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\Lambda^d \sim \left\{ \begin{array} {lc }(-\mu)^{d/2-1}, & 2<d<4 \\

\Lambda^d \sim \left\{ \begin{array} {lc }(-\mu)^{d/2-1}, & 2<d<4 \\

\\ \mu \ln(-\mu/kT), & d=4 \\ \\ -\mu , & d>4 . \end{array}\right. </math>|3={{EquationRef|19}}}}

\\ \mu \ln(-\mu/kT), & d=4 \\ \\ -\mu , & d>4 . \end{array}\right. </math>|3={{EquationRef|19}}}}

<math>d>4</math> the singular <math>(-\mu)^{d/2-1}</math> is still present but is dominated by the mean-field <math>-\mu\ .</math>

<math>d>4</math> the singular <math>(-\mu)^{d/2-1}</math> is still present but is dominated by the mean-field <math>-\mu\ .</math>

 

This behavior is reflected again in the R'''enormalization-group theory''' [19-21].  In the simplest cases there are two competing fixed points for  the renormalization-group flows, one of them associated with <math>d</math>-dependent  

This behavior is reflected again in the [[renormalization-group theory]] [19-21].  In the simplest cases there are two competing fixed points for  the renormalization-group flows, one of them associated with <math>d</math>-dependent  

exponents that satisfy both the <math>d</math>-independent scaling relations and

exponents that satisfy both the <math>d</math>-independent scaling relations and

the hyperscaling relations, the other with the <math>d</math>-independent  

the hyperscaling relations, the other with the <math>d</math>-independent  

as though it were a single spin in a re-scaled model.  Each block is

as though it were a single spin in a re-scaled model.  Each block is

of finite size so the spins in its interior contribute only analytic

of finite size so the spins in its interior contribute only analytic

terms to the free energy of the system. The part of the free-energy

terms to the free energy of the system. The part of the free-energy

density (free energy per spin) that carries the critical-point

density (free energy per spin) that carries the critical-point

singularities and their exponents comes from the interactions between

singularities and their exponents comes from the interactions between blocks. Let this free-energy density be <math>f(t,H)\ ,</math> a function of temperature through <math>t=T/T_c-1</math> and of the magnetic field <math>H\ .</math>  The correlation length is the same in the re-scaled picture as in the original, but measured as a number of lattice spacings it is smaller in the former by the factor <math>L\ .</math>  Thus, the re-scaled model is effectively further from its critical point than the original was from its; so with <math>H</math> and <math>t</math> both going to 0 as the critical point is approached, the effective <math>H</math> and <math>t</math>

blocks. Let this free-energy density be <math>f(t,H)\ ,</math> a

in the re-scaled model are <math>L^xH</math> and <math>L^yt</math> with positive exponents <math>x</math> and <math>y\ ,</math> so increasing with <math>L\ .</math>  From the point of view of the original model the contribution to the singular part of the free energy made by the spins in each block is <math>L^df(t,H)\ ,</math> while that same quantity, from the point of the view of the re-scaled model, is <math>f(L^yt, L^xH)\ .</math>  Thus,  

function of temperature through <math>t=T/T_c-1</math> and of the

magnetic field <math>H\ .</math>  The correlation length is the same in

the re-scaled picture as in the original, but measured as a number of

lattice spacings it is smaller in the former by the factor

<math>L\ .</math>  Thus, the re-scaled model is effectively further from

its critical point than the original was from its; so with

<math>H</math> and <math>t</math> both going to 0 as the critical

point is approached, the effective <math>H</math> and <math>t</math>

in the re-scaled model are <math>L^xH</math> and <math>L^yt</math>

with positive exponents <math>x</math> and <math>y\ ,</math> so

increasing with <math>L\ .</math>  From the point of view of the

original model the contribution to the singular part of the free

energy made by the spins in each block is <math>L^df(t,H)\ ,</math>

while that same quantity, from the point of the view of the re-scaled

model, is <math>f(L^yt, L^xH)\ .</math>  Thus,  

:<math>\label{eq:20}

:<math>\label{eq:20}

f(L^yt,

f(L^yt,

L^xH) \equiv L^df(t,H); </math>

L^xH) \equiv L^df(t,H); </math>

 

i.e., by \eqref{eq:1},

i.e., by \eqref{eq:1}, <math>f(t,H)</math> is a homogeneous function of <math>t</math> and <math>H^{y/x}</math> of degree <math>d/y\ .</math>

<math>f(t,H)</math> is a homogeneous function of <math>t</math> and

<math>H^{y/x}</math> of degree <math>d/y\ .</math>

Therefore, by \eqref{eq:2}, <math>f(t,H)=t^{d/y}

Therefore, by \eqref{eq:2}, <math>f(t,H)=t^{d/y}

\phi(H^{y/x}/t)=H^{d/x}\psi(t/H^{y/x})</math> where <math>\phi</math>

\phi(H^{y/x}/t)=H^{d/x}\psi(t/H^{y/x})</math> where <math>\phi</math> and <math>\psi</math> are functions only of the ratio <math>H^{y/x}/t\ .</math>  At <math>H=0</math> the first of these gives <math>f(t,0)=\phi(0)t^{d/y}\ .</math> But two temperature derivatives of <math>f(t,0)</math> gives the contribution to the heat capacity per spin, diverging as <math>t^{-\alpha}\ ;</math> so <math>d/y=2-\alpha\ .</math>  Also, on the critical isotherm <math>(t=0)\ ,</math> the second relation above gives <math>f(0,H)=\psi(0)H^{d/x}\ .</math>  But the magnetization per spin is <math>-(\partial f/\partial H)_T\ ,</math> vanishing as <math>H^{d/x-1}\ ,</math> so <math>d/x-1=1/\delta\ .</math>  The exponents <math>d/x</math> and <math>d/y</math> have thus been identified in terms of the thermodynamic exponents: the heat-capacity exponent <math>\alpha</math> and the critical-isotherm exponent <math>\delta\ .</math>  In the meantime, again with <math>-(\partial

