Questions: Mean Field Theory

Question 1

The central approximation in mean field theory for the Ising model is:

• A. Ignoring the externally applied magnetic field h to simplify the Hamiltonian
• B. Replacing the fluctuating influence of each spin's neighbors with the average magnetization ⟨σ⟩, converting the coupled many-body problem into independent single-site problems
• C. Assuming spins only interact with next-nearest neighbors rather than nearest neighbors
• D. Linearizing the partition function in the coupling constant J near T_c

Answer: (B)

Mean field theory's core move is decoupling: instead of each spin experiencing the actual, fluctuating spin values of its neighbors (which are correlated and unknown), each spin sees a fixed effective field equal to J times the average magnetization m = ⟨σ⟩. This turns N coupled equations into N identical single-site problems solvable exactly. The cost is that all information about fluctuations and correlations between neighbors is thrown away.

Question 2

A student argues that mean field theory gives its most accurate predictions near the critical temperature T_c, because that is where the large number of correlated spins makes the 'average over neighbors' most reliable. What is wrong?

• A. The student is correct; mean field theory is designed specifically for near-critical behavior
• B. Near T_c, fluctuations are at their maximum and the correlation length diverges — correlations between neighboring spins are strongest precisely there, making the replacement of actual fluctuating neighbors by their average least valid
• C. Mean field theory makes no predictions at or near T_c
• D. The critical temperature is the one point where mean field theory gives exact results

Answer: (B)

This is the key failure mode. Mean field theory averages over neighbor fluctuations and works reasonably when those fluctuations are small and uncorrelated (far from T_c, deep in the ordered or disordered phase). But near T_c, the correlation length diverges — neighboring spins are strongly correlated over long distances, and the actual interaction a spin feels from its neighbors is far from the average. The averaging assumption is worst exactly where it is most needed, which is why mean field critical exponents are quantitatively wrong.

Question 3

Mean field theory incorrectly predicts that a one-dimensional Ising chain undergoes a phase transition at finite temperature, even though the exact solution shows no such transition exists.

• T. True
• F. False

Answer: True

This is one of mean field theory's most dramatic failures. The exact solution of the 1D Ising model (by transfer matrix) shows that thermal fluctuations are so powerful in one dimension that they destroy any ordered phase at any finite temperature — long-range order only exists at T = 0. Mean field theory predicts a finite T_c because it ignores fluctuations entirely, and its T_c = Jz/k scales with the coordination number z. In 1D, z = 2, giving a finite (wrong) prediction. The failure worsens as dimensionality decreases.

Question 4

Mean field theory becomes exact in the limit of infinite spatial dimensions (or infinite-range interactions) because in that limit every spin truly interacts with an infinite number of others, making the central-limit-theorem-like averaging valid.

• T. True
• F. False

Answer: True

In infinite dimensions, each spin has infinitely many neighbors, and by the law of large numbers, the fluctuating sum of neighbor influences converges to its mean — making the mean field approximation exact, not merely approximate. Infinite-range Ising models (where every spin interacts with every other) realize the same physics. This is why mean field theory is exact for the Curie-Weiss model and why it becomes increasingly accurate as dimensionality grows beyond the upper critical dimension (d = 4 for the Ising universality class).

Question 5

Why does mean field theory fail specifically near the critical point, even though it gives qualitatively correct phase diagrams far from it?

Model answer: Because near the critical point, the correlation length diverges — fluctuations from the average extend across the entire system, and neighboring spins are strongly correlated rather than approximately independent. Mean field theory replaces each spin's actual, fluctuating neighbors with their average, which is only valid when neighbors truly fluctuate independently around that average (as they do far from T_c, deep in the ordered or disordered phase). At T_c, this assumption fails maximally: the thing being averaged over (neighbor fluctuations) is exactly what drives the critical behavior, and discarding it produces wrong critical exponents.

This is why the renormalization group is needed: it is a framework that keeps track of fluctuations at all length scales rather than averaging them away. Far from T_c, mean field is a useful first approximation; near T_c, it is qualitatively misleading about the universality class and the scaling behavior that experiments measure.