Critical exponents of the two-dimensional universality classes

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Numbers

class

exponent

value

Ising

$\alpha$, specific heat:

comment: The specific heat diverges logarithmically, $C\propto-\ln|t|$ [1].

Ising

$\beta$, order parameter:

1/8

Ising

$\gamma$, susceptibility:

7/4

Ising

$\delta$, critical isotherm:

Ising

$\nu$, correlation length:

Ising

$\eta$, anomalous dimension:

1/4

3-state Potts

$\alpha$, specific heat:

1/3

3-state Potts

$\beta$, order parameter:

1/9

3-state Potts

$\gamma$, susceptibility:

13/9

3-state Potts

$\delta$, critical isotherm:

3-state Potts

$\nu$, correlation length:

5/6

3-state Potts

$\eta$, anomalous dimension:

4/15

4-state Potts

$\alpha$, specific heat:

2/3

comment: With multiplicative logarithmic corrections, as every 4-state Potts exponent [15].

4-state Potts

$\beta$, order parameter:

1/12

4-state Potts

$\gamma$, susceptibility:

7/6

4-state Potts

$\delta$, critical isotherm:

4-state Potts

$\nu$, correlation length:

2/3

4-state Potts

$\eta$, anomalous dimension:

1/4

percolation

$\alpha$, specific heat:

-2/3

comment: The existence of $\alpha$ is open; the value follows from $\nu=\frac43$ by $2-\alpha=2\nu$ [18].

percolation

$\beta$, order parameter:

5/36

percolation

$\gamma$, susceptibility:

43/18

percolation

$\delta$, critical isotherm:

91/5

percolation

$\nu$, correlation length:

4/3

percolation

$\eta$, anomalous dimension:

5/24

percolation

$\sigma$, cluster cutoff:

36/91

percolation

$\tau$, cluster-size distribution:

187/91

percolation

$d_f$, fractal dimension:

91/48

comment: $d_f=2-\frac{5}{48}$, the fractal dimension of the incipient infinite cluster; $\frac{5}{48}$ is the one-arm exponent [17].

self-avoiding walk

$\alpha$, specific heat:

1/2

self-avoiding walk

$\beta$, order parameter:

5/64

self-avoiding walk

$\gamma$, susceptibility:

43/32

self-avoiding walk

$\delta$, critical isotherm:

91/5

self-avoiding walk

$\nu$, correlation length:

3/4

self-avoiding walk

$\eta$, anomalous dimension:

5/24

self-avoiding walk

$d_f$, fractal dimension:

4/3

comment: $d_f=\frac1\nu$, the fractal dimension of the walk and of the trace of $\mathrm{SLE}_{8/3}$ [19] [20].

Definition

For a universality class of continuous phase transitions in two dimensions [24], with $t=(T-T_c)/T_c$ the reduced temperature and $h$ the field conjugate to the order parameter $M$, the critical exponents are the powers in the asymptotic laws $C\sim|t|^{-\alpha}$ for the specific heat, $M\sim(-t)^{\beta}$, $\chi=\partial M/\partial h\sim|t|^{-\gamma}$, $M\sim h^{1/\delta}$ at $t=0$, $\xi\sim|t|^{-\nu}$ for the correlation length and $\langle\psi(0)\psi(r)\rangle\sim r^{-\eta}$ at $t=0$ for the correlation function of the order parameter. For percolation, with the occupation probability $p$ in place of the temperature, $n_s\sim s^{-\tau}$ is the number of clusters of size $s$ per site at $p_c$, $s_{\max}\sim|p-p_c|^{-1/\sigma}$ the cutoff of the cluster sizes and $M(L)\sim L^{d_f}$ the mass of the critical cluster within distance $L$ of a site; for the self-avoiding walk, with the step fugacity $x$ in place of the temperature, $d_f=1/\nu$ is the fractal dimension of the walk.

Parameters

class

— universality class

exponent

— critical exponent

Formulas

(1)

For the $q$-state Potts class, percolation being $q=1$, $\sqrt q=-2\cos(\pi g)$ with $\frac12\le g\le1$, and $x_T=\frac{3}{2g}-1$, $x_H=\frac{(2g-1)(3-2g)}{8g}$ [6] [7] [8] [10] [11]; $g$ is $\frac23$, $\frac34$, $\frac56$ and $1$ for $q=1,2,3,4$.

