Latest revision as of 05:51, 5 November 2021

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

dimension	$\alpha$	$\beta$	$\gamma$	$\delta$	$\nu$	$\eta$	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

$\alpha$ is known as the heat capacity exponent
$\beta$ is known as the magnetic order parameter exponent
$\gamma$ is known as the susceptibility exponent
$\delta$ is known as the equation of state exponent
$\nu$ 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]

Potts model

Ashkin-Teller[edit]

Ashkin-Teller model

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)

$\beta ={\frac {1}{8}}$

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]:

\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) }

\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[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: $\beta =1/2$ )

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} [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: $\nu$ [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) }

\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[edit]

[1] Michael E. Fisher "Rigorous Inequalities for Critical-Point Correlation Exponents", Physical Review 180 pp. 594-600 (1969)

[Campostrini2002-2] 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

[Campostrini2001-5] 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)

[1]

[2]

[3]

[4]

[5]

@@ Line 1: / Line 1: @@
-{{Stub-general}}
+'''Universality classes''' are groups of [[Idealised models | models]] that have the same set of [[critical exponents]]
-{| border="1"
+:{| border="1"
+|-
+| dimension ||<math>\alpha</math> || <math>\beta</math> || <math>\gamma</math> || <math>\delta</math> ||<math>\nu</math> || <math>\eta</math> || class
+|-
+|  ||    ||   || || ||  || || 3-state Potts
 |-
-| <math>d</math> || <math>n</math> || <math>\sigma</math> || name
+|  ||   ||    || || || ||  ||Ashkin-Teller
 |-
-|  ||   ||   || 3-state Potts
+|  ||  ||    || || || || ||Chiral
 |-
-|  ||   ||   ||Ashkin-Teller
+|  ||   ||    || || || ||  ||Directed percolation
 |-
-|  ||   ||   ||Chiral
+| 2 ||  0 || 1/8  || 7/4 || || 1  || 1/4  ||  2D Ising
 |-
-|  ||   ||   ||Directed percolation
+| 3 ||   0.1096(5)  || 0.32653(10)  ||  1.2373(2)    || 4.7893(8) ||  0.63012(16) || 0.03639(15) ||  3D Ising
 |-
-|  ||   ||   ||Ising
+|  ||    ||    || || || ||   ||Local linear interface
 |-
-|  ||   ||   ||Local linear interface
+| any ||  0 || 1/2   || 1  || 3 || 1/2 || 0 || Mean-field
 |-
-|  ||   ||   ||Mean-field
+|  ||  ||    || || || ||  ||Molecular beam epitaxy
 |-
-|  ||   ||   ||Molecular beam epitaxy
+|  ||   ||   ||  || || ||  ||Random-field
 |-
-|  ||   ||   ||Random-field
+| 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>\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 the anomalous dimension in the critical correlation function.
+=Derivations=
 ==3-state Potts==
+[[Potts model]]
 ==Ashkin-Teller==
+[[Ashkin-Teller model]]
 ==Chiral==
 ==Directed percolation==
 ==Ising==
-The Hamiltonian of the Ising model is
+The [[Hamiltonian]] of the [[Ising model]] is
-\begin{equation}
+<math>
-{\cal H}=\sum{<i,j>}S_iS_j
+ H=\sum_{<i,j>}S_i S_j
-\end{equation}
+</math>
-where $S_i=\pm 1$ and the summation runs over the lattice sites.
-The order parameter is
+where <math>S_i=\pm 1</math> and the summation runs over the lattice sites.
-\begin{equation}
+The [[Order parameters | order parameter]] is
+<math>
 m=\sum_i S_i
-\end{equation}
+</math>
-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
-\begin{equation}
+:<math>
 \alpha=0
-\end{equation}
+</math>
-(In fact, the specific hear diverges logarithmically with the critical temperature)
+(In fact, the [[Heat capacity |specific heat]] diverges logarithmically with the [[Critical points |critical temperature]])
-\begin{equation}
+<math>
 \beta=\frac{1}{8}
-\end{equation}
+</math>
-\begin{equation}
+<math>
 \gamma=\frac{7}{4}
-\end{equation}
+</math>
-\begin{equation}
+<math>
 \delta=15
-\end{equation}
+</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>:
+:<math>
+\nu=0.63012(16)
+</math>
+:<math>
+\alpha=0.1096(5)
+</math>
+:<math>
+\beta= 0.32653(10)
+</math>
+:<math>
+\gamma=1.2373(2)
+</math>
+:<math>
+\delta=4.7893(8)
+</math>
+:<math>
+\eta =0.03639(15)
+</math>
+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==
 ==Mean-field==
+The [[critical exponents]] of are derived as follows <ref>Linda E. Reichl "A Modern Course in Statistical Physics", Wiley-VCH, Berlin 3rd Edition (2009) ISBN 3-527-40782-0 &sect; 4.9.4 </ref>:
+====Heat capacity exponent: <math>\alpha</math>====
+(final result: <math>\alpha=0</math>)
+====Magnetic order parameter exponent: <math>\beta</math>====
+(final result: <math>\beta=1/2</math>)
+====Susceptibility exponent: <math>\gamma</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==
-==See also==
-*[[Critical exponents]]
 ==Random-field==
+==XY==
+For the three dimensional [[XY model]] one has the following [[critical exponents]]<ref name="Campostrini2001" >[http://dx.doi.org/10.1103/PhysRevB.63.214503  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)]</ref>:
+:<math>
+\nu=0.67155(27)
+</math>
+:<math>\alpha = -0.0146(8)</math>
+:<math>
+\beta= 0.3485(2)
+</math>
+:<math>
+\gamma=1.3177(5)
+</math>
+:<math>
+\delta=4.780(2)
+</math>
+:<math>
+\eta =0.0380(4)
+</math>
+=References=
+<references/>
 [[category: Renormalisation group]]

Universality classes: Difference between revisions

Latest revision as of 05:51, 5 November 2021

Contents

Derivations[edit]

3-state Potts[edit]

Ashkin-Teller[edit]

Chiral[edit]

Directed percolation[edit]

Ising[edit]

Local linear interface[edit]

Mean-field[edit]

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]

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]

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]

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]

Correlation length exponent: $\nu$ [edit]

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]

Molecular beam epitaxy[edit]

Random-field[edit]

XY[edit]

References[edit]

Navigation menu

Universality classes: Difference between revisions

Latest revision as of 05:51, 5 November 2021

Derivations[edit]

3-state Potts[edit]

Ashkin-Teller[edit]

Chiral[edit]

Directed percolation[edit]

Ising[edit]

Local linear interface[edit]

Mean-field[edit]

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]

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]

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]

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]

Correlation length exponent: ν {\displaystyle \nu } [edit]

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]

Molecular beam epitaxy[edit]

Random-field[edit]

XY[edit]

References[edit]

Navigation menu

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Correlation length exponent: $\nu$ [edit]