Key Concepts in Critical Phenomena

Why This Matters

Critical phenomena represent one of the most profound achievements in statistical mechanics—the discovery that wildly different physical systems behave identically near their critical points. You're being tested on your ability to understand why a magnet losing its magnetization and a liquid becoming indistinguishable from its vapor share the same mathematical description. This isn't coincidence; it's universality, and it reveals deep truths about how nature organizes itself across scales.

The concepts here—order parameters, critical exponents, scaling laws, and renormalization group theory—form the theoretical backbone for understanding continuous phase transitions. Exam questions will push you beyond memorizing definitions to explaining why mean-field theory fails in low dimensions, how correlation length divergence drives critical behavior, and what universality classes tell us about the irrelevance of microscopic details. Don't just memorize facts—know what physical principle each concept illustrates and how they connect to form a coherent picture of critical behavior.

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Foundations: What Defines a Critical Point

Before diving into the theoretical machinery, you need a solid grasp of what actually happens at a critical point and how we quantify the transition. These foundational concepts set up everything that follows.

Phase Transitions and Critical Points

• Phase transitions mark abrupt changes in physical properties—the system reorganizes from one state of matter to another (solid → liquid, paramagnetic → ferromagnetic)
• Critical points terminate phase boundaries where distinct phases become indistinguishable; here, the system becomes scale-invariant with fluctuations on all length scales
• Thermodynamic quantities show non-analytic behavior at criticality—heat capacity, susceptibility, and compressibility diverge or exhibit singularities that signal the transition

Order Parameters

• Order parameters quantify the degree of broken symmetry—they're zero in the disordered phase and nonzero in the ordered phase
• Magnetization $M$ serves as the classic example: $M = 0$ above the Curie temperature $T_c$, while $M \neq 0$ below it
• The order parameter's behavior near $T_c$ follows $M \sim |T - T_c|^\beta$, where $\beta$ is a critical exponent that encodes universal physics

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Quantifying Critical Behavior: Exponents and Scaling

The remarkable feature of critical phenomena is that the singular behavior follows precise power laws. These exponents aren't random—they're constrained by deep mathematical relationships.

Critical Exponents

• Critical exponents describe power-law singularities as $T \to T_c$: quantities either diverge ($\chi \sim |T - T_c|^{-\gamma}$) or vanish ($M \sim |T - T_c|^\beta$)
• Standard exponents include $\alpha$ (heat capacity), $\beta$ (order parameter), $\gamma$ (susceptibility), $\delta$ (critical isotherm), $\nu$ (correlation length), and $\eta$ (correlation function decay)
• Universality means identical exponents for systems in the same class—the 3D Ising model and uniaxial ferromagnets share exponents despite completely different microscopic physics

Scaling Theory

• Scaling theory unifies critical exponents through scaling relations like $\alpha + 2\beta + \gamma = 2$ (Rushbrooke) and $\gamma = \nu(2 - \eta)$ (Fisher)
• Near criticality, observables follow scaling functions—for example, the equation of state takes the form $M = |t|^\beta f(h/|t|^{\beta\delta})$ where $t = (T - T_c)/T_c$
• Only two independent exponents exist due to scaling relations; knowing $\nu$ and $\eta$ determines all others, dramatically constraining possible critical behavior

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The Theoretical Framework: RG and Mean-Field Approaches

Understanding why universality exists requires theoretical tools that go beyond phenomenology. Renormalization group theory provides the deep explanation, while mean-field theory offers tractable (if limited) approximations.

