Universality classes: Difference between revisions

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(→‎XY: Added the 3D XY model critical exponents)
(Added more data to the table)
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{{Stub-general}}
'''Universality classes''' are groups of [[Idealised models | models]] that have the same set of [[critical exponents]]
{| border="1"
 
:{| border="1"
|-  
|-  
| <math>d</math> || <math>n</math> || <math>\sigma</math> || <math>\alpha</math> || <math>\beta</math> || <math>\gamma</math> || class
| dimension ||<math>\alpha</math> || <math>\beta</math> || <math>\gamma</math> || <math>\delta</math> ||<math>\nu</math> || <math>\eta</math> || class
|-  
|-  
|  ||  || || ||   ||   || 3-state Potts
|  ||    ||  || || || || || 3-state Potts
|-  
|-  
|  ||  || || ||   ||   ||Ashkin-Teller
|  ||  ||   || || || || ||Ashkin-Teller
|-  
|-  
|  ||   ||   || ||   || ||Chiral
|  || ||    || || || || ||Chiral
|-  
|-  
|  ||  || || ||   ||   ||Directed percolation
|  ||  ||   || || || || ||Directed percolation
|-  
|-  
| ||  ||  || 0 || <math>1/8</math>  || <math>7/4</math>  || 2D Ising
| 2 ||  0 || <math>1/8</math>  || <math>7/4</math> || || 1  || 1/4 || 2D Ising
|-  
|-  
| ||  ||  || 0 || <math>1/8</math> || <math>7/4</math>  || 3D Ising
| 3 ||  0.1096(5)  || 0.32653(10) || 1.2373(2)    || 4.7893(8) || 0.63012(16) || 0.03639(15) || 3D Ising
|-  
|-  
|  ||   || || ||   ||  ||Local linear interface
|  ||   ||    || || || ||  ||Local linear interface
|-  
|-  
|  ||  ||  ||0 || <math>1/2</math>  || 1  ||Mean-field
|  ||  0 || <math>1/2</math>  || 1 ||  || || || Mean-field
|-  
|-  
|  ||   ||   || ||   ||  ||Molecular beam epitaxy
|  || ||    || || || ||  ||Molecular beam epitaxy
|-  
|-  
|  ||  ||  || ||   ||   ||Random-field
|  ||  ||  ||  || || || ||Random-field
|-  
|-  
|  ||  ||  ||  ||   ||  || XY
| 3 ||  −0.0146(8) || 0.3485(2) ||  1.3177(5) || 4.780(2) ||0.67155(27) || 0.0380(4) || XY
|}
|}
where
*<math>\alpha</math>  is known as  the [[Critical exponents#Heat capacity exponent| heat capacity exponent]] 
*<math>\beta</math>  is known as the  [[Critical exponents#Magnetic order parameter exponent | magnetic order parameter exponent]]
*<math>\gamma</math> is known as  the [[Critical exponents#Susceptibility exponent |susceptibility exponent ]]
*<math>\nu</math> is known as the [[Critical exponents#Correlation length | correlation length]]
*<math>\eta</math> is known as...
==3-state Potts==
==3-state Potts==
==Ashkin-Teller==
==Ashkin-Teller==
Line 64: Line 73:
\delta=15
\delta=15
</math>
</math>
along with <ref>[http://dx.doi.org/10.1103/PhysRev.180.594 Michael E. Fisher "Rigorous Inequalities for Critical-Point Correlation Exponents", Physical Review '''180''' pp. 594-600 (1969)]</ref>:
:<math>
\nu=1
</math>
:<math>
\eta = 1/4
</math>


In three dimensions, the critical exponents are not known exactly. However, [[Monte Carlo | Monte Carlo simulations]] and  [[Renormalisation group]] analysis provide accurate estimates <ref name="Campostrini2002">[http://dx.doi.org/10.1103/PhysRevE.65.066127 Massimo Campostrini, Andrea Pelissetto, Paolo Rossi, and Ettore Vicari "25th-order high-temperature expansion results for three-dimensional Ising-like systems on the simple-cubic lattice", Physical Review E '''65''' 066127 (2002)]</ref>:
In three dimensions, the critical exponents are not known exactly. However, [[Monte Carlo | Monte Carlo simulations]] and  [[Renormalisation group]] analysis provide accurate estimates <ref name="Campostrini2002">[http://dx.doi.org/10.1103/PhysRevE.65.066127 Massimo Campostrini, Andrea Pelissetto, Paolo Rossi, and Ettore Vicari "25th-order high-temperature expansion results for three-dimensional Ising-like systems on the simple-cubic lattice", Physical Review E '''65''' 066127 (2002)]</ref>:
Line 102: Line 122:
====Susceptibility exponent: <math>\gamma</math>====
====Susceptibility exponent: <math>\gamma</math>====
(final result: <math>\gamma=1</math>)
(final result: <math>\gamma=1</math>)
==Molecular beam epitaxy==
==Molecular beam epitaxy==
==Random-field==
==Random-field==

