Strong zero modes: What they are and what they’re good for Much Ado About MBL One fascinaKng aspect of the recent work on many-‐body localizaKon is the appearance of eigenstate phase transiKons. These transiKons are unconvenKonal (not “thermal”) in that they involve excited states – the ground state need not be aﬀected. One possible example is that spectral staKsKcs (spacing between energy levels) changes from Poisson to GOE. Another kind of eigenstate phase transiKon: Ordered/topological phase Disordered Quantum Ising/Majorana chain with open boundaries (6 sites) Degeneracies occur for all levels! (up to exponenKally small ﬁnite-‐size correcKons) F (-‐1) = 1 -‐1 The eigenstate and order/disorder (topological) transiKons here turn out to occur at the same place. 1 -‐1 The Ising/Majorana Hamiltonian A pictorial representaKon for free boundary condiKons: ψ 1 ψ 2 L−1 ψ2 L L−1 H = − ∑ u2 j−1 σ − ∑ u2 j σ σ x j j=1 2 L−1 z j j=1 = i ∑ ua ψ aψ a+1 a=1 z j+1 ψ 2L Extreme limits: • u2j = 0 (disordered in spin language): • u 2j-‐1 = 0 (ordered in spin language): The fermions on the edges, ψ 1 and ψ 2 L H when 1 u = 0. They commute with H! , do not appear in • These edge modes occur throughout the ordered/topological phase: the operator u u u Ψ =ψ1 + ψ 3 + 1 u2 1 3 u2 u4 ψ5 +… commutes with H up to exponenKally small terms coming from the far edge. Pfeuty 1970; Kitaev 2001 • Note this sKll works when the couplings are disordered. This strong zero mode guarantees the enKre spectrum is degenerate. A strong zero mode Ψ is a normalizable operator obeying exponenKally fast as the system size L → ∞ . F The degeneracies result because {(−1) ,Ψ} = 0 2 Ψ = const Fendley; Jermyn, Mong, Alicea and Fendley; see also review by Alicea and Fendley In Ising Ψ toggles between sectors, and is localized near the edge. [H ,Ψ] → 0 The strong zero mode results in the degeneracies: Ordered/topological phase Disordered maps between the paired levels In the disordered phase the strong zero mode is not normalizable, so no degeneracies F (-‐1) = 1 -‐1 1 -‐1 Norm is essenKal. • In the ordered/topological phase, Ψ is localized near the edge, and normalizable. • For simplicity, look at zero disorder: u2 j−1 ≡ h, u2 j ≡ J ✓ ◆2 h h = 1+ 3+ J J 5 + ... • Because of fermion anKcommutaKon, this squares to the idenKty operator! ✓ ◆2 ✓ ◆4 h h 2 + + ... =1+ J J 1 = for h < J 2 1 (h/J) Is this a free-‐fermionic ﬂuke? There seems to be a belief that if one adds interacKons, this beauKful structure is destroyed in the pure system. However, there is also good evidence that with enough disorder, the resulKng many-‐body localizaKon restores the strong zero mode. Huse et al; Bauer and Nayak; Bahri, Vosk, Altman and Vishwanath