Problem: Consider a Landau-Ginzburg statistical field theory involving a single real scalar field \(\phi(\mathbf x)\) for \(\mathbf x\in\mathbf R^d\) governed by the canonically normalized free energy density:
\[\mathcal F(\phi,\partial\phi/\partial\mathbf x,…)=\frac{1}{2}\biggr|\frac{\partial\phi}{\partial\mathbf x}\biggr|^2+\frac{\phi^2}{2\xi^2}+…\]
Explain what the \(+…\) means, explain which terms have temperature \(T\)-dependence, and explain for which such terms does that \(T\)-dependence actually matter?
Solution: The \(+…\) includes any terms (each with their own coupling constants) compatible with the golden trinity of constraints: locality, analyticity, and symmetry (e.g. a quartic \(g\phi^4\) coupling). The part of the free energy density \(\mathcal F\) before the \(+…\) should be compared to the Lagrangian density \(\mathcal L\) of Klein-Gordon field theory:
\[\mathcal L=\frac{1}{2c^2}\left(\frac{\partial\phi}{\partial t}\right)^2-\frac{1}{2}\biggr|\frac{\partial\phi}{\partial\mathbf x}\biggr|^2-\frac{\phi^2}{2\bar{\lambda}^2}\]
with \(\bar{\lambda}=\hbar/mc\) the reduced Compton wavelength playing a role analogous to the correlation length \(\xi\sim 1/\sqrt{|T-T_c|}\); indeed, this \(T\)-dependence in \(\xi=\xi(T)\) is (usually) the only \(T\)-dependence that matters, even though generically all the other coupling constants will also have \(T\)-dependence.
Problem: Define the (non-standard) notion of “theory space”.
Solution: Roughly speaking, “theory space” is the space of all Landau-Ginzburg statistical field theories \((\mathcal F,k^*)\) (notice it is defined not only by the free energy density \(\mathcal F\) but also the baggage of the UV cutoff \(k^*\); it is an effective field theory). The \(\mathcal F\) part can equivalently be parameterized as a countably \(\infty\)-tuple \((\xi,g,…)\) of the LAS-permitted coupling constants in \(\mathcal F\).
Problem: In broad strokes, describe the sequence of \(3\) steps that comprise a \(\mathbf k\)-space \(\zeta\)-renormalization semigroup transformation from one effective Landau-Ginzburg statistical field theory \((\mathcal F,k^*)\mapsto (\mathcal F_{\zeta},k^*)\) to another with the same UV cutoff \(k^*\) but a new Wilsonian effective free energy \(\mathcal F_{\zeta}\).
Solution: For \(\zeta\in [1,\infty)\), the corresponding \(\mathbf k\)-space \(\zeta\)-RG transformation of \((\mathcal F,k^*)\) is given by the \(3\)-step recipe:
- Coarse-graining \(k^*\mapsto k^*/\zeta\) (blocking in real space/integrating out shells in momentum space)
- Rescaling \(\mathbf k’:=\zeta\mathbf k\) to recover the original UV cutoff \(k^*/\zeta\mapsto k^*\) (this leads to a reciprocal “zooming out” of space \(\mathbf x\mapsto\mathbf x/\zeta\)).
- Rescale fields \(\phi’:=\zeta^{\Delta}\phi\) to make \(\mathcal F_{\zeta}\) canonically normalized with respect to \(\mathcal F\).
Problem: Consider an effective Landau-Ginzburg statistical field theory of a single real scalar field \(\phi(\mathbf x)\in\mathbf R\) with \(\mathbf x\in\mathbf R^d\) whose Fourier transform \(\phi_{\mathbf k}=\int d^d\mathbf x e^{-i\mathbf k\cdot\mathbf x}\phi(\mathbf x)\) is supported only on a ball of radius \(k^*\) (the theory’s UV cutoff). The free energy density corresponds to a free (no pun intended) field:
\[\mathcal F(\phi,\partial\phi/\partial\mathbf x)=\frac{1}{2}\biggr|\frac{\partial\phi}{\partial\mathbf x}\biggr|^2+\frac{\phi^2}{2\xi^2}\]
Perform a \(\mathbf k\)-space \(\zeta\)-renormalization of this theory to find the corresponding Wilsonian effective free energy density \(\mathcal F_{\zeta}\).
