that is the functional of the displacement u ( ξ ), which is defined by Eq. (B1) It is now straightforward to expand the right hand side of Eq. (C4) to the second order in u : Our aim is to expand the functional of Eq. (C1) to the second order in the displacement field, i. e., to the first non-vanishing contribution corresponding to the lin- earized theory. First we explicitly calculate the derivatives in the right hand side of Eq. (C1) and set ξ ′ = ξ . As a result Eq. (C1) reduces to the form Next we consider the covariant tensor g µν (the inverse of g$_{µν}$ ): where n$_{0}$ ( ξ ) = ρ ( ξ , ξ ) is the ground state density, and Making use of the following representation for √ g , we can write the derivative of ln √ g as follows: The last term in Eq. (C8) can be identically represented as follows Note that the coefficient in front of the first term in Eq. (C9) and the corresponding coefficient in first term in Eq. (C8) are naturally combined into the kinetic stress tensor T$_{µν,}$$_{0}$ = 2 K$_{µν}$ − δ$_{µν}$ ∇ $^{2}$n$_{0}$ / 4 m . Hence, inserting Eq. (C9) into Eq. (C8) and integrating by parts terms proportional to ∂$_{ν}$n$_{0}$/ 2 m we arrive at the following final representation for the linearized kinetic energy of the distorted state: which is identical to Eq. (51). It is worth noting a con- venient feature of this representation – the last term in Eq. (C10) vanishes in all 1D systems and for homoge- neous systems with n$_{0}$ = const in any number of dimen- sions. In this appendix we prove Eq. (80) for one-particle systems in the absence of a magnetic field. The matrix elements of the current density operator is where ψ$_{0}$ ,..., ψ$_{n}$ are orthonormal eigenfunctions of the Finally, we substitute Eqs. (C5) and (C7) into Eq. (C2) and keep terms up to the second order in u . The second order contribution to T takes the form Expanding the inverse matrix in Eq. (C6) to the second order and expressing the result in terms of the strain tensor u$_{µν}$ we get 19 T [ u ] = ∫ d ξ g µν { K$_{µν}$ + n$_{0}$ 8 m ( ∂$_{µ}$ ln √ g )( ∂$_{ν}$ ln √ g ) − 1 8 m [ ( ∂$_{µ}$ ln √ g )( ∂$_{ν}$ n$_{0}$ ) + ( ∂$_{ν}$ ln √ g )( ∂$_{µ}$n$_{0}$ ) ] } , (C2) ∂$_{µ}$ ln √ g ≈ ∂$_{µ}$∂$_{α}$u$_{α}$ − ( ∂$_{β}$u$_{α}$ ) ∂$_{µ}$∂$_{α}$u$_{β}$ . (C5) g µν = [ δ$_{µν}$ + ∂$_{µ}$u$_{ν}$ + ∂$_{ν}$u$_{µ}$ + ( ∂$_{µ}$u$_{α}$ )( ∂$_{ν}$ u$_{α}$ ) ] − 1 . (C6) K$_{µν}$ ( ξ ) = 1 2 m [ ∂$_{µ}$$_{∂}$′ $_{ν}$ρ ( ξ , ξ ′$_{)]}$$_{ξ}$$_{=}$$_{ξ}$$_{′}$$_{.}$ (C3) g µν ≈ δ$_{µν}$ − 2 u$_{µν}$ + 4 u$_{µα}$u$_{να}$ − ( ∂$_{µ}$u$_{α}$ )( ∂$_{ν}$ u$_{α}$ ) . (C7) √ g = det ( ∂r$_{α}$ ∂ξ$_{β}$ ) , ∂$_{µ}$ ln √ g = ∂$_{µ}$ ln det ( ∂r$_{α}$ ∂ξ$_{β}$ ) = ∂ξ$_{α}$ ∂r$_{β}$ ∂$_{µ}$ ∂r$_{β}$ ∂ξ$_{α}$ . (C4) T$_{2}$ = ∫ d ξ { K$_{µν}$ [4 u$_{µα}$u$_{να}$ − ( ∂$_{µ}$u$_{α}$ )( ∂$_{ν}$ u$_{α}$ )] + n$_{0}$ 8 m ( ∂$_{µ}$u$_{αα}$ )( ∂$_{µ}$u$_{νν}$ ) + ∂$_{ν}$n$_{0}$ 2 m u$_{µν}$ ∂$_{µ}$u$_{αα}$ + ∂$_{ν}$n$_{0}$ 4 m ( ∂$_{µ}$u$_{α}$ ) ∂$_{ν}$ ( ∂$_{α}$u$_{µ}$ ) } . (C8) ∂$_{ν}$n$_{0}$ 4 m ( ∂$_{µ}$u$_{α}$ ) ∂$_{ν}$ ( ∂$_{α}$u$_{µ}$ ) = − ∇ $^{2}$n$_{0}$ 8 m [4 u$_{µα}$u$_{µα}$ − ( ∂$_{µ}$u$_{α}$ )( ∂$_{µ}$u$_{α}$ )] − ∂$_{ν}$ n$_{0}$ 2 m u$_{µα}$∂$_{ν}$u$_{µα}$ . (C9) T$_{2}$ = ∫ d ξ { 1 2 T$_{µν,}$$_{0}$ [4 u$_{µα}$u$_{να}$ − ( ∂$_{µ}$u$_{α}$ )( ∂$_{ν}$ u$_{α}$ )] + n$_{0}$ 8 m ( ∂$_{µ}$u$_{αα}$ )( ∂$_{µ}$u$_{νν}$ ) + n$_{0}$ 4 m [ ∂$_{µ}$u$_{να}$∂$_{µ}$u$_{να}$ − ∂$_{µ}$u$_{νµ}$∂$_{µ}$u$_{αα}$ ] , (C10) Appendix D: Proof of the orthonormality relation for single-particle transition currents j$_{0}$$_{n}$ ( r ) = 〈 0 | ˆ j ( r ) | n 〉 = − i 2 m [ ψ$_{0}$ ∇ ψ$_{n}$ − ψ$_{n}$ ∇ ψ$_{0}$ ] , (D1)