Critical Exponents

Prerequisites: Statistical Mechanics. Full path (4 pages)

• Why is it interesting?
• Intuitive
• Concrete
• Abstract

Why is it interesting?

As condensed matter physicists were just discovering, when materials that are completely different at the microscopic level are tuned to the critical points at which they undergo phase transitions, they suddenly exhibit the same behaviors and can be described by the exact same handful of numbers. Heat iron to the critical temperature where it ceases to be magnetized, for instance, and the correlations between its atoms are defined by the same “critical exponents” that characterize water at the critical point where its liquid and vapor phases meet. These critical exponents are clearly independent of either material’s microscopic details, arising instead from something that both systems, and others in their “universality class,” have in common. Polyakov and other researchers wanted to find the universal laws connecting these systems. “And the goal, the holy grail of all that, was these numbers,” he said: Researchers wished to be able to calculate the critical exponents from scratch.

Physicists Uncover Geometric ‘Theory Space’

If one is to pick out a single feature that epitomizes the power and successes of RG theory, one can but endorse Gallavotti and Benfatto when they say "it has to be stressed that the possibility of nonclassical critical indices (i.e., of nonzero anomaly $\eta$) is probably the most important achievement of the renormalization group."

Michael Fisher in Conceptual Foundations of Quantum Field Theory, Edited by Cao

Intuitive

Heat a magnet and at a certain temperature, the Curie temperature, it stops being magnetic. Compress a gas at the right temperature and pressure and the difference between liquid and gas disappears. These special points are called critical points. Near a critical point many quantities change as a power of the distance to the critical temperature. The magnetization, for example, goes to zero like the temperature difference raised to some power. These powers are the critical exponents.

What makes them interesting is that they do not depend on the details of the material. Near the critical point the system develops fluctuations on all length scales: patches of aligned spins inside larger patches inside even larger patches. The typical size of these patches, the correlation length, grows without bound as we approach the critical temperature. On such large scales the precise shape of the atoms and the strength of their forces no longer matter. What matters is the number of space dimensions and the symmetry of the quantity that becomes ordered. As a result, the liquid-gas transition of water and the transition in a magnet whose spins can only point up or down have the same critical exponents. Systems that share their exponents form a universality class.

The renormalization group explains why this happens: zooming out step by step, different microscopic systems flow towards the same large-scale description.

For a nice introduction in laymen terms see: Physicists Uncover Geometric ‘Theory Space’

Concrete

Define the reduced temperature $t = (T - T_c)/T_c$ and let $h$ be the field conjugate to the order parameter $m$ (for a magnet, the external magnetic field and the magnetization). The standard critical exponents are defined by the leading behaviour as $t \to 0$:

• specific heat at $h = 0$: $C \sim |t|^{-\alpha}$
• order parameter at $h = 0$, $t < 0$: $m \sim (-t)^{\beta}$
• susceptibility at $h = 0$: $\chi = \partial m/\partial h \sim |t|^{-\gamma}$
• critical isotherm $t = 0$: $m \sim |h|^{1/\delta}$
• correlation length at $h = 0$: $\xi \sim |t|^{-\nu}$
• correlation function at $t = 0$: $G(r) \sim r^{-(d-2+\eta)}$, in $d$ space dimensions.

Mean field example. Landau theory writes the free energy density near $T_c$ as $$f(m) = f_0 + a\,t\,m^2 + b\,m^4 - h\,m, \qquad a, b > 0 .$$ Minimizing at $h = 0$ and $t < 0$ gives $m^2 = -a t/(2b)$, so $\beta = \tfrac12$. At $t = 0$ the condition $4bm^3 = h$ gives $\delta = 3$. For $t > 0$ the linear response $2atm = h$ gives $\chi = 1/(2at)$, so $\gamma = 1$. The specific heat has a jump but no divergence, $\alpha = 0$. Adding a gradient term $c(\nabla m)^2$ gives Ornstein-Zernike correlations with $\nu = \tfrac12$ and $\eta = 0$.

Real values. Mean field theory ignores fluctuations and is correct only above the upper critical dimension, which is $d = 4$ for this class of systems. For the two-dimensional Ising model the exact exponents are $$\alpha = 0 \ (\text{logarithmic}), \quad \beta = \tfrac18, \quad \gamma = \tfrac74, \quad \delta = 15, \quad \nu = 1, \quad \eta = \tfrac14 .$$ In three dimensions the Ising class, which also contains the liquid-gas critical point and uniaxial magnets, has $\beta \approx 0.326$, $\gamma \approx 1.237$, $\nu \approx 0.630$, $\eta \approx 0.036$.

Scaling relations. The exponents are not independent. If the singular part of the free energy is a generalized homogeneous function of $t$ and $h$, and $\xi$ is the only relevant length, one derives $$\alpha + 2\beta + \gamma = 2, \qquad \gamma = \beta(\delta - 1), \qquad \gamma = \nu(2 - \eta), \qquad d\,\nu = 2 - \alpha .$$ (Rushbrooke, Widom, Fisher and Josephson relations.) One checks that the 2D Ising values satisfy all four. The mean field values satisfy the last one, the hyperscaling relation, only for $d = 4$. In the renormalization group the exponents follow from the eigenvalues of the linearized flow at the fixed point: if $t$ scales as $t \to b^{y_t} t$ under a change of length scale by $b$, then $\nu = 1/y_t$, and only two exponents are independent.

The best explanation can be found in Critical point phenomena: universal physics at large length scales by Bruce, A.; Wallace, D. published in the book The new physics, edited by P. Davies

Abstract

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