We consider the spectral gap question for Affleck, Kennedy, Lieb, and Tasaki models defined on decorated versions of simple, connected graphs G. This class of decorated graphs, which are defined by replacing all edges of G with a chain of n sites, in particular includes any decorated multi-dimensional lattice. Using the Tensor Network States approach from [Abdul-Rahman et al., Analytic Trends in Mathematical Physics, Contemporary Mathematics (American Mathematical Society, 2020), Vol. 741, p. 1.], we prove that if the decoration parameter is larger than a linear function of the maximal vertex degree, then the decorated model has a nonvanishing spectral gap above the ground state energy.

One of the most important classes of quantum spin models in the study of topological phases of matter is the family of antiferromagnetic, SU(2)-invariant quantum spin systems introduced by Affleck, Kennedy, Lieb, and Tasaki (AKLT) in Refs. 1 and 2. A fundamental quantity in the characterization of quantum phases is the existence or non-existence of a spectral gap above the ground state energy in the thermodynamic limit. In their seminal work, AKLT proved that their one-dimensional, spin-one chain satisfied the characteristic properties of the Haldane phase,3 including a spectral gap of the finite volume Hamiltonians uniform in the system size. AKLT models on higher dimensional lattices were also introduced, and it was further conjectured that if the spatial dimension and coordination number are sufficiently large, then the model would exhibit Néel order and, hence, be gapless.2 This has been verified analytically for models on Cayley trees with a coordination number of at least five,2,4 and numerical evidence supports the conjecture on three-dimensional lattices.5 

In contrast, the AKLT models on the hexagonal and square lattices were conjectured to be gapped. While it was proved that the AKLT state on the hexagonal lattice does not exhibit Néel order in Ref. 6, the nonvanishing gap was only recently shown in Refs. 7 and 8 using a combination of numerical and analytical techniques. The approach in Ref. 7 uses DMRG with a finite size criterion in the spirit of Knabe,9 while Ref. 8 combines a Lanzcos method with the general theory from Ref. 10 for proving uniform gaps in quantum spin models with Tensor Network States (TNS) ground states on decorated graphs. The TNS method adapts the one-dimensional finitely correlated state approach from Ref. 11 to a particular class of models defined on decorated graphs defined by replacing each edge of a graph G with a chain of n sites. AKLT ground states on decorated lattices are of interest, e.g., as they have been shown to constitute a universal quantum computation resource.12,13 The authors of Ref. 10 applied their theory to show that the AKLT model on the decorated hexagonal lattice was gapped as long as the decoration was sufficiently large. This was then extended in combination with numerical methods to decoration numbers n ≥ 0 in Ref. 14, on the square lattice for decoration parameters n ≥ 2 in Ref. 8, and on the 3D diamond lattice and the 2D kagome lattice for n ≥ 1 in Ref. 15.

Hence, a natural question is whether or not AKLT models on decorated graphs with a higher coordination number are gapped. Concretely, we say a quantum spin model on a connected graph G = (V, E) is uniformly gapped if there exists a sequence of finite subgraphs Gk = (Vk, Ek), k ≥ 1, such that Ek ⊆ Ek+1, Vk ⊆ Vk+1, and ∪kEk = E for which the associated local Hamiltonians satisfy
γ≔infkgap(HGk)>0,
where gap(HGk)>0 is the difference between the ground state and first excited state energies. The AKLT models have a well-defined infinite volume dynamics in the sense of Ref. 16. For such quantum spin models, a positive uniform gap implies a nonvanishing gap in the thermodynamic limit, in the sense that the GNS Hamiltonian associated with any weak-* limit of finite-volume ground states of this sequence has a spectral gap bounded below by γ. See, e.g., Ref. 17 for a precise statement and proof.

In this work, we consider any (possibly infinite) simple graph G = (V, E) such that Δ(G) = supv∈V deg(v) is finite. We show that if all edges of this graph are replaced with a chain of n-sites, then the AKLT model on the decorated graph is uniformly gapped as long as n ≥ n(Δ(G)), where n(Δ(G)) is a linear function of the maximal vertex degree (see Theorem 1 below). Our proof follows from a slight variation of the analytical framework from Ref. 10 that produces tighter bounds on the minimal decoration number. The main quantities for bounding the gap using this method depend on the transfer operators associated with the TNS, which are defined by certain quasi one-dimensional subgraphs of the decorated graph. When the maximal vertex degree is small (i.e., 3 or 4), these quantities can be explicitly computed. However, this becomes nontrivial when Δ(G) is arbitrary. We overcome this challenge by finding the exact singular value decomposition of the transfer operator.

In a recent study18 with a co-author, we proved that the spectral gap of the decorated AKLT model on the hexagonal lattice is stable when n ≥ 5, in the sense that the spectral gap remains positive when the model is perturbed by another sufficiently fast decaying interaction. In terms of the classification of quantum phases, this means that the model belongs to a stable gapped phase. This stability is a result of proving a condition on the ground states known as local topological quantum order (LTQO). It is natural to ask whether this is the case also for the general models considered here, i.e., if it is possible to prove an LTQO condition for a sufficiently large decoration number n for any graph G. We conjecture that this is in fact the case, although we make no claims on the scaling of the minimal decoration number required. The proof of the result of Ref. 18 relies on a representation of the ground states of the AKLT model in terms of a gas of loops.6 The LTQO condition is then a consequence of showing that the cluster expansion for the partition function of the loop model converges. While the loop model would be more complicated for general graphs G (for example, it might allow loops to cross each other, something forbidden in trivalent graphs), increasing the decoration number has the effect of decreasing the weight of each segment of the loop while leaving the other details of the model invariant. Therefore, it is reasonable to expect that a large enough decoration would make the expansion convergent even in the more general case.

The organization of this paper is as follows: in Sec. II, we define and state the spectral gap result for the decorated AKLT models and summarize the modified version of the uniform gap strategy from Ref. 10. A special class of operators, called matching operators, is introduced and studied in Sec. III. We use these operators to establish the necessary SVD in Sec. IV and then prove the main result. In the appendices, we provide helpful combinatorial identities and discuss the differences between the modified TNS approach used in this work and the one proved in Ref. 10. In particular, we prove that the modified version used here produces a tighter bound on the minimal decoration needed to guarantee the decorated model is uniformly gapped.

We consider AKLT models on decorated versions of a (potentially infinite) simple graph G = (V, E) such that
Δ(G)≔supv∈Vdeg(v)<∞.
For each n ≥ 1, the n-decorated graph G(n) = (V(n), E(n)) is defined by adding n additional vertices to each edge e ∈ E (see Fig. 1). The integer n is called the decoration parameter. We will also use n = 0 to denote the original graph.
FIG. 1.
The n = 3 decorated version of a graph G. The subgraph colored in red is Yv.
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The n = 3 decorated version of a graph G. The subgraph colored in red is Yv.

FIG. 1.
The n = 3 decorated version of a graph G. The subgraph colored in red is Yv.
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The n = 3 decorated version of a graph G. The subgraph colored in red is Yv.

Close modal
The AKLT model we consider for any n ≥ 0 is defined as follows: at each vertex v ∈ V(n), we associate a spin-deg(v)/2 particle represented by the local Hilbert space Hv=Cdeg(v)+1⁠, and the interaction for any e = (v, w) ∈ E(n) is the orthogonal projection,
P(z(e)/2)∈B(Hv⊗Hw),
onto the subspace of maximal total spin, i.e., z(e) ≔ deg(v) + deg(w). It is well-known that this defines a frustration-free, nearest–neighbor interaction,19 and we note that the interaction between any two neighboring decorated sites is simply that of the spin-1 AKLT chain.2 
For any v ∈ V, let Yv ⊂ G(n) be the subgraph consisting of the undecorated vertex v and the n · deg(v) sites decorating the edges incident to v (see Fig. 1). Our main result states that as long as the decoration n is sufficiently large, the spectral gap of the finite-volume Hamiltonian
(1)
has a positive lower bound independent of |Λ| for any finite Λ ⊆ G. This result, which we now state, depends on the decreasing function,
(2)

Theorem 1.
Suppose that G = (V, E) is a simple graph such that 3 ≤ Δ(G) < ∞. If n ≥ n(Δ(G)), where
(3)
then there exists γ(Δ(G), n) > 0, such that
(4)

Remark 1.

