Ising critical exponents

This article lists the critical exponents of the ferromagnetic transition in the Ising model. In statistical physics, the Ising model describes a continuous phase transition with scalar order parameter. 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.

	d=2	d=3	d=4	general expression
α	0	0.11008(1)	0	$2-d/(d-\Delta _{\epsilon })$
β	1/8	0.326419(3)	1/2	$\Delta _{\sigma }/(d-\Delta _{\epsilon })$
γ	7/4	1.237075(10)	1	$(d-2\Delta _{\sigma })/(d-\Delta _{\epsilon })$
δ	15	4.78984(1)	3	$(d-\Delta _{\sigma })/\Delta _{\sigma }$
η	1/4	0.036298(2)	0	$2\Delta _{\sigma }-d+2$
ν	1	0.629971(4)	1/2	$1/(d-\Delta _{\epsilon })$
ω	2	0.82966(9)	0	$\Delta _{{\epsilon '}}-d$

From the quantum field theory point of view, the critical exponents can be expressed in terms of scaling dimensions of the local operators $\sigma ,\epsilon ,\epsilon '$ of the conformal field theory describing the phase transition ^[1] (In the Ginzburg-Landau description, these are the operators normally called $\phi ,\phi ^{2},\phi ^{4}$.) 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
$\Delta _{\sigma }$	1/8	0.5181489(10) [2]	1
$\Delta _{\epsilon }$	1	1.412625(10) [2]	2
$\Delta _{{\epsilon '}}$	4	3.82966(9) [3]	4

In d=2, the conformal field theory in question is 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. This theory has been traditionally studied by the renormalization group methods and Monte-Carlo simulations. The estimates following from those techniques, as well as references to the original works, can be found in Refs.^[4] and.^[5]

More recently, a conformal field theory method known as the conformal bootstrap has been applied to the d=3 theory.^{[2][3][6][7][8]} This method gives results in agreement with the older techniques, but up to two orders of magnitude more precise. These are the values reported in the table.

More information on critical exponents may be found at SklogWiki

References

1. John Cardy (1996). Scaling and Renormalization in Statistical Physics. Cambridge University Press. ISBN 978-0-521-49959-0.
2. Kos, Filip; Poland, David; Simmons-Duffin, David; Vichi, Alessandro (14 March 2016). "Precision Islands in the Ising and O(N) Models". arXiv:1603.04436 [hep-th].
3. Komargodski, Zohar; Simmons-Duffin, David (14 March 2016). "The Random-Bond Ising Model in 2.01 and 3 Dimensions". arXiv:1603.04444 [hep-th].
4. Pelissetto, Andrea; Vicari, Ettore (2002). "Critical phenomena and renormalization-group theory". Physics Reports. 368 (6): 549–727. arXiv:cond-mat/0012164. Bibcode:2002PhR...368..549P. doi:10.1016/S0370-1573(02)00219-3.
5. Kleinert, H., "Critical exponents from seven-loop strong-coupling φ4 theory in three dimensions". Physical Review D 60, 085001 (1999)
6. El-Showk, Sheer; Paulos, Miguel F.; Poland, David; Rychkov, Slava; Simmons-Duffin, David; Vichi, Alessandro (2014). "Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents". Journal of Statistical Physics. 157 (4–5): 869–914. arXiv:1403.4545. Bibcode:2014JSP...tmp..139E. doi:10.1007/s10955-014-1042-7.
7. Simmons-Duffin, David (2015). "A semidefinite program solver for the conformal bootstrap". Journal of High Energy Physics. 2015 (6): 1–31. arXiv:1502.02033. Bibcode:2015JHEP...06..174S. doi:10.1007/JHEP06(2015)174. ISSN 1029-8479.
8. Kadanoff, Leo P. (April 30, 2014). "Deep Understanding Achieved on the 3d Ising Model". Journal Club for Condensed Matter Physics.

Books

• Kleinert, H. and Schulte-Frohlinde, V.; Critical Properties of φ⁴-Theories, World Scientific (Singapore, 2001); Paperback ISBN 981-02-4658-7 (also available online) (together with V. Schulte-Frohlinde)