The two-dimensional Ising model

On the square lattice the Ising model has a genuine phase transition at T_c = 2J/[k_B ln(1 + √2)] ≈ 2.269J/k_B, solved exactly by Onsager in 1944. Its exponents, β = 1/8, γ = 7/4 and α = 0 with a logarithmic divergence, are far from mean field. Explore the exact results against simulations, Peierls’ argument for why d = 2 orders while d = 1 cannot, Widom’s scaling ansatz and data collapse, the scaling relations among the exponents, and the Ginzburg criterion that makes d = 4 the upper critical dimension.

Based on lecture notes by Sang Hoon Lee for Chapter 2, following K. Christensen and N. R. Moloney, Complexity and Criticality (2005). Units: k_B = 1 and J = 1. References

Up spins are white and down spins are black, as in the lecture notes. Periodic boundaries.

T/T_c =

T_c

External field H

Lattice

Update rule

Speed

Temperature T (in units of J/k_B): –
Magnetization per spin m: –
Time average of |m|: –
Onsager’s m₀(T) for an infinite lattice: –
Energy per spin ε: –
Sweeps since the last change: –

Magnetization per spin over timeBelow T_c a small lattice flips now and then between +m and −m (dashed: Onsager’s ±m₀); a large one stays put.

Distribution of m over timeOne peak at m = 0 above T_c, two peaks at ±m₀ below it, and a broad distribution at T_c.

Spin–spin correlation function g(r) of this latticeg(r) = ⟨s_is_i+r⟩ − ⟨|m|⟩², averaged over time along both axes. Log–log; dashed: r^{−(d−2+η)} = r^−1/4, the decay at T_c.

Onsager’s exact solution

In zero field the partition function is Z(T, 0) = [2 cosh(2K) e^I]^N with the reduced coupling K = J/k_BT and I = (1/2π)∫^π₀ ln{[1 + √1 − κ² sin²φ]/2} dφ, where κ = 2 sinh(2K)/cosh²(2K). From it follow the spontaneous magnetization (Yang, 1952), the energy and the specific heat:

m₀ = [1 − sinh⁻⁴(2K)]^1/8 ≈ [4√2 ln(1 + √2)]^1/8 [(T_c − T)/T_c]^1/8, ε = −J coth(2K)[1 + (2/π)κ′F(κ)]

c/k_B = (4/π)K² coth²(2K){F(κ) − E(κ) − sech²(2K)[π/2 + κ′F(κ)]}, κ′ = 2 tanh²(2K) − 1

F and E are the complete elliptic integrals of the first and second kind. At T_c, κ = 1 and F diverges logarithmically, while κ′ changes sign, putting a kink in ε. So c ≈ −(8/π)K_c² ln|t| + constant with t = (T − T_c)/T_c: α = 0, but now a logarithmic divergence rather than a jump. χ is not known exactly because Z is not known for H ≠ 0, but χ ≈ Γ^±|t|^−7/4 with Γ⁺ ≈ 0.42420 and Γ⁻ ≈ 0.011254, so Γ⁺/Γ⁻ ≈ 37.69, far from the mean-field 2.

The simulations below sample microstates with the Boltzmann weight (Wolff cluster updates at H = 0) on L × L lattices. For finite N the system stays ergodic, so ⟨m⟩ = 0; they therefore record ⟨|M|⟩/N, and use k_BTχ = (⟨M²⟩ − ⟨|M|⟩²)/N and k_BT²c = (⟨E²⟩ − ⟨E⟩²)/N.

Spontaneous magnetizationDots: ⟨|M|⟩/N for each L. Dashed: Onsager–Yang m₀(T).

m₀ ∝ (−t)^β, β = 1/8Log–log. Line: exact. The dots flatten close to T_c, a finite-size effect once ξ grows past L.

χ ∝ |t|^−γ, γ = 7/4Log–log. Dots above T_c (blue) and below (red) for the largest L. Dashed: Γ^±|t|^−7/4, a factor 37.7 apart.

Energy per spin ε(T)Dashed: exact. It runs from −2J at T = 0 to 0 at T = ∞, with a kink at T_c.

Specific heat c(T)Dashed: exact, with a sharp logarithmic peak at T_c; the simulations are almost indistinguishable away from it.