Our result can easily be generalized to the situation where the decoration varies on different edges. Given a bounded function n:E→N⁠, consider the decorated graph G(n) obtained by adding n(e) additional vertices to each edge e ∈ E. Then the same arguments used to prove Theorem 1 imply that, for each finite Λ ⊂ G, the AKLT Hamiltonian on Λ(n) has a positive spectral gap uniform in Λ as long as mine∈En(e) ≥ n(Δ(G)). Here, Λ(n) is defined as in (1), with Yv being the subgraph consisting of v and all vertices decorating the edges incident to v. The modifications needed to obtain these results are discussed at the end of Sec. IV B.

Several other comments regarding Theorem 1 are listed in the following order:

  1. The constraint Δ(G) ≥ 3 is not necessary, but the case Δ(G) ≤ 2 does not yield any new results. For (undecorated) regular graphs, the case Δ(G) = 2 is the famous AKLT result, while Δ(G) = 1 corresponds to an interaction with commuting terms, which is trivially gapped. Moreover, for any graph with minv deg(v) = 1 and Δ(G) = 2, e.g., a small variation of the one-dimensional finitely correlated states argument from Ref. 4 would also imply a gap.

  2. In the case that Δ(G) = 3, 4, Theorem 1 extends the class of decorated graphs that were studied in Refs. 8, 10, and 14. Moreover, when compared with the previous results that only used analytical techniques, the present result either improves or reproduces the lower bound on n.

  3. Since 1 ≤ f(d) ≤ f(5) ≈ 1.178 51, Theorem 1 proves a positive uniform gap when n is greater than a linear function of Δ(G). However, it is unknown if this bound on minimal decoration is optimal. Since AKLT models on undecorated lattices with large coordination number are expected to exhibit Néel order, it would be interesting to determine the minimal decoration needed to guarantee these models are in a gapped phase.

  4. The lower bound on the spectral gap of the model is only a function of the maximal degree Δ(G). Therefore, we can also apply Theorem 1 to show a uniform spectral gap estimate for a sequence of finite graphs Gk having a uniform maximal degree Δ. This allows us to prove, for example, uniform bounds on the spectral gaps for a sequence of finite volumes with periodic boundary conditions (e.g., Gk=Zk×ν for a fixed ν).

A consequence of Theorem 1 is the following thermodynamic limit result, which is stated with respect to the C*-algebra of (quasi-)local observables:
AG(n)=∑nAΛ̄‖⋅‖,AΛ=⊗verticesv∈ΛB(Hv).

Corollary 2.
Let G be as in Theorem 1, n ≥ n(Δ(G)), and ω:AG(n)→C be any weak-* limit of finite-volume ground states,
ωm(A)=⟨ψm,Aψm⟩∀A∈AΛm(n),
where ψm∈ker(HΛm(n)) is normalized and Λm ↑ G. Then, the spectral gap above the ground state of the corresponding GNS Hamiltonian Hω satisfies
(5)

As discussed in the introduction, this follows immediately from Theorem 1 by well-known arguments. Theorem 1 is proved using a mild modification of the general framework from Ref. 10, which we now review.

As in Ref. 10, Sec. II, rather than estimate the gap of HΛ(n) directly, we instead consider the gap of the coarse-grained Hamiltonian,
H̃Λ(n)=∑v∈ΛPv,Pv=1−Gv,
where Gv is the orthogonal projection onto the ground state space ker(HYv) of HYv⁠. Since each edge e ∈ Λ(n) belongs to at most two subvolumes Yv, it is easy to deduce that [analogous to Ref. 10, Eq. (2.4)],
(6)
Moreover, gap(HYv) only depends on deg(v), and so the assumption Δ(G) < ∞ implies that the above infimum is strictly positive. Therefore, as the two Hamiltonians have the same ground states, proving a uniform gap for H̃Λ(n) implies a uniform gap of HΛ(n)⁠.
Estimating the gap of H̃Λ(n) can then be reduced to bounding the quantity,
(7)
where Gv ∧ Gw is the orthogonal projection onto ran(Gv)∩ran(Gw)=ker(HYv∪Yw) by frustration-freeness. Namely, since {Pv, Pw} ≥ 0 if (v, w) ∉ E and
PvPw+PwPv≥−∥PvPw−Pv∧Pw∥(Pv+Pw),∥PvPw−Pv∧Pw∥=∥GvGw−Gv∧Gw∥,
(see Ref. 11 Lemma 6.3), H̃Λ(n) satisfies the operator inequality,
H̃Λ(n)2=H̃Λ(n)+∑w≠v∈Λ{Pv,Pw}≥H̃Λ(n)−ϵG(n)∑edges(v,w)∈Λ(Pv+Pw).
The final sum above is bounded by Δ(G)H̃Λ(n) as each vertex v belongs to at most Δ(G) edges. Hence, gap(H̃Λ(n))≥1−Δ(G)ϵG(n)⁠, and combining these bounds yields
(8)
Therefore, Theorem 1 immediately follows from showing ϵG(n) < 1/Δ(G) if n ≥ n(Δ(G)). The remainder of this paper is focused on proving this inequality.

AKLT models are the quintessential class of models with TNS ground states. As shown in Ref. 10, Sec. III, the TNS machinery can be used to estimate the norm on the right hand side of (7) for any edge (v, w) ∈ E. Namely, in the situation that the TNS is injective, the norm in (7) is bounded from above by a constant that depends only on the transfer operators associated with various subgraphs of Yv ∪ Yw. Our approach is a slight modification of this framework resulting from using a variation of Ref. 10, Lemma 3.3, which produces a tighter upper bound on (7). This variant (namely, Lemma 18) and the fact that it leads to a better bound are proved in  Appendix B. In this section, we introduce the necessary transfer operators and state the modified bound on this norm.

1. The transfer operators

Label vL and vR as the “left” and “right” vertex, respectively, associated with an edge (vL, vR) ∈ E, and decompose
YvL∪YvR=XL∪Cn∪XR,
where Cn=YvL∩YvR is the chain of n sites decorating the edge (vL, vR), and X#=Yv#\Cn for # ∈ {L, R} (see Fig. 2). The desired transfer operators are those associated with these three regions.
FIG. 2.
The region YvL∪YvR for two vertices of degree d = 6 and decoration n = 4. XL is the region consisting of vL and the 5 decorated edges to its left.
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The region YvL∪YvR for two vertices of degree d = 6 and decoration n = 4. XL is the region consisting of vL and the 5 decorated edges to its left.

FIG. 2.
The region YvL∪YvR for two vertices of degree d = 6 and decoration n = 4. XL is the region consisting of vL and the 5 decorated edges to its left.
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The region YvL∪YvR for two vertices of degree d = 6 and decoration n = 4. XL is the region consisting of vL and the 5 decorated edges to its left.

Close modal
The transfer operator for the center region Cn is the n-fold composition En⁠, where E:M2(C)→M2(C) is the (well known) single vertex spin-1 AKLT transfer operator,
(9)
Here, ρ=1/2⁠, SU = σU/2, and the bra-ket notation is with respect to the Hilbert–Schmidt inner product. The convergence of En to its fixed point is known. Namely,
(10)
Now, let d# = deg(v#) for # ∈ {L, R}, and denote by
EL(dL,n):M2(C)→M2(C)⊗dL−1,ER(dR,n):M2(C)⊗dR−1→M2(C),
the transfer operators associated with XL and XR, respectively. These are the composition of the undecorated transfer operator and n copies of E⊗d#−1⁠,
(11)
See Fig. 2 for a visualization. As HvL=Cd+1 when dL = d, by definition
(12)
where {Wkd:0≤k≤d} are the tensors associated with vL in the TNS representation of the ground states. We construct these using the valence bond state formalism for the AKLT ground states, where we use the convention that the edge(s) to the right-side of a vertex are projected into the singlet state.20 
Identify Hv with the symmetric subspace of d spin-1/2 particles, and let ϕkd∈(C2)⊗d be the normalized symmetric vector with k up spins. These satisfy the recursive formula
(13)
Then, denoting by K=↑↓−↓↑ the singlet tensor, one can take Wkd=KVkd⁠, where
(14)
We note that when d = 2, this produces a scaled version of the 1-dimensional AKLT transfer operator from (9), namely, EL(2,0)=32E⁠. Such a scaling is inconsequential but convenient for producing consistent formulas.
The same procedure works for defining ER(d,0) via the tensors W̃kd associated with vR. This yields W̃kd≔K⊗d−1(Vkd)*=(−1)k−1(Wd−kd)*⁠, which when combined with (9)–(12) implies
(15)
where the adjoint is with respect to the Hilbert–Schmidt inner product.

