Scaling dimensions of the 3D Ising CFT

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Numbers

$\mathcal O$

$\Delta_{\mathcal O}$

$\sigma$:

0.518148806 +/- 0.000000024

comment: The spin primary, odd under spin flip. Table 1 of [1].

$\epsilon$:

1.41262528 +/- 0.00000029

comment: The energy primary, even under spin flip. Table 1 of [1].

$T$:

comment: The conserved stress tensor has spin $2$ and dimension exactly $3$.

$\epsilon^\prime$:

3.82951 +/- 0.00061 †

comment: The next nonidentity even scalar primary. Equation (4.1) of [2], under its even-scalar spectral-gap assumption.

$\sigma^\prime$:

5.262 +/- 0.089 †

comment: The next odd scalar primary. Equation (4.2) of [2], under its odd-scalar spectral-gap assumption.

Definition

The scaling dimension $\Delta_{\mathcal O}$ is the eigenvalue of dilatations of a local primary operator $\mathcal O$ in the unitary three-dimensional Euclidean Ising conformal field theory at its continuum critical point [1].

Parameters

$\mathcal O$

— local primary operator (A local primary of the three-dimensional Ising conformal field theory; descendants are excluded.)

Formulas

(1)

$\langle\mathcal O(x)\mathcal O(0)\rangle=C_{\mathcal O}|x|^{-2\Delta_{\mathcal O}}$ for a scalar primary.

(2)

$\Delta_T=d=3$ for the conserved stress tensor.

(3)

$\nu=\frac{1}{3-\Delta_\epsilon}$, $\eta=2\Delta_\sigma-1$, $\omega=\Delta_{\epsilon^\prime}-3$, $\beta=\frac{\Delta_\sigma}{3-\Delta_\epsilon}$, $\gamma=\frac{3-2\Delta_\sigma}{3-\Delta_\epsilon}$, $\delta=\frac{3-\Delta_\sigma}{\Delta_\sigma}$, and $\alpha=2-\frac{3}{3-\Delta_\epsilon}$.

(4)

$\alpha+2\beta+\gamma=2,\quad\gamma=\beta(\delta-1),\quad\gamma=\nu(2-\eta),\quad2-\alpha=3\nu$. These are the Rushbrooke, Widom, Fisher and Josephson scaling relations, respectively.

(5)

$\alpha+\beta(1+\delta)=2$ is the Griffiths relation, a consequence of the Rushbrooke and Widom relations.

Comments

(6)

All operators selected here are even under spatial parity. The spin field $\sigma$ is the lowest-dimension scalar primary odd under the internal $\mathbb Z_2$ spin-flip symmetry, and the energy field $\epsilon$ is the lowest-dimension nonidentity scalar primary even under that symmetry. The fields $\sigma^\prime$ and $\epsilon^\prime$ are the next scalar primaries in these respective symmetry sectors; a prime does not denote a derivative. The stress tensor $T=T_{\mu\nu}$ has spin $2$ and is even under spin flip [2].

(7)

Scaling dimensions are unchanged when an operator is multiplied by a nonzero constant. In (1), $x\ne0$ is a Euclidean separation and $C_{\mathcal O}>0$ is the two-point normalisation of a Hermitian scalar primary.

(8)

The three-dimensional critical exponents use reduced temperature $t=(T_{\mathrm{temp}}-T_c)/T_c$ and magnetic field $h$: $C\sim|t|^{-\alpha}$ for the singular specific heat, $M\sim(-t)^\beta$ for the spontaneous magnetisation, $\chi\sim|t|^{-\gamma}$ for the susceptibility, $M\sim h^{1/\delta}$ at $t=0$, and $\xi\sim|t|^{-\nu}$ for the correlation length. At criticality, $\langle\sigma(0)\sigma(x)\rangle\sim|x|^{-(1+\eta)}$. The exponent $\omega$ describes the leading spin-flip-even, rotationally invariant irrelevant perturbation and gives corrections proportional to $L^{-\omega}$ at length scale $L$.