The logarithmic divergence of cExact c against |t| on a log scale, above (blue) and below (red) T_c: straight lines with slope −(8/π)K_c² ≈ −0.494 per unit of ln|t|.

Why there is a transition: Peierls’ argument

Ising concluded from the chain that there is no transition in any dimension. Peierls argued otherwise in 1936. Compare a single aligned domain with a lattice containing a second domain of n flipped spins bounded by an interface. In two dimensions the interface costs 2J per broken bond, so the energy grows with its length n, while the entropy is set by the number of interfaces Ω(n). Counting each interface as a walk that never immediately retraces itself, with N possible positions, gives Ω(n) < N3ⁿ, a gross overestimate. For a compact domain of comparable size, n ∝ √N, so

F_2-dom − F_1-dom = 2Jn − k_BT ln Ω(n) > 2J√N − k_BT ln(N3^√N)

The ordered state is stable whenever this is positive, k_BT < 2J/[ln 3 + (ln N)/√N]. As N → ∞, (ln N)/√N → 0, so T_c > 2J/(k_B ln 3) ≈ 1.820: there must be a transition at a nonzero temperature. In one dimension the cost of a droplet is 4J whatever its size and is always outweighed by the entropy k_BT ln[2N(N − 1)].

A droplet of flipped spinsIts interface (red) has n broken bonds and costs 2Jn. Click the lattice for a new droplet.

Peierls bound on T_c against NBlue: 2J/[ln 3 + (ln N)/√N]. It approaches 2/ln 3 ≈ 1.820 (dashed), safely below the exact T_c ≈ 2.269 (red).

Widom’s scaling ansatz

Mean-field theory fails in low dimensions, so a more general framework is needed. With the reduced variables t = (T − T_c)/T_c and h = H/k_BT, the critical point is (t, h) = (0, 0). The free energy splits into a regular part and a singular part responsible for the cusp, and Widom argued that

f_s(t, h) = |t|^2−α 𝓕_±(h/|t|^Δ), m(t, h) = |t|^β 𝓜_±(h/|t|^Δ) for t → 0^±, h → 0

Δ is the gap exponent and 𝓜_± are odd scaling functions for t > 0 and t < 0. As x = h/|t|^Δ → 0, 𝓜₊ → 0 and 𝓜₋ → ± a nonzero constant; as x → ±∞, 𝓜_± ∝ sign(x)|x|^1/δ, which requires Δ = βδ. So if m/|t|^β is plotted against h/|t|^Δ, data at many t and h fall onto just two curves. The page simulates a 64 × 64 lattice at 10 temperatures and 4 fields with Metropolis updates, which allow a field. Move the exponents away from β = 1/8 and Δ = 15/8 and the collapse breaks.

Raw data: m against hLog–log. Each line is one t: above T_c (blue shades) and below (red shades).

Collapse: m/|t|^β against h/|t|^ΔBlue: t > 0 traces 𝓜₊. Red: t < 0 traces 𝓜₋. Dashed: slope 1/δ = 1/15 at large x.

Δ = βδ

Differentiating the ansatz gives the other quantities: m from ∂f_s/∂h gives 2 − α − Δ = β; χ from ∂²f_s/∂h² gives 2 − α − 2Δ = −γ; and c = |t|^−α𝒞_±(h/|t|^Δ). The correlation function follows a similar form, g(r, t, h) ∝ r^{−(d−2+η)}𝒢_±(r/ξ, h/|t|^Δ), with 𝒢 constant for r ≪ ξ and ∝ (r/ξ)^{[η+(d−3)]/2}e^−r/ξ for r ≫ ξ. In one dimension η = 1 and 𝒢 = e^−r/ξ: no power-law part at all.

Scaling relations

The six exponents are not independent: they all describe the same non-analytic free-energy surface near its cusp. From Δ = βδ, 2 − α − Δ = β and 2 − α − 2Δ = −γ follow β + γ = βδ (Widom) and α + 2β + γ = 2 (Rushbrooke). The sum rule k_BTχ = ∫g dr gives γ = ν(2 − η) (Fisher), and f_s ∝ ξ^−d gives the hyperscaling relation 2 − α = νd (Josephson), valid only for d ≤ 4. So only two exponents are independent. Pick ν, η and d and the rest follow:

Non-universal T_c, universal exponents

Like p_c in percolation, T_c depends on the lattice, even in mean-field theory where T_c ∝ z. The exponents depend only on the dimension.