2. Bounding ϵG(n) via transfer operator quantities

The desired estimate on ϵG(n) for G = (V, E) is a consequence of bounding
ϵ(vL,vR)(n)≔‖GvLGvR−GvL∧GvR‖∀(vL,vR)∈E,
by a constant that only depends on three types of quantities. Namely, the convergence rate a(n) from (10), the minimal eigenvalues qL(d,n) and qR(d,n)⁠, respectively, of
(16)
and the Schatten-∞ norm of EL(d,n) and ER(d,n)⁠. Here, we recall that, with respect to the Hilbert–Schmidt norm ∥·∥2,
(17)
and ‖ER(d,n)‖∞=‖EL(d,n)‖∞ by (15). Given these values, the following is the mild variation of Ref. 10, Proposition 3.6 that results from replacing Ref. 10, Lemma 3.3 with Lemma 18 (see  Appendix B) everywhere in the former work.

Proposition 3.
Let (vL, vR) be an edge of a simple graph G between vertices with degrees dL and dR, respectively, and define
(18)
If maxbL(dL,n),bR(dR,n),bL(dL,n)bR(dR,n)4a(n)<1, then ϵ(vL,vR)(n)≤δdL,dR(n)⁠, where
(19)

We note that the maximum constraint in the proposition is sufficient to prove that the TNS representation of the ground states associated with YvL,YvR and YvL∪YvR are all injective, see, e.g., Ref. 10, Corollary 3.4, which is necessary for their approach.

Since ρ=1/2⁠, it follows from (15) that
(20)
Hence, the strategy for proving Theorem 1 is to bound b(d, n) so that by (7) and Proposition 3,
(21)
In Lemma 14, we determine the SVD of EL(d,n)⁠, from which both QL(d,n) and ‖EL(d,n)‖∞ are easily calculated. Bounds on the quantities defining b(d, n) that will imply (21) are then proved in Lemma 15. The final details establishing (21) are the content of the proof of Theorem 1 in Sec. IV B.
While this is discussed more thoroughly in  Appendix B, we point out that the main difference between the two approaches comes from using the Schatten-∞ norm in (18) rather than the norm induced by the operator norm, i.e.,
(22)
Two additional comments are in order. First, a(n) is invariant under this change of norms, as an elementary, but tedious, calculation shows
‖En−1ρ‖∞=‖En−1ρ‖op.
Second, the complete positivity of the transfer operator guarantees that
‖E#(d,n)‖op=‖E#(d,n)(1)‖,#∈{L,R}.
Contrary to the statement just before Ref. 10, Eq. (4.18), one can show that ‖EL(d,n)(1)‖≠‖ER(d,n)(1)‖⁠, see, e.g.,  Appendix B. Correcting this small error in the proof of Ref. 10, Theorem 2.2 yields the existence of a gap for n ≥ 4 instead of the claimed n ≥ 3. However, as Proposition 19 illustrates, Proposition 3 gives a tighter bound on the minimal decoration needed to guarantee a spectral gap than the method from Ref. 10. Using the present method recovers the gap result for n = 3.
The goal of this section is to introduce a special collection of eigenvectors of
E⊗d−1:M2(C2)⊗d−1→M2(C2)⊗d−1,
which we refer to as matching operators. These will aid us in calculating the SVD of the decorated transfer operator EL(d,n) from the SVD of the undecorated transfer operator EL(d,0)⁠.

In addition to introducing the matching operators, we show that they form an orthogonal basis (with respect to the Hilbert–Schmidt inner product) for the commutative algebra generated by the spectral projections of the Casimir operator. This follows from proving that the matching operators satisfy certain recursion relations. These relations will also provide a path for explicitly writing individual spectral projections of the Casimir operator in terms of matching operators, which is the key to determining the desired SVD.

To begin, fix m ≥ 2 and recall that the spectral decomposition of the Casimir operator C(m)=(∑i=1mSi)2 acting on (C2)⊗m is
(23)
where Q(m,j)∈M2(C2)⊗m is the orthogonal projection onto the direct sum of all subspaces of total spin j, and
Jm=j0+k:0≤k≤⌊m2⌋,
with j0 = 0 if m is even and j0 = 1/2 otherwise. Moreover, recall that Q(m,j)=Psym(m) is the projection onto the symmetric subspace when j=⌊m2⌋+j0⁠.
The commutative algebra generated by the Casimir operator is then
(24)
which has dimension Jm=⌊m2⌋+1⁠. The goal is to show that the set of matching operators forms an orthogonal basis for Z(m)⁠, which are eigenvalues of E⊗m⁠. To introduce these operators, we first define the notion of a matching.

1. Definition and properties of matchings

Definition 1.
For any 1≤r≤⌊m2⌋⁠, an r-matching of [m] ≔ {1, …, m} is a collection of r unordered pairs [meaning that (a,b) is the same as (b,a)]
p={(a1,b1),…,(ar,br)},ai,bi∈[m],
which are distinct in the sense that ∪ p ≔ {ai, bi: 1 ≤ i ≤ r} has 2r elements. The set of all r-matchings is denoted by Mrm⁠. For consistency, M0m is the set consisting of the empty matching, p = {}.

Example 1.
M14={(i,j)}:1≤i<j≤4 has six elements, and M24 consists of
{(1,2),(3,4)},{(1,3),(2,4)},and{(1,4),(2,3)}.

The number of r-matchings is easily calculated with the multinomial coefficient,
(25)
Here, n!! denotes the double factorial, i.e., the product of all integers from 1 to n having the same parity as n. By convention, 0!! = (−1)!! = 1, and so (25) also holds when r = 0.

For the main recursion result, it will also be important to know how many matchings p∈Mrm contain either both, one, or none of the elements from a single pair (i, j). As such, we introduce the following (possibly empty) sets, which form a partition of Mrm⁠.

Definition 2.
Fix m ≥ 2. For any 0≤r≤⌊m2⌋ and distinct pair i, j ∈ [m], set
Ar(i,j)=p∈Mrm∣(i,j)∈p,Br(i,j)=p∈Mrm∣i∈∪p,j∉∪porj∈∪p,i∉∪p,Cr(i,j)=p∈Mrm∣i,j∈∪p,(i,j)∉p,Dr(i,j)=p∈Mrm∣i,j∉∪p.

Clearly, the cardinality of these sets is independent of the choice of (i, j), and only depends on m and r. To shorten the notation, we will denote by Ar the cardinality of the sets Ar(i, j), and similarly for the others.

Lemma 4.
Fix m > 2. The sets from Definition 2 have the following cardinalities for any r with the convention that Mrm=0 if m < 0 or r < 0:
(26)
(27)
(28)

Proof.

For each set Xr(i, j) from Definition 2, the result follows from constructing an n-to-1 mapping between Xr(i, j) and the appropriate set of matchings.

  1. A bijection between Ar(i, j) and Dr−1(i, j) is given by
    p∈Ar(i,j)↦p\{(i,j)}∈Dr−1(i,j),
    while a bijection between Dr−1(i, j) and Mr−1m−2 results from recognizing Dr−1(i, j) as the set of all r − 1 pairings on the set [m]\{i, j}.
  2. For p ∈ Br(i, j) define the mapping p ↦ p′ by decomposing
    p={(i,x)}∪p′orp={(j,x)}∪p′,
    for some x ∈ [m]\{i, j}. Moreover, after fixing x, the image p′ can be any r − 1 matching on the set [m]\{i, j, x}. As there are m − 2 choices for x and two possible pairings, i.e., (i, x) and (j, x), the result follows.
  3. For p ∈ Cr(i, j), the mapping p ↦ p′ is defined by writing
    p={(x,i),(y,j)}∪p′,
    where x, y ∈ [m]\{i, j}. As before, p′ is an r − 2-matching on [m]\{x, y, i, j}. Since there are (m − 2) (m − 3) distinct choices for x and y, the result follows.□

2. Definition and properties of matching operators

We now introduce the operators of interest: the matching operators.

Definition 3.
Fix m ≥ 2. Then, the r-matching operator Mr(m)∈B((C2)⊗m) is
(29)
for all 1≤r≤⌊m2⌋ and M0(m)=S∅=1⁠.

Note that since the pairs are disjoint, the order of the product in Sp does not matter. Moreover, each matching operator is clearly nonzero since
(30)
Several other important observations are in order.

  1. E⊗m(Mr(m))=3−2rMr(m) since E(1)=1 and E⊗E(S⋅S)=19(S⋅S)⁠.

  2. M1(m) is related to the Casimir operator C(m) via
    (31)
  3. Every matching operator Mr(m) is Hermitian as well as SU(2) and permutation symmetric (since Mrm is permutation invariant). Therefore, Mr(m)∈Z(m) by Schur–Weyl duality. However, we provide an alternative way of verifying this inclusion. Lemma 7 establishes that
    M1(m)⋅Mr(m)=crMr−1(m)+arMr(m)+brMr+1(m)∀r≥0,
    for appropriate coefficients ar, br, cr. As M0(m)=1⁠, repeatedly applying this relation shows that every matching operator can be written as a polynomial of M1(m)⁠, and hence C(m) by (31).

