The Hawking-Unruh phenomenon on graphene

A. Altland and B. Simons, Condensed matter field theory, Cambridge Univ. Press (Cambridge) 2006

2006

Cited alongside, same era.

The indices μ , ν = 0 , 1 , 2 \mu,\nu=0,1,2 respond to diffeomorphisms (Einstein indices), while a , b = 0 , 1 , 2 a,b=0,1,2 respond to flat space transformations (Lorentz indices), η a ​ b = diag ⁡ ( + 1 , − 1 , − 1 ) \eta_{ab}={\rm diag}(+1,-1,-1) , g = det g μ ​ ν \sqrt{g}=\sqrt{\det g_{\mu\nu}} and the diffeomorphic covariant derivative is ∇ μ = ( ∂ μ + 1 2 ω μ b ​ c J b ​ c ) \nabla_{\mu}=(\partial_{\mu}+\frac{1}{2}\omega_{\mu}^{\;bc}J_{bc}) , with J a ​ b = 1 4 ​ [ γ a , γ b ] J^{ab}=\frac{1}{4}[\gamma^{a},\gamma^{b}] , and ω μ b a = e λ a ( δ ν λ ∂ μ + Γ μ ​ ν λ ) E b ν {\omega_{\mu}}^{a}_{\;b}=e^{a}_{\lambda}(\delta^{\lambda}_{\nu}\partial_{\mu}+\Gamma_{\mu\nu}^{\lambda})E^{\nu}_{b} is the spin connection, where Γ μ ​ ν λ \Gamma_{\mu\nu}^{\lambda} is the Christoffel connection, e μ a e^{a}_{\mu} the Vielbein and E a μ E_{a}^{\mu} its inverse, η a ​ b ​ e μ a ​ e ν b = g μ ​ ν \eta_{ab}e^{a}_{\mu}e^{b}_{\nu}=g_{\mu\nu} , e μ a ​ E a ν = δ μ ν e^{a}_{\mu}E_{a}^{\nu}=\delta_{\mu}^{\nu} , e μ a ​ E b μ = δ b a e^{a}_{\mu}E_{b}^{\mu}=\delta_{b}^{a} . Our Riemann tensor is R ρ λ ​ μ ​ ν = ∂ [ ν Γ ρ μ ] λ + Γ ρ [ ν σ Γ σ μ ] λ {R^{\rho}}_{\lambda\mu\nu}=\partial_{[\nu}{\Gamma^{\rho}}_{\mu]\lambda}+{\Gamma^{\rho}}_{[\nu\sigma}{\Gamma^{\sigma}}_{\mu]\lambda} , torsion T μ ​ ν λ = Γ [ μ ​ ν ] λ T^{\lambda}_{\;\;\mu\nu}=\Gamma^{\lambda}_{\;\;[\mu\nu]} is zero. For more expressions see [ 13 ]

Cited in the paper.

The 𝒬 μ {\cal Q}^{\mu} for the sphere are not easy to envisage. Surfaces with constant negative curvature, 𝒦 < 0 {\cal K}<0 (that, due to Hilbert theorem, have singularities [ 20 ] ) are a better candidate because (in spatial isothermal coordinates) d ​ s graphene 2 = r 2 y ~ 2 ​ [ y ~ 2 r 2 ​ d ​ t 2 − d ​ x ~ 2 − d ​ y ~ 2 ] ds^{2}_{\rm graphene}=\frac{r^{2}}{{\tilde{y}}^{2}}\left[\frac{{\tilde{y}}^{2}}{r^{2}}dt^{2}-d{\tilde{x}}^{2}-d{\tilde{y}}^{2}\right] where x ~ , y ~ {\tilde{x}},{\tilde{y}} are the abstract

Cited in the paper.

This is quite a general feature of the power spectrum (not of the expectation value of the number operator!), whose physical origins are not understood as yet, but that clearly depends upon the alternating sign in ( 5

Cited in the paper.