Scaling dimensions of the 3D O(N) CFTs

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Numbers

class

$\mathcal O$

$\Delta_{\mathcal O}$

XY, $N=2$

$\phi$:

0.519088 +/- 0.000022 †

comment: Conservative bootstrap interval from equation (40) of [1].

XY, $N=2$

$s$:

1.51136 +/- 0.00022 †

comment: Conservative bootstrap interval from equation (41) of [1].

XY, $N=2$

$t$:

1.23629 +/- 0.00011 †

comment: Conservative bootstrap interval from equation (42) of [1].

Heisenberg, $N=3$

$\phi$:

0.518936 +/- 0.000067 †

comment: Conservative bootstrap interval from equation (21) of [2].

Heisenberg, $N=3$

$s$:

1.59488 +/- 0.00081 †

comment: Conservative bootstrap interval from equation (22) of [2].

Heisenberg, $N=3$

$t$:

1.20954 +/- 0.00032 †

comment: Conservative bootstrap interval from equation (23) of [2].

Definition

In the unitary three-dimensional Euclidean $O(N)$ Wilson-Fisher conformal field theory, $\Delta_{\mathcal O}$ is the dilatation eigenvalue of a primary $\mathcal O$ [1] [2]. Each entry selects a component of its spectrum, with the universality class specifying $N$.

Parameters

class

— universality class (The Wilson-Fisher universality class of an $N$-component real vector with $O(N)$ symmetry, for an integer $N\geq2$.)

$\mathcal O$

— primary operator (A nonidentity primary operator in the specified conformal field theory; descendants are excluded.)

Formulas

(1)

$\nu=\frac{1}{3-\Delta_s}$, $\eta=2\Delta_\phi-1$, $\beta=\frac{\Delta_\phi}{3-\Delta_s}$, $\gamma=\frac{3-2\Delta_\phi}{3-\Delta_s}$, $\delta=\frac{3-\Delta_\phi}{\Delta_\phi}$, and $\alpha=2-\frac{3}{3-\Delta_s}$.

(2)

$Y_2=3-\Delta_t$ [5].

(3)

$\alpha+2\beta+\gamma=2$, $\gamma=\beta(\delta-1)$, $\gamma=\nu(2-\eta)$, and $2-\alpha=3\nu$ are the Rushbrooke, Widom, Fisher and Josephson relations, respectively. The Griffiths relation $\alpha+\beta(1+\delta)=2$ follows from the Rushbrooke and Widom relations.

(4)

$\langle\phi_i(x)\phi_j(0)\rangle\propto\delta_{ij}|x|^{-(1+\eta)}$.

Comments

(5)

$\phi_i$ is the lowest-dimension scalar primary in the $O(N)$ vector representation. The energy operator $s$ is the lowest-dimension nonidentity scalar primary in the singlet representation, and $t_{ij}$ is the lowest-dimension scalar primary in the symmetric traceless rank-two representation. The indices $i,j=1,\ldots,N$ refer to the internal symmetry, not spatial spin: all three operators have spatial spin zero. For $O(2)$, these representations have charges $1$, $0^+$ and $2$, respectively [1] [2].

(6)

The theory is the continuum limit of the critical $O(N)$ vector model. The XY and Heisenberg universality classes have $N=2$ and $N=3$, respectively. The $N=1$ member is the Ising conformal field theory. Scaling dimensions do not depend on the normalization of the operators.

(7)

Each radius retains the source's conservative bootstrap uncertainty, rather than the smaller uncertainty assigned to an interpolated boundary. These intervals are conditional on spectral assumptions and numerical bootstrap calculations, and are not statistical confidence intervals. The O(3) abstract transposes the labels $\Delta_s$ and $\Delta_\phi$; Table 1 and equations (21)–(23) specify the assignment used here [2].

(8)

The bulk exponents $\alpha,\beta,\gamma,\delta,\nu,\eta$ describe the specific heat, order parameter, susceptibility, critical isotherm, correlation length and anomalous decay, as defined in the three-dimensional critical-exponent table. The exponent $Y_2$ is the renormalization-group eigenvalue of a source coupled to $t_{ij}$ [5].

