Binder cumulant

2010-06-07T21:59:03Z

134.130.4.242: (universality class)

The '''Binder cumulant''' was introduced by [[Kurt Binder]] in the context of [[Finite size effects |finite size scaling]]. It is a quantity that allows
to locate the critical point and critical exponents. For an [[Ising Models |Ising model]] with zero field, it is given by

:<math>U_4 = 1- \frac{\langle m^4 \rangle }{3\langle m^2 \rangle^2 }</math>

where ''m'' is the [[Order parameters |order parameter]], i.e. the magnetization. It is therefore a fourth order cumulant, related to the kurtosis.
In the [[thermodynamic limit]], where the system size <math>L \rightarrow \infty</math>, <math>U_4 \rightarrow 0</math> for <math>T > T_c</math>, and <math>U_4 \rightarrow 2/3</math> for <math>T < T_c</math>. Thus, the function is discontinuous in this limit. An important observation is that the intersection points of the cumulants for different system sizes usually depend only rather weakly on those sizes, providing a convenient estimate for the value of the [[Critical points |critical temperature]]. Caution is needed in identifying the universality class from
the critical value of the Binder cumulant, because that value depends on boundary condition, system shape, and anisotropy of correlations.

==References==
#[http://dx.doi.org/10.1007/BF01293604 K. Binder "Finite size scaling analysis of ising model block distribution functions", Zeitschrift für Physik B Condensed Matter '''43''' pp. 119-140 (1981)]
[[category: Computer simulation techniques]]

Ising model

2010-06-07T21:39:39Z

134.130.4.242: ANNNI model

The '''Ising model''' <ref>[http://dx.doi.org/10.1007/BF02980577 Ernst Ising "Beitrag zur Theorie des Ferromagnetismus", Zeitschrift für Physik A Hadrons and Nuclei '''31''' pp. 253-258 (1925)]</ref> (also known as the '''Lenz-Ising''' model) is commonly defined over an ordered lattice.
Each site of the lattice can adopt two states, <math>S \in \{-1, +1 \}</math>. Note that sometimes these states are referred to as ''spins'' and the values are referred to as ''down'' and ''up'' respectively.
The energy of the system is the sum of pair interactions
between nearest neighbors.

:<math> \frac{U}{k_B T} = - K \sum_{\langle ij \rangle} S_i S_j </math>

where <math>k_B</math> is the [[Boltzmann constant]], <math>T</math> is the [[temperature]], <math> \langle ij \rangle </math> indicates that the sum is performed over nearest neighbors, and
<math> S_i </math> indicates the state of the i-th site, and <math> K </math> is the coupling constant.

For a detailed and very readable history of the Lenz-Ising model see the following references:<ref>[http://dx.doi.org/10.1103/RevModPhys.39.883 S. G. Brush "History of the Lenz-Ising Model", Reviews of Modern Physics '''39''' pp. 883-893 (1967)]</ref>
<ref>[http://dx.doi.org/10.1007/s00407-004-0088-3 Martin Niss "History of the Lenz-Ising Model 1920-1950: From Ferromagnetic to Cooperative Phenomena", Archive for History of Exact Sciences '''59''' pp. 267-318 (2005)]</ref>
<ref>[http://dx.doi.org/10.1007/s00407-008-0039-5 Martin Niss "History of the Lenz–Ising Model 1950–1965: from irrelevance to relevance", Archive for History of Exact Sciences '''63''' pp. 243-287 (2009)]</ref>.
==1-dimensional Ising model==
:''Main article: [[1-dimensional Ising model]]''
The 1-dimensional Ising model has an exact solution.

==2-dimensional Ising model==
The 2-dimensional [[Building up a square lattice |square lattice]] Ising model was solved by [[Lars Onsager]] in 1944
<ref name="Onsager">[http://dx.doi.org/10.1103/PhysRev.65.117 Lars Onsager "Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition", Physical Review '''65''' pp. 117-149 (1944)]</ref>
<ref>[http://dx.doi.org/10.1103/PhysRev.88.1332 M. Kac and J. C. Ward "A Combinatorial Solution of the Two-Dimensional Ising Model", Physical Review '''88''' pp. 1332-1337 (1952)]</ref>
<ref>Rodney J. Baxter "Exactly Solved Models in Statistical Mechanics", Academic Press (1982) ISBN 0120831821 Chapter 7 (freely available [http://tpsrv.anu.edu.au/Members/baxter/book/Exactly.pdf pdf])</ref>
after [[Rudolf Peierls]] had previously shown that, contrary to the one-dimensional case, the two-dimensional model must have a phase transition
<ref>[http://dx.doi.org/10.1017/S0305004100019174 Rudolf Peierls "On Ising's model of ferromagnetism", Mathematical Proceedings of the Cambridge Philosophical Society '''32''' pp. 477-481 (1936)]</ref> <ref>[http://dx.doi.org/10.1103/PhysRev.136.A437 Robert B. Griffiths "Peierls Proof of Spontaneous Magnetization in a Two-Dimensional Ising Ferromagnet", Physical Review A '''136''' pp. 437-439 (1964)]</ref>.
====Critical temperature====
The [[Critical points | critical temperature]] of the 2D Ising model is given by <ref name="Onsager"> </ref>
:<math>\sinh \left( \frac{2S}{k_BT_c} \right) \sinh \left( \frac{2S'}{k_BT_c} \right) =1</math>
where <math>S</math> is the interaction energy in the <math>(0,1)</math> direction, and <math>S'</math> is the interaction energy in the <math>(1,0)</math> direction.
If these interaction energies are the same one has
:<math>k_BT_c = \frac{2S}{ \operatorname{arcsinh}(1)} \approx 2.269 S</math>

====Critical exponents====
The [[critical exponents]] are as follows:
*Heat capacity exponent <math>\alpha = 0</math> (Baxter Eq. 7.12.12)
*Magnetic order parameter exponent <math>\beta = \frac{1}{8}</math> (Baxter Eq. 7.12.14)
*Susceptibility exponent <math>\gamma = \frac{7}{4} </math> (Baxter Eq. 7.12.15)

==3-dimensional Ising model==
Sorin Istrail has shown that the solution of Ising's model cannot be extended into three dimensions for any lattice
<ref>[http://www.sandia.gov/LabNews/LN04-21-00/sorin_story.html Three-dimensional proof for Ising model impossible, Sandia researcher claims to have shown]</ref>
<ref>[http://dx.doi.org/10.1145/335305.335316 Sorin Istrail "Statistical mechanics, three-dimensionality and NP-completeness: I. Universality of intracatability for the partition function of the Ising model across non-planar surfaces", Proceedings of the thirty-second annual ACM symposium on Theory of computing pp. 87-96 (2000)]</ref>
==ANNNI model==
The '''axial next-nearest neighbour Ising''' (ANNNI) model <ref>[http://dx.doi.org/10.1016/0370-1573(88)90140-8 Walter Selke "The ANNNI model — Theoretical analysis and experimental application", Physics Reports '''170''' pp. 213-264 (1988)]</ref> is used to study spatially modulated structures in alloys, adsorbates, ferroelectrics, magnetic systems, and polytypes.
==See also==
*[[Critical exponents]]
*[[Potts model]]
*[[Mean field models]]

==References==
<references/>
[[Category: Models]]

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Binder cumulant

Ising model