PHYS3036 · Condensed Matter and Particle Physics
The Ising Model & Mean-Field Theory
Lectures 12–13 of University of Sydney PHYS3036 turn to the Ising model — spins Sᵢ = ±1 coupled to their neighbours — as the paradigm of a phase transition, and solve it with mean-field theory. Replacing each spin's neighbours by an average field gives a self-consistent magnetisation equation m = tanh(z J m / k_BT), a transition at T_c = zJ/k_B, and — after expanding — a microscopic Landau theory with the same mean-field exponents. The exam asks you to derive the self-consistency, find Tc and state where mean-field fails.
What this chapter covers
- 01The Ising Hamiltonian H = −J Σ_⟨i,j⟩ Sᵢ Sⱼ − h Σᵢ Sᵢ, with spins Sᵢ = ±1 and nearest-neighbour coupling J
- 02Mean-field approximation: replace a spin's neighbours by their average, Sⱼ → m; each spin feels an effective field h_eff = z J m + h (z = coordination number)
- 03Self-consistency (magnetisation) equation: m = tanh((z J m + h)/k_BT)
- 04Critical temperature from linearising tanh at small m: k_B T_c = z J, i.e. T_c = zJ/k_B
- 05Recovering a microscopic Landau theory by expanding the free energy in small m; mean-field gives the same exponents as Landau (β = 1/2, etc.)
- 06Mean-field limitations: ignores correlated fluctuations; fails for d < 4 near Tc; wrongly predicts a finite-T transition for the 1D Ising chain (which has none)
- 07Why mean-field improves with higher dimension / coordination number (more neighbours ⇒ fluctuations relatively smaller)
Mean-field Ising: self-consistency and the critical temperature
- +1Mean-field step: in the interaction of spin i with its z neighbours, replace each neighbour Sⱼ by its average value m = ⟨S⟩. Spin i then sees an effective field h_eff = z J m (plus any external field, here zero). [+1]
- +1A single spin in field h_eff has thermal average ⟨Sᵢ⟩ = tanh(h_eff / k_BT). Self-consistency demands ⟨Sᵢ⟩ = m, so m = tanh(z J m / k_BT). [+1]
- +1Near the transition m is small, so expand tanh(x) ≈ x: m ≈ z J m /(k_BT). A non-zero solution requires the slope on the right to reach 1: z J /(k_BT) = 1 at the onset. [+1]
- +1Hence k_B T_c = z J, i.e. T_c = z J / k_B. Above Tc only m = 0 solves the equation; below Tc the slope exceeds 1 and non-zero ±m solutions appear (the ordered phase). [+1]
- +1For a 2D square lattice z = 4, so T_c = 4J/k_B. This OVERESTIMATES the exact 2D Ising result (Onsager, k_B T_c ≈ 2.27 J) because mean-field ignores fluctuations; it also wrongly predicts a finite Tc for the 1D chain (z = 2), which actually has no finite-temperature transition. [+1]
Key terms
- Ising model
- A lattice model of spins Sᵢ = ±1 with nearest-neighbour coupling, H = −J Σ Sᵢ Sⱼ − h Σ Sᵢ; the paradigm of a phase transition.
- Mean-field theory
- An approximation that replaces the fluctuating neighbours of each spin by their average, reducing the many-body problem to one spin in a self-consistent effective field.
- Self-consistency (magnetisation) equation
- m = tanh(z J m / k_BT) at zero field: the average spin must equal the mean field it is computed from; its non-zero solutions signal order.
- Coordination number (z)
- The number of nearest neighbours of a site (z = 2 for a chain, 4 for a square lattice, 6 for simple cubic); it sets the mean-field Tc = zJ/k_B.
- Critical temperature (Tc)
- The temperature below which a non-zero order parameter appears; in mean-field Ising, k_B Tc = zJ (an overestimate of the true value).
- Mean-field failure (d < 4)
- The breakdown of mean-field/Landau near Tc for spatial dimension below four, where fluctuations dominate; in 1D it wrongly predicts a finite-temperature transition.
The Ising Model & Mean-Field Theory FAQ
What is the mean-field approximation in one sentence?
Replace the fluctuating neighbours of each spin by their average value, so every spin experiences the same self-consistent effective field instead of a fluctuating local environment — turning an intractable many-body problem into a single-spin problem you close by self-consistency.
Where does m = tanh(zJm/k_BT) come from?
A single spin in an effective field h_eff = zJm has thermal average ⟨S⟩ = tanh(h_eff/k_BT). Mean-field then insists that this average equals the m you used to build the field, giving the self-consistent equation m = tanh(zJm/k_BT). Expanding tanh for small m shows a non-zero solution first appears when zJ/(k_BT) = 1, i.e. at Tc = zJ/k_B.
Why is mean-field theory wrong for the 1D Ising chain?
Because it ignores fluctuations, which are strongest in low dimensions. In one dimension a single flipped bond can disorder the whole chain at any non-zero temperature, so there is no finite-Tc transition — yet mean-field, blind to this, predicts Tc = 2J/k_B. Mean-field only becomes reliable in high dimension (d ≥ 4) or large coordination number, where each spin averages over many neighbours.
How is this examined in PHYS3036?
Typically: set up the effective field, derive the self-consistency equation, linearise to get Tc, and then discuss the limitations (overestimated Tc, failure for d < 4, the 1D pathology) and the link to Landau theory. It is a no-formula-sheet derivation, so rehearse it end to end. Confirm the exact emphasis on Canvas and the unit outline.
Exam move
Learn the mean-field routine as a fixed sequence you can reproduce without notes: replace neighbours by m, write the effective field zJm, use ⟨S⟩ = tanh(h_eff/k_BT), impose self-consistency, then linearise to read off Tc = zJ/k_B. Always finish with the honest caveats — mean-field overestimates Tc, fails for d < 4, and wrongly predicts a 1D transition — and connect the small-m expansion to the Landau free energy with its β = 1/2. Because the condensed-matter exam provides no formula sheet, drill this derivation weekly rather than at STUVAC; it is a reliable source of marks and reinforces the Landau chapter. When the self-consistency step feels like a trick, ask Sia to graph m against tanh(zJm/k_BT) and show how the solutions appear as T drops through Tc.
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