Proposition 5.

The set of matching operators {Mr(m)}r=0⌊m2⌋ forms a Hilbert–Schmidt orthogonal basis of Z(m).

Proof.

Since each Mr(m)∈Z(m) is nonzero by (30), it is sufficient to show the set of matching operators is orthogonal, as the number of matching operators is the same as the dimension of Z(m)⁠.

Let r1 ≠ r2. Since each matching operator is Hermitian, it trivially follows that
(32)
Considering (29), on sees that tr(SpSq) = 0 unless ∪ p = ∪q. As ∪p=2r1≠2r2=∪q⁠, this cannot occur. This proves orthogonality.□

We conclude this section by calculating the norm of Mr(m)⁠. To do so, we first recall some useful relationships between spin operators. Let ϵi,j,k denote the Levi-Civita symbol, and define
(33)
which is antisymmetric under transpositions τ of the indices, i.e., Eτ(a),τ(b),τ(c) = −Ea,b,c. Using SiSj=14δi,j1+i2ϵi,j,kSk⁠, the following relations hold:
(34)

Lemma 6.
For all 0≤r≤⌊m2⌋, the Hilbert–Schmidt norm of the r-matching operator satisfies
(35)

Proof.
Since the trace of Ea,b,c over the b-th index is zero, it follows from (34) that
(36)
As indicated in (32),
‖Mr(m)‖22=∑p,q∈Mrm:∪p=∪qtr(SpSq),
where we again use that tr(SpSp) = 0 unless ∪ p = ∪q. Since tr(SpSq) is invariant under permutations of the factors, without loss of generality, assume that ∪ p = ∪q = {1, …, 2r}. Therefore, the norm calculation reduces to
(37)

To calculate ‖Mr(2r)‖22⁠, consider first an arbitrary pair p,q∈Mr2r⁠, which we refer to as perfect matchings over 2r elements. As we now show, pairs (p, q) of perfect matchings of 2r elements are in bijection with permutations π∈S2r of 2r elements that have no odd length cycles, which we denote by S2re⁠.

Given (p, q), the corresponding permutation π is built as follows: let a1 ∈ [2r] be arbitrary, and inductively define a2i to be the element connected to a2i−1 in p, and a2i+1 to be the element connected to a2i in q. Then, by properties of the matching, it must be that there is some finite l such that a2l+1 = a1. Therefore, (a1, …, a2l) is a cycle of length 2l that is added to π. If there are no more elements, then π is the desired permutation. Otherwise, pick an element that has not yet been considered and iterate this process to create a new cycle. After a finite number of steps, the iteration terminates with the desired permutation π.

Vice versa, given π∈S2re⁠, define (p, q) as follows: for each cycle (a1, …, a2l) in π, add (a2i−1, a2i) to p and (a2i, a2i+1) to q, for each i = 0, …, l (where addition is taken modulo 2l). Then p and q are perfect matchings as they both have r disjoint pairs.

Let π∈S2re be the permutation corresponding to a pair p,q∈Mr2r⁠, N(π) the number of cycles of π, and ℓ1, …, ℓN(π) be the length of the cycles. Then by (36),
tr(SpSq)=∏j=1N(π)32ℓj=3N(π)22r,
as ℓ1 + ⋯ + ℓN(π) = 2r. Therefore, we have proved that
(38)
Denoting by h(2r, k) the number of permutations in S2re which have exactly k cycles,21 the last equation can be rewritten in terms of the generating function of h(2r, k). This is computed in Lemma 16, which shows
‖Mr(2r)‖22=122r∑k=1rh(2r,k)3k=122r2r32r̄(2r−1)‼,
where αr̄ is the raising factorial. This evaluates to
2r32r̄=2r3232+1⋯32+r−1=3(3+2)⋯(2r+1)=(2r+1)‼,
which produces
‖Mr(2r)‖22=(2r−1)‼2r2(2r+1).
Inserting this into (37) produces the final result.□

The projection Psym(m) onto the highest-weight spin subspace naturally arises when considering the transfer operator associated with an undecorated AKLT model. The motivation for introducing the operators Mr(m) is that they form a basis for Z(m) and are eigenvectors of the transfer operator E⊗m⁠. If one determines coefficients so that
(39)
then (as we show in Sec. IV), it will be possible to write down the SVD of the transfer operator of the decorated AKLT model from the SVD of the transfer operator of the undecorated model. The goal of this section is to explicitly determine (39). This result will be a consequence of the following recursion relation, which shows how to rewrite the product M1(m)Mr(m) as a sum of matching operators.

Lemma 7.
Fix m ≥ 2, and for all 0≤r≤s≔⌊m2⌋ define
(40)
Then, the matchings operators satisfy
(41)
where, for consistency, one takes M−1(m)=Ms+1(m)=0.

We first prove the main technical result needed to prove Lemma 7, which makes use of the partition elements Xr(i, j) of Mrm from Definition 2.

Lemma 8.
Fix m ≥ 2. Then the following relations hold for any 0≤r≤⌊m2⌋:
(42)
(43)
(44)
(45)
where summations over empty sets are by convention taken to be zero.

We note that for each of the sets from Definition 2, there are one or two possible values of r for which the set is empty. These are precisely the values of r for which the coefficient on the RHS of (42)–(45) is zero, and so the equality holds.

Proof.

By the previous remark, we need only consider the values of r such that Xr(i, j) is nonempty. This is independent of (i, j).

The identity (42) follows from noting that any matching p∈Mrm belongs to precisely r different sets Ar(i, j), namely, those that are associated with the elements (i, j) ∈ p.

For (43), every matching p∈Mrm belongs to precisely 2r(m − 2r) different sets Br(i, j) labeled by taking a pair (i, j) where one element belongs to ∪ p and one element belongs to [m]\∪ p. The result follows.

To establish (44), begin by recalling that that p ∈ Cr(i, j) if and only if i, j ∈ ∪p but (i, j)∉p. Hence, given any fixed p∈Mrm there are exactly 2r(2r−2)2=2r(r−1) distinct choices for the set {i, j} such that p ∈ Cr(i, j). These are obtained by first picking any i ∈ ∪p, then taking an element j ∈ ∪p that belongs to an element (j, k) ∈ p with i ≠ j, k and recalling that (i, j) is unordered.

Finally, the equality in (45) is a consequence of the fact that p ∈ Dr(i, j) if and only if i, j ∈ [m]\∪p. For any r-matching p, there are m−2r2 distinct choices for (i, j).□

Proof of Lemma 7.

We drop the superscript m and set Si,j = Si · Sj to simplify notation.

For the case r = 1, the result is a consequence of breaking up the summation,
(M1)2=∑i<j∑k<lSi,jSk,l,
in terms of |{i, j} ∩ {k, l}|, and then applying the identities from (34). Since Ei,j,k = −Ek,j,i, this yields
(M1)2=∑i<jSi,j2+∑i<k∑j≠i,kSi,j,Sj,k+∑i<j∑k<l:k,l≠i,jSi,jSk,l,=316m21+−12+m−22M1+2M2=316m21+m−32M1+2M2.
Now, fix r > 1. We write the product M1Mr as
(46)
and consider the cases X ∈ {A, B, C, D} separately.
Fix i < j and consider the simplest case, Dr(i, j). Since Dr(i, j) ∋ p ↦ p ∪ (i, j) ∈ Ar+1(i, j) is a bijection,
∑p∈Dr(i,j)Si,jSp=∑p∈Ar+1(i,j)Sp.
Summing over i < j and applying Lemma 8 yields
(47)
For the case p ∈ Ar(i, j), one can write Sp = Si,jSp\(i,j), which by (34) implies
Si,jSp=(Si,j)2Sp\(i,j)=316Sp\(i,j)−12Sp.
Therefore, applying the bijection between Ar(i, j) and Dr−1(i, j) and then Lemma 8 gives
(48)
Now consider Br(i, j), for which one can write
∑p∈Br(i,j)Si,jSp=∑x≠i,j∑p′∈Mr−1m−3Si,jSi,x+Sj,xSp′,
where Mr−1m−3 is identified with the set of r − 1-matchings on [m]\{x, i, j}. Then, applying (34) and using Ei,j,x = −Ej,i,x one finds
Si,jSi,x+Sj,xSp′=14Sj,x+Si,xSp′.
Summing the final expression over all x, p′ and then all i < j produces the final identity,
(49)
where we again apply Lemma 8.
For the case of Cr(i, j), we begin by expanding
(50)
where Mr−2m−4 is identified with all r − 2 pairings on [m]\{i, j, x, y}. Then, by (34),
Si,jSi,xSj,y=14Sj,x−i2Ej,i,xSj,y=116Sx,y−i8Ex,j,y−i2Ej,i,xSj,y.
Expanding Ej,i,x and using SaSb=14δa,b1+i2ϵa,b,cSc⁠, the last term above can be calculated as
−i2Ej,i,xSj,y=−i8Ey,i,x+14∑b,c,d,e∑aϵa,b,cϵa,d,eSib⊗Sje⊗Sxc⊗Syd,=−i8Ey,i,x+14Si,ySj,x−Si,jSx,y,
where we use that the Levi-Civita symbol satisifies ∑aϵa,b,cϵa,d,e = δb,dδc,e − δb,eδc,d.
The analogous calculation holds for Si,jSj,xSi,y by exchanging i and j in the above formulas. Putting all of this together and using again the anti-symmetric property of Ea,b,c, one finds
Si,j(Si,xSj,y+Sj,xSi,y)Sp′=14Si,xSj,y+Sj,xSi,ySp′+18−12Si,jSx,ySp′.
Comparing the first term above with (50), it is clear that summing over all possible x, y and p′ produces
(51)
While for the other term
(52)
where we use p′ ∪ (x, y) ∈ Dr−1(i, j), that each p = {(x1, y1), …, (xr−1, yr−1)} ∈ Dr(i, j) appears exactly r − 1 times on the LHS, and the bijection
Ar(i,j)∋p↦p\(i,j)∈Dr−1(i,j).
Finally, summing (51) and (52) over i < j and applying Lemma 8 shows
(53)