Renormalization Group Theory

• RG analyzes how systems look at different length scales—by systematically "coarse-graining" (integrating out short-wavelength fluctuations), it reveals scale-invariant behavior at criticality
• Fixed points in parameter space correspond to critical points; the flow toward or away from fixed points determines whether parameters are relevant, irrelevant, or marginal
• Universality emerges naturally because different microscopic Hamiltonians flow to the same fixed point—only a few relevant operators matter, explaining why details wash out

Mean-Field Theory and Its Limitations

• Mean-field theory replaces fluctuating neighbors with an average field—each spin feels $\langle M \rangle$ rather than its actual fluctuating environment, yielding tractable self-consistent equations
• It predicts classical exponents ($\beta = 1/2$, $\gamma = 1$, $\nu = 1/2$) that are exact above the upper critical dimension $d_c = 4$ but wrong below it
• Failure in low dimensions stems from ignoring correlations—in $d \leq 4$, critical fluctuations are too strong to average away, and mean-field theory misses the true singular behavior

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Spatial Structure: Correlations and Fluctuations

Critical behavior isn't just about thermodynamic averages—it's fundamentally about how different parts of the system "talk" to each other. Diverging correlation lengths are the hallmark of criticality.

Correlation Functions and Length

• Correlation functions $G(r) = \langle \phi(0)\phi(r) \rangle - \langle \phi \rangle^2$ measure how fluctuations at different points are statistically related
• Away from criticality, correlations decay exponentially: $G(r) \sim e^{-r/\xi}$, where $\xi$ is the correlation length
• At the critical point, $\xi \to \infty$ and correlations decay as a power law: $G(r) \sim r^{-(d-2+\eta)}$, signaling scale invariance

Fluctuations Near Critical Points

• Fluctuations grow dramatically as $T \to T_c$ because the correlation length diverges as $\xi \sim |T - T_c|^{-\nu}$
• Critical opalescence is a visible consequence—density fluctuations near the liquid-gas critical point scatter light, making the fluid appear milky
• Fluctuation-dissipation relations connect susceptibility to fluctuations: $\chi \propto \int G(r) \, d^d r$, explaining why susceptibility diverges when correlations become long-ranged

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The Paradigmatic Model: Ising Systems

No discussion of critical phenomena is complete without the Ising model—it's the "hydrogen atom" of phase transitions. Simple enough to solve (in some cases), rich enough to capture universal physics.

Ising Model

• The model consists of spins $s_i = \pm 1$ on a lattice with Hamiltonian $H = -J\sum_{\langle ij \rangle} s_i s_j - h\sum_i s_i$, where $J$ is the coupling and $h$ is an external field
• Onsager's exact 2D solution (1944) proved a second-order transition exists with non-mean-field exponents ($\beta = 1/8$, $\nu = 1$), revolutionizing the field
• The Ising universality class includes uniaxial ferromagnets, binary alloys, and lattice gases—any system with scalar order parameter and short-range interactions in the same dimension

Universality Classes

• Systems share a universality class if they have the same symmetry of order parameter, dimensionality, and range of interactions—microscopic details are irrelevant
• Major classes include Ising ($Z_2$ symmetry), XY ($O(2)$), Heisenberg ($O(3)$), and percolation—each with distinct critical exponents
• Liquid-gas transitions belong to the Ising class because the density difference $\rho_l - \rho_g$ acts like a scalar order parameter with the same symmetry

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Quick Reference Table

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Self-Check Questions

1. Scaling relations: Given that $\beta = 0.326$ and $\gamma = 1.237$ for the 3D Ising model, use the Rushbrooke equality to estimate $\alpha$. What does a small (or negative) $\alpha$ tell you about the heat capacity singularity?

2. Compare and contrast: Both mean-field theory and renormalization group theory can predict critical exponents. Under what conditions does mean-field theory give correct exponents, and why does it fail otherwise?

3. Universality identification: A binary alloy undergoes an order-disorder transition where atoms of type A preferentially occupy one sublattice below $T_c$. What universality class does this belong to, and what order parameter would you define?

4. Correlation length significance: Explain why the divergence of correlation length $\xi \to \infty$ at the critical point is responsible for both the breakdown of mean-field theory and the phenomenon of critical opalescence.

5. FRQ-style synthesis: The 2D Ising model has $\nu = 1$ and $\eta = 1/4$. Using the Fisher scaling relation, calculate $\gamma$. Then explain physically why susceptibility diverges at the critical point in terms of correlation functions.