Universality classes - Revision history

67.0.226.201: Added short description for \eta

2021-11-05T04:51:39Z

Added short description for \eta

← Older revision		Revision as of 06:51, 5 November 2021
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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==

Proxima Centari: Tweaked style of fractions for consistency/uniformity

2021-04-07T03:45:00Z

Tweaked style of fractions for consistency/uniformity

← Older revision		Revision as of 05:45, 7 April 2021
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	\| \|\| \|\| \|\| \|\| \|\| \|\| \|\|Directed percolation		\| \|\| \|\| \|\| \|\| \|\| \|\| \|\|Directed percolation
	\|-		\|-
	\| 2 \|\| 0 \|\| ~~<math>~~1/8~~</math>~~ \|\| ~~<math>~~7/4~~</math>~~ \|\| \|\| 1 \|\| 1/4 \|\| 2D Ising		\| 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 \|\| ~~<math>~~1/2~~</math>~~ \|\| 1 \|\| 3 \|\| ~~<math>~~1/2~~</math>~~ \|\| 0 \|\| Mean-field		\| any \|\| 0 \|\| 1/2 \|\| 1 \|\| 3 \|\| 1/2 \|\| 0 \|\| Mean-field
	\|-		\|-
	\| \|\| \|\| \|\| \|\| \|\| \|\| \|\|Molecular beam epitaxy		\| \|\| \|\| \|\| \|\| \|\| \|\| \|\|Molecular beam epitaxy

159.149.173.87: I inserted missing mean-field critical exponents (delta, nu, eta) both in the table and in the description below

2020-10-21T11:28:09Z

I inserted missing mean-field critical exponents (delta, nu, eta) both in the table and in the description below

← Older revision		Revision as of 13:28, 21 October 2020
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	\| \|\| \|\| \|\| \|\| \|\| \|\| \|\|Local linear interface		\| \|\| \|\| \|\| \|\| \|\| \|\| \|\|Local linear interface
	\|-		\|-
	\| \|\| 0 \|\| <math>1/2</math> \|\| 1 \|\| \|\| \|\| \|\| Mean-field		\| any \|\| 0 \|\| <math>1/2</math> \|\| 1 \|\| 3 \|\| <math>1/2</math> \|\| 0 \|\| Mean-field
	\|-		\|-
	\| \|\| \|\| \|\| \|\| \|\| \|\| \|\|Molecular beam epitaxy		\| \|\| \|\| \|\| \|\| \|\| \|\| \|\|Molecular beam epitaxy
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	*<math>\beta</math> is known as the [[Critical exponents#Magnetic order parameter exponent \| magnetic order parameter 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>\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>\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>\eta</math> is known as...		*<math>\eta</math> is known as...
	=Derivations=		=Derivations=
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	====Susceptibility exponent: <math>\gamma</math>====		====Susceptibility exponent: <math>\gamma</math>====
	(final result: <math>\gamma=1</math>)		(final result: <math>\gamma=1</math>)
			====Equation of state exponent: <math>\delta</math>====
			(final result: <math>\delta=3</math>)
			====Correlation length exponent: <math>\nu</math>====
			(final result: <math>\nu=1/2</math>)
			====Correlation function exponent: <math>\eta</math>====
			(final result: <math>\eta=0</math>)
	==Molecular beam epitaxy==		==Molecular beam epitaxy==
	==Random-field==		==Random-field==

Carl McBride at 13:01, 26 July 2011

2011-07-26T13:01:54Z

← Older revision		Revision as of 15:01, 26 July 2011
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	*<math>\nu</math> is known as the [[Critical exponents#Correlation length \| correlation length]]		*<math>\nu</math> is known as the [[Critical exponents#Correlation length \| correlation length]]
	*<math>\eta</math> is known as...		*<math>\eta</math> is known as...
			=Derivations=
	==3-state Potts==		==3-state Potts==
	[[Potts model]]		[[Potts model]]
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	\eta =0.0380(4)		\eta =0.0380(4)
	</math>		</math>
			=References=
	==References==
	<references/>		<references/>
	[[category: Renormalisation group]]		[[category: Renormalisation group]]

Carl McBride: Added internal links to the models

2011-07-26T11:21:19Z

Added internal links to the models

← Older revision		Revision as of 13:21, 26 July 2011
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	==3-state Potts==		==3-state Potts==
			[[Potts model]]
	==Ashkin-Teller==		==Ashkin-Teller==
			[[Ashkin-Teller model]]
	==Chiral==		==Chiral==
	==Directed percolation==		==Directed percolation==