Inserting (47)–(49) and (53) into (46) produces the desired expression for M1Mr.□

The recursion from Lemma 7 can be nicely rewritten in terms of an operator-valued vector product. Denoting by s≔⌊m2⌋ and following the convention for spin matrices, let M(m) the vector of s + 1 operators given by
(54)
Then, for any v∈Cs+1⁠, define
(55)
Therefore, Z(m)={v⋅M(m):v∈Cs+1} by Proposition 5. With this notation, Lemma 7 can be rephrased as follows:

Lemma 9.
Fix m ≥ 3, and let B be the (s + 1) × (s + 1) tridiagonal matrix,
(56)
whose entries ar, br, and cr are defined in Lemma 7. Then for each v∈Cs+1,
(57)

Proof.
By Lemma 7,
M1(m)(v⋅M(m))=∑r=0svr(crMr−1(m)+arMr(m)+brMr+1(m))=∑r=0svr′Mr(m),
where,
vr′=c1v1,if r=0,bs−1vs−1+asvs,if r=s,br−1vr−1+arvr+cr+1vr+1,otherwise.
Since a0 = 0, this is equivalent to v′ = Bv.□

The following are two immediate consequences of Lemma 9:

Corollary 10.

Fix m ≥ 3 and define e0=(1,0,…,0)∈Cs+1. The matrix B from Lemma 9 satisfies the following two properties:

  1. q(M1(m))=(q(B)e0)⋅M(m) for any polynomial q∈C[t].

  2. The set {e0, …, es} is a basis for Cs+1⁠, where er = Bre0.

Proof.
Since e0⋅M(m)=1⁠, it follows from Lemma 9 that
(M1(m))p=(Bpe0)⋅M(m),p≥0,
which extends to arbitrary polynomials q by linearity. Moreover, for any 0 ≤ r ≤ s, by the definition of B in Lemma 9, er = (er(0), …, er(s)) where
er(r)=r!ander(p)=0forp>r.
□

From this corollary, an approach for calculating (39) starts to emerge. From (31), it is trivial that
spec(M1(m))=λj≔j(j+1)2−3m8:j∈Jm.
As a consequence, for any polynomial pj such that pj(λj) = 1 and pj(λk) = 0 for all k ≠ j, one has
(58)
Such a polynomial pj is easy to find, e.g., pj(t)=∏i≠jt−λi∏i≠jλj−λi⁠. However, now one needs to evaluate pj(B), and so we turn to considering the spectrum and eigenvectors of B.

Lemma 11.
The spectrum of B and of M1(m) are the same up to multiplicities. If w∈(C2)⊗m is an eigenvector of M1(m), then y∈Cs+1 defined by
(59)
is a left eigenvector of B with the same eigenvalue.

Proof.

Note that spec(M1(m)) has s + 1 distinct real eigenvalues. Hence, one only needs to show that each eigenvalue of M1(m) is an eigenvalue of B with the corresponding left eigenvector.

Let λ∈spec(M1(m)) and w∈(C2)⊗m be a corresponding eigenvector. Then, for all 0 ≤ p ≤ s,
wqp(M1(m))w=0,whereqp(x)=(x−λ)xp.
Defining vp ≔ (B − λ)ep, it follows from Corollary 10 that
qp(M1(m))=(qp(B)e0)⋅M(m)=vp⋅M(m).
Since qp(M1(m))w=0 for all p, and the vectors vp and y have real entries, the above equality implies
(60)
where we use the standard inner product on Cs+1⁠. The vector y is nonzero since y(0) = ‖w‖2 ≠ 0. Hence, (60) is equivalent to
0=y|vp=(Bt−λ)yep∀0≤p≤s.
As {e0, …, es} is a basis for Cs+1⁠, it must be that (Bt − λ)y = 0, i.e., y is a left eigenvector of B with eigenvalue λ.□

While Lemma 11 characterizes the left eigenvectors of B, the next result explains how to obtain the corresponding right eigenvectors.

Lemma 12.
Fix m ≥ 2 and let D=diag(d0,…,ds)∈Ms+1(C) be the diagonal matrix whose entries are given by
(61)
If yr is a left eigenvector of B with eigenvalue λr, then Dyr is a right eigenvector of B with the same eigenvalue. As a consequence,
(62)
and yrDyr′=δr,r′⟨yr|D|yr⟩ for every r, r′ = 0, …, s.

We note that D is a strictly positive matrix since br−1, cr > 0 for all 1 ≤ r ≤ s. As such, the square root and inverse of D are well-defined.

Proof.
Let y be a left eigenvector of B with eigenvalue λ. By definition, it satisfies
(63)
where the cases r = 0 and r = s are interpreted by setting y(−1) = y(r + 1) = 0. Now consider y′ = Dy. Then,
(By′)(r)=br−1y′(r−1)+ary′(r)+cr+1y′(r+1),=br−1dr−1y(r−1)+ardry(r)+cr+1dr+1y(r+1).
The definition of the diagonal elements are such that dr−1/dr = cr/br−1 for 1 ≤ r ≤ s which, considering (63), implies y′ = Dy is a right eigenvector of B as then
(64)
Denote by spec(B) = {λr: 0 ≤ r ≤ s} and let {yr: 0 ≤ r ≤ s} be the corresponding set of left eigenvectors. It is easy to verify that B′ = D−1/2BD1/2 is symmetric, and hence has an orthonormal basis of eigenvectors. Since each eigenvalue of B is simple by Lemma 11, (64) implies that
vr=1∥D1/2yr∥D1/2yr,0≤r≤s,
is an orthonormal eigenbasis of B′. Hence, yrDyr′=δr,r′⟨yr|D|yr⟩⁠, and by the spectral theorem
B=D1/2B′D−1/2=∑r=0sλr1⟨yr|D|yr⟩Dyryr.
□

This final decomposition of B allows us to calculate p(B) for any polynomial p. As such, we can determine the values of the coefficients from (39).

