A. Zee, Quantum Field Theory in a Nutshell · Problem V.3.1

Field Theory Redux

Physics — QFT

Problem

Another important critical exponent

\gamma

is defined by saying that the susceptibility

\chi \equiv (\partial M/\partial H)|_{H=0}

diverges

\sim 1/|T - T_c|^\gamma

T

approaches

T_c

. Determine

\gamma

in Landau-Ginzburg theory. [Hint: Instructively, there are two ways of doing it: (a) Add

-\vec{H} \cdot \vec{M}

to (1) for

\vec{M}

and

\vec{H}

constant in space and solve for

M(H)

. (b) Calculate the susceptibility function

\chi_{ij}(\vec{x} - \vec{y}) \equiv [\partial M_i(\vec{x})/\partial H_j(\vec{y})]|_{H=0}

and integrate over space.]