A classification of gapped Hamiltonians in d = 1 Sven Bachmann Mathematisches Institut Ludwig-Maximilians-Universität München Joint work with Yoshiko Ogata NSF-CBMS school on quantum spin systems Sven Bachmann (LMU) Invariants of ground state phases UAB 2014 1 / 26
Quantum spin systems A lattice Γ of finite dimensional quantum systems (spins), with Hilbert space H Λ = x Λ H x, Λ Γ, finite Observables on Λ Γ: A Λ = L(H Λ ) Local Hamiltonian: a sum of short range interactions Φ(X) = Φ(X) A X H Λ = X Λ Φ(X) The Heisenberg dynamics: τ t Λ(A) = exp(ith Λ )A exp( ith Λ ) Sven Bachmann (LMU) Invariants of ground state phases UAB 2014 2 / 26
States The quasi-local algebra A Γ : A Γ = Λ Γ A Λ State ω: a positive, normalized, linear form on A Γ Finite volume Λ: A Λ = B(H Λ ) and ω(a) = Tr(ρ ω ΛA) where ρ ω Λ is a density matrix Infinite systems Γ: No density matrix in general But: Nets of states ω Λ on A Λ have weak-* accumulation points ω Γ as Λ Γ: states in the thermodynamic limit Sven Bachmann (LMU) Invariants of ground state phases UAB 2014 3 / 26
The Ising model Now: Γ = Z, H x = C 2, spin 1/2, translation invariant Φ: sσx 3 if X = {x} for some x Z, h 0 Φ(X, s) = σx 1 1 σ1 x if X = {x 1, x} for some x Z 0 otherwise N N 1 H [0,N 1] (s) = σx 1σ 1 x 1 s (anisotropic XY chain) x=1 x=0 σ 3 x s 1: Unique ground state, gapped s 1: Two ground states, gapped Decay of correlations: exponential everywhere, polynomial at s c In between: the spectral gap closes: order - disorder QPT Sven Bachmann (LMU) Invariants of ground state phases UAB 2014 4 / 26
Local vs topological order: physics Ordered phases non-unique ground state The usual picture: Local order parameter, e.g. ω(σ 3 0 ) Topological order : Local disorder, for A A X, X Λ, P Λ AP Λ C A 1 Cd(X, Λ) α, C A C, P Λ : The spectral projection associated to the ground state energy Cannot be smoothly deformed to a normal state Positive characterization of topological order (?) Topological degeneracy: if Λ g has genus g, then dimp Λg = f(g) Anyonic vacuum sectors Topological entanglement entropy Sven Bachmann (LMU) Invariants of ground state phases UAB 2014 5 / 26
What is a (quantum) phase transition? A simple answer: A phase transition without temperature but under a continuous change of a parameter: Qualitative change in the set of ground states, parametrized by a coupling Localization-delocalization, parametrized by the disorder Percolation, parametrized by the probability Bifurcations in PDEs Higgs mechanism, parametrized by the coupling... Sven Bachmann (LMU) Invariants of ground state phases UAB 2014 6 / 26
Ground state phases A slightly more precise answer: Consider: A smooth family of interactions Φ(s), s [0, 1] The associated Hamiltonians H Λ (s) = Φ(X, s) X Λ Spectral gap above the ground state energy γ Λ (s) such that { > 0 (s s c ) γ Λ (s) γ(s) C s s c µ (s s c ) QPT Associated singularity of the ground state projection P Λ (s) Basic question: What is a ground state phase? Sven Bachmann (LMU) Invariants of ground state phases UAB 2014 7 / 26
Stability H Λ (s) = X Λ ( Φ(X) + sψ(x) ) If Ψ(X) is local, i.e. Ψ(X) = 0 whenever X Λ c 0, then usually Dynamics τ t Γ,s as a perturbation of τ t Γ,0 Continuity of the spectral gap at s = 0 Local perturbation of ground states Equilibrium states: ω β,s ω β,0 κs as s 0 Return to equilibrium: ω β,s τ t Γ,0 ω β,0 as t For translation invariant perturbations: No general stability results, but Perturbations of classical Hamiltonians Perturbations of frustration-free Hamiltonians Sven Bachmann (LMU) Invariants of ground state phases UAB 2014 8 / 26
