Critical exponents of the three-dimensional universality classes
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Numbers
class
exponent 
critical exponent
Ising, $O(1)$
$\alpha$, specific heat:
0.11008709 +/- 0.00000036
comment: Derived through the scaling relations from [2]; the radius encloses the input error box.
Ising, $O(1)$
$\beta$, order parameter:
0.326418706 +/- 0.000000078
comment: Derived through the scaling relations from [2]; the radius encloses the input error box.
Ising, $O(1)$
$\gamma$, susceptibility:
1.23707550 +/- 0.00000027
comment: Derived through the scaling relations from [2]; the radius encloses the input error box.
Ising, $O(1)$
$\delta$, critical isotherm:
4.78984254 +/- 0.00000028
comment: Derived through the scaling relations from [2]; the radius encloses the input error box.
Ising, $O(1)$
$\nu$, correlation length:
0.62997097 +/- 0.00000012
comment: Published estimate, [2].
Ising, $O(1)$
$\eta$, anomalous dimension:
0.036297612 +/- 0.000000048
comment: Published estimate, [2].
Ising, $O(1)$
$\omega$, leading spatial correction:
0.82951 +/- 0.00061 †
comment: $\omega=\Delta_{\epsilon^\prime}-3$, with $\Delta_{\epsilon^\prime}=3.82951(61)$ from the conditional bootstrap bounds in [3].
XY, $O(2)$
$\alpha$, specific heat:
-0.01507 +/- 0.00021 †
comment: Derived through the scaling relations from [4]; the radius encloses the input error box.
XY, $O(2)$
$\beta$, order parameter:
0.348641 +/- 0.000064 †
comment: Derived through the scaling relations from [4]; the radius encloses the input error box.
XY, $O(2)$
$\gamma$, susceptibility:
1.31779 +/- 0.00020 †
comment: Derived through the scaling relations from [4]; the radius encloses the input error box.
XY, $O(2)$
$\delta$, critical isotherm:
4.77979 +/- 0.00045 †
comment: Derived through the scaling relations from [4]; the radius encloses the input error box.
XY, $O(2)$
$\nu$, correlation length:
0.67169 +/- 0.00007 †
comment: Published estimate, [4].
XY, $O(2)$
$\eta$, anomalous dimension:
0.03810 +/- 0.00008 †
comment: Published estimate, [4].
XY, $O(2)$
$\omega$, leading spatial correction:
0.789 +/- 0.004 †
comment: Published estimate, [4].
Heisenberg, $O(3)$
$\alpha$, specific heat:
-0.13492 +/- 0.00030 †
comment: Derived through the scaling relations from [5]; the radius encloses the input error box.
Heisenberg, $O(3)$
$\beta$, order parameter:
0.369284 +/- 0.000070 †
comment: Derived through the scaling relations from [5]; the radius encloses the input error box.
Heisenberg, $O(3)$
$\gamma$, susceptibility:
1.39635 +/- 0.00024 †
comment: Derived through the scaling relations from [5]; the radius encloses the input error box.
Heisenberg, $O(3)$
$\delta$, critical isotherm:
4.78124 +/- 0.00029 †
comment: Derived through the scaling relations from [5]; the radius encloses the input error box.
Heisenberg, $O(3)$
$\nu$, correlation length:
0.71164 +/- 0.00010 †
comment: Published estimate, [5].
Heisenberg, $O(3)$
$\eta$, anomalous dimension:
0.03784 +/- 0.00005 †
comment: Published estimate, [5].
Heisenberg, $O(3)$
$\omega$, leading spatial correction:
0.759 +/- 0.002 †
comment: Published estimate, [5].
Self-avoiding walk, $O(N)$ as $N\to0$
$\alpha$, specific heat:
0.2372090 +/- 0.0000012 †
comment: Derived through the scaling relations from [6] and [7]; the radius encloses the input error box.
Self-avoiding walk, $O(N)$ as $N\to0$
$\beta$, order parameter:
0.3029190 +/- 0.0000011
comment: Derived through the scaling relations from [6] and [7]; the radius encloses the input error box.
Self-avoiding walk, $O(N)$ as $N\to0$
$\gamma$, susceptibility:
1.15695300 +/- 0.00000095
comment: Monte Carlo estimate, [7].
Self-avoiding walk, $O(N)$ as $N\to0$
$\delta$, critical isotherm:
4.819348 +/- 0.000017
comment: Derived through the scaling relations from [6] and [7]; the radius encloses the input error box.
Self-avoiding walk, $O(N)$ as $N\to0$
$\nu$, correlation length:
0.58759700 +/- 0.00000040
comment: Monte Carlo estimate, [6].
