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Register a new userShear Viscosity at Infinite Coupling: A Field Theory Calculation
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Authors
Paul Romatschke
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I derive an exact integral expression for the ratio of shear viscosity over entropy density $frac{eta}{s}$ for the massless (critical) O(N) model at large N with quartic interactions. The calculation is set up and performed entirely from the field theory side using a non-perturbative resummation scheme that captures all contributions to leading order in large N. In 2+1d, $frac{eta}{s}$ is evaluated numerically at all values of the coupling. For infinite coupling, I find $frac{eta}{s}simeq 0.42(1)times N$. I show that this strong coupling result for the viscosity is universal for a large class of interacting bosonic O(N) models.
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I consider quantum electrodynamics with many electrons in 2+1 space-time dimensions at finite temperature. The relevant dimensionless interaction parameter for this theory is the fine structure constant divided by the temperature. The theory is solvable at any value of the coupling, in particular for very weak (high temperature) and infinitely strong coupling (corresponding to the zero temperature limit). Concentrating on the photon, each of its physical degrees of freedom at infinite coupling only contributes half of the free-theory value to the entropy. These fractional degrees of freedom are reminiscent of what has been observed in other strongly coupled systems (such as N=4 SYM), and bear similarity to the fractional Quantum Hall effect, potentially suggesting connections between these phenomena. The results found for QED3 are fully consistent with the expectations from particle-vortex duality.
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The microscopic formulas for the shear viscosity $eta$, the bulk viscosity $zeta$, and the corresponding relaxation times $tau_pi$ and $tau_Pi$ of causal dissipative relativistic fluid-dynamics are obtained at finite temperature and chemical potential by using the projection operator method. The non-triviality of the finite chemical potential calculation is attributed to the arbitrariness of the operator definition for the bulk viscous pressure.We show that, when the operator definition for the bulk viscous pressure $Pi$ is appropriately chosen, the leading-order result of the ratio, $zeta$ over $tau_Pi$, coincides with the same ratio obtained at vanishing chemical potential. We further discuss the physical meaning of the time-convolutionless (TCL) approximation to the memory function, which is adopted to derive the main formulas. We show that the TCL approximation violates the time reversal symmetry appropriately and leads results consistent with the quantum master equation obtained by van Hove. Furthermore, this approximation can reproduce an exact relation for transport coefficients obtained by using the f-sum rule derived by Kadanoff and Martin. Our approach can reproduce also the result in Baier et al.(2008) Ref. cite{con} by taking into account the next-order correction to the TCL approximation, although this correction causes several problems.
We propose the sparse modeling method to estimate the spectral function from the smeared correlation functions. We give a description of how to obtain the shear viscosity from the correlation function of the renormalized energy-momentum tensor (EMT) measured by the gradient flow method ($C(t,tau)$) for the quenched QCD at finite temperature. The measurement of the renormalized EMT in the gradient flow method reduces a statistical uncertainty thanks to its property of the smearing. However, the smearing breaks the sum rule of the spectral function and the over-smeared data in the correlation function may have to be eliminated from the analyzing process of physical observables. In this work, we demonstrate that the sparse modeling analysis in the intermediate-representation basis (IR basis), which connects between the Matsubara frequency data and real frequency data. It works well even using very limited data of $C(t,tau)$ only in the fiducial window of the gradient flow. We utilize the ADMM algorithm which is useful to solve the LASSO problem under some constraints. We show that the obtained spectral function reproduces the input smeared correlation function at finite flow-time. Several systematic and statistical errors and the flow-time dependence are also discussed.
Recently, non-perturbative approximate solutions were presented that go beyond the well-known mean-field resummation. In this work, these non-perturbative approximations are used to calculate finite temperature equilibrium properties for scalar $phi^4$ theory in two dimensions such as the pressure, entropy density and speed of sound. Unlike traditional approaches, it is found that results are well-behaved for arbitrary temperature/coupling strength, are independent of the choice of the renormalization scale $barmu^2$, and are apparently converging as the resummation level is increased. Results also suggest the presence of a possible analytic cross-over from the high-temperature to the low-temperature regime based on the change in the thermal entropy density.
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