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)

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Work supported by the U.S. National Science Foundation under grant DMS-1813149.

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Outline

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

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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, Hx = Cdx, with dx = degree of x +1.

The dx- dimensional irrep of SU(2) acts on Hx. Let z(e) denote the sum of the degrees of the vertices of the an edge e in G. Then HAKLT

G

=

• edges e in G

P(z(e)/2)

e

, where P(j)

e

denoted the orthogonal projection on the states on the edge e of total spin j. Recall Vj1 ⊗ Vj2 =

j1+j2

• j=|j1−j2|

Vj.

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Simplest and most famous example: the AKLT spin-1 chain

(Affleck-Kennedy-Lieb-Tasaki, 1987-88). G = [1, L] ⊂ Z,

Hx = C3; H[1,L] =

L−1

• x=1

1 31 l + 1 2Sx · Sx+1 + 1 6(Sx · Sx+1)2

• =

L−1

• x=1

P(2)

x,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 · · ·

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AKLT model on hexagonal (honeycomb) lattice

At each vertex sits a spin of magnitude S = 3/2 (Hx = C4). Hamiltonian: HAKLT =

• edges {x,y}

hAKLT

x,y

.

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From: T.-C. Wei, P. Haghnegahdar, and R. Raussendorf,

• Phys. Rev. A 90 (2014), 042333

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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.

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Frustration Freeness

The AKLT Hamiltonians are frustration-free: HG =

• edges {x,y}

hx,y, with hx,y ≥ 0, and ker HG = {0}. Therefore, ker HG is the ground state space. For any cover V1, . . . , Vk of the edges of G, let Vi, 1 ≤ 1 ≤ k, also denote the corresponding subgraphs of G. Then ker HG =

• edges {x,y}

ker hx,y =

k

• i
• {x,y}∈Vi

ker hx,y =

k

• i

ker HVi. If each edge belongs to Vi for at least one and no more than m values of i, and the gap of HVi is bounded below by γV > 0, we have γV m

k

• i=1

Pi ≤ 1 m

k

• i=1

HVi ≤ HG, where Pi is the projection onto (ker HVi)⊥.

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Reduction to a finite-size system problem

Our goal: find γ > 0 such that (HG)2 ≥ γHG i.e., a lower bound for the gap in the spectrum of HG above 0. Because HG is frustration-free it is sufficient to find ˜ γ > 0 for which ( ˜ HG)2 ≥ ˜ γ ˜ HG, with ˜ HG =

k

• i=1

Pi. Then γ ≥ γV m ˜ γ.

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In concreto, for every vertex of degree 3, i.e., every vertex of the simple honeycomb graph (say with periodic b.c.), let Yv be the graph with 3n + 1 vertices centered at v: ˜ HG =

• v∈honeycomb

Pv.

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( ˜ HG)2 = (

• v∈honeycomb

Pv)2 =

• v∈honeycomb

P2

v +

• v=w∈honeycomb

PvPw + PwPv ≥ ˜ HG +

• nearest neighbors v,w

PvPw + PwPv

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).

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Application of the lemma gives, for any G = union of Yv’s: ( ˜ HG)2 ≥ (1 − 3ǫn) ˜ HG, and H2

G ≥ 1

2γY (1 − 3ǫn)HG, with ǫn := (1 l − Pv)(1 l − Pw) − (1 l − Pv) ∧ (1 l − Pw).

Proposition

For the comparison Hamiltonian ˜ HG on the decorated honeycomb graph, we have ǫn ≤ C3−n and, In particular, ǫ3 < .2683.

Theorem

For all n ≥ 3, there exists γn > 0, such that the spectral gap above the ground state of the AKLT model on on the decorated honeycomb graph is at least γn. E.g., γ3 > 0.0289.

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So, everything comes down to estimating ǫn = (1 l − Pv)(1 l − Pw) − (1 l − Pv) ∧ (1 l − Pw). (1 l − Pv) is the projection on to the ground state space of HYv. Call that subspace GYv ⊂ HYv etc. A useful expression for ǫn is ǫn = sup

φ∈GYv ⊗HGR ψ∈HGL⊗GYw φ,ψ⊥GYv ∪Yw , φ,ψ=0

|φ|ψ| φψ

Cn GL GR v w . . . . . . . . . . . . . . .

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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 GG, for G ∈ {Yv, Yw, Yv ∪ Yw, . . .}, of the following form: GG = ranΓG, with ΓG : KG → HG, where KG is an auxiliary space and ΓB has the structure of a Matrix Product State (MPS). Example: for G = Yv ∪ Yw, KG = M2(C) ⊗ M2(C), and for all B ∈ KG, ΓG(B) =

• l,i1,...,in,r

Tr[BT R

r Vin · · · Vi1T L l ]|lL ⊗ |i1, . . . , in ⊗ |rR,

with specific 2 × 2 matrices Vi, 4 × 2 matrices T R

r and 2 × 4

matrices T L

l . KG and the matrices Vi do not depend on n.

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ǫn expresses a geometric property of the ground spaces of the subsystems on Yv and Yw. It is the cosine of the angle between them when ignoring their intersection. The inner product on ranΓG defines a sesquilinear form on KG: ΓG(B), ΓG(C) = B, CG,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, CG,n − 1

2TrQnB∗QnC| ≤ δ(n), with

Qn = 1 l + 4 32n+1S · S, and δ(n) ≤ C3−n.

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Using this structure we can show the following.

Proposition

Let An = 4 3n

• 1 − 8(1+3−2n−1)

3n(1−3−2n)

. Then, for all n ≥ 3, ǫn ≤ An + A2

n

• 1 + 8(1 + 3−2n−1)2

3n(1 − 3−2n)2

• < 1/3.

and ǫ3 < 0.2683.

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A few comments

◮ ǫ1 ∼ 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

• bjects. 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.