Hierarchical growth of entangled states or ‘s-sourcery’ Context I H = ⊗x Hx I H= I P Hx hamiltonian ‘motif’ (rules out many horrible pathologies). support of Hx is localized. families of systems labelled by (linear) system size L: HL with groundstate(s) {|ψL i} x efficiently-findable U which contructs the groundstate from smaller unentangled subsystems : Coarsely-stated, impossible desideratum: ?? |ψL i = U|0i⊗L Warmup example (d = 1, s = 0): adiabatically deform 1d band insulator to product state ? Construct: U = Pe i R1 0 dηH(η) There are two problems with this plan, in general ? Construct: U = Pe i R1 0 dηH(η) 1. (Technical, solvable) Even if H(η) all have gap ≥ ∆ > 0, adiabatic evolution has a nonzero failure probability (per unit time, per unit volume). Solution : [Hastings, Wen] K = −i Find quasilocal K such that i∂η |ψ(η)i = K(η)|ψ(η)i R∞ dtF (t)e iH(η)t ∂η H(η)e −iH(η)t −∞ Produce quasi-local U = e i R1 0 dηK(η) . F (t) odd, rapidly decaying, F̃ (0) = 0, F̃ (ω) = − ω1 , |ω| ≥ ∆. Quasilocal means: P P U = e iK , K = x Kx , Kx = r Kx,r Kx,r supported on disk of radius r , || Kx,r || ≤ e −r 1−d 2. (Crucial, physical) Nontrivial states of matter are defined by the inability to find such a gapped path to a product state! Expanding universe strategy Instead, we are going to grow the system |ψL i → |ψ2L i with local unitaries. U ∼ · · · ◦ U4L0 ←2L0 ◦ U2L0 ←L0 U will in general not have finite depth. but U will have an RG structure. Assumptions: I Raw material: a bath of ‘ancillas’ ⊗|0iM is freely available. I For rigorous results, energy gap ∆ for all excitations. I There may be groundstate degeneracy G (HL ) but the groundstates are locally indistinguishable (a necessary condition for the state to be stable) d = 1, s = 1 example: (Not like crystal growth!) |ψ2L i = U |ψL i ⊗ |0iL . An s-source RG fixed point is a system whose groundstate on (2L)d sites can be made from the groundstate on Ld sites (plus unentangled ancillas ) using a quasilocal unitary. (in d dimensions)   |ψ2L i = U |ψL i · · · |ψL i ⊗|0iM  | {z } s M = Ld (2d − s) How to construct U Construct U by quasiadiabatic evolution : (For s = 1 we must start with s = 1 copy at size L.) Produce quasi-local U . [Hastings, Wen] Reminder: quasilocal means: U = e iK , K = P x Kx , Kx = P r Kx,r Kx,r supported on disk of radius r , || Kx,r || ≤ e −r 1−d Basic property: Recursive entropy bounds: (Uses Small Incremental Entangling result of [Kitaev, Bravyi, van Acoleyn-Marien-Verstraete 2014].) S(2R) ≤ sS(R) + kR d−1 S(2R) ≥ sS(R) − k 0 R d−1 Why is s-source RG fixed point a useful notion? 1. Such a circuit controls the growth of entanglement with system size: Area law theorem: any s ≤ 1 fixed point in d > 1 enjoys an area law for EE of subregions. S(A) ≡ −trρA log ρA ≤ k|∂A| = kR d−1 . s ≥ 2d−1 is required to violate the area law. 2. The groundstate degeneracy satisfies: G (2L) = G (L)s 3. s (smallest possible) is a property of the phase (since by definition an adiabatic path connects any two representatives) =⇒ classification axis. 4. The circuit implies a MERA representation of the groundstate. Many interesting states are s-source fixed points • Mean field symmetry-breaking states • Chern insulators, IQH (s = 0) (s = 1) • Topological states (discrete gauge theory, fractional QH), including chiral ones (s = 1) • Any topological quantum liquid ≡ insensitive to smooth deformations of space ' gapped QFT has s = 1. Why: place it in an expanding universe ds 2 = −dη 2 + a(η)2 d~x 2 Experimental example: QCD I Our universe is expanding, tdoubling ∼ 1010 years. I The QCD gap stays open (mπ , mp > 0). I This is a gapped path from |ψL i to |ψ2L i . I =⇒ ∃ a quasilocal unitary which constructs the QCD groundstate from a small cluster plus ancillas. (i.e. QCD has s = 1 ). This suggests a new approach to simulating its groundstate which is in principle very efficient. Reason to care #3: Classification of gapped states by s ← e.g.: Layers of FQHE or Haah’s cubic code d = 3, s = 2 =⇒ no continuum description! extensive GSD!