David Tong SFT Sheet 2 Solutions: Correlations and Scaling

Source: David Tong, Statistical Field Theory, Example Sheet 2; 6 worked problems.

These prompts paraphrase David Tong's Statistical Field Theory, Example Sheet 2 (November 2018). Problem numbers, starred status, equations, and guidance follow the official sheet.

1 Sheet 2, Problem 1: Ornstein--Zernicke Correlations

The Gaussian two-point function is \[G(x)=\int\frac{d^dk}{(2\pi)^d} \frac{e^{-ik\cdot x}}{\gamma k^2+\mu^2} =\frac{1}{\gamma}\int\frac{d^dk}{(2\pi)^d} \frac{e^{-ik\cdot x}}{k^2+\xi^{-2}}, \qquad \xi^2=\frac{\gamma}{\mu^2}.\tag{1.1}\] Explain why it depends only on $r=|x|$. Establish the parameter integral \[\frac{1}{k^2+\xi^{-2}} =\int_0^\infty dt\,e^{-t(k^2+\xi^{-2})},\tag{1.2}\] perform the Gaussian momentum integral, and express the result, up to an overall factor, as \[G(r)\sim\int_0^\infty dt\,e^{-S(t)}, \qquad S(t)=\frac{r^2}{4t}+\frac{t}{\xi^2}+\frac d2\log t.\tag{1.3}\] Use a saddle-point approximation, stating the saddle equation and quadratic prefactor, in the regimes $r\ll\xi$ and $r\gg\xi$. Recover the Ornstein--Zernicke behaviours \[G(r)\sim r^{-(d-2)}\quad(r\ll\xi), \qquad G(r)\sim\frac{e^{-r/\xi}}{r^{(d-1)/2}}\quad(r\gg\xi).\tag{1.4}\]

Since the free energy respects global $SO(d)$ symmetry, the correlation function must also be invariant under rotation, and hence can only depend on $r$. The parameter integral is easy to see, and plugging into the correlation function, we find \[\begin{aligned} G(r) &\sim\int d^dk \int_0^\infty dt \, e^{-i\mathbf{k}\cdot \mathbf{x} - t(k^2 +\xi^{-2})}\\ &=\int d^dk \int_0^\infty dt \, e^{-(\sqrt{t}\mathbf{k} + i\mathbf{x}/2\sqrt{t})^2 - t\xi^{-2} - r^2/4t}\\ &\sim\int^\infty_0dt \, \frac{e^{- t\xi^{-2} - r^2/4t}}{t^{d/2}}. \end{aligned}\tag{1.5}\] The saddle point is at \[S'(t_*) = 0 \implies t_* = \frac{r^2}{d\pm \sqrt{d^2 + 4r^2/\xi^2}}.\tag{1.6}\] Since $t>0$, we take the plus sign. The correlation function becomes \[G(r) \sim \int^\infty_0 dt \, e^{-S(t_*) - (t-t_*)^2S''(t_*)/2}\sim \frac{e^{-S(t_*)}}{\sqrt{S''(t_*)}}.\tag{1.7}\] For $r\ll \xi$, we have $t_* \sim r^2$, and hence \[G(r) \sim r^{-(d-2)}.\tag{1.8}\] For $r \gg \xi$, we have $t_* \sim r\xi$, and hence \[G(r) \sim \frac{e^{-r/\xi}}{r^{(d-1)/2}}.\tag{1.9}\]

2 Sheet 2, Problem 2: The Two-Point Function as a Green Function

Verify that the correlation function satisfies \[(-\gamma\nabla^2+\mu^2)\langle\phi(x)\phi(0)\rangle \sim\delta(x),\tag{2.1}\] and explain its physical interpretation.

3 Sheet 2, Problem 5*: Scaling in a Lifshitz Theory

Separate $x=(x,\mathbf y)$ with $\mathbf y\in\mathbb R^{d-1}$, and consider the free energy \[F[\phi]=\int d^dx\left[ \frac12(\partial_x\phi)^2 +\frac12(\nabla_{\!y}^{2}\phi)^2 +\frac12\mu_0^2\phi^2\right].\tag{3.1}\] Write it in Fourier space using momenta $(k,\mathbf q)$. Under $k'=\zeta k$, $\mathbf q'=\zeta^a\mathbf q$, and $\phi'_{\mathbf{k}'}=\zeta^{-b}\phi_\mathbf{k}$, determine $a,b$ by normalising both gradient terms, and find the running $\mu^2(\zeta)$. In position space, obtain the Gaussian scaling dimension $\Delta_\phi$. For an interaction $\int d^dx\,g_n\phi^{2n}$, find the dimension of $g_n$ and show that $\phi^4$ is relevant for $d<7$ and irrelevant for $d>7$.

