Shear Viscosity/Entropy Ratio in Quantum Systems

• Shear viscosity to entropy density ratio is a dimensionless measure that quantifies momentum diffusion relative to microscopic state count, serving as a benchmark for perfect fluidity.
• Theoretical analyses leverage holographic models and Kubo formulas, linking shear viscosity to retarded correlators and spectral densities in quantum many-body systems.
• Experimental and lattice studies reveal that this ratio approaches a universal lower bound, highlighting its significance in characterizing quantum criticality and phase transitions.

The shear viscosity coefficient to entropy density ratio, typically denoted as $\eta/s$, is a central dimensionless measure of fluidity in quantum many-body systems, extensively studied in both condensed matter and high-energy physics. This ratio quantifies the efficacy of momentum transport (dissipative diffusion) relative to the number of accessible microscopic states (entropy density) and has emerged as a universal diagnostic in the search for “perfect fluids,” quantum critical transport, and the phenomenology of the quark–gluon plasma.

1. Definition and Physical Significance

Shear viscosity $\eta$ characterizes the relaxation of transverse momentum under a velocity gradient: for a hydrodynamic velocity field $u^i$, the stress tensor acquires a correction $T^{xy} = -\eta\,\partial_x u^y$ (Chiofalo et al., 7 Sep 2025). In quantum field theory, $\eta$ is extracted from the low-frequency, low-momentum limit of the retarded Green's function of the stress tensor via the Kubo formula: $$\eta = \lim_{\omega\to0}\frac{1}{\omega}\,\mathrm{Im}\,G^R_{T^{xy}T^{xy}}(\omega, \mathbf{k}=0).$$ The entropy density $s$ is defined thermodynamically as $s = (ε + P)/T$ for energy density $\epsilon$ and pressure $P$ at temperature $T$ (Horváth et al., 2015, 0704.1801). The ratio $\eta/s$ is dimensionless (in natural units) and functions as a “quantum Reynolds number.” Small $\eta/s$ implies the system is highly efficient at dissipating momentum compared to the number of available states; in the limit $\eta/s \to 0$, transport is as “perfect” as quantum mechanics allows.

2. Theoretical Frameworks and Kubo Relations

Theoretical treatments of $\eta/s$ are rooted in linear response and spectral representations. The retarded correlator is linked to the spectral density $\rho(\omega,\mathbf{p})$ of the relevant stress-tensor channel: $$\eta = \lim_{\omega\to0} \frac{1}{\omega} \rho_{T^{12}T^{12}}(\omega,0),$$ where $\rho$ encodes the density of states and the structure of excitations, including quasi-particles, finite-width resonances, and continuum multiparticle contributions (Horváth et al., 2015, Jakovac, 2011). For systems with well-defined quasi-particles, kinetic theory approaches based on Boltzmann transport and relaxation time approximations provide

$$\eta \sim \int d^3p\, \tau(p)\, p^4\, f_\mathrm{eq}(p),$$

with $\tau(p)$ the mean free path or lifetime; the entropy $s$ typically follows from equilibrium statistical mechanics.

The robustness of $\eta/s$ in the presence of strongly interacting, short-lived, or continuum-dominated spectral functions is a subject of ongoing paper. Notably, broad spectral distributions can lower $\eta/s$ well below quasi-particle estimates, reflecting enhanced many-body scattering and non-quasi-particle dynamics (Horváth et al., 2015, Jakovac, 2011).

3. The KSS Bound and Holography

A seminal result from the AdS/CFT correspondence is the Kovtun–Son–Starinets (KSS) lower bound: for a large class of strongly coupled quantum field theories with Einstein gravity duals,

$\frac{\eta}{s} \geq \frac{1}{4\pi}$

in units $\hbar = k_B = 1$ (Chiofalo et al., 7 Sep 2025, Cremonini, 2011). This bound is saturated for $\mathcal{N}=4$ supersymmetric Yang-Mills at infinite 't Hooft coupling and large $N$ limit via the identification of $\eta$ with the low-energy absorption cross-section of bulk gravitons and $s$ with the Bekenstein–Hawking entropy density of black-brane horizons. The universality of $1/(4\pi)$ as a lower bound arises from minimal coupling and regularity at the horizon in the bulk; more generally, in isotropic geometries, only the mixed-index shear tensor to entropy density ratio, $\eta^j_{\ i}{}^j_{\ i}/s$, remains universal (Mamo, 2012).

Modifications beyond Einstein gravity, such as higher-derivative (e.g., Gauss–Bonnet) corrections, finite $\lambda$ or $N$, or anisotropy, yield

$$\frac{\eta}{s} = \frac{1}{4\pi}(1 + \Delta[\lambda,N,\mathrm{curvature},\ldots])$$

where $\Delta$ can be negative, allowing for controlled violations of the KSS bound; however, physical consistency demands (from causality, positivity of energy, etc.) that $\eta/s$ remains positive and often imposes new, albeit weaker, lower bounds such as $4/(25\pi)$ in certain higher-derivative models (Cremonini, 2011, Chiofalo et al., 7 Sep 2025).

