Broken translation invariance in quasifree fermionic correlations out of equilibrium

Abstract

Using the C * algebraic scattering approach to study quasifree fermionic systems out of equilibrium in quantum statistical mechanics, we construct the nonequilibrium steady state in the isotropic XY chain whose translation invariance has been broken by a local magnetization and analyze the asymptotic behavior of the expectation value for a class of spatial correlation observables in this state. The effect of the breaking of translation invariance is twofold. Mathematically, the finite rank perturbation not only regularizes the scalar symbol of the invertible Toeplitz operator generating the leading order exponential decay but also gives rise to an additional trace class Hankel operator in the correlation determinant. Physically, in its decay rate, the nonequilibrium steady state exhibits a left mover-right mover structure affected by the scattering at the impurity.

A Rigorous Scattering Approach to Quasifree Fermionic Systems out of Equilibrium

“…by means of the unitary operatorf :ĥ →h defined in [5] on the momentum spaceĥ as (fϕ)(e) := [ϕ(arccos(e)), ϕ(− arccos(e))]/( √ 2π 4 1 − e 2 ). Since we want the XY Hamiltonian H to become the operator acting through multiplication by the free variable in H :=h ⊕h (i.e., we want to make use of the multiplication operator version of the spectral theorem), we extendf to F :=f ⊕θξf because of the 2nd factor in H, where, for all a ∈ L(h) and all A ∈ L(H), we setã :=fâf * ∈ L(h) and A := F A F * ∈ L( H), and we note that (θη)(e) = σ 1 η(e) and (ξη)(e) = σ 1 η(−e) for all η ∈h.…”

“…We know from stationary scattering theory that, if the limit of ε(R e−iε (H)G, R e−iε (H γ )F )/π for ε → 0 + exists for all F, G ∈ H and almost all e ∈ R (where here and in the following, the set of full measure in R may depend on F and G), the limit and the integration in (55) can be interchanged, the limit for ε → 0 + of the integrand in (55) equals the limit of In order to verify the existence in question, we use (58) and replace G in (58) by the term (ε/π)R e−iε (H)G. Since stationary scattering theory also guarantees the existence of the limits lim ε→0 + (G, R e±iε (H)F ) for almost all e ∈ R (the existence argument holds for any Hamiltonian), and since (ε/π)R e+iε (H)R e−iε (H) = (R e+iε (H) − R e−iε (H))/(2πi) due to the 1st resolvent identity, the limits of the 1st, 2nd, and 4th term on the right hand side of (58) exist. As for the 3rd term, we know from [5] that, for all e ∈ (−1, 1) and all x ∈ Z, the limit α e−i0 (x) := lim ε→0 + α e−iε (x) exists and has the form…”

“…In order to verify the existence in question, we use (58) and replace G in (58) by the term (ε/π)R e−iε (H)G. Since stationary scattering theory also guarantees the existence of the limits lim ε→0 + (G, R e±iε (H)F ) for almost all e ∈ R (the existence argument holds for any Hamiltonian), and since (ε/π)R e+iε (H)R e−iε (H) = (R e+iε (H) − R e−iε (H))/(2πi) due to the 1st resolvent identity, the limits of the 1st, 2nd, and 4th term on the right hand side of (58) exist. As for the 3rd term, we know from [5] that, for all e ∈ (−1, 1) and all x ∈ Z, the limit α e−i0 (x) := lim ε→0 + α e−iε (x) exists and has the form…”

“…by means of the unitary operator f : ĥ → h defined in [5] on the momentum space ĥ as ( fϕ)(e) := [ϕ(arccos(e)), ϕ(− arccos(e))]/( √ 2π 4 1 − e 2 ). Since we want the XY Hamiltonian H to become the operator acting through multiplication by the free variable in H := h ⊕ h (i.e., we want to make use of the multiplication operator version of the spectral theorem), we extend f to F := f ⊕ θ ξf because of the 2nd factor in H, where, for all a ∈ L(h) and all A ∈ L(H), we set ã := fâ f * ∈ L( h) and A := F A F * ∈ L( H), and we note that ( θη)(e) = σ 1 η(e) and ( ξη)(e) = σ 1 η(−e) for all η ∈ h. Hence, since ( ĥϕ)(k) = cos(k)ϕ(k) for all ϕ ∈ ĥ, the XY Hamiltonian H indeed becomes the desired multiplication operator under conjugation with F because ( hη)(e) = eη(e) and ( θ ξhθ ξη)(e) = −eη(e) for all η ∈ h, where we used that ( θ ξ) * = θ ξ and that, for all η = [η 1 , η 2 ] ∈ h, the adjoint of f has the form…”

On a quantum phase transition in a steady state out of equilibrium

“…In [7], we determined the action in momentum space of the wave operator (38) on the completely localized orthonormal Kronecker basis {δ x } x∈Z of h using the stationary scheme of scattering theory and the weak abelian form of the wave operator. Recall that the plane wave function is related to the Kronecker function by e x = fδ x for all x ∈ Z, where we used the notations introduced after (50).…”