9.1: The Program of Renormalization

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\)
\( \newcommand\Vpi

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\)
\( \newcommand\Vvarpi

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\)
\( \newcommand\Vrho

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\)
\( \newcommand\Vvarrho

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\)
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\)
\( \newcommand\Vtau

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\( \newcommand\Vupsilon

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\)
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\)
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\)
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\)
\( \newcommand\VTheta{\vec\Theta}\)
\( \newcommand\VLambda{\vec\Lambda}\)
\( \newcommand\VXi{\vec\Xi}\)
\( \newcommand\VPi{\vec\Pi}\)
\( \newcommand\VSigma{\vec\Sigma}\)
\( \newcommand\VUps{\vec\Upsilon}\)
\( \newcommand\VPhi{\vec\Phi}\)
\( \newcommand\VPsi{\vec\Psi}\)
\( \newcommand\VOmega{\vec\Omega}\)
\( \newcommand\BA{\mib A}\)
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\( \newcommand\BC{\mib C}\)
\( \newcommand\BD{\mib D}\)
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\( \newcommand\BG{\mib G}\)
\( \newcommand\BH{\mib H}\)
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\( \newcommand\Bxhi{\raise.35ex\hbox{$\Bchi$}}\)
\( \newcommand\RGamma{ \Gamma}\)
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\( \newcommand\Ra{ a}\)
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\( \newcommand\Rf{ f}\)
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\( \newcommand\Rh{ h}\)
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\( \newcommand { m}\)
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\( \newcommand\BBA{\boldsymbol\RA}\)
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\( \def\tket#1{|  #1 \rangle}\)
\( \def\tbra#1{\langle #1|}\)
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\( \def\texpect#1#2#3{\langle #1 |   #2  |  #3 \rangle}\)
\( \def\sket#1{|  \, #1 \,  \rangle}\)
\( \def\sbra#1{\langle \,  #1 \, |}\)
\( \def\sbraket#1#2{\langle \, #1  \, |  \, #2 \,  \rangle}\)
\( \def\sexpect#1#2#3{\langle \, #1 \, | \,  #2  \, | \, #3 \, \rangle}\)
\(\def\ket#1{\big| \, #1\, \big\rangle}\)
\( \def\bra#1{\big\langle \, #1 \, \big|}\)
\( \def\braket#1#2{\big\langle \, #1\, \big| \,#2 \,\big\rangle}\)
\( \def\expect#1#2#3{\big\langle\, #1\, \big|\, #2\, \big| \,#3\, \big\rangle}\)
\( \newcommand\pz{\partial}\)
\( \newcommand\pzb{\bar{\partial}}\)
\( \newcommand\svph{\vphantom{\int}}\)
\( \newcommand\vph{\vphantom{\sum_i}}\)
\( \newcommand\bvph{\vphantom{\sum_N^N}}\)
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\( \newcommand\Htil{\tilde H}\)
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\( \newcommand\oc{\omega_\Rc}\)

\(\newcommand \gtwid{\approx}\)

\( \newcommand\index{\textsf{ind}}\)
\( \newcommand\csch{\,{ csch\,}}\)
\( \newcommand\ctnh{\,{ ctnh\,}}\)
\( \newcommand\ctn{\,{ ctn\,}}\)
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\( \def\tmapright#1{\xrightarrow \limits^{#1}}\)
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\( \newcommand\Rep{\textsf{Re}\,}\)
\( \newcommand\Imp{\textsf{Im}\,}\)
\( \newcommand\ncdot{\!\cdot\!}\)
\( \def\tmapright#1{ \smash{\mathop{\hbox to 35pt{\rightarrowfill}}\limits^{#1}}\ }\)
\( \def\bmapright#1{ \smash{\mathop{\hbox to 35pt{\rightarrowfill}}\limits_{#1}}\ }\)
\( \newcommand\bsqcap{\mbox{\boldmath{$\sqcap$}}}\)