(2)

For the dilute $O(n)$ class, the self-avoiding walk being $n=0$, $n=-2\cos(\pi g)$ with $1\le g\le2$, and $x_T=\frac4g-2$, $x_H=\frac{(3g-2)(2-g)}{8g}$ [9] [11]; $g$ is $\frac32$ for the walk and $\frac43$ for $n=1$, where the two families of formulas give the same Ising exponents.

(3)

$\nu=\frac{1}{2-x_T}$, $\alpha=2-2\nu$, $\beta=\nu x_H$, $\gamma=\nu(2-2x_H)$, $\delta=\frac{2-x_H}{x_H}$ and $\eta=2x_H$, the exponents in $d=2$ of a class whose thermal and magnetic operators have scaling dimensions $x_T$ and $x_H$ [21].

(4)

$\alpha+2\beta+\gamma=2$, $\gamma=\beta(\delta-1)$, $\gamma=\nu(2-\eta)$ and $2\nu=2-\alpha$: the scaling relations of Rushbrooke, Widom and Fisher and the hyperscaling relation $d\nu=2-\alpha$ at $d=2$ [25].

(5)

$d_f=2-\frac{\beta}{\nu}$, $\tau=1+\frac{2}{d_f}=2+\frac{\beta}{\beta+\gamma}$ and $\sigma=\frac{1}{\nu d_f}=\frac{1}{\beta+\gamma}$ for percolation [26], and $d_f=\frac1\nu$ for the self-avoiding walk.

(6)

$\kappa=\frac4g$, and the curves of $\mathrm{SLE}_\kappa$ have Hausdorff dimension $1+\frac{\kappa}{8}$ [20]: $\frac43$ for the self-avoiding walk, which is its $d_f$, and $\frac74$ for the hull of a percolation cluster.

Comments

(7)

The exponents are those of the standard definitions [25]: $\alpha$ is negative where the specific heat stays finite at $T_c$ and only a derivative diverges, as for percolation, and $\alpha=0$ for the Ising model stands for a logarithmic divergence, $C\propto-\ln|t|$; $\delta$ is defined on the critical isotherm by $h\propto M^{\delta}$; $\eta$ by the decay $r^{-(d-2+\eta)}$ of the order-parameter correlation function at $T_c$, which in $d=2$ is $r^{-\eta}$. For every class here the exponents are the same on both sides of the transition, and hyperscaling holds, so that only two of the six are independent (4): the six exponents of a class are determined by the scaling dimensions $x_T$ of its thermal and $x_H$ of its magnetic operator (3), and the class is characterised by its Coulomb-gas coupling $g$ [11] (1) (2), or equivalently by the parameter $\kappa=4/g$ of the Schramm–Loewner evolution that describes its interfaces [21] (6).

(8)

The Ising class is that of the Ising model [27], whose critical couplings on lattices are in the table of Ising critical couplings; it is the $q=2$ Potts model and the $n=1$ member of the $O(n)$ family, and its exponents are those of the minimal model $M(4,3)$ of conformal field theory [13] [14], with $x_T=2h_{2,1}=1$ and $x_H=2h_{1,2}=\frac18$. The $q$-state Potts model [28] has a continuous transition for $q\le4$ [4] [5], the 3-state class being the minimal model $M(6,5)$ [12] with $x_T=2h_{2,1}=\frac45$ and $x_H=2h_{3,3}=\frac{2}{15}$, and the 4-state class the endpoint $q=4$, where every power law carries a multiplicative logarithmic correction [15]; the 4-state Potts point is also a point of the Ashkin–Teller line. Percolation is the $q\to1$ limit of the Potts model in its Fortuin–Kasteleyn cluster form [26], with $p$ in place of the temperature; its thresholds on lattices are in the table of percolation thresholds. The self-avoiding walk [29] is the $n\to0$ limit of the $O(n)$ model, with the step fugacity $x$ in place of the temperature and the critical point at $x_c=1/\mu$, $\mu$ the connective constant of the lattice. The interfaces of the five classes are described by $\mathrm{SLE}_\kappa$ with $\kappa=3$ for the Ising spin clusters and $\frac{16}{3}$ for its Fortuin–Kasteleyn clusters, $\frac{24}{5}$ for the 3-state Potts clusters, $4$ for the 4-state Potts clusters, $6$ for percolation and $\frac83$ for the self-avoiding walk [21].

