Ineffective Theory

The Spectral Density Function of a Correlator

The correlation function, at temperature $\beta^{-1}$, of an operator $\mathcal O$ is defined to be $$ C(t) \equiv \langle \mathcal O(t) \mathcal O(0)\rangle = \mathop{\mathrm{Tr}} e^{-\beta H} e^{i H t} \mathcal O e^{-i H t} \mathcal O \text, $$ where $H$ is the Hamiltonian and we can safely ignore the normalization of $e^{-\beta H}$ (because we’ll only ever consider the one temperature).

This is often referred to specifically as the Minkowski or real-time correlation function (sometimes the retarded Green’s function if you’re feeling fancy), to contrast with the Euclidean/imaginary-time correlation function, which is much easier to extract in lattice calculations. $$ C_E(\tau) \equiv \langle \mathcal O(\tau) \mathcal O(0)\rangle = \mathop{\mathrm{Tr}} e^{-\beta H} e^{\tau H} \mathcal O e^{-\tau H} \mathcal O $$ The Euclidean correlator, written this way, is defined only for $0 \le \tau \le \beta$.

These two correlators are obviously connected by analytic continuation: $C_E(\tau) = C(-i \tau)$. In fact they can both be obtained from a physically sensible underlying object termed the spectral density (or spectral function, or spectral density function to be safe).

First let’s look at the imaginary-time correlator. I’ll pretend we have a finite number of states all at different energies, since it doesn’t change any of the physics and allows me to use a manageable notation. The correlator can be expanded into a double sum of states like so: $$ C_E(\tau) = \sum_{E,E'} e^{-\beta E}e^{\tau E}e^{-\tau E'} \big|\langle E| \mathcal O |E' \rangle \big|^2 = \sum_{E,\omega} e^{-\beta E} e^{-\tau\omega} \big|\langle E| \mathcal O |E+\omega \rangle \big|^2 $$

Now let’s treat the real-time correlator in the same way. The expression is not very different, of course. $$ C(t) = \sum_{E,\omega} e^{-\beta E} e^{-i t \omega} \big|\langle E| \mathcal O |E+\omega \rangle \big|^2 $$

From the similarity of the expressions for $C(t)$ and $C_E(\tau)$, we see that it is reasonable to pull out a function $f(\omega)$, whose Fourier transform is the real-time correlator, and whose Laplace transform is the imaginary-time correlator. $$ f(\omega) \equiv \sum_E e^{-\beta E} \big|\langle E |\mathcal O | E + \omega\rangle\big|^2 $$

The original Euclidean correlation function was symmetric under $\tau \rightarrow \beta-\tau$. The symmetry manifests in $f(\omega)$ as $$ f(-\omega) = e^{-\beta\omega} f(\omega)\text. $$

It’s convenient not to work directly with $f(\omega)$, but rather a symmetrized form of it defined only on $\mathbb R_+$. $$ \rho(\omega) = \sum_E e^{-\beta E}\left[\big|\langle E | \mathcal O | E + \omega\rangle\big|^2 e^{-\beta\omega/2} + \big|\langle E | \mathcal O | E - \omega\rangle\big|^2 e^{\beta\omega/2}\right] $$ In terms of $\rho(\omega)$ or $f(\omega)$, the correlation function is given by $$ C(t) = \int_{-\infty}^\infty d\omega\; e^{-it\omega} f(\omega)= \int_0^\infty d\omega\; \cosh\Big(\frac{\beta\omega}{2} - i t \omega\Big) \rho(\omega) $$ Note, of course, that $\rho(\omega)$ is also a function of the temperature and chosen operator.

I haven’t actually said what the spectral function is yet! That’s because some people call $f(\cdot)$ the spectral function, some people call $\rho(\cdot)$ the spectral function, and there seem to be at least two other conventions in the literature that I can’t quite work out. Broadly speaking, any object that looks like, and contains the same information as, $\rho(\cdot)$, will be referred to as “the spectral function”. The key point is that it contains the underlying information about the operator from which the two-point function can be reconstructed, either in real or imaginary time.