Lattice	z	k_BT_c/J
d = 1, line	2	0
d = 2, hexagonal	3	2/ln(2 + √3) ≈ 1.519
d = 2, square	4	2/ln(1 + √2) ≈ 2.269
d = 2, triangular	6	4/ln 3 ≈ 3.641
d = 3, diamond	4	≈ 2.70
d = 3, simple cubic	6	≈ 4.5115
d = 3, body-centered cubic	8	≈ 6.40
d = 3, face-centered cubic	12	≈ 9.79
Mean field	z	z

Exponent	Quantity	d = 1 *	d = 2	d = 3	d ≥ 4	Mean field
α	c(t, 0) ∝ \|t\|^−α	2 − 2/k	0 (log)	0.111(2)	0	0 (jump)
β	m(t, 0) ∝ (−t)^β	0	1/8	0.3262(13)	1/2	1/2
γ	χ(t, 0) ∝ \|t\|^−γ	2/k	7/4	1.237(3)	1	1
δ	m(0, h) ∝ sign(h)\|h\|^1/δ	∞	15	4.792(18)	3	3
ν	ξ(t, 0) ∝ \|t\|^−ν	2/k	1	0.6297(8)	1/2	1/2
η	g(r, t, 0) ∝ r^{−(d−2+η)}𝒢_±(r/ξ, 0)	1	1/4	0.036(5)	0	0

* Using the reduced “temperature” t = exp(−kJ/k_BT) with an arbitrary constant k > 0, since T_c = 0 in one dimension. These exponents are somewhat artificial: they describe power laws in this t rather than in T − T_c, and α, γ and ν depend on k. Values in d = 3 are numerical, with the uncertainty in the last digits in brackets.

The Ginzburg criterion and the upper critical dimension

Mean-field theory agrees with the exponents for d ≥ 4, which suggests that fluctuations stop mattering in high dimensions. The energy neglected by mean field is ΔE = −J Σ_⟨ij⟩(s_i − ⟨s_i⟩)(s_j − ⟨s_j⟩), and by the sum rule |⟨ΔE⟩| = (Jz/2)k_BTχ per spin. For mean field to hold it must be small compared with the mean-field energy (Jz/2)Nm², that is Nk_BTχ ≪ (Nm)², or simply √⟨M²⟩ − ⟨M⟩² ≪ ⟨M⟩. Within a correlation volume ξ^d ∝ |t|^−νd this ratio scales as (−t)^{νd − γ − 2β}, which stays small as t → 0⁻ only if

d > (γ + 2β)/ν

With the mean-field values β = 1/2, γ = 1 and ν = 1/2 this means d > 4: d_u = 4 is the upper critical dimension, above which mean-field theory gives the correct exponents. In three or fewer dimensions the criterion is bound to fail close enough to T_c, although mean field can still be a good approximation in practice, as for type I superconductors (Ginzburg, Nobel Prize 2003).

Relative fluctuations with mean-field exponents(−t)^{d/2 − 2} against −t, log–log, for d = 2 (red), 3, 4 (dark) and 5 (blue). For d < 4 the fluctuations blow up as t → 0⁻ and mean field is inconsistent; at d = 4 they stay marginal; above, they vanish.

References

Key reference. K. Christensen and N. R. Moloney, Complexity and Criticality, Imperial College Press, London (2005), Chapter 2.

L. Onsager, “Crystal statistics. I. A two-dimensional model with an order-disorder transition,” Phys. Rev. 65, 117 (1944).
C. N. Yang, “The spontaneous magnetization of a two-dimensional Ising model,” Phys. Rev. 85, 808 (1952).
H. A. Kramers and G. H. Wannier, “Statistics of the two-dimensional ferromagnet. Part I,” Phys. Rev. 60, 252 (1941).
R. Peierls, “On Ising’s model of ferromagnetism,” Proc. Cambridge Philos. Soc. 32, 477 (1936).
B. Widom, “Equation of state in the neighborhood of the critical point,” J. Chem. Phys. 43, 3898 (1965).
K. Huang, Statistical Mechanics, 2nd ed., Wiley (1987), Chapters 14 and 15.
N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller and E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087 (1953).
U. Wolff, “Collective Monte Carlo updating for spin systems,” Phys. Rev. Lett. 62, 361 (1989).
R. J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press (1982).