Automorphic equivalence Definition. Two gapped H, H are in the same phase if there is s Φ(s), C 0 and piecewise C 1, with Φ(0) = Φ, Φ(1) = Φ the Hamiltonians H(s) are uniformly gapped inf γ Λ(s) γ > 0 Λ Γ,s [0,1] The set of ground states on Γ: S Γ (s). Then there exists a continuous family of automorphism α s 1,s 2 Γ of A Γ α s 1,s 2 Γ S Γ (s 2 ) = S Γ (s 1 ) α s 1,s 2 Γ is local: satisfies a Lieb-Robinson bound Now: Invariants of the equivalence classes? Sven Bachmann (LMU) Invariants of ground state phases UAB 2014 9 / 26
Frustration-free Hamiltonians in d = 1 Now Γ = Z, and H x H = C n Consider spaces {G N } N N such that G N H N and G N = N m x=0 for some m N; intersection property H x G m H (N m x) Natural positive translation invariant interaction: G m projection onto G m { τ x (1 G m ) X = [x, x + m 1] Φ(X) = 0 otherwise By the intersection property: KerH [1,N] = G N, parent Hamiltonian Sven Bachmann (LMU) Invariants of ground state phases UAB 2014 10 / 26
Matrix product states Consider B = (B 1,..., B n ), B i M k and two projections p, q M k A CP map M k M k : B B n,k (p, q) if Ê B (a) = n B µ abµ 1. Spectral radius of ÊB is 1 and a non-degenerate eigenvalue 2. No peripheral spectrum: other eigenvalues have λ < 1 3. e B and ρ B : right and left eigenvectors of ÊB : pe B p and qρ B q invertible A map Γ k,b N,p,q : pm kq H N : Γ k,b N,p,q (a) = n µ 1,...µ N =1 µ=1 Tr(paqB µ N B µ 1 )ψ µ1 ψ µn Sven Bachmann (LMU) Invariants of ground state phases UAB 2014 11 / 26
Gapped parent Hamiltonian Notation: and parent Hamiltonian H k,b N,p,q. ( ) G k,b N,p,q = Ran Γ k,b N,p,q H N Proposition. Assume that G k,b N,p,q satisfies the intersection property. Then i. H k,b N,p,q is gapped ii. S Z (H,p,q) k,b = { ω B } iii. Let d L = dim(p), d R = dim(q). There are affine bijections: E (M dl ) S (, 1] (H k,b,p,q), E (M dr ) S [0, ) (H k,b,p,q) i.e. Unique ground state on Z, edge states determined by p, q Sven Bachmann (LMU) Invariants of ground state phases UAB 2014 12 / 26
Bulk state Given B, for A A {x}, n E B A(b) := ψ µ, Aψ ν B µ bbν µ,ν=1 Note: ÊB = E B 1 (b). ( ) ω (A B x A y ) = ρ B E B A x E B A y (e B ) ω B (Φ k,b m,p,q(x)) = 0: Ground state Exponential decay of correlations if σ(êb ) \ {1} {z C : z < 1} ( ) ω (A B x 1 y x 1 A y ) = ρ B E B A (ÊB ) y x 1 E B B(e) Sven Bachmann (LMU) Invariants of ground state phases UAB 2014 13 / 26
Edge states Note: ω (A B x A y ) extends to Z: ( ) ( ) ρ B E B A x E B A y (e B ) = ρ B E B 1 E B A x E B A y (E B 1 (e B )) For the same E B, ω B ϕ(a 0 A x ) := ϕ ( ( ) (pe B p) 1/2 p E B A 0 E B A x (e B ) p(pe B p) 1/2) for any state ϕ on pm k p, and ω B ϕ(φ k,b m,p,q(x)) = 0 These extend to the right, but not to the left: S [0, ) (H k,b,p,q) E(M dl ) Sven Bachmann (LMU) Invariants of ground state phases UAB 2014 14 / 26
A complete classification Theorem. Let H := H,p,q k,b and H := H k,b,p,q as in the proposition, with associated (d L, d R ), resp. (d L, d R ). Then, H H (d L, d R ) = (d L, d R) Remark: No symmetry requirement Proof by explicit construction of a gapped path of interactions Φ(s): on the fixed chain with A {x} = B(C n ) constant finite range translation invariant (no blocking) Sven Bachmann (LMU) Invariants of ground state phases UAB 2014 15 / 26
Bulk product states Very simple representatives of each phase: Proposition. Let n 3, and (d L, d R ) N 2. Let k := d L d R. There exists B and projections p, q in Mat k (C) such that dimp = d L, dimq = d R B B n,k (p, q) G k,b m,p,q satisfy the intersection property the unique ground state ω B of the Hamiltonian H k,b,p,q on Z is the pure product state ω B (A x A y ) = y ψ 1, A i ψ 1 i=x Sven Bachmann (LMU) Invariants of ground state phases UAB 2014 16 / 26
Example: the AKLT model SU(2)-invariant, antiferromagnetic spin-1 chain Nearest-neighbor interaction b 1 H[a,b] AKLT = x=a [ 1 2 (S x S x+1 ) + 1 6 (S x S x+1 ) 2 + 1 ] b 1 = 3 x=a where P (2) x,x+1 is the projection on the spin-2 space of D 1 D 1 Uniform spectral gap γ of H [a,b], γ > 0.137194 H AKLT = H 2,B,1,1 with B B 3,2(1, 1) ( 1/3 0 B 1 = 0 1/3 ), B 2 = the AKLT model belongs to the phase (2, 2) P (2) x,x+1 ( ) ( ) 0 2/3, B 3 = 0 0 0 0 2/3 0 Sven Bachmann (LMU) Invariants of ground state phases UAB 2014 17 / 26
