Entanglement Viscosities in QFT

• Entanglement viscosities are defined as shear and bulk viscosities emerging from quantum entanglement across the Rindler horizon, leading to thermal behavior at the Unruh temperature.
• Kubo formulas and spectral representations underpin their derivation, with explicit calculations yielding a global η/s ratio of 1/(4π) that mirrors the KSS bound.
• These coefficients connect conformal anomalies to emergent hydrodynamics, suggesting experimental probes in high-energy collisions and condensed matter analogs.

Entanglement viscosity refers to hydrodynamic-like transport coefficients—specifically, shear and bulk viscosities—emerging in the vacuum of quantum field theory (QFT) when the theory is restricted to a subsystem separated by a horizon, typically the Rindler horizon in Minkowski space. In this context, the Minkowski vacuum, as seen by an observer under constant acceleration, behaves as a thermal ensemble at the Unruh temperature and exhibits finite viscosity. The origin and universal properties of these "entanglement viscosities" are rooted in quantum entanglement across the horizon, and their mathematical structure is controlled by basic principles such as linear response theory, spectral representations, and quantum anomalies (Lapygin et al., 25 Feb 2025, Prokhorov, 5 Jan 2026).

1. Definition and Physical Context

Entanglement viscosities arise when correlations across an event horizon, such as the Rindler horizon, induce dissipative responses in the subsystem accessible to an accelerated observer. In the Unruh effect, an observer with proper acceleration $a$ in Minkowski space perceives the vacuum as a thermal state at $T_U = a/(2\pi)$. The causal boundary imposed by the Rindler horizon effectively partitions the global vacuum, and entanglement between the accessible and inaccessible regions leads to nonzero retarded correlators of stress–energy components. These correlators, in linear-response (Kubo) theory, yield finite viscosity coefficients—even for free fields. The "entanglement viscosity" is thus defined as the shear viscosity of the Unruh thermal bath, which is a direct consequence of entanglement rather than thermalization from interactions (Lapygin et al., 25 Feb 2025).

2. Kubo Formulas and Spectral Representation

The formalism for entanglement viscosities employs standard Kubo formulas, generalized to curved backgrounds:

$$\eta_{\text{ent}} = \lim_{\omega \to 0} \frac{1}{\omega} \operatorname{Im} G^R_{T^{xy},T^{xy}}(\omega, \mathbf{k}=0)$$

$$\zeta_{\text{ent}} = \lim_{\omega \to 0} \frac{1}{9\omega} \operatorname{Im} G^R_{T^{ii},T^{jj}}(\omega, \mathbf{k}=0)$$

where $G^R$ is the retarded two-point function of energy–momentum tensor components. In the coordinate representation for the Rindler wedge, the viscosities at distance $\rho$ from the horizon take the spectral-integral forms:

$$\eta_{\text{ent}}(\rho) = k_{d}\, \rho \int_0^{\infty} d\mu\, c^{(2)}(\mu)\, \mu^2 K_0(\mu \rho)$$

$$\zeta_{\text{ent}}(\rho) = \frac{2k_{d}\, \rho}{(d-1)^2} \int_0^{\infty} d\mu\, c^{(0)}(\mu)\, \mu^2 K_0(\mu \rho)$$

Here $K_0$ is the modified Bessel function, $c^{(0,2)}(\mu)$ are spectral densities for spin-0 and spin-2 channels, and $k_d$ is a geometric constant. Positivity of $c^{(0,2)}(\mu)$ (implied by unitarity of the full QFT) guarantees non-negative shear and bulk entanglement viscosities, linking these coefficients to thermodynamic irreversibility for the Rindler-wedge subsystem. This spectral foundation closely parallels c-theorems in RG flows, where positivity of two-point functions underlies monotonically decreasing central charges (Prokhorov, 5 Jan 2026).

3. Explicit Results: Spin Dependence and Entropy Density

For specific free massless fields, direct calculation gives explicit expressions for shear viscosity and entropy density, both locally and globally. Key results for the entanglement viscosities and entropies (with UV stretched horizon cutoff $\rho = \epsilon$):

Field	$\eta$	$s$	$\eta/s$ (global)
Massless scalar	$1/(1440\pi^2\epsilon^2)$	$1/(180\pi\epsilon^2)$	$1/(4\pi)$
Massless Dirac	$1/(240\pi^2\epsilon^2)$	$1/(60\pi\epsilon^2)$	$1/(4\pi)$
Massless photon	$1/(120\pi^2\epsilon^2)$	$1/(30\pi\epsilon^2)$	$1/(4\pi)$

In each case, the viscosity and entropy are calculated from first principles: $\eta$ via the stress-tensor Kubo formula and $s$ via the thermodynamic relation $s_{\text{loc}} = (\partial p/\partial T)_a$ at the Unruh temperature. The global (area-averaged) ratio $\eta/s = 1/(4\pi)$ exactly saturates the Kovtun-Son-Starinets (KSS) bound, as demonstrated by explicit integration (Lapygin et al., 25 Feb 2025, Prokhorov, 5 Jan 2026).

