Ising Critical Exponents: Scaling Relations

Main result

The six standard critical exponents $\alpha, \beta, \gamma, \delta, \nu, \eta$ of a second-order phase transition are not independent. Given the Widom scaling hypothesis that the singular part of the free-energy density is a generalised homogeneous function of the reduced temperature and magnetic field,

\[f_{\rm sing}(t, h) \;=\; |t|^{2 - \alpha}\,\Phi\!\left(\frac{h}{|t|^{\Delta_{h}}}\right), \tag{0}\]

four exact scaling relations follow:

\[\boxed{\; \begin{aligned} \alpha + 2\beta + \gamma &= 2 & &\text{(Rushbrooke, 1963)}\\ \gamma &= \beta(\delta - 1) & &\text{(Widom, 1965)}\\ \gamma &= \nu(2 - \eta) & &\text{(Fisher, 1969)}\\ 2 - \alpha &= d\,\nu & &\text{(Josephson, 1967; valid for }d \le 4\text{)} \end{aligned} \;}\]

Four relations among six exponents $\Rightarrow$ two independent exponents. For the 2D Ising universality class the two independent exponents can be taken to be the pair $(\Delta_{\sigma}, \Delta_{\varepsilon}) = (1/8, 1)$ of CFT scaling dimensions (see ising-cft-primary-operators), and all six thermodynamic exponents follow:

\[\alpha = 0\ (\log),\quad \beta = \tfrac{1}{8},\quad \gamma = \tfrac{7}{4},\quad \delta = 15,\quad \nu = 1,\quad \eta = \tfrac{1}{4}.\]

All four scaling relations above are satisfied exactly by this tuple — the QAtlas test suite verifies this using Rational{Int} arithmetic (no floating-point slack) in test/standalone/test_universality_exponents.jl.

Setup

Six critical exponents

Define six critical exponents in terms of the singular behaviour of thermodynamic quantities near a second-order phase transition at $T = T_{c}$, with reduced temperature $t = (T - T_{c})/T_{c}$ and applied field $h$:

The full set $\{\alpha, \beta, \gamma, \delta, \nu, \eta\}$ looks a priori independent, but the scaling hypothesis forces four algebraic relations among them.

Widom scaling hypothesis

Hypothesis (Widom 1965). Near $T_{c}$, the singular part of the free-energy density has the generalised homogeneous form

\[f_{\rm sing}(t, h) \;=\; |t|^{2 - \alpha}\, \Phi\!\bigl(h\,|t|^{-\Delta_{h}}\bigr), \tag{1}\]

with $\Phi(\cdot)$ a smooth scaling function and a single gap exponent $\Delta_{h}$ fixed by the two relations we derive below.

The two-parameter scaling (1) is the microscopic statement that $(t, h)$ renormalise as $(b^{y_{t}} t, b^{y_{h}} h)$ under a rescaling $\mathbf{r} \to b\mathbf{r}$, with exponents $y_{t}, y_{h}$ determined by the CFT scaling dimensions of the thermal ($\varepsilon$) and magnetic ($\sigma$) relevant operators:

\[y_{t} = d - \Delta_{\varepsilon},\qquad y_{h} = d - \Delta_{\sigma}. \tag{2}\]

(The scaling dimension of a relevant coupling is $y = d - \Delta_{\rm operator}$ — see ising-cft-primary-operators Step 5.) In 2D Ising, $\Delta_{\varepsilon} = 1$ gives $y_{t} = 1$ and $\Delta_{\sigma} = 1/8$ gives $y_{h} = 15/8$.

The scaling form (1) captures the consequences of this renormalisation-group structure without assuming the specific microscopic operator content.

Goal

Derive the four scaling relations (Rushbrooke, Widom, Fisher, Josephson) from (1) by differentiating the singular free energy and matching the power-law behaviours of the resulting thermodynamic quantities.