Solution: Work with the free energy \(F=\int d^d\mathbf x\mathcal F\) itself instead of just its density \(\mathcal F\):
\[F[\phi]=\frac{1}{2}\int_{|\mathbf k|<k^*}\frac{d^d\mathbf k}{(2\pi)^d}\left(|\mathbf k|^2+\frac{1}{\xi^2}\right)|\phi_{\mathbf k}|^2\]
- Partition the support \(|\mathbf k|<k^*\) of \(\phi_{\mathbf k}\) into \(|\mathbf k|<k^*/\zeta\) and \(k^*/\zeta<|\mathbf k|<k^*\) and based on this \(\zeta\), piecewise decompose \(\phi_{\mathbf k}=\phi^{<}_{\mathbf k}+\phi^{>}_{\mathbf k}\). Then one has an instance of the freshman’s dream (thanks to the disjoint supports of \(\phi^{<}_{\mathbf k}\) and \(\phi^{>}_{\mathbf k}\)):
\[|\phi_{\mathbf k}|^2=|\phi^{<}_{\mathbf k}+\phi^{>}_{\mathbf k}|^2=|\phi^{<}_{\mathbf k}|^2+|\phi^{>}_{\mathbf k}|^2\]
So:
\[F[\phi]=\frac{1}{2}\int_{|\mathbf k|<k^*/\zeta}\frac{d^d\mathbf k}{(2\pi)^d}\left(|\mathbf k|^2+\frac{1}{\xi^2}\right)|\phi^{<}_{\mathbf k}|^2+\frac{1}{2}\int_{k^*/\zeta<|\mathbf k|<k^*}\frac{d^d\mathbf k}{(2\pi)^d}\left(|\mathbf k|^2+\frac{1}{\xi^2}\right)|\phi^{>}_{\mathbf k}|^2\]
\[=F[\phi^<_{\zeta}]+F[\phi^>_{\zeta}]\]
In this case the partition function factorizes:
\[Z=\int\mathcal D\phi e^{-\beta F[\phi]}=\int\mathcal D\phi^{<}_{\zeta} e^{-\beta F[\phi^{<}_{\zeta}]}\int\mathcal D\phi^{>}_{\zeta}e^{-\beta F[\phi^{>}_{\zeta}]}=Z^{>}_{\zeta}\int\mathcal D\phi^{<}_{\zeta} e^{-\beta F[\phi^{<}_{\zeta}]}\]
where the measures are \(\mathcal D\phi^{<}_{\zeta}=\prod_{|\mathbf k|<k^*/\zeta}d\phi^{<}_{\mathbf k}\) and \(\mathcal D\phi^{>}_{\zeta}=\prod_{k^*/\zeta<|\mathbf k|<k^*}d\phi^{>}_{\mathbf k}\). The constant \(Z^>_{\zeta}\) doesn’t affect the physics, being absorbed as a constant shift into the Wilsonian effective free energy \(F_{\zeta}[\phi^{<}_{\zeta}]=F[\phi^{<}_{\zeta}]-k_BT\ln Z^{>}_{\zeta}\).
2. Rescaling \(\mathbf k’:=\zeta\mathbf k\), the Wilsonian effective free energy becomes:
\[F_{\zeta}[\phi^{<}_{\zeta}]=\frac{1}{2}\int_{|\mathbf k’|<k^*}\frac{d^d\mathbf k’}{(2\pi)^d}\zeta^{-d}\left(\zeta^{-2}|\mathbf k’|^2+\frac{1}{\xi^2}\right)|\phi^{<}_{\mathbf k’/\zeta}|^2\]
3. Rescaling \(\phi^{<\prime}_{\mathbf k’}:=\zeta^{\Delta}\phi^{<}_{\mathbf k’/\zeta}\), the Wilsonian effective free energy becomes:
\[F_{\zeta}[\phi^{<}_{\zeta}]=\frac{1}{2}\int_{|\mathbf k’|<k^*}\frac{d^d\mathbf k’}{(2\pi)^d}\zeta^{-d}\left(\zeta^{-2}|\mathbf k’|^2+\frac{1}{\xi^2}\right)\zeta^{-2\Delta}|\phi^{<\prime}_{\mathbf k’}|^2\]
so in order to canonically normalize the gradient term, one requires \(\Delta=-(d+2)/2\). This leads to the desired Wilsonian effective free energy density:
\[\mathcal F_{\zeta}(\phi^{<\prime}_{\zeta},\partial\phi^{<\prime}_{\zeta}/\partial\mathbf x)=\frac{1}{2}\biggr|\frac{\partial\phi^{<\prime}_{\zeta}}{\partial\mathbf x}\biggr|^2+\zeta^2\frac{(\phi^{<\prime}_{\zeta})^2}{2\xi^2}\]
Thus, by construction the gradient coupling is marginal (i.e. \(\zeta\)-independent) while the quadratic coupling is relevant because \(\zeta^2\to\infty\) as \(\zeta\to\infty\).