Revision as of 12:17, 26 July 2011

Universality classes are groups of models that have the same set of critical exponents

dimension Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \alpha } Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \eta} class
3-state Potts
Ashkin-Teller
Chiral
Directed percolation
2 0 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/8} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 7/4} 1 1/4 2D Ising
3 0.1096(5) 0.32653(10) 1.2373(2) 4.7893(8) 0.63012(16) 0.03639(15) 3D Ising
Local linear interface
0 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/2} 1 Mean-field
Molecular beam epitaxy
Random-field
3 −0.0146(8) 0.3485(2) 1.3177(5) 4.780(2) 0.67155(27) 0.0380(4) XY

where

  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} is known as the heat capacity exponent
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta} is known as the magnetic order parameter exponent
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma} is known as the susceptibility exponent
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu} is known as the correlation length
  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \eta} is known as...

3-state Potts

Ashkin-Teller

Chiral

Directed percolation

Ising

The Hamiltonian of the Ising model is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H=\sum_{<i,j>}S_i S_j }


where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_i=\pm 1} and the summation runs over the lattice sites.

The order parameter is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m=\sum_i S_i }

In two dimensions, Onsager obtained the exact solution in the absence of a external field, and the critical exponents are

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha=0 }

(In fact, the specific heat diverges logarithmically with the critical temperature)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta=\frac{1}{8} }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma=\frac{7}{4} }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta=15 }

along with [1]:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu=1 }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \eta = 1/4 }


In three dimensions, the critical exponents are not known exactly. However, Monte Carlo simulations and Renormalisation group analysis provide accurate estimates [2]:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu=0.63012(16) }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha=0.1096(5) }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta= 0.32653(10) }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma=1.2373(2) }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta=4.7893(8) }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \eta =0.03639(15) }

with a critical temperature of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_BT_c = 4.51152786~S } [3]. In four and higher dimensions, the critical exponents are mean-field with logarithmic corrections.

Local linear interface

Mean-field

The critical exponents of are derived as follows [4]:

Heat capacity exponent: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha}

(final result: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha=0} )

Magnetic order parameter exponent: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta}

(final result: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta=1/2} )

Susceptibility exponent: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma}

(final result: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma=1} )

Molecular beam epitaxy

Random-field

XY

For the three dimensional XY model one has the following critical exponents[5]:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu=0.67155(27) }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha = -0.0146(8)}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta= 0.3485(2) }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma=1.3177(5) }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta=4.780(2) }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \eta =0.0380(4) }

References

  1. ↑ Michael E. Fisher "Rigorous Inequalities for Critical-Point Correlation Exponents", Physical Review 180 pp. 594-600 (1969)
  2. ↑ Massimo Campostrini, Andrea Pelissetto, Paolo Rossi, and Ettore Vicari "25th-order high-temperature expansion results for three-dimensional Ising-like systems on the simple-cubic lattice", Physical Review E 65 066127 (2002)
  3. ↑ A. L. Talapov and H. W. J Blöte "The magnetization of the 3D Ising model", Journal of Physics A: Mathematical and General 29 pp. 5727-5733 (1996)
  4. ↑ Linda E. Reichl "A Modern Course in Statistical Physics", Wiley-VCH, Berlin 3rd Edition (2009) ISBN 3-527-40782-0 § 4.9.4
  5. ↑ Massimo Campostrini, Martin Hasenbusch, Andrea Pelissetto, Paolo Rossi, and Ettore Vicari "Critical behavior of the three-dimensional XY universality class" Physical Review B 63 214503 (2001)
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