Self-avoiding walk, $O(N)$ as $N\to0$
$\eta$, anomalous dimension:
0.0310434 +/- 0.0000030 †
comment: Derived through the scaling relations from [6] and [7]; the radius encloses the input error box.
Self-avoiding walk, $O(N)$ as $N\to0$
$\omega$, leading spatial correction:
0.899 +/- 0.015 †
comment: Derived as $\omega=\Delta_1/\nu$ from $\Delta_1=0.528(8)$ and $\nu=0.58759700(40)$ in [6].
Ordinary percolation
$\alpha$, specific heat:
-0.6295 +/- 0.0054 †
comment: Derived through the scaling relations from [8]; the radius encloses the input error box.
Ordinary percolation
$\beta$, order parameter:
0.4181 +/- 0.0011 †
comment: Derived through the scaling relations from [8]; the radius encloses the input error box.
Ordinary percolation
$\gamma$, susceptibility:
1.7933 +/- 0.0041 †
comment: Derived through the scaling relations from [8]; the radius encloses the input error box.
Ordinary percolation
$\delta$, critical isotherm:
5.2894 +/- 0.0023 †
comment: Derived through the scaling relations from [8]; the radius encloses the input error box.
Ordinary percolation
$\nu$, correlation length:
0.8765 +/- 0.0018 †
comment: Published estimate, [8]. The statistical and correction-fit uncertainties are added.
Ordinary percolation
$\eta$, anomalous dimension:
-0.04602 +/- 0.00034 †
comment: Published estimate, [8]. The statistical and correction-fit uncertainties are added.
Ordinary percolation
$\omega$, leading spatial correction:
1.62 +/- 0.13 †
comment: Published estimate, [8]. This is the spatial correction exponent from that study; other determinations show substantial dispersion [9].
Definition
The critical-exponent set describes singular power laws and their leading correction at a three-dimensional continuous transition [1] [17]. Each entry is one component of this set for a specified universality class.
Parameters
class
—   universality class
exponent
—   critical-exponent component
Formulas
(1)
$C_{\mathrm{sing}}\sim |t|^{-\alpha}$, $M(t,0)\sim(-t)^\beta$ as $t\uparrow0$, $\chi\sim|t|^{-\gamma}$, $M(0,h)\sim\operatorname{sgn}(h)|h|^{1/\delta}$, $\xi\sim|t|^{-\nu}$ and $G(r)\sim r^{-(1+\eta)}$ at criticality [1].
(2)
$\alpha+2\beta+\gamma=2$, $\gamma=\beta(\delta-1)$, $\gamma=\nu(2-\eta)$ and $3\nu=2-\alpha$ are the Rushbrooke, Widom, Fisher and hyperscaling relations. Consequently $(1+\eta)\delta=5-\eta$ [1].
(3)
For the Ising class, let $\Delta_\sigma$, $\Delta_\epsilon$ and $\Delta_{\epsilon^\prime}$ be the scaling dimensions of the leading odd scalar, the leading even scalar and the leading irrelevant even scalar primary, respectively. Then $\nu=1/(3-\Delta_\epsilon)$, $\eta=2\Delta_\sigma-1$, $\omega=\Delta_{\epsilon^\prime}-3$, $\beta=\Delta_\sigma/(3-\Delta_\epsilon)$, $\gamma=(3-2\Delta_\sigma)/(3-\Delta_\epsilon)$, $\delta=(3-\Delta_\sigma)/\Delta_\sigma$ and $\alpha=2-3/(3-\Delta_\epsilon)$ [2] [3].
(4)
$c_n\sim A\mu^n n^{\gamma-1}$, $\langle R_n^2\rangle\sim Dn^{2\nu}$, and a correction proportional to $n^{-\Delta_1}$ has $\Delta_1=\omega\nu$, where $A$ and $D$ are nonuniversal amplitudes [6] [7].
Comments
(5)
The classes describe ordinary bulk transitions with short-range interactions in three Euclidean dimensions, in the thermodynamic limit. The isotropic critical fixed points are intended.
(6)
For a thermal transition, $t=(T-T_c)/T_c$ is the reduced temperature, $h$ is the field conjugate to the order parameter $M$, $C_{\mathrm{sing}}$ is the singular part of the specific heat, $\chi=\partial M/\partial h$ is the susceptibility, $\xi$ is the correlation length, and $G(r)$ is the connected order-parameter correlation function at separation $r$. The power laws in (1) define the exponents; $\beta$ refers to the ordered side, and $\gamma$ and $\nu$ may be defined on the disordered side. A negative $\alpha$ describes a finite specific heat with a singular correction, rather than a diverging total specific heat [1].