4. Local Structure and Universal Scaling

The local ratio $\eta_{\text{loc}}/s_{\text{loc}}(\rho)$ is governed by a universal scaling function depending only on the dimensionless proper distance from the stretched horizon, $x = \rho/\epsilon$. The scaling function,

$$f(x) = \frac{x^4\left[x^4 + 4x^2 - 5 - 4(2x^2+1)\ln x\right]}{4\pi(x^2-1)^4}$$

interpolates between universal limiting values:

• At the stretched horizon ($\rho = \epsilon$, $x = 1$): $\eta_{\text{loc}}/s_{\text{loc}} = 1/(8\pi)$
• Far from the horizon ($\rho \gg \epsilon$, $x \to \infty$): $\eta_{\text{loc}}/s_{\text{loc}} \to 3/(4\pi)$

The function $f(x)$ is independent of spin for spin 0, $1/2$, and $1$ free fields, and encodes the transition from the horizon region ("membrane surface") to the bulk Rindler wedge. At distances $x \gtrsim 1.66$, $f(x)$ exceeds the global KSS value of $1/(4\pi)$ (Lapygin et al., 25 Feb 2025).

5. Relation to Conformal Anomalies and Quantum Field Theory Structure

For four-dimensional conformal field theories, the spectral densities in the viscosity formula have universal forms, with the vanishing of bulk viscosity as required by conformal invariance ($c^{(0)}(\mu)\propto \mu^2\delta(\mu)$). The shear viscosity is directly proportional to the central charge $C_T$, which itself is related to the Weyl anomaly coefficient $a$: $C_T = (640/\pi^2)a$. The entanglement viscosity in flat space is then a direct probe of the conformal anomaly:

$$\eta_{\text{ent}}(\rho) = 8 a \alpha^3\quad(\rho = 1/\alpha)$$

This provides a novel manifestation of the quantum gravitational anomaly in observable transport coefficients without requiring dynamical gravity or strong interactions (Prokhorov, 5 Jan 2026).

6. Thermodynamic Irreversibility and Physical Implications

The positivity of entanglement viscosities, traced to the positivity of spectral functions, guarantees non-decreasing entropy production in the accessible subsystem, paralleling the second law of thermodynamics. The physical interpretation is that tracing over degrees of freedom hidden behind the horizon induces dissipation-like effects in the remaining subsystem, manifest as viscosity. This mechanism is entirely kinematical and does not depend on real interactions, supporting the perspective that vacuum entanglement can endow spacetime with emergent hydrodynamic properties.

Globally, $\eta/s = 1/(4\pi)$ mirrors the bound found via AdS/CFT duality for strongly coupled quantum fluids, despite arising here from free fields in flat spacetime. Locally, the horizon region ("stretched membrane") displays an even smaller ratio, suggesting possible sub-KSS behavior in the microscopic "membrane" regime, with implications for black hole microphysics and the membrane paradigm (Lapygin et al., 25 Feb 2025).

7. Experimental Prospects and Broader Connections

Potential avenues for realizing or probing entanglement viscosities include:

• Ultra-relativistic heavy-ion collisions, where extreme accelerations may generate Unruh-like environments and allow for phenomenological inference of viscous damping linked to anomaly coefficients.
• Engineered acceleration in cavity QED or atomic systems, enabling detection of viscous correlations in Unruh radiation.
• Solid-state analogs in systems such as Weyl or Dirac semimetals subjected to strain or rotation, mimicking horizon physics.

A plausible implication is that entanglement viscosities provide a robust, anomaly-linked signature of emergent irreversibility and could serve as a probe of fundamental QFT structure in both high-energy and condensed matter settings (Prokhorov, 5 Jan 2026).

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References

• "Viscosity, entanglement and acceleration" (Lapygin et al., 25 Feb 2025)
• "Entanglement Viscosity: from Unitarity to Irreversibility in Accelerated Frames" (Prokhorov, 5 Jan 2026)