Ising critical exponents explained

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

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.

				general expression
	0	0.11008(1)	0	2- d/(d - \Delta _{\epsilon)}
	1/8	0.326419(3)	1/2	\Delta _{\sigma/(d-\Delta} _{\epsilon)}
	7/4	1.237075(10)	1	(d -2 \Delta _{\sigma)/(d-\Delta} _{\epsilon)}
	15	4.78984(1)	3	(d - \Delta _{\sigma)/\Delta} _{\sigma}
	1/4	0.036298(2)	0	2 \Delta _{\sigma} - d +2
	1	0.629971(4)	1/2	1 /(d - \Delta _{\epsilon)}
	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
\Delta _{\epsilon'}	4	3.82966(9) ^[3] ^[4]	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. 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.^{[5] [6]} and.^{[7] [8]}

More recently, a conformal field theory method known as the conformal bootstrap has been applied to the d=3 theory.^{[2] [9] [10] [11]} 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.

See also

• Universality class
• XY model

Books

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

External links

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

Notes and References

1. Book: John Cardy. Scaling and Renormalization in Statistical Physics. 1996. Cambridge University Press. 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. 1603.04436. 10.1007/JHEP08(2016)036. 2016. 8. 36. Journal of High Energy Physics. 2016JHEP...08..036K. 119230765 .
3. Komargodski. Zohar. Simmons-Duffin. David. 14 March 2016. The Random-Bond Ising Model in 2.01 and 3 Dimensions. 1603.04444. 10.1088/1751-8121/aa6087. 50. 15. Journal of Physics A: Mathematical and Theoretical. 154001. 2017JPhA...50o4001K. 34925106 .
4. Reehorst . Marten . 2022-09-21 . Rigorous bounds on irrelevant operators in the 3d Ising model CFT . Journal of High Energy Physics . 2022 . 9 . 177 . 10.1007/JHEP09(2022)177 . 2111.12093 . 244527272 . 1029-8479.
5. Pelissetto . Andrea . Vicari, Ettore . Critical phenomena and renormalization-group theory . Physics Reports . 368 . 6 . 2002 . 549–727 . 10.1016/S0370-1573(02)00219-3. cond-mat/0012164 . 2002PhR...368..549P . 119081563 .
6. [Hagen Kleinert|Kleinert, H.]
7. Balog . Ivan . Chate, Hugues . Delamotte, Bertrand . Marohnic, Maroje . Wschebor, Nicolas . Convergence of Non-Perturbative Approximations to the Renormalization Group . Phys. Rev. Lett. . 123 . 2019 . 240604. 1907.01829 .
8. De Polsi . Gonzalo . Balog, Ivan . Tissier, Matthieu . Wschebor, Nicolas . Precision calculation of critical exponents in the O(N) universality classes with the nonperturbative renormalization group . Phys. Rev. E . 101 . 2020 . 042113. 1907.01829 .
9. 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. 10.1007/s10955-014-1042-7. El-Showk. Sheer. 1403.4545 . 2014JSP...157..869E . 39692193 .
10. Simmons-Duffin. David. 2015. A semidefinite program solver for the conformal bootstrap. Journal of High Energy Physics. 2015. 6. 174 . 10.1007/JHEP06(2015)174. 1029-8479. 1502.02033 . 2015JHEP...06..174S . 35625559 .
11. Web site: Kadanoff . Leo P. . Deep Understanding Achieved on the 3d Ising Model . Journal Club for Condensed Matter Physics . April 30, 2014 . July 18, 2015 . https://web.archive.org/web/20150722062827/http://www.condmatjournalclub.org/?p=2384 . July 22, 2015 . dead .