\( \def\pabc#1#2#3{\left({\pz #1\over\pz #2}\right)\ns_{\!\!#3}}\)
\( \def\spabc#1#2#3{\big({\pz #1\over\pz #2}\big)\ns_{\!#3}}\)
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\( \def\rabc#1#2#3#4{(\pz #1,\pz #2)\over (\pz #3,\pz #4)}\)
\( \newcommand\subA{\ns_\ssr{A}}\)
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\( \newcommand\subBC{\ns_\ssr{BC}}\)
\( \newcommand\subCD{\ns_\ssr{CD}}\)
\( \newcommand\subDA{\ns_\ssr{DA}}\)
\( \def\lmapright#1{\ \ \smash{\mathop{\hbox to 55pt{\rightarrowfill}}\limits^{#1}}\ \ }\)
\( \def\enth#1{\RDelta {\textsf H}^0_\Rf[{ #1}]}\)
\( \newcommand\longrightleftharpoons{ \mathop{\vcenter{\hbox{\ooalign{\raise1pt\hbox{$\longrightharpoonup\joinrel$}\crcr  \lower1pt\hbox{$\longleftharpoondown\joinrel$}}}}}}\)
\( \newcommand\longrightharpoonup{\relbar\joinrel\rightharpoonup}\)
\( \newcommand\longleftharpoondown{\leftharpoondown\joinrel\relbar}\)
\( \newcommand\cds{\,\bullet\,}\)
\( \newcommand\ccs{\,\circ\,}\)
\( \newcommand\nsub{_{\vphantom{\dagger}}}\)
\( \newcommand\rhohat{\hat\rho}\)
\( \newcommand\vrhhat{\hat\vrh}\)
\( \newcommand\impi{\int\limits_{-\infty}^\infty\!\!\!}\)
\( \newcommand\brangle{\big\rangle}\)
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\( \newcommand\vet{\tilde\ve}\)
\( \newcommand\zbar{\bar z}\)
\( \newcommand\ftil{\tilde f}\)
\( \newcommand\XBE{\RXi\ns_\ssr{BE}}\)
\( \newcommand\XFD{\RXi\ns_\ssr{FD}}\)
\( \newcommand\OBE{\Omega\ns_\ssr{BE}}\)
\( \newcommand\OFD{\Omega\ns_\ssr{FD}}\)
\( \newcommand\veF{\ve\ns_\RF}\)
\( \newcommand\kF{k\ns_\RF}\)
\( \newcommand\kFu{k\ns_{\RF\uar}}\)
\( \newcommand\SZ{\textsf Z}}\) \( \newcommand\kFd{k\ns_{\RF\dar}\)
\( \newcommand\muB{\mu\ns_\ssr{B}}\)
\( \newcommand\mutB{\tilde\mu}\ns_\ssr{B}\)
\( \newcommand\xoN{\Bx\ns_1\,,\,\ldots\,,\,\Bx\ns_N}\)
\( \newcommand\rok{\Br\ns_1\,,\,\ldots\,,\,\Br\ns_k}\)
\( \newcommand\xhiOZ{\xhi^\ssr{OZ}}\)
\( \newcommand\xhihOZ

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\)
\( \newcommand\jhz{\HJ(0)}\)
\( \newcommand\nda{\nd_\alpha}\)
\( \newcommand\ndap{\nd_{\alpha'}}\)
\( \newcommand\labar

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A statistical mechanical system is defined by a set of degrees of freedom and by a set of coupling constants \(\{K\ns_\alpha\}\). The degrees of freedom can be discrete, such as Ising spins \(\sigma\ns_i\), or continuous, such as a field \(\phi(\Br)\). Additionally, each such system possesses a microscopic length scale \(\ell\). For discrete, lattice-based systems, this length scale is simply the lattice spacing: \(\ell=a\). For continuous systems, we can define a microscopic length scale by imposing a cutoff \(\RLambda\) on the wavevectors we integrate over in all Fourier transforms. That is, we replace \[\int\!\!{d^d\!k\over (2\pi)^d}\>F(\Bk) \longrightarrow \int\!\!{d^d\!k\over (2\pi)^d}\>F(\Bk)\,\gla(\Bk)\ ,\] where \(F(\Bk)\) is any function and \(\gla(\Bk)\) is the cutoff function. The simplest such case to imagine is a sharp cutoff which is isotropic in wavevector, \(\gla(\Bk)=\RTheta(\RLambda-|\Bk|)\). Other cutoff schemes, however, are possible, including ‘soft cutoffs’ where \(\gla(\Bk)\) is smooth. The microscopic length scale is then \(\ell\sim\RLambda^{-1}\), which is the smallest distance in real space over which the system can independently fluctuate.

The idea behind renormalization is that we can successively winnow degrees of freedom from a system in some exact or approximate way, and in so doing we generate a new version of the system, at a different length scale \(\ell'>\ell\), and with different couplings \(\{K'_\alpha\}\). We then iterate this procedure. The result is a set of equations which tell us how the couplings behave under a change of the microscopic length scale. As we shall see, the fixed points of this procedure – where couplings do not change under a change of length scale – are critical points. Such a fixed point is defined by a set of couplings \(\{K^*_\alpha\}\).

If we denote by \(\CR\ns_b\) the renormalization procedure \[\CR\ns_b\big(\ell\,,\{K\ns_\alpha\}\big) = \big(\ell'\,,\{K'_\alpha\}\big)\ ,\] where \(\ell'=b\,\ell\), then we have the composition law \(\CR\ns_b\,\CR\ns_{b'}=\CR\ns_{b+b'}\). The set of transformations \(\{\CR\ns_b\}\) is collectively referred to as the renormalization group (RG) because of this mathematical structure. It is somewhat of a misnomer, however, since the transformations are only defined for \(b\ge 1\), which means that there is no inverse operation, and hence no true group structure¹. Nevertheless we shall use the RG terminology because it has become universally accepted in the literature.