and <math>\psi</math> are functions only of the ratio

f/\partial H)_T</math> the magnetization per spin, the homogeneity of form of <math>f(t,H)</math> in \eqref{eq:20} is equivalent to that of <math>H(t,M)</math> in \eqref{eq:7}, from which the scaling laws <math>\gamma=\beta(\delta-1)</math> and <math>\alpha +

<math>H^{y/x}/t\ .</math>  At <math>H=0</math> the first of these gives

<math>f(t,0)=\phi(0)t^{d/y}\ .</math> But two temperature derivatives

of <math>f(t,0)</math> gives the contribution to the heat capacity per

spin, diverging as <math>t^{-\alpha}\ ;</math> so

<math>d/y=2-\alpha\ .</math>  Also, on the critical isotherm

<math>(t=0)\ ,</math> the second relation above gives

<math>f(0,H)=\psi(0)H^{d/x}\ .</math>  But the magnetization per spin is

<math>-(\partial f/\partial H)_T\ ,</math> vanishing as

<math>H^{d/x-1}\ ,</math> so <math>d/x-1=1/\delta\ .</math>  The exponents

<math>d/x</math> and <math>d/y</math> have thus been identified in

terms of the thermodynamic exponents: the heat-capacity exponent

<math>\alpha</math> and the critical-isotherm exponent

<math>\delta\ .</math>  In the meantime, again with <math>-(\partial

f/\partial H)_T</math> the magnetization per spin, the homogeneity of

form of <math>f(t,H)</math> in \eqref{eq:20} is equivalent to

that of <math>H(t,M)</math> in \eqref{eq:7}, from which the

scaling laws <math>\gamma=\beta(\delta-1)</math> and <math>\alpha +

2\beta + \gamma =2</math> are known to follow.

2\beta + \gamma =2</math> are known to follow.

A related argument yields the scaling law \eqref{eq:10} for the

A related argument yields the scaling law \eqref{eq:10} for the correlation function <math>h(r,t)\ ,</math> with <math>H=0</math> again for simplicity.  In the re-scaled model, <math>t</math> becomes <math>L^yt\ ,</math> as before, while <math>r</math> becomes <math>r/L\ .</math>  There may also be a factor, say <math>L^p</math> with some exponent <math>p\ ,</math> relating the magnitudes of the original and rescaled functions; thus,  

correlation function <math>h(r,t)\ ,</math> with <math>H=0</math> again

for simplicity.  In the re-scaled model, <math>t</math> becomes

<math>L^yt\ ,</math> as before, while <math>r</math> becomes

<math>r/L\ .</math>  There may also be a factor, say <math>L^p</math>

with some exponent <math>p\ ,</math> relating the magnitudes of the

original and rescaled functions; thus,  

:<math>\label{eq:21}

:<math>\label{eq:21}

h(r,t) \equiv

h(r,t) \equiv

L^{p}h(r/L,L^yt); </math>

L^{p}h(r/L,L^yt); </math>

i.e., <math>h(r,t)</math> is homogeneous of

 

degree <math>p</math> in <math>r</math> and <math>t^{-1/y}\ .</math>

i.e., <math>h(r,t)</math> is homogeneous of degree <math>p</math> in <math>r</math> and <math>t^{-1/y}\ .</math> Then from the alternative form \eqref{eq:2} of the property of homogeneity,  

Then from the alternative form \eqref{eq:2} of the property of

homogeneity,  

:<math>\label{eq:22}

:<math>\label{eq:22}

h(r,t)\equiv r^p G(r/t^{-1/y}) </math>

h(r,t)\equiv r^p G(r/t^{-1/y}) </math>

The block-spin picture thus yields the critical-point scaling of the

thermodynamic and correlation functions, and both the

 

<math>d</math>-independent and <math>d</math>-dependent relations

The block-spin picture thus yields the critical-point scaling of the thermodynamic and correlation functions, and both the <math>d</math>-independent and <math>d</math>-dependent relations among the scaling exponents. The essence of this picture is confirmed in the renormalization-group theory [19,20].

among the scaling exponents. The essence of this picture is confirmed

in the renormalization-group theory [19,20].

== References ==

== References ==

* Cesar A. Hidalgo R. and Albert-Laszlo Barabasi (2008) [[Scale-free networks]]. Scholarpedia, 3(1):1716.

* Cesar A. Hidalgo R. and Albert-Laszlo Barabasi (2008) [[Scale-free networks]]. Scholarpedia, 3(1):1716.

==See also==

==See also==