Universality class#Ising model

Critical exponents are defined in terms of the variation of certain physical properties of the system near its phase transition point. These physical properties will include its reduced temperature $\backslash tau$, its order parameter measuring how much of the system is in the "ordered" phase, the specific heat, and so on.

• The exponent $\backslash alpha$ is the exponent relating the specific heat C to the reduced temperature: we have $C\; =\; \backslash tau^\{-\backslash alpha\}$. The specific heat will usually be singular at the critical point, but the minus sign in the definition of $\backslash alpha$ allows it to remain positive.
• The exponent $\backslash beta$ relates the order parameter $\backslash Psi$ to the temperature. Unlike most critical exponents it is assumed positive, since the order parameter will usually be zero at the critical point. So we have $\backslash Psi\; =\; |\backslash tau|^\{\backslash beta\}$.
• The exponent $\backslash gamma$ relates the temperature with the system's response to an external driving force, or source field. We have $d\backslash Psi/dJ\; =\; \backslash tau^\{-\backslash gamma\}$, with J the driving force.
• The exponent $\backslash delta$ relates the order parameter to the source field at the critical temperature, where this relationship becomes nonlinear. We have $J\; =\; \backslash Psi^\backslash delta$ (hence $\backslash Psi\; =\; J^\{1/\backslash delta\}$), with the same meanings as before.
• The exponent $\backslash nu$ relates the size of correlations (i.e. patches of the ordered phase) to the temperature; away from the critical point these are characterized by a correlation length $\backslash xi$. We have $\backslash xi\; =\; \backslash tau^\{-\backslash nu\}$.
• The exponent $\backslash eta$ measures the size of correlations at the critical temperature. It is defined so that the correlation function scales as $r^\{-d+2-\backslash eta\}$.
• The exponent $\backslash sigma$, used in percolation theory, measures the size of the largest clusters (roughly, the largest ordered blocks) at 'temperatures' (connection probabilities) below the critical point. So $s\_\{\backslash max\}\; \backslash sim\; (p\_c\; -\; p)^\{-1/\backslash sigma\}$.
• The exponent $\backslash tau$, also from percolation theory, measures the number of size s clusters far from $s\_\{\backslash max\}$ (or the number of clusters at criticality): $n\_s\; \backslash sim\; s^\{-\backslash tau\}\; f(s/s\_\{\backslash max\})$, with the $f$ factor removed at critical probability.

For symmetries, the group listed gives the symmetry of the order parameter. The group $\backslash mathrm\{Dih\}\_n$ is the dihedral group, the symmetry group of the n-gon, $S\_n$ is the n-element symmetric group, $\backslash mathrm\{Oct\}$ is the octahedral group, and $O(n)$ is the orthogonal group in n dimensions. 1 is the trivial group.

This section 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 $\backslash mathbb\{Z\}\_2$ 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.

From the quantum field theory point of view, the critical exponents can be expressed in terms of scaling dimensions of the local operators $\backslash sigma,\backslash epsilon,\backslash epsilon\text{'}$ of the conformal field theory describing the phase transition{{cite book|first=John |last=Cardy |author-link=John Cardy |title=Scaling and Renormalization in Statistical Physics|url=https://books.google.com/books?id=Wt804S9FjyAC|date=1996|publisher=Cambridge University Press|isbn=978-0-521-49959-0}} (In the Ginzburg–Landau description, these are the operators normally called $\backslash phi,\backslash phi^2,\backslash phi^4$.) 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:

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.{{Cite journal |last1=Kos |first1=Filip |last2=Poland |first2=David |last3=Simmons-Duffin |first3=David |last4=Vichi |first4=Alessandro |date=14 March 2016 |title=Precision Islands in the Ising and O(N) Models |journal=Journal of High Energy Physics |volume=2016 |issue=8 |pages=36 |arxiv=1603.04436 |bibcode=2016JHEP...08..036K |doi=10.1007/JHEP08(2016)036 |s2cid=119230765}}{{Cite journal|last2=Paulos|first2=Miguel F.|last3=Poland|first3=David|last4=Rychkov|first4=Slava|last5=Simmons-Duffin|first5=David|last6=Vichi|first6=Alessandro|date=2014|title=Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents|journal=Journal of Statistical Physics|volume=157|issue=4–5|pages=869–914|doi=10.1007/s10955-014-1042-7|last1=El-Showk|first1=Sheer|arxiv = 1403.4545 |bibcode = 2014JSP...157..869E |s2cid=39692193 }}{{Cite journal|last=Simmons-Duffin|first=David|date=2015|title=A semidefinite program solver for the conformal bootstrap|journal=Journal of High Energy Physics|volume=2015|issue=6|page=174 |doi=10.1007/JHEP06(2015)174|issn=1029-8479|arxiv = 1502.02033 |bibcode = 2015JHEP...06..174S |s2cid=35625559 }}{{cite web

