Loading [MathJax]/extensions/TeX/boldsymbol.js
Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Physics LibreTexts

9.4: Scaling Variables

( \newcommand{\kernel}{\mathrm{null}\,}\)




























































































































































































































































































































\( \newcommand\Dalpha

ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[1], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.04:_Scaling_Variables), /content/body/p[1]/span, line 1, column 23
\)
\( \newcommand\Dbeta
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[2], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.04:_Scaling_Variables), /content/body/p[1]/span, line 1, column 23
\)
\( \newcommand\Dgamma
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[3], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.04:_Scaling_Variables), /content/body/p[1]/span, line 1, column 23
\)
\( \newcommand\Ddelta
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[4], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.04:_Scaling_Variables), /content/body/p[1]/span, line 1, column 23
\)
\( \newcommand\Depsilon
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[5], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.04:_Scaling_Variables), /content/body/p[1]/span, line 1, column 23
\)
\( \newcommand\Dvarepsilon
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[6], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.04:_Scaling_Variables), /content/body/p[1]/span, line 1, column 23
\)
\( \newcommand\Dzeta
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[7], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.04:_Scaling_Variables), /content/body/p[1]/span, line 1, column 23
\)
\( \newcommand\Deta
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[8], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.04:_Scaling_Variables), /content/body/p[1]/span, line 1, column 23
\)
\( \newcommand\Dtheta
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[9], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.04:_Scaling_Variables), /content/body/p[1]/span, line 1, column 23
\)
\( \newcommand\Dvartheta
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[10], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.04:_Scaling_Variables), /content/body/p[1]/span, line 1, column 23
\)
\( \newcommand\Diota
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[11], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.04:_Scaling_Variables), /content/body/p[1]/span, line 1, column 23
\)
\( \newcommand\Dkappa
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[12], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.04:_Scaling_Variables), /content/body/p[1]/span, line 1, column 23
\)
\( \newcommand\Dlambda
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[13], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.04:_Scaling_Variables), /content/body/p[1]/span, line 1, column 23
\)





\( \newcommand\Dvarpi
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[14], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.04:_Scaling_Variables), /content/body/p[1]/span, line 1, column 23
\)











\( \newcommand\DGamma
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[15], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.04:_Scaling_Variables), /content/body/p[1]/span, line 1, column 23
\)
\( \newcommand\DDelta
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[16], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.04:_Scaling_Variables), /content/body/p[1]/span, line 1, column 23
\)
\( \newcommand\DTheta
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[17], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.04:_Scaling_Variables), /content/body/p[1]/span, line 1, column 23
\)









































































\( \newcommand\Vmu
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[18], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.04:_Scaling_Variables), /content/body/p[1]/span, line 1, column 23
\)
\( \newcommand\Vnu
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[19], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.04:_Scaling_Variables), /content/body/p[1]/span, line 1, column 23
\)
\( \newcommand\Vxi
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[20], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.04:_Scaling_Variables), /content/body/p[1]/span, line 1, column 23
\)
\( \newcommand\Vom
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[21], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.04:_Scaling_Variables), /content/body/p[1]/span, line 1, column 23
\)
\( \newcommand\Vpi
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[22], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.04:_Scaling_Variables), /content/body/p[1]/span, line 1, column 23
\)
\( \newcommand\Vvarpi
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[23], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.04:_Scaling_Variables), /content/body/p[1]/span, line 1, column 23
\)
\( \newcommand\Vrho
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[24], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.04:_Scaling_Variables), /content/body/p[1]/span, line 1, column 23
\)
\( \newcommand\Vvarrho
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[25], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.04:_Scaling_Variables), /content/body/p[1]/span, line 1, column 23
\)
\( \newcommand\Vsigma
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[26], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.04:_Scaling_Variables), /content/body/p[1]/span, line 1, column 23
\)
\( \newcommand\Vvarsigma
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[27], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.04:_Scaling_Variables), /content/body/p[1]/span, line 1, column 23
\)
\( \newcommand\Vtau
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[28], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.04:_Scaling_Variables), /content/body/p[1]/span, line 1, column 23
\)
\( \newcommand\Vupsilon
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[29], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.04:_Scaling_Variables), /content/body/p[1]/span, line 1, column 23
\)
\( \newcommand\Vphi
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[30], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.04:_Scaling_Variables), /content/body/p[1]/span, line 1, column 23
\)
\( \newcommand\Vvarphi
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[31], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.04:_Scaling_Variables), /content/body/p[1]/span, line 1, column 23
\)
\( \newcommand\Vchi
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[32], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.04:_Scaling_Variables), /content/body/p[1]/span, line 1, column 23
\)
\( \newcommand\Vpsi
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[33], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.04:_Scaling_Variables), /content/body/p[1]/span, line 1, column 23
\)
\( \newcommand\Vomega
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[34], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.04:_Scaling_Variables), /content/body/p[1]/span, line 1, column 23
\)
\( \newcommand\VGamma
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[35], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.04:_Scaling_Variables), /content/body/p[1]/span, line 1, column 23
\)
\( \newcommand\VDelta
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/p[1]/span[36], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.04:_Scaling_Variables), /content/body/p[1]/span, line 1, column 23
\)

















