PHYS3036 · Condensed Matter and Particle Physics
Topological Transitions & the 2D XY Model
Lectures 14–15 of University of Sydney PHYS3036 cover a transition with no local order parameter: the 2D XY model, whose continuous symmetry (Mermin–Wagner) forbids conventional long-range order. Order is instead destroyed by vortices — topological defects — unbinding at the Berezinskii–Kosterlitz–Thouless (BKT) transition. A single energy-versus-entropy free-energy argument gives the critical temperature T_c = πJ/(2k_B). This is a signature exam question (it appears on both available past papers), worked with no formula sheet.
What this chapter covers
- 01The 2D XY model: H = −J Σ_⟨i,j⟩ cos(θᵢ − θⱼ), spins are planar angles θ; continuous rotational symmetry
- 02Continuum limit (cos x ≈ 1 − x²/2): H = E₀ + (J/2)∫dr (∇θ)², with E₀ = −2JN on a square lattice
- 03Vortices as topological defects: ∮ dr·∇θ = 2π n (winding number n = ±1); higher |n| cost more energy and are ignored
- 04Cylindrically-symmetric vortex |∇θ| = 1/r; single-vortex energy E = E₀ + πJ ln(L/a) + E_core (L = system size, a = core size)
- 05Vortex position entropy S = k_B ln[(L/a)²] (≈ (L/a)² possible core locations)
- 06Free energy F = E − TS = (πJ − 2k_BT) ln(L/a) ⇒ BKT critical temperature T_c = πJ/(2k_B)
- 07Low-T phase: bound vortex–antivortex pairs; high-T phase: proliferation of free vortices; a transition with no spontaneous symmetry breaking / no local order parameter
The BKT transition temperature from a single-vortex free energy
- +1Energy of one vortex (size-dependent part): E = πJ ln(L/a). This grows with system size, so a single vortex is very costly in a large system. [+1]
- +1Entropy from the ≈ (L/a)² possible core positions: S = k_B ln[(L/a)²] = 2 k_B ln(L/a). This also grows with system size. [+1]
- +1Free energy of introducing one vortex: F = E − TS = πJ ln(L/a) − T·2k_B ln(L/a) = (πJ − 2k_B T) ln(L/a). [+1]
- +1Because ln(L/a) → +∞ as L → ∞, the sign of the prefactor decides everything. If πJ − 2k_B T > 0 (low T) then F → +∞: isolated vortices are forbidden. If πJ − 2k_B T < 0 (high T) then F → −∞: free vortices are favoured and proliferate. [+1]
- +1The changeover is at πJ − 2k_B Tc = 0 ⇒ Tc = πJ/(2k_B). Below Tc the only vortices are tightly bound vortex–antivortex pairs (finite energy); above Tc free vortices unbind and proliferate, destroying the quasi-order. [+1]
Key terms
- 2D XY model
- A lattice of planar spins (angles θ) with H = −J Σ cos(θᵢ − θⱼ); its continuous symmetry forbids conventional long-range order in two dimensions (Mermin–Wagner).
- Vortex
- A topological defect in the spin-angle field around which θ winds by 2π n; the winding number n = ±1 is a conserved topological charge, with |∇θ| = 1/r for a symmetric vortex.
- Winding number
- The integer n in ∮ dr·∇θ = 2π n counting how many times the spin angle wraps around a closed loop; it labels vortices (+1) and antivortices (−1).
- BKT transition
- The Berezinskii–Kosterlitz–Thouless transition at Tc = πJ/(2k_B), driven by the unbinding of vortex–antivortex pairs rather than by a local order parameter.
- Vortex core (a)
- The small central region where the continuum |∇θ| = 1/r picture breaks down; its size a sets the short-distance cutoff and contributes a core energy E_core.
- Topological transition
- A transition (like BKT) that changes the topological character of the configurations — bound vs free defects — without spontaneously breaking a symmetry or switching on a local order parameter.
Topological Transitions & the 2D XY Model FAQ
Why can't the 2D XY model order the usual way?
Because of the Mermin–Wagner theorem: a continuous symmetry cannot break to give true long-range order at finite temperature in two dimensions, since gapless spin-wave (Goldstone) fluctuations destroy it. So the XY model has no conventional ordered phase and no local order parameter — order is limited by topological defects instead, which is what makes the BKT transition special.
Where does Tc = πJ/(2k_B) come from?
From an energy-versus-entropy balance for a single vortex. Its energy grows as πJ ln(L/a) with system size, while the entropy from its ≈ (L/a)² possible positions grows as 2k_B ln(L/a). The free energy is F = (πJ − 2k_BT) ln(L/a), and the sign of the prefactor flips at Tc = πJ/(2k_B): below it isolated vortices cost infinite free energy, above it they proliferate.
What physically changes at the BKT transition?
The vortices unbind. Below Tc vortices exist only as tightly bound vortex–antivortex pairs, which do not disrupt the quasi-long-range order; above Tc thermal energy frees them, and the proliferation of independent vortices destroys the order. No symmetry is spontaneously broken and no local order parameter switches on — the transition is topological.
Why is BKT an exam favourite in PHYS3036?
Because it packages the whole condensed-matter toolkit — continuum limit, topological defects, an energy–entropy free-energy argument — into one self-contained derivation, and it appears on the available past papers. Expect to reproduce the continuum Hamiltonian, the vortex energy and entropy, the free energy and Tc, plus sketches of vortex/antivortex spin fields, all with no formula sheet. Confirm the paper's structure on Canvas and the unit outline.
Exam move
Treat the BKT derivation as a set-piece you can reproduce cold, because it recurs on the past papers and carries several marks. Rehearse the chain: continuum limit via cos x ≈ 1 − x²/2, the symmetric vortex |∇θ| = 1/r, the single-vortex energy πJ ln(L/a), the positional entropy 2k_B ln(L/a), the free energy (πJ − 2k_BT) ln(L/a), and Tc = πJ/(2k_B). Practise drawing a vortex and an antivortex and checking their ±1 winding, and be ready to say the transition breaks no symmetry (tie it back to Mermin–Wagner). Since no formula sheet is provided, drill it weekly rather than in a STUVAC rush. When the ln(L/a) cancellation feels slippery, ask Sia to walk the energy-versus-entropy balance line by line.
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