(9)

Each interval retains the quoted uncertainty of its source. These are conditional conformal-bootstrap bounds, not experimental standard deviations or unconditional proofs of the Ising spectrum. The bounds for the primed operators were obtained separately and do not specify a joint confidence region. The spectral assumptions are stated in the rigour details.

References

[1]

C.-H. Chang et al., Bootstrapping the 3d Ising stress tensor, Journal of High Energy Physics 03 (2025), 136. (arXiv) (doi)

[2]

M. Reehorst, Rigorous bounds on irrelevant operators in the 3d Ising model CFT, Journal of High Energy Physics 09 (2022), 177. (arXiv) (doi)

[3]

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)

[4]

D. Simmons-Duffin, The lightcone bootstrap and the spectrum of the 3d Ising CFT, Journal of High Energy Physics 03 (2017), 086. (arXiv) (doi)

Links

[5]

Primary source: Bootstrapping the 3d Ising stress tensor, with numerical ancillary data

Similar tables

T472 — the three-dimensional bulk critical exponents, related to the Ising scaling dimensions by (3)

Critical_exponents_of_the_two-dimensional_universality_classes — the two-dimensional counterparts of the critical exponents associated with these dimensions

Critical_couplings_of_the_Ising_model_on_lattices — the lattice-dependent critical points whose continuum limit gives the Ising conformal field theory

Data properties

Entries are of type: real number

How well the digits are known: assumed-bound

Table is complete: no (it holds the two lowest-dimension nonidentity scalar primaries in each spin-flip sector with even spatial parity, and the conserved stress tensor, a standard set governing the leading critical behaviour and its first scalar corrections; the identity, higher primaries and descendants are not tabulated)

Sources of data: [1], [2]

How they were obtained:

The generator transcribes published conformal-bootstrap determinations, retaining each source's centre and stated radius. It does not repeat a bootstrap optimisation. These dimensions are not known to 100 significant digits. The exact exception is $\Delta_T=3$, fixed by conservation of the stress tensor.

For $\sigma$ and $\epsilon$, Table 1 of [1] gives the bounds at derivative order $\Lambda=43$ using mixed correlators of $\sigma$, $\epsilon$ and $T$; source data are available through [5]. Its Table 6 assumes dimension at least $3$ for every other scalar in the sectors probed, dimension at least $4$ for the next spin-flip-even, spatial-parity-even spin-$2$ primary, and twist $\tau=\Delta-\ell\geq1+10^{-6}$ in the remaining sectors, where $\ell$ is spin.

For $\epsilon^\prime$, Table 1 and equation (4.1) of [2] give $3.82951(61)$ at $\Lambda=31$, assuming that the next even scalar after $\epsilon^\prime$ has dimension greater than $6$ and that the next odd scalar after $\sigma$ has dimension greater than $3$. For $\sigma^\prime$, equation (4.2) gives $5.262(89)$ at $\Lambda=19$, assuming that the next odd scalar after $\sigma^\prime$ has dimension greater than $6$ and that the next even scalar after $\epsilon$ has dimension greater than $3$. Both calculations use unitarity and the mixed $\sigma,\epsilon$ crossing equations. The wider conditional bounds are used here in preference to the narrower nonrigorous extremal-functional estimates in [4].

The attached checks independently parse the parenthesised source uncertainties and compare all four transcriptions as exact rational intervals. They test scalar unitarity, the relevant and irrelevant assignments, the exact stress-tensor dimension, the conversion to $\eta$ and $\nu$ against Table 2 of [1], and all five scaling relations in the Formulas section. The interval method is first tested on the exact two-dimensional Ising exponents and on an intentionally inconsistent control.

As independent numerical comparisons, the implied $\nu$, $\eta$ and $\omega$ overlap the finite-size Monte Carlo estimates $0.63002(10)$, $0.03627(10)$ and $0.832(6)$ in [3]. Both primed dimensions overlap the separate extremal-functional estimates in Table 2 of [4]; those estimates have nonrigorous uncertainties and share the conformal-bootstrap framework. All seven implied Ising exponents are consistent with T472; this last comparison shares source literature and is a consistency check, not an independent determination. The same numerical checks are run on the values read back from this draft.