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December 3, 2007

QGP on the Lattice

I was chatting with John Harris, the former spokesman for the STAR Collaboration, a few weeks ago, when he mentioned to me some recent lattice calculations of η/s\eta/s, the ratio of shear viscosity to entropy density, in pure SU(3)SU(3) Yang-Mills.

This sounded really interesting.

The claim-to-fame of AdS/QGP is that it allows you to compute transport coefficients in a strongly-coupled gauge theory, something that lattice techniques are supposed to be no good at.

Most famous of all is the calculation of η/s\eta/s, which is exactly 14π\tfrac{1}{4\pi} in conformally-invariant AdS/CFT backgrounds, and 14π(1+O(1/λ))\tfrac{1}{4\pi}\left(1+O(1/\lambda)\right) in non-conformally-invariant backgrounds (with the leading 1/λ1/\lambda correction being positive). The quark gluon plasma studied at RHIC is the least viscous fluid known to man. STAR reports η/s0.1\eta/s \lesssim 0.1, more than 100 times smaller than that of water.

It would be pretty cool if lattice calculations could reproduce this. So what’s the trick?

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 η/s\eta/s from their measurements. But extracting η\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, OO, define the spectral function ρ O(ω,p)=d 4xe i(ωtpx)[O(t,x),O(0)] th \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 th=1Z(T)Tr(e βH){\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

(1)ρ O(ω,p) * =ρ O(ω,p) reality ρ O(ω,p)/ω 0 positivity ρ O(ω,p) =ρ O(ω,p) symmetry\begin{aligned} {\rho_O(\omega,\vec{p})}^* &= \rho_O(\omega,\vec{p}) &\text{reality}\\ \rho_O(\omega,\vec{p})/\omega &\geq 0&\text{positivity}\\ \rho_O(-\omega,\vec{p}) &= - \rho_O(\omega,\vec{p}) & \text{symmetry} \end{aligned}

A Kubo formula relates the shear viscosity, η\eta, to the ρ0\rho\to 0 limit of the spectral function for a component of the traceless part of the stress tensor1

(2)η=lim ω012ωρ T 12(ω,0)\eta = \lim_{\omega\to 0} \frac{1}{2\omega}\rho_{T_{12}}(\omega,0)

There’s nothing special about T 12T_{12}; Meyer actually uses 12(T 11T 22)\tfrac{1}{2}(T_{11}-T_{22}).

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

(3)G O E(τ,x)=O(τ,x),O(0) EG^E_O(\tau,\vec{x}) = {\langle O(\tau,\vec{x}),O(0)\rangle}_E

for 0τ<β0\leq \tau \lt \beta by

(4)d 3xG O E(τ,x)= 0 dω2πK(τ,ω)ρ O(ω,0)\int d^3x G^E_O(\tau,\vec{x}) = \int_0^\infty \frac{d\omega}{2\pi} K(\tau,\omega) \rho_O(\omega,0)

via the kernel2 K(τ,ω)=cosh(ω(τβ/2))sinh(βω/2) 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(τ)d 3xG T 12 E(τ,x)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 ω0\omega\to 0 limit, (2), to recover the viscosity.

Unfortunately, on the lattice, we’re faced with the task of reconstructing a continuous function, ρ(ω)\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(τ)=G(βτ)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-ω\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-ω\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 ρ(ω)12πρ T 12(ω,0)=Aπ[γ(mω) 2+γ 2γ(m+ω) 2+γ 2] \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-ω\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 η\eta, by integrating both sides of (4) 0 β/2G(τ)dτ= 0 ρ(ω)ωdω \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 ρ(ω)\rho(\omega) for ω>ω c\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 ρ(ω)ω\frac{\rho(\omega)}{\omega} between 00 and ω c\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 ω=0\omega=0, which make a negligible contribution to the integral, but a large correction to η\eta. Including more independent data points makes the problem better, but doesn’t make it go away.


1 The AdS/CFT computation also proceeds via Kubo’s formula. But there, the two-point function of the stress tensor is related to the cross section for absorbing a graviton in the dual gravitational description.

2 To relate this form to the more familiar expression involving a sum over Matsubara frequencies, ω n=2πn/β\omega_n = 2\pi n/\beta, one uses the identity ne iω nτsiω n=βe sτ1e βs \sum_n \frac{e^{-i\omega_n\tau}}{s - i \omega_n} = \beta \frac{e^{-s\tau}}{1-e^{-\beta s}}

Posted by distler at December 3, 2007 11:55 PM

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8 Comments & 0 Trackbacks

Re: QGP on the Lattice

The small euclidean-time extent of many finite-temperature lattices is a well-known problem. Anisotropic lattices with a much smaller lattice spacing in the time direction help, but of course there’s still the question of cost.