\[F[\phi_\mathbf{k}] = \int \frac{d^dk}{(2\pi)^d}\frac{1}{2}(k^2 + \mathbf{q}^4 + \mu_0^2) \phi_\mathbf{k}\phi_{-\mathbf{k}} = \int \frac{d^dk'}{(2\pi)^d}\zeta^{-1-a(d-1)}\frac{1}{2}(\zeta^{-2}k'^2 + \zeta^{-4a}\mathbf{q}^4 + \mu^2_0)\zeta^{2b}\phi'_{-\mathbf{k}'}\phi_{\mathbf{k}'}'.\tag{3.2}\] In order to normalise both gradient terms, we must have \[a = \frac{1}{2}, \quad b = \frac{d+5}{4}.\tag{3.3}\] Hence, we find \[\mu^2(\zeta) = \zeta^2\mu_0^2.\tag{3.4}\] In position space, we have \[\phi'(\mathbf{x}') = \int\frac{d^dk'}{(2\pi)^d}\phi_{\mathbf{k}'}'e^{ik'\cdot x'} = \zeta^{1+a(d-1)-b}\int \frac{d^dk}{(2\pi)^d}\phi_\mathbf{k}e^{ ik\cdot x} = \zeta^{\Delta_\phi}\phi(\mathbf{x}) \implies \Delta_\phi = \frac{d-3}{4}.\tag{3.5}\] For an interaction, we have \[\int d^dx g_n \phi^{2n}(\mathbf{x}) = \int d^dx'\zeta^{1+a(d-1)}g_n\zeta^{-2n\Delta_\phi}\phi'^{2n}(\mathbf{x}') = \int d^dx'g_n(\zeta)\phi'^{2n}(\mathbf{x}'),\tag{3.6}\] which implies \[g_n(\zeta) = \zeta^{-n(d-3)/2 +d/2+1/2}g_n.\tag{3.7}\] For $n=2$, we have \[g_2(\zeta) = \zeta^{-(d-7)/2}g_2,\tag{3.8}\] and hence it is relevant for $d<7$.

4 Sheet 2, Problem 6: Scalar--Gauge Coupling

For a complex scalar and gauge field, use \[F[\psi,A_i]=\int d^dx\left[ \frac14F_{ij}F^{ij} +|\partial_i\psi-ieA_i\psi|^2+\mu^2|\psi|^2\right], \qquad F_{ij}=\partial_iA_j-\partial_jA_i.\tag{4.1}\] Find the critical dimension $d_c$ below which the coupling $e$ is relevant and above which it is irrelevant. Discuss what may happen at $d=d_c$. With a term $g|\psi|^4$, the $d=3$ case is the Ginzburg--Landau theory of a superconductor.

5 Sheet 2, Problem 7: Emergent Rotational Symmetry

A microscopic cubic lattice in $d$ dimensions has a local order parameter $\phi(x)$ with $\phi\mapsto-\phi$ symmetry. Identify the lowest-order interaction allowed by the lattice rotations but forbidden by full $SO(d)$ rotations. Use scaling to explain why full rotational invariance can emerge at long distances.

6 Sheet 2, Problem 8: Wick's Identity for a Gaussian Ensemble

Let $G$ be a symmetric, positive-definite, invertible $n\times n$ matrix, and set \[\langle f(\phi)\rangle =\frac1{\mathcal N}\int_{\mathbb R^n}d^n\phi\, f(\phi)e^{-\phi\cdot G^{-1}\phi/2}, \qquad \mathcal N=\det(2\pi G)^{1/2}.\tag{6.1}\] Prove $\langle\phi_a\phi_b\rangle=G_{ab}$ and, for constant $B_a$, \[\left\langle e^{B_a\phi_a}\right\rangle =\exp\!\left[\frac12B_a\langle\phi_a\phi_b\rangle B_b\right].\tag{6.2}\] Expand both sides to obtain all three pairings in $\langle\phi_a\phi_b\phi_c\phi_d\rangle$. Show that odd moments vanish and give the general even-moment pairing formula.