(9)

For percolation [26] the order parameter is the strength $P_\infty\sim(p-p_c)^{\beta}$ of the infinite cluster, the susceptibility is the mean cluster size $S\sim|p-p_c|^{-\gamma}$, $\alpha$ is defined by the singular part $|p-p_c|^{2-\alpha}$ of the number of clusters per site, $\delta$ by the response $P_\infty\sim h^{1/\delta}$ to a ghost field $h$ at $p_c$, $\nu$ by the correlation length and $\eta$ by the probability $r^{-\eta}$ that two sites at distance $r$ are connected at $p_c$. The number of clusters of size $s$ per site at $p_c$ is $n_s\sim s^{-\tau}$, the cutoff of the cluster sizes away from $p_c$ is $s_{\max}\sim|p-p_c|^{-1/\sigma}$, and the mass of the incipient infinite cluster within a distance $L$ of a site is $M(L)\sim L^{d_f}$, so that $d_f=2-\frac{5}{48}$ with $\frac{5}{48}$ the exponent of the probability that a site is connected to distance $L$, the one-arm exponent [17]. The related dimensions of the cluster's hull, $\frac74$, of its external perimeter, $\frac43$, and of its backbone are not here.

(10)

For the self-avoiding walk [29] the exponents describe the number $c_n\sim A\mu^nn^{\gamma-1}$ of $n$-step walks from a site and their mean squared end-to-end distance $\langle R_n^2\rangle\sim Dn^{2\nu}$, and $d_f=1/\nu$ is the fractal dimension of the walk, equal to the Hausdorff dimension $\frac43$ of the trace of $\mathrm{SLE}_{8/3}$ [20]. $\alpha$, $\delta$ and $\eta$ are the exponents of the $O(n)$ model at $n\to0$; $\alpha$ also counts the $n$-step self-avoiding polygons, $p_n\sim B\mu^nn^{\alpha-3}$, and $\eta=2x_H$ where $x_H=\frac{5}{48}$ is the dimension of the one-leg operator, the same value as the one-arm exponent of percolation.

(11)

The Ising exponents are theorems: $\alpha=0$ and $\nu=1$ from Onsager's free energy [1], $\beta=\frac18$ from Yang's spontaneous magnetisation [2], $\eta=\frac14$ and $\gamma=\frac74$ from the correlation functions [3], and $\delta=15$ from the two-sided bound $\langle\sigma_0\rangle_{\beta_c,h}\asymp h^{1/15}$ of Camia, Garban and Newman on the magnetisation in a field [23], proved from the Griffiths–Hurst–Sherman inequality and the Russo–Seymour–Welsh theorem of the random-cluster model together with Wu's asymptotics of the two-point function. For percolation, $\beta$, $\gamma$, $\nu$ and $\eta$ are theorems for site percolation on the triangular lattice [18], combining Kesten's scaling relations [16], the one-arm and four-arm exponents of the Schramm–Loewner evolution [17] and Smirnov's proof of Cardy's formula; the existence of $\alpha$ is open, and $\delta$, $\sigma$, $\tau$ and $d_f$ follow from the four by the scaling relations. The Potts exponents were conjectured from the Coulomb gas, the thermal ones by den Nijs [6] and the magnetic ones by Nienhuis, Riedel and Schick [7] and by Pearson [8], derived by Nienhuis [10] and confirmed by conformal field theory [12]; they are accepted as exact and are not theorems. The self-avoiding-walk exponents are Nienhuis's conjecture [9] and would follow from the scaling limit of the walk being $\mathrm{SLE}_{8/3}$ [19]; none of the walk's exponents is a theorem, and the one exact critical quantity proved on this branch is the connective constant $\mu=\sqrt{2+\sqrt2}$ of the honeycomb lattice [22].

