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