A family of two-dimensional AKLT models with a spectral gap above - PowerPoint PPT Presentation

1 Venice, 22 August 2019 A family of two-dimensional AKLT models with a spectral gap above the ground state Bruno Nachtergaele (UC Davis) Joint work with Houssam Abdul-Rahman (U Arizona), Marius Lemm (Harvard), Angelo Lucia (Caltech), Amanda Young (TU Munich) Work supported by the U.S. National Science Foundation 1 under grant DMS-1813149.

2 Outline ◮ AKLT models ◮ Decorated lattices ◮ Frustration freeness ◮ Reduction to a finite size problem ◮ Matrix Product and Tensor Networks States ◮ A few comments

3 AKLT models Affleck, Kennedy, Lieb, and Tasaki (1987-88), introduced a class of nearest neighbor Hamiltonians on regular lattices, later generalized by Kirillov and Korepin (1989) to general graphs G . For each x ∈ G , H x = C d x , with d x = degree of x +1. The d x - dimensional irrep of SU (2) acts on H x . Let z ( e ) denote the sum of the degrees of the vertices of the an edge e in G . Then � P ( z ( e ) / 2) H AKLT = , G e edges e in G where P ( j ) denoted the orthogonal projection on the states on e the edge e of total spin j . Recall j 1 + j 2 � V j 1 ⊗ V j 2 = V j . j = | j 1 − j 2 |

4 Simplest and most famous example: the AKLT spin-1 chain (Affleck-Kennedy-Lieb-Tasaki, 1987-88) . G = [1 , L ] ⊂ Z , H x = C 3 ; L − 1 L − 1 � 1 l + 1 2 S x · S x +1 + 1 � P (2) � � 6( S x · S x +1 ) 2 H [1 , L ] = 3 1 = x , x +1 x =1 x =1 In the limit of the infinite chain, the ground state is unique, has a finite correlation length, and there is a non-vanishing gap in the spectrum above the ground state (Haldane phase). Ground state is frustration free (Valence Bond Solid state (VBS), aka Matrix Product State (MPS), aka Finitely Correlated State (FCS))., and has String Order ( den Nijs-Rommelse 1989 ): ground states are linear combinations of · · · 0100101100010000101 · · ·

5 AKLT model on hexagonal (honeycomb) lattice At each vertex sits a spin of magnitude S = 3 / 2 ( H x = C 4 ). Hamiltonian: H AKLT = � h AKLT . x , y edges { x , y }

6

7 From: T.-C. Wei, P. Haghnegahdar, and R. Raussendorf, Phys. Rev. A 90 (2014), 042333

8 The new result: gap of AKLT on n -decorated honeycomb. E.g.: 2-decorated hexagonal lattice: Theorem ( arXiv:1901.09297 ) For all n ≥ 3 , there exist γ > 0 , such that spectral gap above the ground state (0) of the AKLT model on an n-decorated hexagonal lattice is bounded below by γ .

9 Frustration Freeness The AKLT Hamiltonians are frustration-free: � H G = h x , y , with h x , y ≥ 0 , and ker H G � = { 0 } . edges { x , y } Therefore, ker H G is the ground state space. For any cover V 1 , . . . , V k of the edges of G , let V i , 1 ≤ 1 ≤ k , also denote the corresponding subgraphs of G . Then k k � � � � ker H G = ker h x , y = ker h x , y = ker H V i . edges { x , y } i { x , y }∈ V i i If each edge belongs to V i for at least one and no more than m values of i , and the gap of H V i is bounded below by γ V > 0, we have k k γ V P i ≤ 1 � � H V i ≤ H G , m m i =1 i =1 where P i is the projection onto (ker H V i ) ⊥ .

10 Reduction to a finite-size system problem Our goal: find γ > 0 such that ( H G ) 2 ≥ γ H G i.e., a lower bound for the gap in the spectrum of H G above 0. Because H G is frustration-free it is sufficient to find ˜ γ > 0 for which k H G ) 2 ≥ ˜ ( ˜ γ ˜ H G , with ˜ � H G = P i . i =1 Then γ ≥ γ V m ˜ γ.

