Ising critical exponents

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 ${\displaystyle \mathbb {Z} _{2}}$ 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.

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

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

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". Journal of High Energy Physics. 2016 (8): 36. arXiv: 1603.04436 . Bibcode:2016JHEP...08..036K. doi:10.1007/JHEP08(2016)036. S2CID   119230765.
3. Komargodski, Zohar; Simmons-Duffin, David (14 March 2016). "The Random-Bond Ising Model in 2.01 and 3 Dimensions". Journal of Physics A: Mathematical and Theoretical. 50 (15): 154001. arXiv: 1603.04444 . Bibcode:2017JPhA...50o4001K. doi:10.1088/1751-8121/aa6087. S2CID   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. arXiv: 2111.12093 . doi:10.1007/JHEP09(2022)177. ISSN   1029-8479. S2CID   244527272.
5. 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. S2CID   119081563.
6. Kleinert, H., "Critical exponents from seven-loop strong-coupling φ4 theory in three dimensions". Physical Review D 60, 085001 (1999)
7. Balog, Ivan; Chate, Hugues; Delamotte, Bertrand; Marohnic, Maroje; Wschebor, Nicolas (2019). "Convergence of Non-Perturbative Approximations to the Renormalization Group". Phys. Rev. Lett. 123: 240604. arXiv: 1907.01829 .
8. De Polsi, Gonzalo; Balog, Ivan; Tissier, Matthieu; Wschebor, Nicolas (2020). "Precision calculation of critical exponents in the O(N) universality classes with the nonperturbative renormalization group". Phys. Rev. E. 101: 042113. arXiv: 1907.01829 .
9. 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...157..869E. doi:10.1007/s10955-014-1042-7. S2CID   39692193.
10. Simmons-Duffin, David (2015). "A semidefinite program solver for the conformal bootstrap". Journal of High Energy Physics. 2015 (6): 174. arXiv: 1502.02033 . Bibcode:2015JHEP...06..174S. doi:10.1007/JHEP06(2015)174. ISSN   1029-8479. S2CID   35625559.
11. Kadanoff, Leo P. (April 30, 2014). "Deep Understanding Achieved on the 3d Ising Model". Journal Club for Condensed Matter Physics. Archived from the original on July 22, 2015. Retrieved July 18, 2015.

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)

External links

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