4. Models Beyond Quasi-Particle Approximations and Variational Bounds

Effective field theory treatments in the extended quasi-particle framework explicitly relate both $\eta$ and $s$ to the full spectral function $\rho(\omega,p)$, capturing the effects of broad, multi-particle continua (Horváth et al., 2015, Jakovac, 2011). In such models, formulas for these observables are

$$s = T^{-3} \int dp\, g(p,T)\, \chi_s(p/T), \quad \eta = T^{-3} \int dp\, g(p,T)^2\, \lambda_\eta(p/T)$$

for effective mass function $g(p,T)$ and known weights $\chi_s, \lambda_\eta$. Minimizing $\eta$ at fixed $s$ yields a non-universal bound

$\frac{\eta}{s} \geq \frac{1}{48}\frac{s}{T^3}$

which is set by the entropy density. The presence of strong continuum admixture (as in large-width Lorentzian resonances or multi-particle spectral cuts) monotonically depresses $\eta/s$, and in the limit of dominant continuum support, the ratio can be driven arbitrarily small, violating the KSS bound (Horváth et al., 2015, Jakovac, 2011).

Explicitly, for a Lorentzian spectral function $\rho_L(\omega,p)$ of width $\gamma$, the ratio

$$\frac{\eta_L}{s_L} = \frac{5}{4\pi^2}\frac{\gamma}{T} + \frac{1}{5}\frac{T}{\gamma},$$

with a global minimum $1/\pi$, which can fall below $1/(4\pi)$ as continuum weight increases.

5. Lattice QCD, Nuclear, and Experimental Insights

Nonperturbative calculations in pure-gauge SU(2) and SU(3) lattice gauge theories using multi-level algorithms and spectral-reconstruction methods yield

$\frac{\eta}{s}\sim 0.10-0.13$

in a temperature range $T \sim 1.2$–$1.65\,T_c$, close to the KSS value (0704.1801, Astrakhantsev et al., 2015). These reflect the near-“perfect” fluidity of non-Abelian plasmas just above deconfinement, consistent with the small $\eta/s$ needed to explain collective flow in heavy-ion collisions.

In nuclear systems, empirical extraction from giant dipole resonance widths, the extended statistical multifragmentation model, and kinetic transport frameworks finds $\eta/s$ minima near $2$–$4$ (in units of $\hbar /4\pi k_B$) for hot nuclei and nuclear matter close to the liquid–gas critical point (Dang, 2011, Pal, 2010, Magner et al., 2016). Kinetic-theory descriptions also observe minima of $\eta/s$ at the phase transition, with the temperature dependence tracking that of ordinary fluids (e.g., water) near criticality (Pal, 2010, Magner et al., 2016, Chen et al., 2010).

In the context of the Hagedorn spectrum, inclusion of exponentially many heavy states leads to a sharp increase of $s$ and rapid suppression of $\eta/s$ near $T_H$, approaching, and in the infinite-spectrum limit formally violating, $1/(4\pi)$ (Rais et al., 2019).

Ultracold Fermi gases at unitarity provide another venue: local measurements show a shallow minimum $\eta/s \sim 0.4\,(\hbar/k_B)$ at the superfluid transition, above but of the same order as the KSS bound (Joseph et al., 2014).

6. Behavior Near Phase Transitions and Limitations

A pronounced minimum in $\eta/s$ is commonly observed in systems near phase transitions, such as the nuclear liquid–gas or the QCD chiral/deconfinement crossover (Horváth et al., 2015, Pal, 2010, Chen et al., 2010). However, this behavior is not universal; counterexamples exist where decoupled sectors or weakly interacting modes do not produce a minimum at the critical point (Chen et al., 2010). The existence and sharpness of the minimum depend on the strength of coupling between the order-parameter modes and the transport channels.

Temperature, density, and compositional dependence (e.g., in baryon-rich hadronic matter) can strongly modify the location and depth of the minimum. In models with first-order transitions, discontinuities in $\eta/s$ can occur. For extremely broad spectral functions or strong continuum, the ratio can be further suppressed at low temperature, suggesting no strictly universal quantum lower bound (Horváth et al., 2015, Jakovac, 2011).

7. Universal Versus Non-Universal Ratios

Recent work clarifies that only specific tensor components of the viscosity-to-entropy ratio remain universal even in holographic theories: the mixed-index shear, $\eta^j_{\ i}{}^j_{\ i}/s$, remains fixed at $1/(4\pi)$ for any isotropic metric, while other components, such as $\eta^{ijij}/s$ or $\eta_{ijij}/s$, depend on details of the bulk geometry and are non-universal (Mamo, 2012). This distinction is crucial when comparing holographic predictions to real systems, especially in the presence of anisotropy or finite-size (compact) effects.

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In summary, the shear viscosity coefficient to entropy density ratio $\eta/s$ serves as a fundamental fluidity benchmark linking transport properties, spectral structure, and many-body correlations. Its canonical lower bound of $1/(4\pi)$ is saturated in holographic Einstein gravity but can be violated or saturated from above or below depending on higher-derivative corrections, the presence of broad spectral continua, multiparticle excitations, or crossover physics. Lattice, nuclear, cold-atom, and condensed-matter experiments approach this bound but typically remain above it, underscoring the profound influence of microscopic many-body structure on macroscopic transport. Recent studies highlight that any truly universal bound must be qualified by the regime and the tensor structure examined, and that with sufficiently strong continuum spectral weight, no universal lower bound can be enforced by first principles alone (Horváth et al., 2015, Jakovac, 2011, Cremonini, 2011, Mamo, 2012, Chiofalo et al., 7 Sep 2025).