Carl McBride: Added more data to the table

2011-07-26T11:17:27Z

Added more data to the table

← Older revision		Revision as of 13:17, 26 July 2011
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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==
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	\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>:
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	====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==

Carl McBride: /* XY */ Added the 3D XY model critical exponents

2011-07-26T10:44:21Z

XY: Added the 3D XY model critical exponents

@@ Line 106: / Line 106: @@
 ==Random-field==
 ==XY==
 ==References==
 <references/>
 [[category: Renormalisation group]]

Carl McBride: Added XY universality class sub-section

2011-07-26T10:39:31Z

Added XY universality class sub-section

← Older revision		Revision as of 12:39, 26 July 2011
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	\|-		\|-
	\| \|\| \|\| \|\| \|\| \|\| \|\|Random-field		\| \|\| \|\| \|\| \|\| \|\| \|\|Random-field
			\|-
			\| \|\| \|\| \|\| \|\| \|\| \|\| XY
	\|}		\|}
	==3-state Potts==		==3-state Potts==
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	==Molecular beam epitaxy==		==Molecular beam epitaxy==
	~~==See also==~~
	*[[Critical exponents]]
	==Random-field==		==Random-field==
			==XY==
	==References==		==References==
	<references/>		<references/>
	[[category: Renormalisation group]]		[[category: Renormalisation group]]

Carl McBride: /* 3-dimensional Ising model */ Cited a newer reference

2011-07-26T10:32:13Z

3-dimensional Ising model: Cited a newer reference

← Older revision		Revision as of 12:32, 26 July 2011
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	</math>		</math>

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

	:<math>		:<math>
	\nu=0.~~6298~~ (5)		\nu=0.63012(16)
	</math><ref name="Hasenbusch">[http://dx.doi.org/10.1103/PhysRevB.59.11471 M. Hasenbusch, K. Pinn and S. Vinti "Critical exponents of the three-dimensional Ising universality class from finite-size scaling with standard and improved actions", Physical Review B '''59''' pp. 11471-11483 (1999)]</ref>		</math>

	:<math>		:<math>
	\alpha=0.~~108~~(5)		\alpha=0.1096(5)
	</math> <ref name="Kolesik"> [http://dx.doi.org/10.1016/0378-4371(94)00302-A Miroslav Kolesik and Masuo Suzuki "Accurate estimates of 3D Ising critical exponents using the coherent-anomaly method", Physica A: Statistical and Theoretical Physics '''215''' pp. 138-151 (1995)]</ref>		</math>

	:<math>		:<math>
	\beta= 0.~~3269~~(6)		\beta= 0.32653(10)
	</math> <ref name="Talapov">[http://dx.doi.org/10.1088/0305-4470/29/17/042 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)]</ref>		</math>

	:<math>		:<math>
	\gamma=1.~~237~~(4)		\gamma=1.2373(2)
	</math~~><ref name="Kolesik"> </ref~~>		</math>

	:<math>		:<math>
	\delta=4.77(5)		\delta=4.7893(8)
	</math~~><ref name="Kolesik"> </ref~~>		</math>

	:<math>		:<math>
	\eta =0.~~0366~~(8)		\eta =0.03639(15)
	</math~~><ref name="Hasenbusch"> </ref~~>		</math>

	with a critical temperature of <math>k_BT_c = 4.51152786~S </math><ref ~~name=~~"~~Talapov~~"> </ref>. In four and higher dimensions, the critical exponents are mean-field with logarithmic corrections.		with a critical temperature of <math>k_BT_c = 4.51152786~S </math><ref>[http://dx.doi.org/10.1088/0305-4470/29/17/042 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)]</ref>. In four and higher dimensions, the critical exponents are mean-field with logarithmic corrections.

	==Local linear interface==		==Local linear interface==

Carl McBride: /* Ising */

2011-07-26T10:19:01Z

Ising

← Older revision		Revision as of 12:19, 26 July 2011
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	In two dimensions, Onsager obtained the exact solution in the absence of a external field, and the [[critical exponents]] are		In two dimensions, Onsager obtained the exact solution in the absence of a external field, and the [[critical exponents]] are
	<math>
			:<math>
	\alpha=0		\alpha=0
	</math>		</math>

	(In fact, the [[Heat capacity \|specific heat]] diverges logarithmically with the [[Critical points \|critical temperature]])		(In fact, the [[Heat capacity \|specific heat]] diverges logarithmically with the [[Critical points \|critical temperature]])