| last = Kadanoff

| first = Leo P.

| author-link = Leo Kadanoff

| title = Deep Understanding Achieved on the 3d Ising Model

| website = Journal Club for Condensed Matter Physics

| date = April 30, 2014

| url = http://www.condmatjournalclub.org/?p=2384

| access-date = July 18, 2015

| archive-url = https://web.archive.org/web/20150722062827/http://www.condmatjournalclub.org/?p=2384

| archive-date = July 22, 2015

| url-status = usurped

}} These are the values reported in the tables. Renormalization group methods,{{cite journal |last=Pelissetto |first=Andrea |author2=Vicari, Ettore |year=2002 |title=Critical phenomena and renormalization-group theory |journal=Physics Reports |volume=368 |pages=549–727 |arxiv=cond-mat/0012164 |bibcode=2002PhR...368..549P |doi=10.1016/S0370-1573(02)00219-3 |s2cid=119081563 |number=6}}Kleinert, H., [http://www.physik.fu-berlin.de/~kleinert/279/279.pdf "Critical exponents from seven-loop strong-coupling φ4 theory in three dimensions".] Physical Review D 60, 085001 (1999){{cite journal |last=Balog |first=Ivan |author2=Chate, Hugues |author3=Delamotte, Bertrand |author4=Marohnic, Maroje |author5=Wschebor, Nicolas |year=2019 |title=Convergence of Non-Perturbative Approximations to the Renormalization Group |journal=Phys. Rev. Lett. |volume=123 |issue=24 |pages=240604 |doi=10.1103/PhysRevLett.123.240604 |pmid=31922817 |arxiv=1907.01829 |bibcode= 2019PhRvL.123x0604B|s2cid=}}{{cite journal |last=De Polsi |first=Gonzalo |author2=Balog, Ivan |author3=Tissier, Matthieu |author4=Wschebor, Nicolas |year=2020 |title=Precision calculation of critical exponents in the O(N) universality classes with the nonperturbative renormalization group |journal=Phys. Rev. E |volume=101 |issue=24 |pages=042113 |doi=10.1103/PhysRevLett.123.240604 |pmid=31922817 |arxiv=1907.01829 |bibcode= 2019PhRvL.123x0604B|s2cid=}} Monte-Carlo simulations,{{Cite journal |last=Hasenbusch |first=Martin |date=2010 |title=Finite size scaling study of lattice models in the three-dimensional Ising universality class |url=https://journals.aps.org/prb/abstract/10.1103/PhysRevB.82.174433 |journal=Physical Review B |volume=82 |issue=17 |page=174433 |doi=10.1103/PhysRevB.82.174433|arxiv=1004.4486 |bibcode=2010PhRvB..82q4433H }} and the fuzzy sphere regulator{{Cite journal |last=Zhu |first=Wei |date=2023 |title=Uncovering Conformal Symmetry in the 3D Ising Transition: State-Operator Correspondence from a Quantum Fuzzy Sphere Regularization |url=https://journals.aps.org/prx/abstract/10.1103/PhysRevX.13.021009 |journal=Physical Review X |volume=13 |issue=2 |page=021009 |doi=10.1103/PhysRevX.13.021009|arxiv=2210.13482 |bibcode=2023PhRvX..13b1009Z }} give results in agreement with the conformal bootstrap, but are several orders of magnitude less accurate.

{{Reflist}}

• [http://www.sklogwiki.org/SklogWiki/index.php/Universality_classes Universality classes] from Sklogwiki
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Category:Critical phenomena

Category:Renormalization group

Category:Scale-invariant systems