18 This work was supported by DOE grant DE-FG02- 05ER46203 (GV) and DE-AC52-06NA25396 (JT) and by the IKERBASQUE Foundation. GX was supported by NSF of China under Grant No. 10704066 and 10974181. IVT acknowledges funding by the Span- ish MEC (FIS2007-65702-C02-01), “Grupos Consolida- dos UPV/EHU del Gobierno Vasco” (IT-319-07), and the European Community through e-I3 ETSF project (Con- tract Number 211956). GV gratefully acknowledges the kind hospitality of the ETSF in San Sebastian where this work was completed. We thank Dr. Stefano Pittalis for his help in calculating an plotting the curves shown in Fig. 1, and Dr. Paola Gori-Giorgi for kindly providing the code for calculating the structure factor of the elec- tron gas. From the evaluation of Eq. (20) at the Euclidean met- rics g$_{µν}$ = δ$_{µν}$ we get 18,19 where and Here ˆ Ψ( r ) is the field operator, is the diagonal two-particle density operator, and w ( r ) is the interaction potential. In this appendix we derive an identity which is used in Sec. II A to identify the right hand sides of Eqs. (36) and (38). Namely, we consider a functional S [ g$_{µν}$ ] of the following metric tensor and prove that the following equality holds where ρ ( ξ , ξ ′ ) is the exact ground state one-particle den- sity matrix, and g $^{µν}$( ξ ) and g ( ξ ) are, respectively, the inverse and the determinant of the metric tensor g$_{µν}$ ( ξ ) We start with the general nonlinear expression for the kinetic energy T [ u ] in the elastic approximation [see Eq. (50)] In this appendix we derive the linearized form of the kinetic energy Eqs. (51) for the instantaneously distorted ground state. Finally, to make a connection to Eqs. (36) and (38) we set S [ g$_{µν}$ ] = ˆ H$_{u}$ [ g$_{µν}$ ] and take the expectation value of Eq. (B2) in the state | ˜ ψ ( t ) 〉 . Then, linearizing with re- spect the displacement u ( ξ ), we find that the right hand side of Eq. (B2) becomes identical to the right hand side of Eq. (36), while the left hand side of Eq. (B2) is exactly equal to the right hand side of Eq. (38). The direct comparison of Eqs. (B3) and (B6) proves the announced identity (B2). Performing the partial integration in the right hand side of Eq. (B5), and using the symmetry of the tensor g$_{µν}$ we reduce Eq. (B5) to the following form Hence the variation of S can be also written as On other hand, the variation of r$_{µ}$ ( ξ ) induces the follow- ing variation of the metric tensor: g$_{µν}$ ( ξ ) ↦→ g$_{µν}$ ( ξ ) + δg$_{µν}$ ( ξ ), where To prove Eq. (B2) we consider a small variation of the function r$_{µ}$ ( ξ ): r$_{µ}$ ( ξ ) ↦→ r$_{µ}$ ( ξ )+ δr$_{µ}$ ( ξ ). The correspond- ing variation of the functional S takes the form Note that the identity relates the functional derivative of S with respect to the displacement to the functional derivative with respect to the metric/deformation tensor, which physically means a connection of the force to the stress. VII. ACKNOWLEDGEMENTS Appendix A: Stress tensor operator Appendix B: Derivation of the force identity Eq. (38) Appendix C: Derivation of Eq.(51) ˆ P$_{µν}$ = ˆ T$_{µν}$ + ˆ W$_{µν}$ , (A1) ˆ T$_{µν}$ = 1 2 m { ( ∂$_{µ}$ ˆ Ψ † )( ∂$_{ν}$ ˆ Ψ) + ( ∂$_{ν}$ ˆ Ψ † )( ∂$_{µ}$ ˆ Ψ) − 1 2 ∇ $^{2}$ˆ nδ$_{µν}$ } , (A2) ˆ W$_{µν}$ = − 1 2 ∫ d r ′ r ′ $_{µ}$r ′ ν r ′ ∂w ( r ′ ) ∂r ′ × ∫ 1 0 dλ ˆ ρ$_{2}$ ( r + λ r ′ , r − (1 − λ ) r ′ ) . (A3) ˆ ρ$_{2}$ ( r , r ′$_{) =}$ ˆ $_{Ψ}$†$_{(}$ r ) ˆ $_{Ψ}$†$_{(}$ r ′$_{)}$ ˆ Ψ( r ′$_{)}$ ˆ Ψ( r ) (A4) δS = ∫ d ξ δS δr$_{µ}$ δr$_{µ}$ ( ξ ) (B3) δg$_{µν}$ = ∂δr$_{α}$ ∂ξ$_{µ}$ ∂r$_{α}$ ∂ξ$_{ν}$ + ∂r$_{α}$ ∂ξ$_{µ}$ ∂δr$_{α}$ ∂ξ$_{ν}$ (B4) δS = ∫ d ξ δS δg$_{µν}$ δg$_{µν}$ ( ξ ) = d ξ δS δg$_{µν}$ ( ∂δr$_{α}$ ∂ξ$_{µ}$ ∂r$_{α}$ ∂ξ$_{ν}$ + ∂r$_{α}$ ∂ξ$_{µ}$ ∂δr$_{α}$ ∂ξ$_{ν}$ ) (B5) δS = − ∫ d ξ 2 ∂ ∂ξ$_{α}$ ( ∂r$_{µ}$ ∂ξ$_{β}$ δS δg$_{αβ}$ ) δr$_{µ}$ ( ξ ) (B6) g$_{µν}$ ( ξ , t ) = ∂r$_{α}$ ∂ξ$_{µ}$ ∂r$_{α}$ ∂ξ$_{ν}$ , r$_{α}$ ( ξ ) = ξ$_{α}$ + u$_{α}$ ( ξ ) (B1) δS δr$_{µ}$ = − 2 ∂ ∂ξ$_{α}$ ( ∂r$_{µ}$ ∂ξ$_{β}$ δS δg$_{αβ}$ ) ≡ ∂ ∂ξ$_{α}$ ( ∂r$_{µ}$ ∂ξ$_{β}$ √ gP αβ ) (B2) T [ u ] = 1 2 m ∫ d ξ √ gg $^{µν}$∂$_{µ}$ $_{∂}$′ $_{ν}$[ $_{g}$− 1 / $^{4}$( r ) $_{g}$− 1 / $^{4}$( ξ ′$_{)}$$_{ρ}$$_{(}$ ξ , ξ ′$_{)]}$$_{ξ}$$_{=}$$_{ξ}$$_{′}$ , (C1)