Programs

(P1)

Sage

g = 2/3                                                   # percolation: sqrt(q) = -2 cos(pi g) at q = 1
xT, xH = 3/(2*g) - 1, (2*g - 1)*(3 - 2*g)/(8*g)
nu = 1/(2 - xT)
[2 - 2*nu, nu*xH, nu*(2 - 2*xH), (2 - xH)/xH, nu, 2*xH]   # alpha, beta, gamma, delta, nu, eta
[1/(nu*(2 - xH)), 1 + 2/(2 - xH), 2 - xH]                 # sigma, tau, d_f

References

[1]

L. Onsager, Crystal statistics. I. A two-dimensional model with an order-disorder transition, Physical Review 65 (1944), 117–149.

[2]

C. N. Yang, The spontaneous magnetization of a two-dimensional Ising model, Physical Review 85 (1952), 808–816.

[3]

T. T. Wu, B. M. McCoy, C. A. Tracy and E. Barouch, Spin-spin correlation functions for the two-dimensional Ising model: exact theory in the scaling region, Physical Review B 13 (1976), 316–374.

[4]

R. J. Baxter, Potts model at the critical temperature, Journal of Physics C 6 (1973), L445–L448.

[5]

F. Y. Wu, The Potts model, Reviews of Modern Physics 54 (1982), 235–268.

[6]

M. P. M. den Nijs, A relation between the temperature exponents of the eight-vertex and q-state Potts model, Journal of Physics A 12 (1979), 1857–1868.

[7]

B. Nienhuis, E. K. Riedel and M. Schick, Magnetic exponents of the two-dimensional q-state Potts model, Journal of Physics A 13 (1980), L189–L192.

[8]

R. B. Pearson, Conjecture for the extended Potts model magnetic eigenvalue, Physical Review B 22 (1980), 2579–2580.

[9]

B. Nienhuis, Exact critical point and critical exponents of O(n) models in two dimensions, Physical Review Letters 49 (1982), 1062–1065.

[10]

B. Nienhuis, Critical behavior of two-dimensional spin models and charge asymmetry in the Coulomb gas, Journal of Statistical Physics 34 (1984), 731–761.

[11]

B. Nienhuis, Coulomb gas formulation of two-dimensional phase transitions, in Phase Transitions and Critical Phenomena, volume 11, eds C. Domb and J. L. Lebowitz, Academic Press, London, 1987, 1–53.

[12]

Vl. S. Dotsenko, Critical behaviour and associated conformal algebra of the Z3 Potts model, Nuclear Physics B 235 (1984), 54–74.

[13]

A. A. Belavin, A. M. Polyakov and A. B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nuclear Physics B 241 (1984), 333–380.

[14]

D. Friedan, Z. Qiu and S. Shenker, Conformal invariance, unitarity, and critical exponents in two dimensions, Physical Review Letters 52 (1984), 1575–1578.

[15]

J. Salas and A. D. Sokal, Logarithmic corrections and finite-size scaling in the two-dimensional 4-state Potts model, Journal of Statistical Physics 88 (1997), 567–615. (arXiv)

[16]

H. Kesten, Scaling relations for 2D-percolation, Communications in Mathematical Physics 109 (1987), 109–156.

[17]

G. F. Lawler, O. Schramm and W. Werner, One-arm exponent for critical 2D percolation, Electronic Journal of Probability 7 (2002), paper 2, 1–13. (arXiv)

[18]

S. Smirnov and W. Werner, Critical exponents for two-dimensional percolation, Mathematical Research Letters 8 (2001), 729–744. (arXiv)

[19]

G. F. Lawler, O. Schramm and W. Werner, On the scaling limit of planar self-avoiding walk, in Fractal Geometry and Applications: a Jubilee of Benoît Mandelbrot, Part 2, Proceedings of Symposia in Pure Mathematics 72, American Mathematical Society, Providence, 2004, 339–364. (arXiv)

[20]

V. Beffara, The dimension of the SLE curves, Annals of Probability 36 (2008), 1421–1452. (arXiv)

[21]

J. Cardy, SLE for theoretical physicists, Annals of Physics 318 (2005), 81–118. (arXiv)

[22]

H. Duminil-Copin and S. Smirnov, The connective constant of the honeycomb lattice equals $\sqrt{2+\sqrt2}$, Annals of Mathematics 175 (2012), 1653–1665. (arXiv)

