The Ising model and the mean-field approximation · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

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Science · Thermal & Statistical Physics · ages 23-24
Name ______________________   Date ____________

  1. An Ising lattice has z = 6 and J/k_B = 5 K. What critical temperature does zero-field mean field predict?

    • 60 K
    • 5 K
    • 30 K

  2. What replaces each neighbouring spin when constructing the Ising mean field?

    • The most common spin configuration
    • The mean neighbouring spin m
    • The lowest-energy neighbouring spin

  3. At zero field, Ising mean field has stable nonzero solutions below its predicted critical temperature.

    Circle one:   True   False

  4. At zero field, z = 6 and J/(k_B T) = 1/3. Which equation must the mean spin satisfy?

    • m = tanh(m/2)
    • m = tanh(2m)
    • m = 2tanh(m)

  5. For h = 0 and T < T_c, how many intersections of y=m and y=tanh(T_c m/T) are there, including unstable ones?

    • Three: zero and a nonzero pair
    • Two: only the nonzero pair
    • One: only the zero solution

  6. Mean field predicts a positive T_c for the one-dimensional nearest-neighbour Ising chain. Which neglected effect helps explain why this prediction fails?

    • A missing factor in the definition of z
    • A failure to allow two spin values
    • Correlations between different spins

  7. A student rejects mean-field leading critical exponents for every short-range Ising system, including five dimensions. Which qualification is needed?

    • Short-range systems need different exponents in every dimension
    • Leading mean-field exponents can hold for d > 4
    • The upper critical dimension removes every finite-size effect

  8. In leading zero-field mean-field scaling below T_c, m ∝ τ^(1/2), where τ = (T_c−T)/T_c. If τ falls from 0.04 to 0.01, how does positive m change?

    • It becomes half as large
    • It becomes one quarter as large
    • It remains at its earlier value
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Answer key

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  1. 30 K · The zero-field slope criterion gives T_c = zJ/k_B. Here T_c = 6 × 5 K = 30 K.
  2. The mean neighbouring spin m · Each neighbour is replaced by the mean spin m. The z neighbours then contribute zJm to the local field energy.
  3. True · For a > 1, the tanh curve crosses the line at nonzero roots ±m*. Their iteration slopes a(1−m*²) are below 1, so small perturbations shrink and the roots are locally stable.
  4. m = tanh(2m) · Substitute into m = tanh(zJm/(k_B T)). The coefficient of m inside tanh is 6 × 1/3 = 2.
  5. Three: zero and a nonzero pair · There are three intersections: −m*, 0 and +m*. The zero root is unstable below T_c, but it remains an algebraic solution.
  6. Correlations between different spins · Mean field omits correlations between spin fluctuations. The exact infinite nearest-neighbour chain at zero field has no ordered phase at positive temperature.
  7. Leading mean-field exponents can hold for d > 4 · Four is the upper critical dimension for short-range Ising criticality. Above it, leading critical exponents take mean-field values.
  8. It becomes half as large · The ratio is m_new/m_old = (0.01/0.04)^(1/2) = 1/2. The order parameter becomes half as large in the leading approximation.
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