References

[1]

S. M. Chester, W. Landry, J. Liu, D. Poland, D. Simmons-Duffin, N. Su and A. Vichi, Carving out OPE space and precise O(2) model critical exponents, Journal of High Energy Physics 06 (2020), 142, equations (40)–(42). (arXiv) (doi)

[2]

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, Table 1 and equations (21)–(23). (arXiv) (doi)

[3]

M. Hasenbusch, Monte Carlo study of an improved clock model in three dimensions, Physical Review B 100 (2019), 224517. (arXiv) (doi)

[4]

M. Hasenbusch, Monte Carlo study of a generalized icosahedral model on the simple cubic lattice, Physical Review B 102 (2020), 024406. (arXiv) (doi)

[5]

M. Hasenbusch and E. Vicari, Anisotropic perturbations in three-dimensional O(N)-symmetric vector models, Physical Review B 84 (2011), 125136, Table II. (arXiv) (doi)

Similar tables

T473 — the operator dimensions of the N=1 member, the three-dimensional Ising conformal field theory

T472 — the three-dimensional bulk critical exponents associated with these operator dimensions

Critical_exponents_of_the_two-dimensional_universality_classes — critical-exponent data in two dimensions for comparison across dimension and universality class

Critical_couplings_of_the_Ising_model_on_lattices — critical points of the N=1 Ising member on lattices, complementing the N=2 and N=3 continuum data here

Data properties

Entries are of type: real number

How well the digits are known: assumed-bound

Table is complete: no (it holds $\Delta_\phi$, $\Delta_s$ and $\Delta_t$ for $N=2$ and $N=3$, the XY and Heisenberg classes, which describe superfluid criticality and isotropic magnets and have precise mixed-correlator bootstrap determinations; higher primaries and other values of $N$ are not tabulated)

Sources of data: [1], [2]

How they were obtained:

The generator transcribes six published intervals. It does not rerun the bootstrap optimization or Monte Carlo simulations. No entry is known to 100 significant digits, and no exact entries are included. Each row identifies its source equation. The conservative dimension bounds at derivative order $\Lambda=43$ are retained, rather than the narrower interpolated estimates.

For $N=2$, [1], section 4.2 and Table 3, assumes that $s,\phi,t$ are the only relevant scalars in their respective representations; other scalars in those representations and charge-four scalars have dimension at least $3$, and charge-three scalars have dimension at least $1$. The conserved current and stress tensor obey Ward identities, with a twist gap of $10^{-6}$ for the remaining spinning operators. For $N=3$, [2], section 4.1, makes the analogous assumptions on $s,\phi,t$, sets the leading rank-four scalar dimension at least $2$, and uses the conserved current, stress tensor, Ward identities and a $10^{-6}$ twist gap.

The assumed bounds adopt the conservative outer regions of these bootstrap studies: convex hulls of excluded points in Delaunay cells that meet the boundary of the allowed region. Their coordinate ranges give the six quoted intervals. Treating these intervals as bounds assumes that the sampled outer regions enclose the physical solution and that finite precision, conformal-block approximations and the finite spin sets do not invalidate the numerical exclusions. The computational settings are specified in appendix B of each paper. The intervals remain conditional on these spectral and numerical assumptions.

Independent checks compare every interval with Monte Carlo determinations: $\eta$ and $\nu$ from [3] and [4] imply $\Delta_\phi=(1+\eta)/2$ and $\Delta_s=3-1/\nu$, while $Y_2$ from [5] implies $\Delta_t=3-Y_2$. All six pairs overlap at the quoted uncertainties. Exact rational controls check the interval transformations. The five scaling relations are verified symbolically, then on the full interval boxes for both classes. The derived leading exponents also overlap the corresponding entries of T472. The checks are repeated on the values read back from the database. These checks establish consistency and transcription, not independent precision at the level of the bootstrap calculation.