Calculation

Step 1 — Consequences of the scaling form (1) for $M, \chi, C$

From the thermodynamic definitions:

Order parameter. $M = -(\partial f/\partial h)_{t}$. At $h = 0$, differentiating (1) once,

\[M(t, 0) \;=\; -\,|t|^{2 - \alpha - \Delta_{h}}\,\Phi'(0)\cdot \mathrm{sgn}(t),\]

vanishing at $t = 0$ and scaling as $|t|^{2 - \alpha - \Delta_{h}}$ for $t < 0$ (below $T_c$). Comparing to the definition $M \sim (-t)^{\beta}$:

\[\boxed{\;\beta \;=\; 2 - \alpha - \Delta_{h}.\;} \tag{3}\]

Susceptibility. $\chi = -(\partial^{2} f/\partial h^{2})_{t}$ at $h = 0$:

\[\chi(t, 0) \;=\; -\,|t|^{2 - \alpha - 2\Delta_{h}}\,\Phi''(0).\]

So

\[|t|^{-\gamma} \;\sim\; |t|^{2 - \alpha - 2\Delta_{h}} \quad\Longrightarrow\quad \boxed{\;\gamma \;=\; 2\Delta_{h} + \alpha - 2.\;} \tag{4}\]

Specific heat. $C = -T(\partial^{2} f/\partial t^{2})_{h}$ at $h = 0$:

\[C(t, 0) \;\sim\; |t|^{-\alpha},\]

matching the definition by construction — the exponent $\alpha$ in (1) is precisely the specific-heat exponent (this is why the $2 - \alpha$ prefactor is used in the scaling form).

Critical isotherm. At $t = 0$, the scaling variable $h|t|^{-\Delta_{h}} \to \infty$, and $\Phi(x) \sim x^{p}$ for some power $p$ as $x \to \infty$. Requiring $f$ to be finite at $t = 0$ fixes $p = (2 - \alpha)/\Delta_{h}$, so

\[f(0, h) \;\sim\; h^{(2 - \alpha)/\Delta_{h}}.\]

Then $M(0, h) = -(\partial f/\partial h)_{t = 0} \sim h^{(2 - \alpha)/\Delta_{h} - 1}$, comparing to $M \sim h^{1/\delta}$:

\[\frac{1}{\delta} \;=\; \frac{2 - \alpha}{\Delta_{h}} - 1 \;=\; \frac{2 - \alpha - \Delta_{h}}{\Delta_{h}} \;=\; \frac{\beta}{\Delta_{h}}, \quad\Longrightarrow\quad \boxed{\;\Delta_{h} = \beta\,\delta.\;} \tag{5}\]

The gap exponent $\Delta_{h}$ is thus determined by the product $\beta\,\delta$, sometimes called the "gap" between the order- parameter and critical-isotherm exponents.

Step 2 — Rushbrooke: $\alpha + 2\beta + \gamma = 2$

Add (3) and (4):

\[\beta + \gamma = (2 - \alpha - \Delta_{h}) + (2\Delta_{h} + \alpha - 2) = \Delta_{h}.\]

Using (5) $\Delta_{h} = \beta\,\delta$:

\[\beta + \gamma \;=\; \beta\,\delta, \quad\Longrightarrow\quad \gamma \;=\; \beta(\delta - 1).\]

That is the Widom relation (we'll return to it). Meanwhile, re-arranging (3) directly gives

\[\Delta_{h} \;=\; 2 - \alpha - \beta.\]

Substitute into (4) $\gamma = 2\Delta_{h} + \alpha - 2 = 2(2 - \alpha - \beta) + \alpha - 2 = 2 - \alpha - 2\beta$, i.e.

\[\boxed{\;\alpha + 2\beta + \gamma \;=\; 2.\;} \tag{RUSHBROOKE}\]

2D Ising check: $0 + 2\cdot\tfrac{1}{8} + \tfrac{7}{4} = 0 + \tfrac{1}{4} + \tfrac{7}{4} = 2$. ✓

Historical note: Rushbrooke 1963 originally proved the inequality $\alpha + 2\beta + \gamma \ge 2$ from thermodynamic stability; the scaling hypothesis saturates the bound to an equality.

