Critical Exponents in Phase Transitions

The Language of Criticality: Power Laws

Near a continuous phase transition, the most important variable is not the temperature $T$ itself, but how far we are from the critical temperature $T_c$. We therefore work with the dimensionless reduced temperature:

The critical point is located at $t=0$. A continuous phase transition is characterized by the singular behavior of thermodynamic functions as $t\to 0$. This singular behavior takes the form of power laws, and the exponents in these laws are the critical exponents.

Let's begin with a quick overview of the main exponents we will be defining.

Now, we will proceed to define each of these exponents formally and discuss their physical significance. We will use the language of a magnetic system (order parameter is magnetization $m$, external field is $h$), but these definitions are general.

The Critical Exponents

The critical exponents describe the leading singular term of a physical quantity near the critical point. The general form is $F(t)\sim |t{|}^{\lambda }$, where the exponent $\lambda$ is formally defined by the limit:

Let's examine the six most important exponents that characterize the physics of critical points.

1. Specific Heat Exponent: $\alpha$

The specific heat at constant external field, $C_h$, measures the system's ability to absorb heat. Near $T_c$, it behaves as:

• If $\alpha >0$, the specific heat diverges at the critical point, signaling massive energy fluctuations. This is the case for the superfluid transition in ${}^4$He.
• If $\alpha =0$, the specific heat may have a logarithmic divergence (e.g., $C_h\sim \ln |t|$), as in the 2D Ising model, or a finite "cusp".
• If $\alpha <0$, the specific heat remains finite at $T_c$ but has a cusp, with its derivative diverging.

The exponents above and below $T_c$ are often distinguished as $\alpha$ ($t>0$) and ${\alpha }^{\prime }$ ($t<0$). For most systems, it is found that $\alpha ={\alpha }^{\prime }$.

2. Order Parameter Exponent: $\beta$

This exponent describes how the order parameter spontaneously appears as the system is cooled below $T_c$ in zero field.

Note that this is defined only for $t<0$, as the spontaneous magnetization is zero for $t>0$. The exponent $\beta$ governs the shape of the order parameter curve you saw in the last lesson as it approaches $T_c$. A smaller $\beta$ implies a sharper onset of order.

3. Susceptibility Exponent: $\gamma$

The generalized susceptibility, $\chi$, measures the linear response of the order parameter to an infinitesimal applied field. For a magnet, this is the magnetic susceptibility.

The divergence of the susceptibility ($\gamma >0$) is a hallmark of a continuous transition. It indicates that an infinitesimally small field can induce a large order parameter. This is a direct consequence of the large-scale, correlated fluctuations near $T_c$. Again, we can distinguish $\gamma$ and ${\gamma }^{\prime }$ for $t>0$ and $t<0$, but they are generally equal.

4. Critical Isotherm Exponent: $\delta$

This exponent describes the non-linear relationship between the order parameter and the external field precisely at the critical temperature.

At any other temperature, for small fields, the response is linear ($m\propto h$). At $T_c$, this linear response breaks down, and $\delta$ quantifies this non-linearity.

5. Correlation Length Exponent: $\nu$

The previous exponents describe bulk thermodynamic properties. The next two describe the spatial structure of fluctuations. The correlation length, $\xi$, is the characteristic length scale over which the fluctuations of the order parameter are correlated. Away from $T_c$, the correlation function typically decays exponentially, $G(r)\sim e^{-r/\xi }$. As we approach the critical point, $\xi$ diverges:

The divergence of the correlation length is perhaps the most fundamental property of a critical point. It means that fluctuations are correlated over all length scales, leading to the singular behavior of macroscopic observables. This scale-invariance is the key idea behind the renormalization group.

6. Correlation Function Exponent: $\eta$

Exactly at the critical point, where $\xi$ is infinite, the exponential decay of correlations is replaced by a much slower power-law decay. The exponent $\eta$ describes this decay. For large distances $r$, the correlation function $G(r)=⟨m(\mathbf{r})m(0){⟩}_c$ behaves as:

where $d$ is the spatial dimensionality of the system. The term $d-2$ is the behavior expected from a simple random walk (or free field theory), so $\eta$ is the "anomalous dimension" that measures the deviation from this simple picture due to interactions. Your background in QFT will make this concept of anomalous dimensions familiar.

Universality and Mean-Field Exponents

A remarkable discovery of 20th-century physics is the concept of universality: systems with vastly different microscopic details can exhibit the exact same set of critical exponents. For example, the liquid-gas transition in carbon dioxide and the ferromagnetic transition in an Ising magnet share the same critical exponents.

Universality classes are determined not by the microscopic Hamiltonian, but by:

1. The spatial dimensionality ($d$) of the system.
2. The symmetry of the order parameter (e.g., scalar for Ising, vector for Heisenberg).

This is why we can learn about a complex fluid by studying a simple spin model.

As a point of reference, let's recall the predictions from mean-field theory, which you encountered in your previous course and our earlier discussions on Landau theory.

By ignoring fluctuations, mean-field theory predicts a set of exponents that are independent of dimension:

• $\alpha =0$ (a finite jump in specific heat)
• $\beta =1/2$
• $\gamma =1$
• $\delta =3$
• $\nu =1/2$
• $\eta =0$

These values provide a baseline but are quantitatively incorrect for systems in low dimensions ($d\le 4$), where fluctuations, which MFT neglects, become dominant. The discrepancy between these classical exponents and experimental/exact values is what drove the development of the renormalization group.

Here is a table comparing MFT to the exactly solved 2D Ising model and experimental values for a 3D magnet, illustrating universality and the failure of MFT.

Table: Comparison of critical exponents from mean-field theory, the exact solution of the 2D Ising model, and numerical/experimental results for 3D systems.

Conclusion

In this lesson, we have defined the language used to describe the singular physics near a continuous phase transition. By quantifying the non-analytic behavior of thermodynamic and correlation functions, we have established a precise framework for studying critical phenomena.

Key Takeaways:

• Near a continuous transition, physical quantities exhibit power-law singularities as a function of the reduced temperature $t=(T-T_c)/T_c$.
• These singularities are characterized by a set of critical exponents:
  • $\alpha$: Specific heat, $C_h\sim |t{|}^{-\alpha }$
  • $\beta$: Order parameter, $m\sim (-t{)}^{\beta }$ for $t<0$
  • $\gamma$: Susceptibility, $\chi \sim |t{|}^{-\gamma }$
  • $\delta$: Critical isotherm, $h\sim |m{|}^{\delta }$ at $t=0$
  • $\nu$: Correlation length, $\xi \sim |t{|}^{-\nu }$
  • $\eta$: Correlation function at criticality, $G(r)\sim r^{-(d-2+\eta )}$
• The concept of universality states that systems with the same dimensionality and order parameter symmetry share the same critical exponents, regardless of their microscopic details.
• Mean-field theory provides a first approximation for the exponents but fails quantitatively in dimensions where fluctuations are important ($d\le 4$).

You might wonder if these six exponents are all independent. It turns out they are not. Thermodynamic identities and deeper theoretical principles constrain them. In our next lesson, we will introduce the scaling hypothesis, a powerful idea which posits that the singular part of the free energy is a special type of homogeneous function. From this single assumption, we will derive a series of "scaling relations" that connect the exponents, revealing that only two of them are truly independent.