About the proof B B n,k (p, q) {ÊB ω B Γ k,b N,p,q G k,b N,p,q H,p,q, k,b and by the proposition Gap(ÊB ) Gap(H k,b,p,q) Given B B n,k (p, q), B B n,k (p, q ), construct a path of gapped parent Hamiltonians H k,b(s),p(s),q(s) by embedding M k M k and interpolating interpolating p(s), q(s): dimensions interpolating B(s), keeping spectral properties of ÊB(s) Need pathwise connectedness of a certain subspace of (M k ) n Sven Bachmann (LMU) Invariants of ground state phases UAB 2014 18 / 26
Primitive maps Ê B = i.e. {B µ } are the Kraus operators n B µ Bµ µ=1 The spectral gap condition: Perron-Frobenius Irreducible positive map = 1. Spectral radius r is a non-degenerate eigenvalue 2. Corresponding eigenvector e > 0 3. Eigenvalues λ with λ = r are re 2πiα/β, α Z/βZ A primitive map is an irreducible CP map with β = 1 Lemma. ÊB is primitive iff there exists m N such that span {B µ1 B µm : µ i {1,..., n}} = M k Sven Bachmann (LMU) Invariants of ground state phases UAB 2014 19 / 26
Primitive maps How to construct paths of primitive maps? Consider Y n,k := with the choice { B : B 1 = k α=1 } λ α e α e α, and B 2 e α, e β 0 (λ 1,..., λ k ) Ω := {λ i 0, λ i λ j, λ i /λ j λ k /λ l } Then, e α e β span {B µ1 B µm : µ i {1, 2}} for m 2k(k 1) + 3. Problem reduced to the pathwise connectedness of Ω C k Use transversality theorem Sven Bachmann (LMU) Invariants of ground state phases UAB 2014 20 / 26
Consequences What we obtain: B(s) B n,k (1, 1) B n,k (p(s), q(s)) i.e. a good ÊB(s) For those: Γ k,b m,p(s),q(s) is injective dimgk,b m,p(s),q(s) = d Rd L G k,b m,p(s),q(s) satisfy the intersection property i.e. a good path H k,b,p(s),q(s) Sven Bachmann (LMU) Invariants of ground state phases UAB 2014 21 / 26
Remarks More work at s = 0, 1, where the given B, B / Y n,k : Why is it hard? Because dimh = n is fixed Simpler problem for n k 2, i.e. by allowing periodic interactions Interaction range: m min = max{m, m, k 2 + 1, (k ) 2 + 1} all in all: (d L, d R ) = (d L, d R ) is sufficient (d L, d R ) = (d L, d R ) necessary: H H implies S [0, ) = S [0, ) α [0, ), S (, 1] = S (, 1] α (, 1] and α is bijective Sven Bachmann (LMU) Invariants of ground state phases UAB 2014 22 / 26
Concrete representatives 1 S 0 (λ, d) = λ 0 1..., S 0 1 +(d) =... 1, λ d 1 0 Let B 1 = S 0 (λ R, d R ) S 0 (λ L, d L ) B 2 = S + (d R ) S 0 (λ L, d L ) B 3 = S 0 (λ R, d R ) S + (d L ) B i = 0 if i 3. Properties: B d R 2 = 0, B d L 3 = 0, and B 1B 2 = λ R B 2B 1, B 1B 3 = λ L B 3B 1, B 2B 3 = ( λl λ R ) B 3B 2. Sven Bachmann (LMU) Invariants of ground state phases UAB 2014 23 / 26
Concrete spectrum Simple consequence: Ê B = D + N R + N L with D = B 1 B1 diagonal, N R = B 2 B2, N L = B 3 B3, nilpotent, and DN R = λ 2 R N RD, DN L = λ 2 L N LD, N R N L = (λ R /λ L ) 2 N L N R. Then, Spectral gap if λ L, λ R 1 σ(êb ) = σ(d) Sven Bachmann (LMU) Invariants of ground state phases UAB 2014 24 / 26
Product vacuum in the bulk Vectors? Recall Γ k,b N,p,q (a) = n µ 1,...µ N =1 Tr(paqB µ N B µ 1 )ψ µ1 ψ µn The product B µ1 B µn can have at most d R 1 B 2 s, and d L 1 B 1 s, so { G k,b N,p,q = span Γ k,b N,p,q (pbα 2 B β 3 }α=0,...,d q) R 1,β=0,...,d L 1 for α = β = 0, product vacuum: Γ k,b N,p,q (1) = Tr(p1q(B 1) m )ψ 1 ψ 1 Sven Bachmann (LMU) Invariants of ground state phases UAB 2014 25 / 26
Conclusions Construction of gapped Hamiltonians from frustration-free states Unique ground state on Z Tunable number of edge states Edge index: Complete classification by the number of edge states (no symmetry) No bulk index: each phase has a representative with a pure product state on Z Many-body localization? Sven Bachmann (LMU) Invariants of ground state phases UAB 2014 26 / 26