Step 3 — Widom: $\gamma = \beta(\delta - 1)$

From the derivation above, the relation $\beta + \gamma = \beta\,\delta$ is just a rewriting of $\gamma = \beta(\delta - 1)$:

\[\boxed{\;\gamma \;=\; \beta(\delta - 1).\;} \tag{WIDOM}\]

2D Ising check: $\tfrac{1}{8}\cdot(15 - 1) = \tfrac{14}{8} = \tfrac{7}{4}$. ✓

Alternative derivation from equation of state. On the critical isotherm $t = 0$, the equation of state reads $M \sim h^{1/\delta}$ from (5). Off the critical isotherm at $t < 0$, integrating the $\chi \sim |t|^{-\gamma}$ over $h \sim 0$ gives $M \sim \chi\,h \sim |t|^{-\gamma}\,h$ for small $h$. Matching the small-$h$ and large-$h$ regimes through the scaling form fixes $\gamma = \beta(\delta - 1)$; the detailed derivation uses the short-distance behaviour of $\Phi$.

Step 4 — Fisher: $\gamma = \nu(2 - \eta)$

The susceptibility is the integrated connected correlator

\[\chi \;=\; \int d^{d}\mathbf{r}\,\bigl\langle s(\mathbf{r})\, s(\mathbf{0})\bigr\rangle_{c}.\]

Near $T_{c}$ at $h = 0$, the correlator has the scaling form

\[\bigl\langle s(\mathbf{r})\,s(\mathbf{0})\bigr\rangle_{c} \;=\; |\mathbf{r}|^{-(d - 2 + \eta)}\,\Psi(|\mathbf{r}|/\xi),\]

with $\Psi$ a cutoff function that decays rapidly for $|\mathbf{r}| \gg \xi$. Substituting and switching to the dimensionless variable $u = |\mathbf{r}|/\xi$,

\[\chi \;=\; \int\,|\mathbf{r}|^{-(d - 2 + \eta)}\,\Psi(|\mathbf{r}|/\xi) \,d^{d}\mathbf{r} \;=\; \xi^{2 - \eta}\int u^{-(d - 2 + \eta)}\,\Psi(u)\, u^{d - 1}\,du\]

\[= \xi^{2 - \eta}\int u^{1 - \eta}\,\Psi(u)\,du \;\sim\; \xi^{2 - \eta}.\]

Using $\xi \sim |t|^{-\nu}$,

\[\chi \sim |t|^{-\nu(2 - \eta)},\]

and comparing to $\chi \sim |t|^{-\gamma}$:

\[\boxed{\;\gamma \;=\; \nu(2 - \eta).\;} \tag{FISHER}\]

2D Ising check: $1\cdot(2 - 1/4) = 7/4$. ✓

Step 5 — Josephson (hyperscaling): $2 - \alpha = d\,\nu$

The singular part of the free-energy density scales as the inverse of the correlation volume times a characteristic energy:

\[f_{\rm sing}(t, 0) \;\sim\; \frac{k_{B} T}{\xi^{d}(t)} \;\sim\; \xi^{-d} \;\sim\; |t|^{d\nu}.\]

This is the "hyperscaling" assumption: each correlation volume contributes order $k_{B}T$ of free energy, and there are $\xi^{-d}$ such volumes per unit volume.

On the other hand, from (1) at $h = 0$ the scaling form gives $f_{\rm sing}(t, 0) \sim |t|^{2 - \alpha}$. Matching the two:

\[|t|^{2 - \alpha} \;\sim\; |t|^{d\nu} \quad\Longrightarrow\quad \boxed{\;2 - \alpha \;=\; d\,\nu.\;} \tag{JOSEPHSON}\]

2D Ising check ($d = 2$): $2 - 0 = 2 \cdot 1 = 2$. ✓

Validity regime. The hyperscaling argument assumes that $\xi^{-d}$ is the only relevant IR cutoff on the free energy. This fails above the upper critical dimension $d_{c} = 4$ (for the Ising universality class), because at $d > 4$ mean-field exponents take over and the "hyperscaling" relation $2 - \alpha = d\nu$ would predict $2 - 0 = d/2$, i.e. $d = 4$ — any higher and the relation breaks. Above $d_{c}$ the mean-field values $\alpha = 0, \beta = 1/2, \gamma = 1, \delta = 3, \nu = 1/2, \eta = 0$ satisfy (RUSHBROOKE), (WIDOM), (FISHER) but violate (JOSEPHSON) in $d > 4$.

