Ising critical exponents

Template:Short description This article lists the critical exponents of the ferromagnetic transition in the Ising model. In statistical physics, the Ising model is the simplest system exhibiting a continuous phase transition with a scalar order parameter and ${\mathbb{Z}}_2$ symmetry. The critical exponents of the transition are universal values and characterize the singular properties of physical quantities. The ferromagnetic transition of the Ising model establishes an important universality class, which contains a variety of phase transitions as different as ferromagnetism close to the Curie point and critical opalescence of liquid near its critical point.

	Template:Math	Template:Math	Template:Math	general expression
Template:Math	0	0.11008708(35)	0	$2-d/(d-{\mathrm{\Delta }}_{ϵ})$
Template:Math	1/8	0.32641871(75)	1/2	${\mathrm{\Delta }}_{\sigma }/(d-{\mathrm{\Delta }}_{ϵ})$
Template:Math	7/4	1.23707551(26)	1	$(d-2{\mathrm{\Delta }}_{\sigma })/(d-{\mathrm{\Delta }}_{ϵ})$
Template:Math	15	4.78984254(27)	3	$(d-{\mathrm{\Delta }}_{\sigma })/{\mathrm{\Delta }}_{\sigma }$
Template:Math	1/4	0.036297612(48)	0	$2{\mathrm{\Delta }}_{\sigma }-d+2$
Template:Math	1	0.62997097(12)	1/2	$1/(d-{\mathrm{\Delta }}_{ϵ})$
Template:Math	2	0.82966(9)	0	${\mathrm{\Delta }}_{{ϵ}^{\prime }}-d$

From the quantum field theory point of view, the critical exponents can be expressed in terms of scaling dimensions of the local operators $\sigma ,ϵ,{ϵ}^{\prime }$ of the conformal field theory describing the phase transition (In the Ginzburg–Landau description, these are the operators normally called $\varphi ,{\varphi }^2,{\varphi }^4$.) These expressions are given in the last column of the above table, and were used to calculate the values of the critical exponents using the operator dimensions values from the following table:

	d=2	d=3	d=4
${\mathrm{\Delta }}_{\sigma }$	1/8	0.518148806(24)	1
${\mathrm{\Delta }}_{ϵ}$	1	1.41262528(29)	2
${\mathrm{\Delta }}_{{ϵ}^{\prime }}$	4	3.82966(9)	4

In d=2, the two-dimensional critical Ising model's critical exponents can be computed exactly using the minimal model $M_{3,4}$. In d=4, it is the free massless scalar theory (also referred to as mean field theory). These two theories are exactly solved, and the exact solutions give values reported in the table.

The d=3 theory is not yet exactly solved. The most accurate results come from the conformal bootstrap. These are the values reported in the tables. Renormalization group methods, Monte-Carlo simulations, and the fuzzy sphere regulator give results in agreement with the conformal bootstrap, but are several orders of magnitude less accurate.

Based on the numerical conformal bootstrap results, Ning Su conjectured in 2019 that ${\mathrm{\Delta }}_{\sigma }=\frac{\mathrm{\Gamma }(1/6)+4}{\mathrm{\Gamma }(1/24)-5}=0.51814880858\dots$ in d=3. As of 2024, this conjecture is still compatible with the most precise numerical bootstrap results.

See also

• Universality class
• XY model

Books

• Kleinert, H. and Schulte-Frohlinde, V.; Critical Properties of φ⁴-Theories, World Scientific (Singapore, 2001); Paperback Template:ISBN (also available online) (together with V. Schulte-Frohlinde)

External links

• A discussion of critical exponents in general at the Statistical Mechanics Wiki