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{
"Title": "Scaling dimensions of the 3D Ising CFT",
"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 CITE{Chang2025}.",
"Parameters": {
"operator": {
"type": "Symbolic",
"title": "local primary operator",
"display": "$\\mathcal O$",
"constraints": "A local primary of the three-dimensional Ising conformal field theory; descendants are excluded.",
"values": {
"sigma": "$\\sigma$",
"epsilon": "$\\epsilon$",
"T": "$T$",
"epsilon-prime": "$\\epsilon^\\prime$",
"sigma-prime": "$\\sigma^\\prime$"
}
}
},
"Comments": {
"operators": "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 CITE{Reehorst2022}.",
"normalisation": "Scaling dimensions are unchanged when an operator is multiplied by a nonzero constant. In CITE{two-point}, $x\\ne0$ is a Euclidean separation and $C_{\\mathcal O}>0$ is the two-point normalisation of a Hermitian scalar primary.",
"exponents": "The HREF{T472}[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 $L$.",
"uncertainties": "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."
},
"Formulas": {
"two-point": "$\\langle\\mathcal O(x)\\mathcal O(0)\\rangle=C_{\\mathcal O}|x|^{-2\\Delta_{\\mathcal O}}$ for a scalar primary.",
"stress-tensor": "$\\Delta_T=d=3$ for the conserved stress tensor.",
"dimensions-exponents": "$\\nu=\\frac{1}{3-\\Delta_\\epsilon}$, $\\eta=2\\Delta_\\sigma-1$, $\\omega=\\Delta_{\\epsilon^\\prime}-3$, $\\beta=\\frac{\\Delta_\\sigma}{3-\\Delta_\\epsilon}$, $\\gamma=\\frac{3-2\\Delta_\\sigma}{3-\\Delta_\\epsilon}$, $\\delta=\\frac{3-\\Delta_\\sigma}{\\Delta_\\sigma}$, and $\\alpha=2-\\frac{3}{3-\\Delta_\\epsilon}$.",
"scaling-relations": "$\\alpha+2\\beta+\\gamma=2,\\quad\\gamma=\\beta(\\delta-1),\\quad\\gamma=\\nu(2-\\eta),\\quad2-\\alpha=3\\nu$. These are the Rushbrooke, Widom, Fisher and Josephson scaling relations, respectively.",
"griffiths": "$\\alpha+\\beta(1+\\delta)=2$ is the Griffiths relation, a consequence of the Rushbrooke and Widom relations."
},
"Similar tables": [
{
"table": "HREF{T472}",
"relation": "the three-dimensional bulk critical exponents, related to the Ising scaling dimensions by CITE{dimensions-exponents}"
},
{
"table": "HREF{Critical_exponents_of_the_two-dimensional_universality_classes}",
"relation": "the two-dimensional counterparts of the critical exponents associated with these dimensions"
},
{
"table": "HREF{Critical_couplings_of_the_Ising_model_on_lattices}",
"relation": "the lattice-dependent critical points whose continuum limit gives the Ising conformal field theory"
}
],
"Links": {
"BootstrapData": {
"title": "Primary source: Bootstrapping the 3d Ising stress tensor, with numerical ancillary data",
"url": "https://arxiv.org/abs/2411.15300"
}
},
"References": {
"Chang2025": {
"bib": "C.-H. Chang et al., Bootstrapping the 3d Ising stress tensor, Journal of High Energy Physics 03 (2025), 136.",
"arxiv": "2411.15300",
"doi": "10.1007/JHEP03(2025)136"
},
"Reehorst2022": {
"bib": "M. Reehorst, Rigorous bounds on irrelevant operators in the 3d Ising model CFT, Journal of High Energy Physics 09 (2022), 177.",
"arxiv": "2111.12093",
"doi": "10.1007/JHEP09(2022)177"
},
"Hasenbusch2010": {
"bib": "M. Hasenbusch, A finite size scaling study of lattice models in the three-dimensional Ising universality class, Physical Review B 82 (2010), 174433.",
"arxiv": "1004.4486",
"doi": "10.1103/PhysRevB.82.174433"
},
"SimmonsDuffin2017": {
"bib": "D. Simmons-Duffin, The lightcone bootstrap and the spectrum of the 3d Ising CFT, Journal of High Energy Physics 03 (2017), 086.",
"arxiv": "1612.08471",
"doi": "10.1007/JHEP03(2017)086"
}
},
"Tags": [
"statistical mechanics",
"physics"
],
"Keywords": [
"3D Ising",
"conformal field theory",
"CFT",
"conformal bootstrap",
"scaling dimension",
"spin field",
"energy operator",
"stress tensor",
"irrelevant operators"
],
"Data properties": {
"type": "R",
"rigour": "assumed-bound",
"complete": "no",
"complete-note": "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": [
"CITE{Chang2025}",
"CITE{Reehorst2022}"
],
"rigour details": "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.\n\nFor $\\sigma$ and $\\epsilon$, Table 1 of CITE{Chang2025} gives the bounds at derivative order $\\Lambda=43$ using mixed correlators of $\\sigma$, $\\epsilon$ and $T$; source data are available through CITE{BootstrapData}. 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.\n\nFor $\\epsilon^\\prime$, Table 1 and equation (4.1) of CITE{Reehorst2022} 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 CITE{SimmonsDuffin2017}.\n\nThe 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 CITE{Chang2025}, 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.\n\nAs 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 CITE{Hasenbusch2010}. Both primed dimensions overlap the separate extremal-functional estimates in Table 2 of CITE{SimmonsDuffin2017}; those estimates have nonrigorous uncertainties and share the conformal-bootstrap framework. All seven implied Ising exponents are consistent with HREF{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."
},
"Display properties": {
"number-header": "$\\Delta_{\\mathcal O}$"
},
"Numbers": {
"sigma": {
"number": "0.518148806 +/- 0.000000024",
"comment": "The spin primary, odd under spin flip. Table 1 of CITE{Chang2025}."
},
"epsilon": {
"number": "1.41262528 +/- 0.00000029",
"comment": "The energy primary, even under spin flip. Table 1 of CITE{Chang2025}."
},
"T": {
"number": "3",
"comment": "The conserved stress tensor has spin $2$ and dimension exactly $3$."
},
"epsilon-prime": {
"number": "3.82951 +/- 0.00061",
"comment": "The next nonidentity even scalar primary. Equation (4.1) of CITE{Reehorst2022}, under its even-scalar spectral-gap assumption."
},
"sigma-prime": {
"number": "5.262 +/- 0.089",
"comment": "The next odd scalar primary. Equation (4.2) of CITE{Reehorst2022}, under its odd-scalar spectral-gap assumption."
}
}
}