2D Ising Correlation Function
The spin-spin correlation functions for the two-dimensional Ising model
is known exactly at zero external field. It is expressed in terms of
integrals of Painlevé functions which, while of fundamental importance
in many fields of physics, are not provided in most software environments. There
are also exact results known at finite field at the critical temperature.
This site provides data, software, and supplemental documentation for
the manuscript XXX. Here we post useful implementations
of exact results and numerical simulations.
Analytic results
A python implementation for the
universal scaling function for the two-dimensional Ising
correlation function at zero magnetic field (both
F+ for T>Tc and
F-, the (disconnected) correlation function for
T<Tc), together with the
particular Painlevé function needed,
Numerical snapshots
To estimate the scaling function at non-zero magnetic fields, we generated
an ensemble of uncorrelated snapshots of the Ising model at various fields and
temperatures (chosen to form a contour of constant 'radius' in Schofield
coordinates), as follows
Numerical correlation functions
From these, we extract correlation functions as a function of radius.
(We average the 2D radii (i,j) into distance bins around the nearest integer to
distance sqrt(i*i+j*j).) The following tgz file
contains all our numerical
correlation functions as txt files with triples (distance, correlation function,
standard deviation):
Corr1024.tgz.
CorrPickle.tgz (Python pickle format)
The names of the files give the temperatures and fields of the simulations.
The error estimates in the above correlation functions are the diagonal
entries of their covariance matrices. The entire covariance matrices, used
for our fits, is given in
CorrVar.tgz (Python pickle format)
Asymptotic correlation length
Here we provide an interpolation between three exact results for the
exponential decay length of the correlation function near the critical
temperature and field of the 2D Ising model.
XXX
Interpolated universal scaling form
The universal scaling form for non-zero field is fit well by a suitable
interpolation between the forms F+ and F-. We generate
this scaling form using an approximate scaling form for the
susceptibility and the correlation length (the exponential decay of the
connected correlation function). These are written as a polynomial
in the Schofield coordinate Θ
(The correlation length is fit to the three exact values at Θ=0, 1, and
Θc ≈ 1.08, corresponding to T>Tc,
T=Tc, and T<Tc.)
One can systematically improve on the universal scaling form for small
distances, larger fields, and farther from the critical temperature by
adding analytic and singular corrections to scaling. For the 2D Ising model,
the leading singular corrections to scaling are believed to vanish.
YJ's Git repository
References
- The Painlevé form for the scaling functions is found in
McCoy and Wu XXX.
- The approximate scalng form for the susceptibility, and the
values of the analytic corrections to scaling, are from
Caselle XXX.
Last Modified: July 22, 2013
This work supported by the Division of Materials Research of the U.S. National
Science Foundation, through grant NSF DMR 1312160.
Statistical Mechanics: Entropy, Order Parameters, and Complexity,
now available at
Oxford University Press
(USA,
Europe).