at the fixed point: $M_{ij}=\frac{\partial S_i^{\prime }}{\partial S_j}{|}_{S^{\ast }}$.

The behavior of the flow is now governed by the eigenvalues and eigenvectors of the matrix $M$. Let the eigenvectors be $O_i$ and the corresponding eigenvalues be ${\lambda }_i$. If we express the initial deviation $\delta S$ as a linear combination of these eigenvectors, $\delta S={\sum }_i{g}_i{O}_i$, the flow becomes particularly simple:
$\delta S^{\prime }={\sum }_i{g}_i(M\cdot O_i)={\sum }_i(g_i{\lambda }_i)O_i$
The coefficient of each eigenvector is simply multiplied by its eigenvalue. Since the RG transformation is a semi-group, applying it successively corresponds to scaling by $b$. This implies that the eigenvalues must take the form:
${\lambda }_i=b^{y_i}$
The exponents $y_i$ are known as the RG exponents or scaling dimensions of the operators $O_i$. They are the crucial quantities that determine the stability of the fixed point and the critical exponents.

• If $y_i>0$, then ${\lambda }_i>1$. The component of the deviation along the eigenvector $O_i$ grows with each RG step. This is a relevant direction.
• If $y_i<0$, then ${\lambda }_i<1$. The component shrinks. This is an irrelevant direction.
• If $y_i=0$, then ${\lambda }_i=1$. The component is unchanged at linear order. This is a marginal direction.

For a critical point to be physically accessible by tuning a small number of experimental parameters (like temperature), its fixed point must have only a few relevant directions.

From RG Exponents to Critical Exponents

The power of this formalism lies in its ability to connect the abstract exponents $y_i$ to the measurable critical exponents $\alpha ,\beta ,\gamma ,\delta ,\nu ,\eta$. The two most important relevant operators correspond to the reduced temperature $t=(T-T_c)/T_c$ and the external magnetic field $h$. Let's call their RG exponents $y_t$ and $y_h$.

1. Correlation Length Exponent $\nu$:
   The correlation length $\xi$ scales with temperature as $\xi \sim |t{|}^{-\nu }$. Under an RG transformation, lengths are rescaled as $L\to L/b$, so the correlation length must transform as $\xi (S)=b\cdot \xi (S^{\prime })$. For temperature, we have $t^{\prime }=b^{y_t}t$. Combining these gives:
   $|t{|}^{-\nu }\sim \xi (t)=b\cdot \xi (t^{\prime })=b\cdot |t^{\prime }{|}^{-\nu }=b\cdot |b^{y_t}t{|}^{-\nu }=b^{1-\nu y_t}|t{|}^{-\nu }$
   For this to hold, the exponent of $b$ must be zero: $1-\nu y_t=0$. This yields the fundamental relationship:
   $\nu =\frac{1}{y_t}$

2. Other Exponents via the Scaling Hypothesis:
   The singular part of the free energy density, $f_s(t,h)$, transforms as $f_s(S)=b^{-d}{f}_s(S^{\prime })$, where the $b^{-d}$ factor comes from the change in volume. Substituting the transformations for $t$ and $h$, we get:
   $f_s(t,h)=b^{-d}{f}_s(b^{y_t}t,b^{y_h}h)$
   This is precisely the scaling hypothesis. By choosing $b$ such that $b^{y_t}|t|=1$, we can derive all the exponent relations you may have seen from scaling theory.

   • Specific Heat ($\alpha$): From $C\sim \frac{{\partial }^2{f}_s}{\partial t^2}$, we get the hyperscaling relation $2-\alpha =d\nu$.
   • Magnetization ($\beta$): From $m=-\frac{\partial f_s}{\partial h}$, we get $\beta =\frac{d-y_h}{y_t}=(d-y_h)\nu$.
   • Susceptibility ($\gamma$): From $\chi =\frac{\partial m}{\partial h}$, we get $\gamma =\frac{2y_h-d}{y_t}=(2y_h-d)\nu$.
   • Critical Isotherm ($\delta$): At $t=0$, $m\sim h^{1/\delta }$, which gives $\delta =\frac{y_h}{d-y_h}$.

