Universality classes: Difference between revisions
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| || || || || || || ||Directed percolation | | || || || || || || ||Directed percolation | ||
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| 2 || 0 || | | 2 || 0 || 1/8 || 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 | | 3 || 0.1096(5) || 0.32653(10) || 1.2373(2) || 4.7893(8) || 0.63012(16) || 0.03639(15) || 3D Ising | ||
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| || || || || || || ||Local linear interface | | || || || || || || ||Local linear interface | ||
|- | |- | ||
| any || 0 || | | any || 0 || 1/2 || 1 || 3 || 1/2 || 0 || Mean-field | ||
|- | |- | ||
| || || || || || || ||Molecular beam epitaxy | | || || || || || || ||Molecular beam epitaxy | ||
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*<math>\delta</math> is known as the [[Critical exponents#Equation of state exponent |equation of state exponent ]] | *<math>\delta</math> is known as the [[Critical exponents#Equation of state exponent |equation of state exponent ]] | ||
*<math>\nu</math> is known as the [[Critical exponents#Correlation length | correlation length exponent]] | *<math>\nu</math> is known as the [[Critical exponents#Correlation length | correlation length exponent]] | ||
*<math>\eta</math> is known as | *<math>\eta</math> is known as the anomalous dimension in the critical correlation function. | ||
=Derivations= | =Derivations= | ||
==3-state Potts== | ==3-state Potts== | ||
Latest revision as of 05:51, 5 November 2021
Universality classes are groups of models that have the same set of critical exponents
dimension class 3-state Potts Ashkin-Teller Chiral Directed percolation 2 0 1/8 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 any 0 1/2 1 3 1/2 0 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
- is known as the heat capacity exponent
- is known as the magnetic order parameter exponent
- is known as the susceptibility exponent
- is known as the equation of state exponent
- is known as the correlation length 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 \eta} is known as the anomalous dimension in the critical correlation function.
Derivations[edit]
3-state Potts[edit]
Ashkin-Teller[edit]
Chiral[edit]
Directed percolation[edit]
Ising[edit]
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 \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 \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) }
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[edit]
Mean-field[edit]
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} [edit]
(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} [edit]
(final result: )
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} [edit]
(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} )
Equation of state 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 \delta} [edit]
(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 \delta=3} )
Correlation length exponent: [edit]
(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 \nu=1/2} )
Correlation function 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 \eta} [edit]
(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 \eta=0} )
Molecular beam epitaxy[edit]
Random-field[edit]
XY[edit]
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 \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[edit]
- ↑ Michael E. Fisher "Rigorous Inequalities for Critical-Point Correlation Exponents", Physical Review 180 pp. 594-600 (1969)
- ↑ 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)
- ↑ 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)
- ↑ Linda E. Reichl "A Modern Course in Statistical Physics", Wiley-VCH, Berlin 3rd Edition (2009) ISBN 3-527-40782-0 § 4.9.4
- ↑ 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)