(7)
The positive exponent $\omega$ describes the leading irrelevant perturbation that preserves the internal symmetry of the critical point: its correction is proportional to $L^{-\omega}$ on length scale $L$, or to $|t|^{\omega\nu}$ in thermal scaling. Corrections caused by lattice anisotropy, boundaries and dynamics are different quantities. The isotropic Heisenberg fixed point is meant here, rather than the nearby cubic fixed point [5].
(8)
For percolation, choose $t=p_c-p$, where $p$ is the occupation probability and the threshold $p_c$ is the critical point. The analogues of $M$, $\chi$ and $G(r)$ are the infinite-cluster density, the mean finite-cluster size and pair connectedness. The ordered side is $p>p_c$, so the $\beta$ law in (1) applies as $p\downarrow p_c$, equivalently $t\uparrow0$ from $t<0$. The singular part of the cluster-number density scales as $|t|^{2-\alpha}$, and a ghost field $h$ defines the critical-isotherm response. The values use the internally consistent $\nu$ and $\eta$ estimates from Table 5 of [8]; the remaining leading exponents follow by scaling. Its $\omega=1.62(13)$ is one finite-size determination: the literature has substantial dispersion and the quoted uncertainty is not a consensus bound [9].
(9)
The self-avoiding-walk class is the dilute $O(N)$ continuation as $N\to0$. If $c_n$ is the number of $n$-step walks from a fixed site and $R_n$ their end-to-end distance, then $\gamma$ and $\nu$ have the meanings in (4), with $\mu$ the connective constant. The other leading exponents are those of this continuation; a thermodynamic magnetisation of a single walk is not being measured. The walk-length correction exponent $\Delta_1$ is distinct from the spatial exponent $\omega$ [6] [7]. Unitarity assumptions used for the Ising, XY and Heisenberg conformal theories are not imposed on this continuation or on percolation.
(10)
The estimates are specific published determinations, not averages over the literature. A directly transcribed value retains its source uncertainty; derived entries carry an outward-rounded image of the input error box. Statistical correlations are not supplied, and these boxes are neither simultaneous confidence regions nor unconditional rigorous bounds. Bootstrap estimates remain conditional on the spectrum assumptions of the cited studies [2] [3].
(11)
The Heisenberg estimate of $\beta$ derived from [5] has a quoted error interval that does not overlap the earlier $0.3689(3)$ determination in [12]. These are different numerical estimates; the source uncertainties are retained without enlargement. The leading exponent set is also compatible with the independent conformal-bootstrap bounds of [16].
References
[1]
A. Pelissetto and E. Vicari, Critical phenomena and renormalization-group theory, Physics Reports 368 (2002), 549–727. (arXiv) (doi)
[2]
C.-H. Chang et al., Bootstrapping the 3d Ising stress tensor, Journal of High Energy Physics 03 (2025), 136. (arXiv) (doi)
[3]
M. Reehorst, Rigorous bounds on irrelevant operators in the 3d Ising model CFT, Journal of High Energy Physics 09 (2022), 177. (arXiv) (doi)
[4]
M. Hasenbusch, Monte Carlo study of an improved clock model in three dimensions, Physical Review B 100 (2019), 224517. (arXiv) (doi)
[5]
M. Hasenbusch, Monte Carlo study of a generalized icosahedral model on the simple cubic lattice, Physical Review B 102 (2020), 024406. (arXiv) (doi)
[6]
N. Clisby and B. Dünweg, High-precision estimate of the hydrodynamic radius for self-avoiding walks, Physical Review E 94 (2016), 052102. (arXiv) (doi)
[7]
N. Clisby, Scale-free Monte Carlo method for calculating the critical exponent gamma of self-avoiding walks, Journal of Physics A: Mathematical and Theoretical 50 (2017), 264003. (arXiv) (doi)
[8]
H. G. Ballesteros, L. A. Fernández, V. Martín-Mayor, A. Muñoz Sudupe, G. Parisi and J. J. Ruiz-Lorenzo, Scaling corrections: site percolation and Ising model in three dimensions, Journal of Physics A: Mathematical and General 32 (1999), 1–13. (arXiv) (doi)
[9]
M. Borinsky, J. A. Gracey, M. V. Kompaniets and O. Schnetz, Five-loop renormalization of phi-cubed theory with applications to the Lee–Yang edge singularity and percolation theory, Physical Review D 103 (2021), 116024. (arXiv) (doi)
[10]
M. Hasenbusch, A finite size scaling study of lattice models in the three-dimensional Ising universality class, Physical Review B 82 (2010), 174433. (arXiv) (doi)
[11]
M. Campostrini, M. Hasenbusch, A. Pelissetto and E. Vicari, The critical exponents of the superfluid transition in helium-4, Physical Review B 74 (2006), 144506. (arXiv) (doi)
[12]