Theorem 13.
For any m ≥ 2, the projection onto the symmetric subspace Psym(m) is
(65)

Proof.
Let s=⌊m2⌋ and λs=maxspec(M1(m))⁠. Since Psym(m)=Q(m,m/2)⁠, it follows from (58) that
Psym(m)=p(M1(m))=(p(B)e0)⋅M(m),
where p is such that p(λs) = 1 and p(λ) = 0 for all λ∈spec(M1(m))\{λs}⁠. Therefore, c(m) (r) = (p(B)e0) (r), and so by Lemma 12,
p(B)e0=1⟨ys|D|ys⟩Dysys|e0=ys(0)⟨ys|D|ys⟩Dys.
The proof is completed by determining ys, ys′≔Dys⁠, and their scalar product.
Since w=↑…↑ is an eigenvector of M1(m) associated with λs, Lemma 11 implies that the corresponding left eigenvector of B is defined by
(66)
where we have used (30) and (25). In particular, ys(0) = 1.
Now, by the definition of D, ys′(0)=1 and ys′(r)=drys(r) for 1 ≤ r ≤ s, where
dr=∏i=1rbi−1ci=42rr!(2r+1)‼∏i=1r2(m−2i+2)(m−2i+1).
This can be further simplified by recognizing ∏i=1r(m−2i+2)(m−2i+1)=m!(m−2r)!⁠, which yields
dr=42r(2r+1)‼⋅2rr!(m−2r)!m!=42r(2r+1)‼(2r−1)‼m2r.
Therefore, ys′(r)=4r(2r+1)!! and
(67)
The last equality can be seen, e.g., by first using 12r+1m2r=1m+1m+12r+1 and then applying m+12r+1=m2r+1+m2r⁠. Combining these produces the desired result,
p(B)e0(r)=ys′(r)ys(0)⟨ys|D|ys⟩=m+12m4r(2r+1)‼.
□

The proof of Theorem 1 follows from Proposition 3 together with a sufficiently tight upper bound on ‖EL(d,n)‖∞ and lower bound on qL(d,n)⁠. Recalling that QL=EL(d,n)(1)⁠, both of these bounds will be a consequence of writing
(68)
in terms of its singular value decomposition. The goal of Sec. IV A is to determine this and obtain the desired bounds. The proof of Theorem 1 is given in Sec. IV B.
Recall that ρ=1/2⁠. Since EL(d,n):M2(C)→M2(C)⊗d−1 and B≔{1,σX,σY,σZ} is an orthogonal basis with respect to the Hilbert–Schmidt inner product, one can write
(69)
What is special in this case (and is proved in Lemma 14 below) is that
EL(d,n)(1),EL(d,n)(σX),EL(d,n)(σY),EL(d,n)(σZ)⊆M2(C)⊗d−1,
is also an orthogonal set. Therefore, up to constants, (69) is the desired SVD. This result is stated with respect to the following matrices, which depend on a parameter α∈C⁠,
(70)
(71)
Above, U ∈ {X, Y, Z} and the notation (Mr(d−2))[d−1]\{j} denotes the matching operator associated with the d − 2 indices that remain after removing j from [d − 1].

Lemma 14
(SVD of EL(d,n)⁠). Let d ≥ 3, n ≥ 0, and set αn = 2(−1/3)n. Then,
(72)
The set {VA/2(d−1)(αn):A∈B} is orthogonal, and the singular values of EL(d,n) are
(73)

Proof.
Notice that it is sufficient to prove the result for n = 0, as E(SU)=−13SU for each U = X, Y, Z implies
E⊗d−1(Mr(d−1))=132rMr(d−1),E⊗d−1(SU⊗Mr(d−2))=−132r+1SU⊗Mr(d−2),
from which the general case follows from (68). To simplify notation, we drop the superscript and write EL for EL(d,0) in the remainder of this proof.
Considering (69), we proceed by calculating the Choi matrix τ corresponding to EL as a simple, but tedious, calculation shows
(74)
Using the definition of EL (12) and the recursion relation (13), one quickly finds
(75)
where the last equality follows from Theorem 13. We use properties of the set of matchings to rewrite τ in the latter form from (74).
Fix 0 ≤ r ≤ ⌊d/2⌋. For each matching p∈Mrd either (j, d) ∈ p for some 1 ≤ j ≤ d − 1, or d∉ ∪ p (in which case p∈Mrd−1⁠). Therefore, Mrd can be partitioned as
(76)
where Mr−1d−2 is interpreted as the set of matchings on [d − 1]\{j}. The corresponding matching operator then factorizes as
(77)
where π(j,d−1) is the permutation that swaps the tensor factors d − 1 and j, and any previously undefined matching operators are taken to be zero, e.g., M−1(d−2)=0⁠.
Substituting (77) into (75), one then finds
(78)
Consider the two summations above separately. Inserting the value of c(d)(r) from Theorem 13, the first summation can be rewritten as
(79)
The bound change in the first equality follows since either ⌊d2⌋=⌊d−12⌋ or ⌊d2⌋>d−12⁠, in which case the matching operator M⌊d2⌋(d−1)=0⁠.
For the remaining terms from (78), applying the relation KSUK*=−SŪ and again inserting c(d) (r) shows
(80)
where for the last line, we have used that
π(j,d−1)[Mr−1(d−2)⊗(Sd−1⋅Sd̄)]=Mr−1(d−2)[d−1]\{j}⊗(Sj⋅Sd̄).
Inserting (79) and (80) into (78) and comparing them with (74) establishes (72).
To see that the set of operators {VA/2(d−1)(2):A∈B} forms an orthogonal set, fix p={(i1,j1),…,(ir,jr)}∈Mrm and recall that by (29),
Sp=∑U1,…,Ur⊗k=1rSikUk⊗SjkUk,
is a sum of simple tensors, each of which has an even number of the spin operator SU for all U = X, Y, Z. As the r-th matching operator is the sum of all such Sp, by (70), Vρ(d−1)(2) is also a sum of simple tensors, each of which has an even number of the spin operator SU for all U. The same argument shows that VSU(d−1)(2) is a sum of simple tensors with an odd number of SU, and an even number of SV for V ≠ U. Since B is an orthogonal basis of M2(C)⁠, the orthogonality claim is a consequence of these observations. The set of singular values then follows from normalizing (69) appropriately.□

We now produce the necessary bounds on qL(d,n) and ‖EL(d,n)‖∞ to prove Theorem 1.

Lemma 15.
Fix d ≥ 3. The minimal eigenvalue qL(d,n) of QL(d,n)=EL(d,n)(1) satisfies
(81)
In particular, QL(d,n) is invertible for any n≥ln(d−1)ln(3)−ln(ln(3))ln(3)+12. For such n, one also has
(82)

By Lemma 14, it is simple to calculate the spectrum of QL(d,n) and ‖EL(d,n)‖∞ directly for small d using Lemma 6 and (89). Here, it is also convenient to use (23) and (31). For 1 ≤ d ≤ 4, this produces the values in Table I. A similar calculation can be performed for other small values of d by applying the recursion relation from Lemma 9 to write Mr(d−1) as a polynomial in M1(d−1) and again invoking the relationship to the Casimir operator.

TABLE I.

Exact values of qL(d,n) and ‖EL(d,n)‖∞ for 1 ≤ d ≤ 4.

dqL(d,n)‖EL(d,n)‖∞
1 2 2 
2 32 32 
3 1 − 3−2n 2(1+3−4n−1)1/2 
4 58(1−3−2n) 54(1+3−4n)1/2 
dqL(d,n)‖EL(d,n)‖∞
1 2 2 
2 32 32 
3 1 − 3−2n 2(1+3−4n−1)1/2 
4 58(1−3−2n) 54(1+3−4n)1/2 
View Large
Furthermore, using the recursion relation one can show that the eigenvalue of Mr(d−1) corresponding to the subspace of minimal spin j0 is
(83)
from which (70) implies that
(84)
We conjecture that this is qL(d,n)⁠, which has been verified with the help of a computer algebra system for d ≤ 30. This would imply that QL(d,n) is invertible when n ≥ 1 for all values of d. It also implies that the function f(d) in Theorem 1 can be improved, although doing so would not change the asymptotic scaling of n(Δ(G)).