\newcommand\BI{\mib I}}










































\)










































































\newcommand { M}

























\newcommand { m}














































}


















\( \newcommand\tcb{\textcolor{blue}\)
\( \newcommand\tcr{\textcolor{red}\)



































1$#1_$






















































































\newcommand\SZ{\textsf Z}} \( \newcommand\kFd{k\ns_{\RF\dar}\)

\newcommand\mutB{\tilde\mu}\ns_\ssr{B}



\( \newcommand\xhihOZ
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/span[1], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.04:_Scaling_Variables), /content/body/p[1]/span, line 1, column 23
\)



\( \newcommand\labar
ParseError: invalid DekiScript (click for details)
Callstack:
    at (Template:MathJaxArovas), /content/body/div/span[2], line 1, column 1
    at template()
    at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/09:_Renormalization/9.04:_Scaling_Variables), /content/body/p[1]/span, line 1, column 23
\)





















We’ve seen how an RG transformation acts on the (infinite) set of couplings which define the Hamiltonian of a system. We found \BK'=\BCR\ns_b(\BK). If \xi(\BK) is the correlation length in units of the lattice spacing, then since each RG step involves a rescaling by a factor b, we must have \xi(\BK)=b\,\xi(\BK') = b^2\,\xi(\BK'') = \cdots \ .

A fixed point of the transformation \BCR\ns_b is a set of couplings \BK^* such that \BCR\ns_b(\BK^*)=\BK^*\ . Linearizing \BCR\ns_b(\BK) about the fixed point, we have K'_\alpha-K^*_\alpha = \sum_\beta Q\ns_{\alpha\beta}\,(K\ns_\beta-K^*_\beta) \qquad,\qquad Q\ns_{\alpha\beta}={\pz K'_\alpha\over \pz K\ns_\beta}\bigg|_{\BK^*}\ . The matrix Q\ns_{\alpha\beta} is real but not necessarily symmetric. We define the left eigenvectors of Q, \phi^{(i)}_\alpha, such that \sum_\alpha \phi^{(i)}_\alpha \, Q\ns_{\alpha\beta}=\lambda\ns_i \, \phi^{(i)}_\beta\ . The scaling variable u\ns_i is then defined as u\ns_i\equiv\sum_\alpha \phi^{(i)}_\alpha (K\ns_\alpha-K^*_\alpha)\ . It should now be apparent that under an RG transformation, we have u'_i=\sum_\alpha \phi^{(i)}_\alpha (K'_\alpha-K^*_\alpha) = \sum_{\alpha,\beta}\phi^{(i)}_\alpha\ , Q\ns_{\alpha\beta} (K\ns_\beta-K^*_\beta) =\lambda\ns_i \sum_\alpha \phi^{(i)}_\alpha (K_\alpha-K^*_\alpha) =\lambda\ns_i u\ns_i\ . We say that u\ns_i\ {is}\ \begin{cases} \hbox{\it relevant} & {if}\ \lambda\ns_i > 1 \\ \hbox{\it irrelevant} & {if}\ \lambda\ns_i < 1 \\ \hbox{\it marginal} & {if}\ \lambda\ns_i = 1 \ . \end{cases} Under renormalization, relevant scaling variables flow away from the fixed point, while irrelevant scaling variables flow toward the fixed point. For marginal variables, one must go to higher order, beyond the above linearization, to determine whether the flow is away from (marginally relevant) or toward (marginally irrelevant) the fixed point.

clipboard_e6dbd1e9fde327bc3772cc519fed48698.png

clipboard_e0a8e62e5aeb642c1594b8b4d7d05e1e7.png


This page titled 9.4: Scaling Variables is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Daniel Arovas.

Support Center

How can we help?

9.3: Block Spin Transformation
9.5: RSRG on Hierarchical Lattices The