Ising Model and 2D CFT Fixed Points

The Ising Model: From Lattice Spins to Criticality

The Ising model is a cornerstone of statistical physics. In its simplest 2D form, it describes a grid of "spins," each of which can point either up ($s_i=+1$) or down ($s_i=-1$). The energy of a configuration depends on an interaction between neighboring spins and an external magnetic field.

As you know from your background in statistical mechanics, at a specific critical temperature, $T_c$, and zero external magnetic field, the system undergoes a second-order phase transition. At this point, the correlation length diverges, and the system loses any intrinsic length scale.

Let's watch a short clip that visualizes this phenomenon.

The self-similarity at $T_c$ is the physical manifestation of scale invariance. As we discussed, this scale invariance at a critical point implies that the system is located at a fixed point of the Renormalization Group (RG) flow. All microscopic models within the "Ising universality class"—whether they are magnets, liquid-vapor systems, or binary alloys—will flow to this same fixed point when tuned to criticality.

The Fixed Point as a Conformal Field Theory

Since the fixed point is scale-invariant and we are in two dimensions, we expect it to be described by a Conformal Field Theory. This is not just an approximation; the long-distance physics of the critical Ising model is a 2D CFT.

The task is to identify which CFT it is. The answer is that the 2D critical Ising model is described by the simplest non-trivial unitary CFT, known as the $m=3$ minimal model. This theory is uniquely defined by a single number: its central charge.

CFT Data of the Critical Ising Model:

• Central Charge: $c=1/2$
• Primary Operator Content: The theory has a finite set of primary fields. The three with the lowest scaling dimensions are:
  1. Identity Operator, $1$: Holomorphic and anti-holomorphic weights $(h,\bar{h})=(0,0)$. This operator is present in every CFT.
  2. Spin Field, $\sigma (z,\bar{z})$: $(h,\bar{h})=(1/16,1/16)$. This corresponds to the fundamental spin variable in the lattice model.
  3. Energy Density Field, $ϵ(z,\bar{z})$: $(h,\bar{h})=(1/2,1/2)$. This corresponds to the local energy density of the lattice model.

Let's now read a detailed summary of these facts.

From CFT Data to Physical Observables

The power of this CFT description is that it allows us to directly compute universal physical quantities, such as critical exponents. Recall that for a primary operator $\varphi$ with scaling dimension $\Delta =h+\bar{h}$, its two-point function is fixed by conformal symmetry to be:

Comparing this to the definition of the critical exponent $\eta$ from statistical mechanics, $⟨s_i{s}_j⟩\sim |r_{ij}{|}^{-(d-2+\eta )}$, we see that for the spin field in $d=2$, $2{\Delta }_{\sigma }=\eta$.

Let's apply this to our operators:

1. Spin-Spin Correlator:
   The spin field $\sigma$ has ${\Delta }_{\sigma }=h+\bar{h}=1/16+1/16=1/8$.
   Therefore, the two-point function scales as:

   $$⟨\sigma (x)\sigma (0)⟩\sim \frac{1}{|x{|}^{2(1/8)}}=\frac{1}{|x{|}^{1/4}}$$

   This immediately gives us the critical exponent $\eta =1/4$, matching Onsager's 1944 exact solution.

2. Energy-Energy Correlator:
   The energy field $ϵ$ has ${\Delta }_{ϵ}=h+\bar{h}=1/2+1/2=1$.
   Its two-point function scales as:

   $$⟨ϵ(x)ϵ(0)⟩\sim \frac{1}{|x{|}^{2(1)}}=\frac{1}{|x{|}^2}$$

This demonstrates the predictive power of the CFT framework. Once the operator content and their dimensions are known, the critical exponents for correlation functions are immediately determined.

Solving the Theory: A Glimpse of the Bootstrap

Beyond just identifying exponents, the CFT framework provides a constructive way to "solve" the theory—that is, to compute all correlation functions. This is achieved through the bootstrap approach, which uses consistency conditions to determine the theory's data.

A key piece of data is the set of fusion rules, which dictate how operators combine in an Operator Product Expansion (OPE). For the Ising model, the rules are:

The first rule tells us that the OPE of two spin fields will contain contributions from the identity and the energy field. This structure, combined with the power of the infinite-dimensional Virasoro algebra in 2D, is so constraining that it allows for the calculation of OPE coefficients like $C_{\sigma \sigma ϵ}$, and ultimately, all n-point functions. The text from LINK that you read earlier sketches how this is done for the four-point function, leading to $C_{\sigma \sigma ϵ}=1/2$. This computational power, without ever needing to refer back to the original lattice model, is the hallmark of the CFT approach.

Conclusion

Today we have grounded the abstract principles of universality and RG flow in the concrete, physical example of the 2D Ising model. We have seen that this celebrated model, at its critical point, is not just described by but is a specific, solvable Conformal Field Theory.

Key Takeaways:

• The critical point of the 2D Ising model is a fixed point of the RG flow, exhibiting scale invariance.
• This fixed point is precisely the $c=1/2$ unitary minimal model of 2D CFT.
• The primary operators of this CFT correspond to physical observables in the Ising model: the spin field $\sigma$ ($\Delta =1/8$) and the energy field $ϵ$ ($\Delta =1$).
• The scaling dimensions of these CFT operators directly determine the critical exponents of the Ising model's correlation functions, such as $\eta =2{\Delta }_{\sigma }=1/4$.
• The consistency of the CFT (fusion rules, crossing symmetry) provides a path to fully solve the theory, demonstrating the power of the bootstrap approach.

This concludes our module on the connection between CFT, critical phenomena, and the Renormalization Group. You now have a solid understanding of why CFT is so central to modern theoretical physics: it provides the universal language for describing systems at criticality.

In our next module, we will dive deeper into the unique features of two-dimensional CFTs, starting with the origin of the infinite-dimensional conformal symmetry that makes models like the critical Ising theory so beautifully solvable.