Critical exponents, scaling and universality · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.
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Why magnets and fluids share numbers

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

  1. At an Ising critical point with zero ordering field, what happens to the correlation length in the ideal infinite-system limit?

    • It diverges in the thermodynamic limit
    • It must stay equal to one lattice spacing
    • It falls to zero as long-range order appears

  2. In m ≈ B(−t)^β below an Ising-type critical point at zero field, what does β describe?

    • The divergence of ξ as the critical point is approached
    • The microscopic temperature scale set by the coupling
    • The power-law vanishing of m below the critical point

  3. Different materials can share critical exponents when their critical points belong to the same universality class.

    Circle one:   True   False

  4. For an illustrative mean-field law at zero field, ξ ∝ |t|^(−1/2). On the same side of T_c in the leading scaling regime, |t| falls from 0.09 to 0.01. What is ξ_new/ξ_old?

    • 1/3
    • 9
    • 3

  5. Which magnetic critical point is the standard match for a three-dimensional short-range liquid-gas critical point with a scalar order parameter?

    • A 3D short-range, three-component magnet
    • A 3D short-range, scalar Ising-like magnet
    • A 1D nearest-neighbour, scalar Ising magnet

  6. Which description gives the actions in a renormalisation step?

    • Integrate short scales, rescale lengths, update couplings
    • Erase correlations at every scale and hold couplings fixed
    • Increase microscopic detail while leaving lengths unchanged

  7. Two materials share critical exponents. A student concludes that their critical temperatures must also be equal. Is that conclusion justified?

    • Yes; matching exponents fixes every temperature scale
    • Yes; universal exponents force the same atomic species
    • No; T_c depends on microscopic couplings

  8. Two short-range magnets are both three-dimensional. One has a scalar order parameter and the other has a three-component rotational order parameter. Must their exponents match?

    • Yes; equal dimension fixes exponents for all interactions
    • Not necessarily; order-parameter symmetry differs
    • Yes; component count changes only the critical temperature
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Answer key

For grown-ups. Fold this page away before handing over the rest.

Why magnets and fluids share numbers W1-mt_EaJOjyvjjj-s1

  1. It diverges in the thermodynamic limit · The ideal correlation length diverges. Finite sample size or a nonzero ordering field limits the growth observed in a real system.
  2. The power-law vanishing of m below the critical point · β is the power in the order-parameter scaling law. As t approaches zero from below, m vanishes according to (−t)^β.
  3. True · Universality allows different microscopic systems to share long-distance critical exponents when they belong to the same class.
  4. 3 · The ratio is (0.01/0.09)^(−1/2) = (1/9)^(−1/2) = 3. The correlation length triples.
  5. A 3D short-range, scalar Ising-like magnet · The standard match is a three-dimensional short-range Ising-like magnet with one scalar order parameter, not a general magnet with freely rotating moments.
  6. Integrate short scales, rescale lengths, update couplings · Renormalisation removes short-distance degrees of freedom from the explicit description, rescales lengths and tracks the changed effective couplings.
  7. No; T_c depends on microscopic couplings · Critical temperature is nonuniversal. Materials can share exponents while having different microscopic couplings, critical temperatures and scaling amplitudes.
  8. Not necessarily; order-parameter symmetry differs · Equal dimension does not fix the universality class. Different order-parameter components and symmetries can give different critical exponents.
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