11 In concreto, for every vertex of degree 3, i.e., every vertex of the simple honeycomb graph (say with periodic b.c.), let Y v be the graph with 3 n + 1 vertices centered at v : ˜ � H G = P v . v ∈ honeycomb

12 ( ˜ � H G ) 2 P v ) 2 = ( v ∈ honeycomb � � P 2 = v + P v P w + P w P v v ∈ honeycomb v � = w ∈ honeycomb ˜ � ≥ H G + P v P w + P w P v nearest neighbors v , w Lemma ( Fannes-N-Werner 1992 ) For any pair of orthogonal projections E and F one has: EF + FE ≥ −� EF − E ∧ F � ( E + F ) � EF − E ∧ F � = � ( 1 l − E )( 1 l − F ) − ( 1 l − E ) ∧ ( 1 l − F ) � .

13 Application of the lemma gives, for any G = union of Y v ’s: G ≥ 1 H G ) 2 ≥ (1 − 3 ǫ n ) ˜ ( ˜ H G , and H 2 2 γ Y (1 − 3 ǫ n ) H G , with ǫ n := � ( 1 l − P v )( 1 l − P w ) − ( 1 l − P v ) ∧ ( 1 l − P w ) � . Proposition For the comparison Hamiltonian ˜ H G on the decorated honeycomb graph, we have ǫ n ≤ C 3 − n and, In particular, ǫ 3 < . 2683 . Theorem For all n ≥ 3 , there exists γ n > 0 , such that the spectral gap above the ground state of the AKLT model on on the decorated honeycomb graph is at least γ n . E.g., γ 3 > 0 . 0289 .

14 So, everything comes down to estimating ǫ n = � ( 1 l − P v )( 1 l − P w ) − ( 1 l − P v ) ∧ ( 1 l − P w ) � . ( 1 l − P v ) is the projection on to the ground state space of H Y v . Call that subspace G Y v ⊂ H Y v etc. A useful expression for ǫ n is |� φ | ψ �| ǫ n = sup � φ �� ψ � φ ∈G Yv ⊗H GR ψ ∈H GL ⊗G Yw φ,ψ ⊥G Yv ∪ Yw , φ,ψ � =0 . . . . . . C n . . . G L G R v w . . . . . .

15 The ground states AKLT models, the case at hand included, are examples of Tensor Network States (TNS). Concretely, this means that we have a representation of the spaces G G , for G ∈ { Y v , Y w , Y v ∪ Y w , . . . } , of the following form: G G = ran Γ G , with Γ G : K G → H G , where K G is an auxiliary space and Γ B has the structure of a Matrix Product State (MPS). Example: for G = Y v ∪ Y w , K G = M 2 ( C ) ⊗ M 2 ( C ), and for all B ∈ K G , � Tr[ BT R r V i n · · · V i 1 T L Γ G ( B ) = l ] | l � L ⊗ | i 1 , . . . , i n � ⊗ | r � R , l , i 1 ,..., i n , r with specific 2 × 2 matrices V i , 4 × 2 matrices T R r and 2 × 4 matrices T L l . K G and the matrices V i do not depend on n .

16 ǫ n expresses a geometric property of the ground spaces of the subsystems on Y v and Y w . It is the cosine of the angle between them when ignoring their intersection. The inner product on ran Γ G defines a sesquilinear form on K G : � Γ G ( B ) , Γ G ( C ) � = � B , C � G , n The key properties are the following: for n ≥ 2 one has (i) �· , ·� G , n is non-degenerate (hence defines an inner product); (ii) |� B , C � G , n − 1 2 Tr Q n B ∗ Q n C | ≤ δ ( n ), with 4 Q n = 1 l + 3 2 n +1 S · S , and δ ( n ) ≤ C 3 − n .

17 Using this structure we can show the following. Proposition Let 4 A n = � . � 1 − 8(1+3 − 2 n − 1 ) 3 n 3 n (1 − 3 − 2 n ) Then, for all n ≥ 3 , 1 + 8(1 + 3 − 2 n − 1 ) 2 � � ǫ n ≤ A n + A 2 < 1 / 3 . n 3 n (1 − 3 − 2 n ) 2 and ǫ 3 < 0 . 2683 .

18 A few comments ◮ ǫ 1 ∼ 0 . 478 > 1 / 3. So the approach does not work for n = 1. ◮ Our estimates are not good enough to prove a gap for the case n = 2, but it is possible they can be improved. ◮ The method generalizes to AKLT models in many other ‘decorated’ lattices; relies on the calculation of eigenvalues, norms, etc of a set of finite-dimensional objects. See, e.g., N. Pomata and T.-C. Wei, arXiv:1905.01275 (uses numerical calculation). ◮ The original 1988 conjecture of Kennedy, Lieb, and Tasaki for the AKLT model on the honeycomb graph remains open.