Step 6 — All six exponents from $(\Delta_{\sigma}, \Delta_{\varepsilon})$

From ising-cft-primary-operators Step 5, the CFT scaling dimensions give:

\[\eta = 2\Delta_{\sigma} + 2 - d,\qquad \nu = \frac{1}{d - \Delta_{\varepsilon}},\qquad \beta = \nu\,\Delta_{\sigma}.\]

Using Widom and Rushbrooke, the remaining three exponents are

\[\gamma = \beta(\delta - 1),\qquad \alpha = 2 - 2\beta - \gamma,\qquad \delta = \frac{d + 2 - \eta}{d - 2 + \eta}.\]

For 2D Ising ($d = 2, \Delta_{\sigma} = 1/8, \Delta_{\varepsilon} = 1$):

• \[\eta = 2\cdot 1/8 + 2 - 2 = 1/4\]

• \[\nu = 1/(2 - 1) = 1\]

• \[\beta = 1\cdot 1/8 = 1/8\]

• \[\delta = (2 + 2 - 1/4)/(2 - 2 + 1/4) = (15/4)/(1/4) = 15\]

• \[\gamma = (1/8)\cdot 14 = 7/4\]

• \[\alpha = 2 - 2\cdot 1/8 - 7/4 = 2 - 1/4 - 7/4 = 0\]

  (log)

All six 2D Ising exponents are determined by the two CFT scaling dimensions $(1/8, 1)$. This is the minimal-data statement of the 2D Ising universality class.

Step 7 — Algebraic verification (Rational{Int})

The four scaling relations, the six exponents, and their pairwise consistency can all be verified in exact arithmetic — QAtlas does this in test/standalone/test_universality_exponents.jl using Rational{Int}:

using Test

(α, β, γ, δ, ν, η) = (0//1, 1//8, 7//4, 15//1, 1//1, 1//4)
d = 2

@test α + 2β + γ == 2            # Rushbrooke
@test γ == β * (δ - 1)           # Widom
@test γ == ν * (2 - η)           # Fisher
@test 2 - α == d * ν             # Josephson

All four assertions pass exactly (no floating-point tolerance needed). The same test is extended in QAtlas to the other universality classes stored in src/universalities/*.jl (percolation, Potts-$q$, KPZ, $O(N)$ models) with the appropriate exponent tuples.

Step 8 — Limiting-case consistency checks

(i) Mean-field exponents ($d \ge d_{c} = 4$). The Landau-theory values

\[(\alpha, \beta, \gamma, \delta, \nu, \eta)_{\rm MF} \;=\; (0,\, \tfrac{1}{2},\, 1,\, 3,\, \tfrac{1}{2},\, 0)\]

satisfy:

• Rushbrooke: $0 + 1 + 1 = 2$ ✓
• Widom: $1 = \tfrac{1}{2}\cdot 2$ ✓
• Fisher: $1 = \tfrac{1}{2}\cdot 2$ ✓
• Josephson: $2 - 0 = 4\cdot\tfrac{1}{2} = 2$ ✓ at $d = 4$; fails for $d > 4$ (predicts $d\nu = d/2 \ne 2$).

Above $d_{c} = 4$, Josephson requires $d\nu = 2$ with $\nu = 1/2$, so only $d = 4$. For $d > 4$ one must drop the hyperscaling assumption, replacing $\xi^{-d}$ with $\xi^{-d_{c}}$ in the free-energy argument — a classical dangerous irrelevant variable effect (Fisher 1983).

(ii) 3D Ising (numerical bootstrap / Monte Carlo). With $d = 3$ and the conformal-bootstrap exponents (Kos–Poland–Simmons- Duffin 2016, Phys. Rev. D 93, 036404): $\beta = 0.32642$, $\gamma = 1.23708$, $\nu = 0.62997$, $\eta = 0.03631$, $\alpha \approx 0.110$, $\delta \approx 4.79$.