Proof.
Note that Lemma 14 and (69) imply QL(d,n)=d+12d−1Vρ(d−1)(αn)⁠, and define
RL≔Vρ(d−1)(αn)−1=∑r=1⌊d−12⌋αn2r(2r+1)‼Mr(d−1).
Hence,
(85)
To bound ‖RL‖, first use ‖S⋅S‖=34 and (25) to bound the operator norm,
(86)
Then, since αn2=4⋅3−2n⁠, the operator norm of RL is bounded by
(87)
Hence, (81) holds from substituting (87) into the lower bound from (85). That QL(d,n) is invertible follows from using d−1r≤(d−1)rr! to further bound (87) by
13∑r=1d−113(n−1/2)rd−1r≤ed−13n−1/23,
which is less than one when n>ln(d−1)ln(3)−ln(ln(3))ln(3)+12⁠.
For the largest singular value of EL(d,n)⁠, Lemma 14 implies that for any choice of U,
(88)
In the case of Vρ(d−1)(αn)⁠, applying the mutual orthogonality of the matching operators and Lemma 6 it is straightforward to calculate
(89)
where in the last equality, we have substituted αn = 2 · (−3)−n.
In the case of VSU(d−1)(αn)⁠, first define
WSU(d−1)(r)≔∑j=1d−1SjU⊗(Mr(d−2))[d−1]\{j},0≤r≤⌊d−22⌋.
As the matching operators are mutually orthogonal, these are also mutually orthogonal with respect to the Hilbert–Schmidt norm. Therefore,
‖VSU(d−1)(αn)‖22=∑r=0⌊d−22⌋αn2r+1(2r+3)‼2‖WSU(d−1)(r)‖22.
The permutation invariance of WSU(d−1)(r)⁠, moreover, implies
‖WSU(d−1)(r)‖22=d−12‖Mr(d−2)‖22+d−12Sd−1U⊗Mr(d−2),Sd−2U⊗(Mr(d−2))[d−1]\{d−2}.
The remaining inner product can be calculated by once again using (77) to decompose the matching operators over an appropriate index and then invoking the orthogonality of the basis B⁠. This produces the recursive formula
‖WSU(d−1)(r)‖22=d−12‖Mr(d−2)‖22+(d−1)!22(d−3)!‖WSU(d−3)(r−1)‖22.
Iterating this identity r-times and again applying Lemma 6, one deduces
(90)
where in the last equality, we have substituted the value for αn and used
∑s=0r(2s+1)‼(2s)‼=(2r+3)‼3(2r)‼.
Comparing term by term (89) with (90), one finds that the largest singular value corresponds to ‖Vρ(d−1)(αn)‖2 as long as n+1≥ln(d−12)/ln(9)⁠. Then, arguing similarly to (87),
‖Vρ(d−1)(αn)‖22=2d−1∑r=0⌊d−12⌋3−4rn2r+1d−12r≤2d−11+(1+3−2n)d−1−13.
Inserting this into (88) produces (82).□

Proof of Theorem 1.
Let G = (V, E) be any simple graph such that Δ(G)=supv∈Vdeg(v)≥3⁠. As discussed in Sec. II, Theorem 1 follows immediately from proving,
ϵG(n)≔sup(vL,vR)∈Eϵ(vL,vR)(n)<1Δ(G).
Set d# = deg(v#) for # ∈ {L, R}. Then Proposition 3 shows that, as long as the maximum assumption is satisfied, ϵG(n)≤sup(vL,vR)∈EδdL,dR(n)⁠, where
δdL,dR(n)=4a(n)1(1−b(dL,n))(1−b(dR,n))+4a(n)+b(dL,n)b(dR,n)(1−b(dL,n))(1−b(dR,n)),
and b(d,n)=4a(n)‖EL(d,n)‖∞qL(d,n) by Ref. 20.
Now, define the function
c(d,n)=4a(n)⋅‖EL(d,n)‖∞qL(d,n),1≤d≤4,2d/2−13[2+(1+3−2n)d−1]4−(1+3−n+1/2)d−1d>4.
The constraint n ≥ n(Δ(G)) is sufficient to guarantee that the denominator above is strictly positive by Lemma 15. It is then easy to check using the values from Table I that (for n fixed) this function is increasing in d for 1 ≤ d ≤ Δ(G). Finally, by Lemma 15
b(d#,n)≤c(d#,n)≤c(Δ(G),n),#∈{L,R}.
To ease notation, set Δ(G) = D. Two key bounds follow.
The first is that
maxb(dL,n),b(dR,n),b(dL,n)b(dR,n)4a(n)≤3n4c(D,n)2=2D3n−1⋅2+(1+3−2n)D−1[4−(1+3−n+1/2)D−1]2.
The last fraction in the final expression above is decreasing in n. As n > ln(2)D/ln(3) for all D ≥ 3, this is bounded by
(91)
which is at most one for all n≥ln(2)D+ln(f(D))ln(3)⁠. This implies that the maximum assumption from Proposition 3 is satisfied for any edge (vL, vR) ∈ E when n ≥ n(Δ(G)).
The second key bound is that
(92)
where the last bound uses c(D, n) < 2/3n/2 by (91). The final expression is decreasing in n. Since n > ln(2)D/ln(3), one finds ϵn < 1/D for all D ≥ 5. In the case of D = 3 and D = 4, one can use the values from Table I to exactly evaluate (92). This yields ϵG(n) < 1/D when n ≥ D. This completes the proof.□

Let us now discuss the case in which the decoration number is a function of the edge, i.e., when each edge e is decorated with n(e) vertices for some n:E→N⁠. Let n = mine∈E n(e). We claim that the result of Theorem 1 still holds for the more generalized decorated model so long as n ≥Δ(G).

Fix a pair (vL, vR) of adjacent sites in G with degrees dL and dR. We show that ϵ(vL,vR)(n)≔∥GvLGvR−GvL∧GvR∥≤δdL,dR(n)⁠, where δdL,dR(n) is again the function from Proposition 3.

Enumerate the edges that are incident to vL but not to vR as {eL1,…,eLdL−1}⁠, and similarly the ones that are incident to vR but not to vL as {eR1,…,eRdR−1}⁠. Then the transfer operators corresponding to the regions XL and XR are, respectively,
ELn=En(eL1)⊗⋯⊗En(eLdL−1)◦EL(dL,0)=En(eL1)−n⊗⋯⊗En(eLdL−1)−n◦EL(dL,n),
and
ERn=ER(dR,0)◦En(eR1)⊗⋯⊗En(eLdR−1)=ER(dR,n)◦En(eR1)−n⊗⋯⊗En(eRdR−1)−n.
From the fact that ∥·∥∞ is sub-multiplicative and ∥E∥∞=1⁠, it immediately follows that
∥ELn∥∞≤∥EL(dL,n)∥∞,∥ERn∥∞≤∥ER(dR,n)∥∞.
Now let
QLn=ELn(1),QRn=(ERn)*(ρ),
and let qLn and qRn be their minimal eigenvalues, respectively. Then we have that
QLn=En(eL1)−n⊗⋯⊗En(eLdL−1)−n(QL(dL,n)),with QL(dL,n)=EL(dL,n)(1).
If qL(dL,n) is the minimal eigenvalue of QL(dL,n)⁠, then QL(dL,n)≥qL(dL,n)1 and from the positivity of the transfer operators maps, consequently,
QLn≥qL(dL,n)En(eL1)−n⊗⋯⊗En(eLdL−1)−n(1)=qL(dL,n)1,
as E(1)=1⁠. Hence, qLn≥qL(dL,n)⁠. A similar calculation for QRn shows that qRn≥qR(dR,n)⁠.
The final step is to adapt the proof of Proposition 3 to this case. Denote by
bL(e,n)=4a(n(e))∥ELn∥∞qLn,bR(e,n)=2a(n(e))∥ERn∥∞qRn,
where e = (vL, vR). From the previous discussions and the fact that a(n) is monotone decreasing, we see that these quantities are not larger than the ones corresponding to the same graph with decoration n, namely, bL(dL, n) and bR(dR, n). From this is follows that ϵ(vL,vR)(n) is upper bounded by δdL,dR(n)⁠.

A.L. was supported by Grant Nos. PID2020-113523GB-I00 and CEX2019-000904-S, funded by MCIN/AEI/10.13039/501100011033, by Grant No. RYC2019-026475-I, funded by MCIN/AEI/10.13039/501100011033 and “ESF Investing in your future,” and by Comunidad de Madrid (Grant No. QUITEMAD-CM, Ref. No. S2018/TCS-4342). A.Y. was supported by the DFG under Grant No. EXC-2111–390814868. The authors acknowledge the support of the Erwin Schrödinger International Institute for Mathematics and Physics (ESI), where part of this work was carried out during the “Tensor Networks: Mathematical Structures and Novel Algorithms” workshop. They also thank Bruno Nachtergaele for his helpful discussions during the development of this work, as well as the reviewers whose careful assessments of our work led to improvements in our results and proofs.

The authors have no conflicts to disclose.

All authors contributed equally to this work.

Angelo Lucia: Conceptualization (equal); Formal analysis (equal); Writing – original draft (equal); Writing – review & editing (equal). Amanda Young: Conceptualization (equal); Formal analysis (equal); Writing – original draft (equal); Writing – review & editing (equal).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

We now state and prove the result used to calculate ∑k=1rh(2r,k)3k in Lemma 6.

Lemma 16.
Fix r ≥ 1 and let h(2r, k) denote the number of permutations in S2re with exactly k cycles. Then
(A1)
where yr̄≔y(y+1)⋯(y+r−1) denotes the raising factorial.