[23]

F. Camia, C. Garban and C. M. Newman, The Ising magnetization exponent on $\mathbb{Z}^2$ is $1/15$, Probability Theory and Related Fields 160 (2014), 175–187. (arXiv)

Links

[24]

Wikipedia: Universality class

[25]

Wikipedia: Critical exponent

[26]

Wikipedia: Percolation critical exponents

[27]

Wikipedia: Ising model

[28]

Wikipedia: Potts model

[29]

Wikipedia: Self-avoiding walk

Similar tables

Site and bond percolation thresholds of lattices — the critical points whose neighbourhood the percolation exponents describe

Critical couplings of the Ising model on lattices — the critical points of the Ising class on lattices

Hausdorff dimension of some fractals — the fractal dimensions here, $\frac{91}{48}$ and $\frac43$, are Hausdorff dimensions of random fractals, the incipient percolation cluster and the trace of $\mathrm{SLE}_{8/3}$

Rational numbers — holds the small rationals $0$, $1$, $\frac12$, $\frac14$, $\frac13$, $\frac23$, $-\frac23$, $\frac34$ and $\frac43$ that are also exponents here

Data properties

Entries are of type: rational number

How well the digits are known: exact

Sources of data: [11], [24]

Table is complete: no (it holds $\alpha$, $\beta$, $\gamma$, $\delta$, $\nu$ and $\eta$ for the Ising, 3-state Potts, 4-state Potts, percolation and self-avoiding-walk classes, with $\sigma$, $\tau$ and $d_f$ for percolation and $d_f$ for the walk; the Potts classes at other $q\le4$, the $O(n)$ classes at other $n$ and the tricritical classes are not here, nor are the three-dimensional classes, whose exponents are not known exactly)

How they were obtained:

Every entry is a rational computed from the Coulomb-gas coupling $g$ of its class, $\frac34$, $\frac56$, $1$ and $\frac23$ on the Potts branch and $\frac32$ on the $O(n)$ branch, through the scaling dimensions $x_T$ and $x_H$ and the relations in $d=2$, every division between Sage rationals; the scaling relations of Rushbrooke, Widom, Fisher and Josephson are asserted on every class before a value is returned, the percolation relations for $\sigma$, $\tau$ and $d_f$ on that class, and the Ising exponents are required to come out the same from $q=2$ and from $n=1$.

Exactness is not the same as proof: the Ising values and the percolation values of $\beta$, $\gamma$, $\nu$ and $\eta$ are theorems, and the 3-state and 4-state Potts values, the self-avoiding-walk values and the percolation $\alpha$ are exact conjectures of the Coulomb gas and conformal field theory, universally accepted and confirmed numerically. Before any entry was written, all 30 exponents $\alpha$ to $\eta$ were compared with the table on Wikipedia's page on universality classes and $\sigma$, $\tau$, $d_f$ with its page on percolation exponents; the thermal and magnetic dimensions of the Ising and 3-state Potts classes with the Kac formula of the minimal models, $h_{2,1}=\frac12$ and $h_{1,2}=\frac{1}{16}$ for $M(4,3)$ and $h_{2,1}=\frac25$ and $h_{3,3}=\frac{1}{15}$ for $M(6,5)$, and the thermal dimensions of percolation and of the walk with $h_{2,1}=\frac58$ and $h_{1,3}=\frac13$ at $c=0$, a computation sharing nothing with the Coulomb gas; the percolation values with the theorem of Smirnov and Werner and the one-arm exponent of Lawler, Schramm and Werner; $d_f=\frac{91}{48}$ with a Monte Carlo measurement of the largest cluster of site percolation on the triangular lattice at $p_c=\frac12$, which gave $1.88$ between $L=128$ and $L=2048$ with a control at $p=0.65$ returning $2.00$, and $\tau$ with the cluster-size distribution there, which gave $1.98$ to $2.01$ depending on the range fitted; and $\gamma=\frac{43}{32}$ and $\nu=\frac34$ of the self-avoiding walk with an exact enumeration of square-lattice walks to 16 steps, whose ratio estimates $1.340$ and $0.730$ rise towards the values, the ordinary random walk as a control returning exactly $1$ and $\frac12$.