Rushbrooke: $0.110 + 2\cdot 0.32642 + 1.23708 = 1.99992 \approx 2$. ✓ Widom: $0.32642\cdot(4.79 - 1) \approx 1.237$ vs $\gamma = 1.237$. ✓ Fisher: $0.62997\cdot(2 - 0.03631) = 0.62997\cdot 1.96369 = 1.23708$ vs $\gamma = 1.23708$. ✓ Josephson: $2 - 0.110 = 1.890$ vs $3\cdot 0.62997 = 1.890$. ✓

All four relations hold to the accuracy of the bootstrap determination (5–6 significant figures).

(iii) Scaling-relation completeness. Given the four relations above, any two exponents determine the other four. The "canonical" choices are either $(\alpha, \nu)$ (thermodynamic: specific-heat + correlation-length), or $(\Delta_{\sigma}, \Delta_{\varepsilon})$ (CFT: magnetic + thermal scaling dimensions). The latter is more fundamental because $\Delta_{\sigma}, \Delta_{\varepsilon}$ are the defining scaling dimensions of the RG fixed-point operator algebra, whereas $\alpha, \nu$ are derived measurable quantities.

References

• B. Widom, Equation of state in the neighborhood of the critical point, J. Chem. Phys. 43, 3898 (1965). Original scaling-form hypothesis (1) and derivation of (WIDOM).
• G. S. Rushbrooke, On the thermodynamics of the critical region for the Ising problem, J. Chem. Phys. 39, 842 (1963). Thermodynamic-stability inequality $\alpha + 2\beta + \gamma \ge 2$; saturated to equality by scaling.
• M. E. Fisher, Rigorous inequalities for critical-point correlation exponents, Phys. Rev. 180, 594 (1969). Derivation of (FISHER) from integrated correlator.
• B. D. Josephson, Inequality for the specific heat. I. Derivation, Proc. Phys. Soc. 92, 269 (1967) and II. Application to critical phenomena, ibid. 276. Hyperscaling inequality $2 - \alpha \ge d\nu$; saturated to equality by scaling.
• L. P. Kadanoff, W. Götze, D. Hamblen, R. Hecht, E. A. S. Lewis, V. V. Palciauskas, M. Rayl, J. Swift, D. Aspnes, J. Kane, Static phenomena near critical points: theory and experiment, Rev. Mod. Phys. 39, 395 (1967). Comprehensive review.
• M. E. Fisher, Scaling, universality and renormalization group theory, in Critical Phenomena, Springer Lecture Notes in Physics 186 (1983). Upper critical dimension and dangerous irrelevant variables; breakdown of hyperscaling at $d > 4$.
• F. Kos, D. Poland, D. Simmons-Duffin, Precision islands in the Ising and O(N) models, Phys. Rev. D 93, 036404 (2016). Precise 3D Ising exponents from the conformal bootstrap; used for the 3D check in Step 8.
• J. Cardy, Scaling and Renormalization in Statistical Physics, Cambridge University Press (1996), Ch. 3. Textbook treatment of scaling relations from the RG point of view.

Used by

• Ising universality class — the exponent table is stored in src/universalities/Ising2D.jl using Rational{Int} values that satisfy all four scaling relations exactly; similarly for the 3D and mean-field entries.
• ising-cft-primary-operators — provides the two CFT scaling dimensions $(\Delta_{\sigma}, \Delta_{\varepsilon}) = (1/8, 1)$ from which all six thermodynamic exponents are derived in Step 6 above.
• yang-magnetization-toeplitz — microscopic derivation of $\beta = 1/8$ from the Toeplitz determinant, cross-checked against $\beta = \nu\,\Delta_{\sigma}$ here.
• Percolation universality, Potts universality, KPZ universality, O(N) universality, Mean-field universality — same four scaling relations, different exponent tuples; QAtlas verifies each with Rational{Int} arithmetic.