The entire set of static critical exponents can be determined if we can just calculate the two relevant RG exponents, $y_t$ and $y_h$.

A Concrete Example: The 2D Ising Model

Let's apply this machinery to a concrete problem. While an exact real-space RG for the 2D Ising model is intractable, approximate schemes can be very illuminating. The Niemeijer-van Leeuwen approximation for the Ising model on a 2D triangular lattice is a classic example.

The following video walks through this calculation. It first derives the recursion relation for the coupling $K$, finds the non-trivial fixed point $K^{\ast }$, and then linearizes the flow around this point to find the eigenvalues ${\lambda }_t$ and ${\lambda }_h$.

From the video, the approximate calculation yields:

• ${\lambda }_t\approx 1.62$, which gives $y_t=\frac{\ln (1.62)}{\ln (\sqrt{3})}\approx 0.88$.
  • This implies $\nu =1/y_t\approx 1.13$.
• ${\lambda }_h\approx 3/\sqrt{2}\approx 2.12$, which gives $y_h=\frac{\ln (2.12)}{\ln (\sqrt{3})}\approx 1.4$.

The exact results for the 2D Ising model are $y_t=1$ (so $\nu =1$) and $y_h=15/8=1.875$. While the approximation isn't perfect, it correctly captures the existence of two relevant operators and gives exponents that are in the right ballpark—a remarkable success for a first-order, uncontrolled approximation.

The Wilson-Fisher Fixed Point and the $ϵ$-Expansion

Real-space RG methods are powerful conceptually but often difficult to improve systematically. The momentum-space RG, pioneered by Kenneth Wilson, provides a controlled perturbative approach. The key idea, which you have seen in your QFT courses, is to analyze the theory in $d=4-ϵ$ dimensions, where $ϵ$ is a small parameter.

In $d=4-ϵ$, the ${\varphi }^4$ coupling $g$ is slightly relevant. The competition between this relevance and the interaction effects (which tend to decrease the coupling) leads to a new, non-trivial fixed point at a small value of the coupling, $g^{\ast }\sim ϵ$. This is the Wilson-Fisher fixed point. Because $g^{\ast }$ is small, we can use perturbation theory to analyze the flow around it and calculate the RG exponents $y_t$ and $y_h$ as a series in $ϵ$.

This calculation is a cornerstone of modern theoretical physics. It shows that near $d=4$, the critical exponents deviate from their mean-field values in a way that can be systematically calculated. For instance, to first order in $ϵ$:
$y_t=2-\frac{ϵ}{3}⟹\nu =\frac{1}{y_t}\approx \frac{1}{2}+\frac{ϵ}{12}$
This provides a controlled way to compute the critical exponents of the 3D Ising model (where $ϵ=1$), and the results are surprisingly accurate.

Conclusion

Today we have put the "group" in Renormalization Group on a solid mathematical footing. We've seen that the behavior of the RG flow near a fixed point can be understood by linearizing the transformation and analyzing its eigenvalues.

Key Takeaways:

• The flow near a fixed point $S^{\ast }$ is described by the linear transformation $\delta S^{\prime }=M\cdot \delta S$.
• The eigenvalues of the matrix $M$ take the form ${\lambda }_i=b^{y_i}$, where $y_i$ are the RG exponents.
• The sign of $y_i$ determines whether the corresponding operator is relevant ($y_i>0$), irrelevant ($y_i<0$), or marginal ($y_i=0$).
• The critical exponents of a phase transition are universal functions of the relevant RG exponents, primarily $y_t$ and $y_h$. We found explicit relations like $\nu =1/y_t$.
• We saw this procedure applied in both an approximate real-space RG for the 2D Ising model and the systematic $ϵ$-expansion for ${\varphi }^4$ theory.

Preview of the Next Lesson:

Now that we have the quantitative tool to calculate the RG exponents $y_i$, we can circle back and formally classify fixed points based on the number and signs of these exponents. We will delve deeper into the physical interpretation of stable, unstable, and marginal fixed points and see how the entire global structure of the RG flow diagram is dictated by the properties of these special points.