Note that what Meyer uses at the end of his paper sounds very much like a version of MEM, even though his description does not mention it.

As for the behaviour of the spectral function near zero frequency, at least in a MEM approach one should presumably use a default model that excludes such a peak. Of course, if the true spectral function should have such a peak (which would not appear to be terribly likely, though), the GIGO (garbage in, garbage out) principle hits, but at least one would obtain a low Bayesian probability for the model in this case (assuming the lattice data would allow to get any information on the peak; if it is beyond reach, it is beyond reach).

More on MEM for transport properties (conductivity in this case) can be found in hep-lat/0703008; more on problems with MEM at small temporal lattice extent in hep-lat/0510026.

Posted by: Georg on December 5, 2007 4:47 AM | Permalink | Reply to this

MEM

As for the behaviour of the spectral function near zero frequency, at least in a MEM approach one should presumably use a default model that excludes such a peak.

I would agree with you, except for the fact that perturbation theory produces just such a peak (and, indeed, the viscosity goes to \infty at zero coupling).

Normally, in the absence of a better idea, one would use the perturbative result as the default model. If you did that here, I suspect that you could very well find a decently high Bayesian probability for the resulting model. But it would yield a “large”, rather than a small value for η/s\eta/s.

Posted by: Jacques Distler on December 5, 2007 10:25 AM | Permalink | PGP Sig | Reply to this

Re: MEM

Well, yes, but that peak in perturbation theory is due to free gluons appearing in asymptotic states. We know that that is not the case in the real world. At low energies (and we are talking about the low-energy limit of the spectral function here), perturbation theory is not a particularly good guide in QCD.

Posted by: Georg on December 5, 2007 11:50 AM | Permalink | Reply to this

Re: QGP on the Lattice

Until a few years ago, our thinking on this issue was guided by perturbative QCD. In perturbative QCD the spectral density has a narrow peak near the origin, and extracting the shear viscosity from euclidean lattice data is essentially impossible. Now it seems that the (strong coupling) AdS/CFT prediction may be a much better (though far from perfect) guide to QCD near T_c. In AdS/CFT the spectral function is smooth, and extracting the shear viscosity appears feasible, even with the resources available today. So AdS/CFT gives lattice QCD a chance. (This does not mean that there is not a lot of work done be done beyond the “pioneering” papers that you mention.)

Posted by: Thomas on December 5, 2007 11:54 AM | Permalink | Reply to this

Re: QGP on the Lattice

Until a few years ago, our thinking on this issue was guided by perturbative QCD. In perturbative QCD the spectral density has a narrow peak near the origin … Now it seems that the (strong coupling) AdS/CFT prediction may be a much better (though far from perfect) guide to QCD near T cT_c. In AdS/CFT the spectral function is smooth, and extracting the shear viscosity appears feasible, even with the resources available today. So AdS/CFT gives lattice QCD a chance.

Which is to say that one should use the AdS/CFT result as the default model in an MEM analysis, rather than some bogus Breit-Wigner (or whatever) form.

But, either way, you’re putting in some ab-initio knowledge about the behaviour of the spectral function near zero, that you didn’t have before AdS/CFT came along.

That’s not necessarily bad, unless you’re determined to see this as a “competition” between AdS/CFT and lattice methods.

Posted by: Jacques Distler on December 5, 2007 12:32 PM | Permalink | PGP Sig | Reply to this

Re: QGP on the Lattice

>> is the least viscous fluid known to man

Does superfluid helium have a well-defined eta/s ratio? If so, what is it?

Thanks,

Posted by: Martin on December 7, 2007 10:27 AM | Permalink | Reply to this

Liquid Helium

Does superfluid helium have a well-defined eta/s ratio? If so, what is it?

Liquid helium, below the λ\lambda-point, is described as a 2-component liquid.

Some fraction of the helium atoms have condensed to the ground state and, therefore, cease to contribute to the thermodynamics (both η\eta and ss vanish).

The remaining, “normal,” component of the fluid has η/s\eta/s considerably greater than the QGP.

Posted by: Jacques Distler on December 7, 2007 11:43 AM | Permalink | PGP Sig | Reply to this

Re: Liquid Helium

“Some fraction of the helium atoms have condensed to the ground state and, therefore, cease to contribute to the thermodynamics (both ç and s vanish).”

Thanks for this explanation. Am I right to think that for the condensed fraction, where both ç and s vanish, one cannot sensibly define their ratio?

Posted by: Martin on December 9, 2007 1:02 PM | Permalink | Reply to this

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