M. Campostrini, M. Hasenbusch, A. Pelissetto, P. Rossi and E. Vicari, Critical exponents and equation of state of the three-dimensional Heisenberg universality class, Physical Review B 65 (2002), 144520. (arXiv) (doi)
[13]
N. Clisby, Accurate estimate of the critical exponent nu for self-avoiding walks via a fast implementation of the pivot algorithm, Physical Review Letters 104 (2010), 055702. (arXiv) (doi)
[14]
M. V. Kompaniets and E. Panzer, Minimally subtracted six-loop renormalization of O(n)-symmetric phi-fourth theory and critical exponents, Physical Review D 96 (2017), 036016. (arXiv) (doi)
[15]
X. Xu, J. Wang, J.-P. Lv and Y. Deng, Simultaneous analysis of three-dimensional percolation models, Frontiers of Physics 9 (2014), 113–119. (arXiv) (doi)
[16]
S. M. Chester, W. Landry, J. Liu, D. Poland, D. Simmons-Duffin, N. Su and A. Vichi, Bootstrapping Heisenberg magnets and their cubic instability, Physical Review D 104 (2021), 105013. (arXiv) (doi)
Links
Similar tables
Critical_exponents_of_the_two-dimensional_universality_classes —   the two-dimensional counterparts of the leading critical exponents
Critical_couplings_of_the_Ising_model_on_lattices —   the lattice-dependent critical points whose neighbourhood the Ising exponents describe
Site_and_bond_percolation_thresholds_of_lattices —   the critical occupation probabilities whose neighbourhood the percolation exponents describe
Connective_constants_of_lattices —   the exponential growth rates that accompany the walk-count exponent
Data properties
Entries are of type: real number
Table is complete: no (it holds all seven components $\alpha,\beta,\gamma,\delta,\nu,\eta,\omega$ for the Ising, XY, Heisenberg, self-avoiding-walk and ordinary percolation classes, which are standard bulk short-range universality classes; other symmetries, multicritical points and disordered classes are not covered)
Sources of data: [2], [3], [4], [5], [6], [7], [8]
How they were obtained:

The generator transcribes published numerical determinations; it does not rerun a Monte Carlo simulation or a conformal-bootstrap optimisation. These estimates are not known to 100 significant digits. The source precision is retained, and every entry states whether it is transcribed or derived.

more
  • Ising: $\nu$ and $\eta$ are from Table 2 of [2], using mixed correlators of the spin, energy and stress-tensor operators. The correction exponent is $\Delta_{\epsilon^\prime}-3$ from Table 1 and equation (4.1) of [3]. That result assumes a sparse scalar spectrum, including no further even scalar primary below dimension 6.
  • XY and Heisenberg: $\nu$, $\eta$ and $\omega$ are finite-size Monte Carlo estimates from the improved clock model in [4] and the generalized icosahedral model in [5], respectively.
  • Self-avoiding walks: $\nu$ and $\Delta_1$ are from the pivot-algorithm simulations in [6]; $\gamma$ is from the independent scale-free simulation in [7].
  • Percolation: Table 5 of [8] gives $\nu=0.8765(16)(2)$ and $\eta=-0.04602(27)(7)$, where the second uncertainty is due to the fitted correction exponent. Their radii are the sums $0.0018$ and $0.00034$. The spatial correction exponent is the study's $\omega=1.62(13)$.

The four remaining leading components are calculated from $(\nu,\eta)$; for walks, $\alpha,\beta,\delta,\eta$ are calculated directly from $(\nu,\gamma)$ and $\omega=\Delta_1/\nu$. Exact rational arithmetic finds the extrema at the corners of each input box. Radii are rounded upward to two significant figures after allowing for rounding of the centre. This preserves numerical containment of the source error boxes, not a physical guarantee that the unknown exponents lie inside them.

The attached checks test a known two-dimensional Ising control and a deliberately inconsistent control, finite widths, all five scaling relations for every class, a common consistent point for each leading-exponent set, and all seven Ising relations with the published scaling dimensions. A separate comparison uses the Heisenberg bootstrap bounds in Table 1 of [16]. Independent literature comparisons cover the Ising Monte Carlo estimates in [10], all six leading XY and Heisenberg components in [11] and [12], and the XY correction exponent. For walks they cover $\nu$ and $\Delta_1$ in [13] and $\gamma$ and $\eta$ from the six-loop resummation in [14]. For percolation they cover the thermal and magnetic eigenvalues in [15]. The older Heisenberg $\beta$ comparison does not overlap and is recorded as a discrepancy; its physical cause is not established here. The Heisenberg and percolation correction exponents are transcribed from their primary sources, without an independent recomputation. The generator and these checks are also run on the stored entries.