Proof.
We compute this generating function using the exponential formula for labeled combinatorial structures.22 Let dr = (r − 1)! be the number of cyclic permutations of [r], and set
D(x)=∑r≥0d2rx2r(2r)!=log1(1−x2)1/2.
This is the deck enumerator function of the exponential family of even length cycles. The corresponding hand enumerator is then
(A2)
where h(0, 0) = 1 by convention. The exponential formula (see Ref. 22, Theorem 3.4.1) states that D(x) and H(x,y) are related via
H(x,y)=eyD(x)=(1−x2)−y2.
Expanding (1−x2)−y2 with the generalized binomial series produces
H(x,y)=∑r≥0−y2r(−1)rx2r=∑r≥0y2y2+1⋯y2+r−1x2rr!,=∑r≥0y2r̄(2r)!r!x2r(2r)!=∑r≥0y2r̄2r(2r−1)‼x2r(2r)!,
which when compared with (A2) implies the result.□

The aim of this section is twofold. The first is to present the modifications to Ref. 10 that are needed to obtain Proposition 3. The second is to compare the two approaches in the case of the decorated AKLT models, from which we will conclude that the new approach produces a better result for the minimal decoration needed to guarantee a spectral gap for this model. As we only consider decorated lattices in this section, to simplify the notation set,
E#=E#(d#,n),q#=q#(d#,n).

Let us recall the bound on ϵn from Ref. 10, Proposition 3.6 applied to our setting:

Proposition 17
(Ref. 10 , Proposition 3.6). For any edge (vL, vR) of a simple graph G with degrees dL and dR, repsectively, let
(B1)
If maxbLop(n),bRop(n),bLRop(n)<1, then ϵn(vL,vR)≤δop(n)⁠, where
(B2)

1. The alternate inner product bound

The Proof of Proposition 17 relies on the estimate contained in Ref. 10, Lemma 3.3. Here, we prove a variation of that result, from which the estimate of Proposition 3 follows.

For each Λ∈{YvL∪YvR,YvL,YvR}⁠, let KΛ be the virtual matrix space for the ground states of Λ in the TNS representation, namely,
KYvL∪YvR=M2dL−1×2dR−1(C),KYvL=M2dL−1×2(C),andKYvR=M2×2dR−1(C).
These spaces parameterize the ground states in the following sense: there exists linear maps ΓΛ:KΛ→HΛ such that ranΓΛ is exactly the subspace of ground states on region Λ [for an explicit definition in terms of the tensors of the TNS representation, see Ref. 10, Eq. (3.4)].
We define positive semi-definite Hermitian form on KΛ⁠, denoted by ⟨·, ·⟩Λ, via
(B3)
(B4)
(B5)
These are inner products as long as QL=EL(1) and QR=(ER)*(ρ) are (strictly) positive-definite, which by Lemma 15 holds for the decorated AKLT models when n≥ln(d−1)ln(3)−ln(ln(3))ln(3)+12⁠. In this case, one can also verify that the maps ΓΛ are injective. They also satisfy the following approximation bound, which is a variant of Ref. 10, Lemma 3.3.

Lemma 18.
Let Λ∈{YvL∪YvR,YvL,YvR}. Then for any B,C∈KΛ,
(B6)
where the constants are defined by
(B7)

We discuss the differences between this and Ref. 10, Lemma 3.3 and then give the proof. First, since the virtual bound dimension is D = 2 for the decorated AKLT models, the constant prefactor in the bound from Ref. 10, Lemma 3.3 is obtained from replacing CΛ with
CYvL∪YvR′=2‖EL‖op‖ER‖op,CYvL′=2‖EL‖op,andCYvR′=2‖ER‖op,
where ‖·‖op denotes the norm induced from the operator norm from Ref. 21.
The other difference is that the bound in Ref. 10, Lemma 3.3 is given in terms of the operator norm of B and C instead of the Hilbert–Schmidt norm. However, in the previous work, the operator norm is immediately bounded from above using one of the norms induced by (B3)–(B5). The Hilbert–Schmidt norm satisfies the same bound,
(B8)
where B∈KΛ for the appropriate choice of Λ∈{YvL∪YvR,YvL,YvR}⁠. The above should be compared with Ref. 10, Eq. (3.28). Here, ρmin ≔ min spec(ρ) = 1/2.

Using instead Lemma 18 and (B8) in Ref. 10, all arguments run as stated with the small modification of replacing CΛ′ with CΛ. This results in Proposition 3 from Sec. II C 1.

Proof.

Similar to the proof of Ref. 10, Lemma 3.3, we prove the result for Λ=YvL∪YvR as the other two cases follow from simple modifications of this case.

Let B={0,1} be an orthonormal basis of C2⁠. Then,
(B9)
[see Ref. 10, Eq. (3.25)]. Furthermore, the identity
(B10)
can easily be seen from simplifying the RHS.
Substituting (B9) and (B10) into the LHS of (B6), and recalling that Eα,β=αβ is an orthonormal basis with respect to the Hilbert–Schmidt norm, one finds
(B11)
where we have used cyclicity of the trace and the Cauchy-Schwarz inequality, denoted by ‖·‖p the Schatten p-norm, and introduced the map SB,C:M2dL−1(C)→M2dR−1(C) defined by SB,C(A) ≔ B*AC.
Recalling the diagonalization of E:M2(C)→M2(C) from (9), one finds
‖En−|1⟩⟨ρ|‖2≤dim(M2(C))‖En−|1⟩⟨ρ|‖∞=2a(n)
Then, repeatedly applying the generalized Hölder inequality ‖RT‖2 ≤ ‖R‖∞‖T‖2 to (B11) produces
(B12)
The final result is then a consequence of substituting ‖SB,C‖2=‖B‖22‖C‖22⁠, which holdsfrom calculating
‖SB,C‖22=∑γ,δ=12dL−1B*|γδ|C,B*|γδ|C2=‖B‖22‖C‖22
where |γ⟩:1≤γ≤2dL−1 denotes any orthonormal basis of C2dL−1⁠.□

2. Comparing the two bounds for decorated AKLT models

To compare the two approaches, it is sufficient to assume that G is a simple, regular graph, i.e., d = deg(v) for all vertices. Therefore, recalling Proposition 3 and Proposition 17, the two approaches show that the AKLT model on G(n) is uniformly gapped if
(B13)
where for the approach of the present work, we redefine the quantities from Proposition 3 as δ∞(n) = δd,d(n), and
(B14)
We prove that, in the case of decorated AKLT models, the conditions from (B13) are necessarily satisfied for the approach used in the present work if they are satisfied with the approach from Ref. 10.

Proposition 19.
Suppose that G = (V, E) is a regular, simple graph such that deg(v) = d for all v ∈ V. Then, for any n≥ln(d−1)ln(3)−ln(ln(3))ln(3)+12, one has bLR∞(n)<bLRop(n) and
(B15)
Said differently, δ∞(n) < δop(n).

The proof will make use of the following function, which is even and strictly increasing for x ≥ 0,
(B16)

Proof of Proposition 19.
To begin, we produce formulas of the necessary quantities for the comparison. Recall that αn = 2(−3)−n. Then by (15),
(B17)
Here, we have invoked Lemma 15, and used (89) and (B16) for the last equality.
On the other hand, the norm induced by the operator norm satisfies ‖E#‖op=‖E#(1)‖ since E# is a completely positive map. Therefore, considering (12),
(B18)
while by Lemma 14,
(B19)
The lower bound in (B19) follows from using the definition of Vρ(d−1)(αn) and (66) to calculate
‖Vρ(d−1)(αn)‖≥⟨↑⋯↑|Vρ(d−1)(αn)|↑⋯↑⟩=Fd−1(|αn|/2).
We can now compare the desired quantities. Considering (B1) and (B14), it is clear that bLR∞(n)<bLRop(n) if and only if ‖EL‖∞‖ER‖∞<2‖EL‖op‖ER‖op⁠. From (B17)–(B19), it is easily deduced that this inequality holds since
Fd−1(αn2/4)=Fd−1(9−n)<2Fd−1(3−n)=2Fd−1(|αn|/2).
Now consider (B15), and note that (15) implies QR=12QL and bL∞(n)=bR∞(n)⁠. Therefore, by (B14) and (B17),
(1−bL∞(n))(1−bR∞(n))=1−bL∞(n)=1−a(n)(d+1)2d−3qL⋅2d−2Fd−1(αn2/4).
On the other hand, applying (B1), (B18), and (B19),
(1−bLop(n))(1−bRop(n))≤1−bLop(n)+bRop(n)2≤1−a(n)(d+1)2d−3qL(Fd−1(αn/2)+2d−2).
As Fd−1(αn/2)>0⁠, (B15) is a consequence of using nk=n−1k+n−1k−1 for 1 ≤ k ≤ n − 1 to bound
Fd−1(αn2/4)<Fd−1(1)<∑r=0⌊d−12⌋d−12r=∑r=0d−2d−2r=2d−2.
□

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