Musings

I finally got around to looking at the papers by Harvey Meyer and Sakai and Nakamura, and I’m a little underwhelmed. They do report very low values for $\eta/s$ from their measurements. But extracting $\eta$ from the data requires some rather dubious assumptions.

The basic point is that there are Kubo formulas which relate transport coefficients to quantities calculable in equilibrium Statistical Mechanics (which is what the lattice is good at).

For any gauge-invariant (bosonic) Hermitian local operator, $O$, define the spectral function $$\rho_O(\omega,\vec{p}) = \int d^4 x\, e^{i(\omega t -\vec{p}\cdot \vec{x})} {\langle [O(t,\vec{x}), O(0)]\rangle}_{th}$$ where ${\langle \dots\rangle}_{th}= \tfrac{1}{Z(T)}Tr\left(e^{-\beta H}\dots \right)$ is the real-time thermal correlation function. The spectral function obeys

A Kubo formula relates the shear viscosity, $\eta$, to the $\rho\to 0$ limit of the spectral function for a component of the traceless part of the stress tensor¹

There’s nothing special about $T_{12}$; Meyer actually uses $\tfrac{1}{2}(T_{11}-T_{22})$.

The same spectral function, in turn, is related to the Euclidean 2-point function

for $0\leq \tau \lt \beta$ by

via the kernel² $$K(\tau,\omega) = \frac{\cosh(\omega(\tau-\beta/2))}{\sinh(\beta\omega/2)}$$

So the program sounds simple:

1. Measure the Euclidean 2-point $G(\tau)\equiv \int d^3 x G^E_{T_{12}}(\tau,\vec{x})$, on the lattice.
2. Use it to reconstruct the spectral function (4).
3. Study the $\omega\to 0$ limit, (2), to recover the viscosity.

Unfortunately, on the lattice, we’re faced with the task of reconstructing a continuous function, $\rho(\omega)$, from a (small number of) discrete measurements of the lattice 2-point function. The lattices used by Meyer and by Sakai and Nakamura have 8 lattice points in the Euclidean time direction. Because of the symmetry $G(\tau)= G(\beta-\tau)$, this means there are only 4 independent data points.

One way to proceed is to assume some functional form for the spectral function, write down a family of trial functions with a few free parameters, and try to fit the data by adjusting those parameters.

The large-$\omega$ behaviour of the spectral function may well be captured by perturbation theory (and so we know the functional form). But we’re interested in the small-$\omega$ behaviour and it’s not at all clear what functional form one should assume there.

Sakai and Nakamura fit their data to a Breit-Wigner ansatz $$\rho(\omega)\equiv \tfrac{1}{2\pi}\rho_{T_{1 2}}(\omega,0) = \frac{A}{\pi} \left[\frac{\gamma}{(m-\omega)^2 +\gamma^2}-\frac{\gamma}{(m+\omega)^2 +\gamma^2}\right]$$ Meyer studies a variety of functional forms (and points out that Sakai and Nakamura’s Breit-Wigner ansatz doesn’t agree with perturbation theory at large-$\omega$).

There is a very fancy technique, called the Maximum Entropy Method for constructing a best-fit for the spectral function from a finite set of lattice measurements, using Bayesian Statistics (see Georg von Hippel’s post). But it’s not what these guys use (apparently, it requires a much finer lattice in the Euclidean time direction).

And I really wonder whether even MEM techniques are well-adapted to distinguishing the behaviour of the spectral function near zero.

To see the nature of the difficulty, consider the following very crude method for obtaining an upper bound on $\eta$, by integrating both sides of (4) $$\int_0^{\beta/2} G(\tau) d\tau = \int_0^\infty \frac{\rho(\omega)}{\omega} d\omega$$ The LHS involves only the crudest information, namely the sum of our measurements. If you assume you know $\rho(\omega)$ for $\omega\gt \omega_c$ (say, because you expect it to be given accurately by perturbation theory), then you have an upper bound on the integrated value of $\frac{\rho(\omega)}{\omega}$ between $0$ and $\omega_c$. Because of positivity (1), this tells you something about the behaviour near zero, but it doesn’t tell you much. You can’t exclude a narrow spike near $\omega=0$, which make a negligible contribution to the integral, but a large correction to $\eta$. Including more independent data points makes the problem better, but doesn’t make it go away.