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.

## I. INTRODUCTION

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.

*G*= (

*V*,

*E*) is uniformly gapped if there exists a sequence of finite subgraphs

*G*

_{k}= (

*V*

_{k},

*E*

_{k}),

*k*≥ 1, such that

*E*

_{k}⊆

*E*

_{k+1},

*V*

_{k}⊆

*V*

_{k+1}, and ∪

_{k}

*E*

_{k}=

*E*for which the associated local Hamiltonians satisfy

*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*) = sup_{v∈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 study^{18} 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.

## II. MAIN RESULT AND PROOF STRATEGY

### A. The uniform gap for the decorated models

*G*= (

*V*,

*E*) such that

*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.

*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,

*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}

*v*∈

*V*, let

*Y*

_{v}⊂

*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

*G*. This result, which we now state, depends on the decreasing function,

*Suppose that*

*G*= (

*V*,

*E*)

*is a simple graph such that*3 ≤ Δ(

*G*) <

*∞*

*. If*

*n*≥

*n*(Δ(

*G*)),

*where*

*then there exists*

*γ*(Δ(

*G*),

*n*) > 0,

*such that*

Our result can easily be generalized to the situation where the decoration varies on different edges. Given a bounded function $n:E\u2192N$, 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 min

_{e∈E}

**(**

*n**e*) ≥

*n*(Δ(

*G*)). Here, Λ

^{(n)}is defined as in (1), with

*Y*

_{v}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:

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 min_{v}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.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*.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.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*G*_{k}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\xd7\nu $ for a fixed*ν*).

*C**-algebra of (quasi-)local observables:

*Let*

*G*

*be as in Theorem*1

*,*

*n*≥

*n*(Δ(

*G*))

*, and*$\omega :AG(n)\u2192C$

*be any weak-**

*limit of finite-volume ground states,*

*where*$\psi m\u2208ker(H\Lambda m(n))$

*is normalized and*Λ

_{m}↑

*G*

*. Then, the spectral gap above the ground state of the corresponding GNS Hamiltonian*

*H*

_{ω}

*satisfies*

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.

### B. Reduction to a quasi one-dimensional system

*G*

_{v}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

*Y*

_{v}, it is easy to deduce that [analogous to Ref. 10, Eq. (2.4)],

*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 $H\u0303\Lambda (n)$ implies a uniform gap of $H\Lambda (n)$.

*G*

_{v}∧

*G*

_{w}is the orthogonal projection onto $ran(Gv)\u2229ran(Gw)=ker(HYv\u222aYw)$ by frustration-freeness. Namely, since {

*P*

_{v},

*P*

_{w}} ≥ 0 if (

*v*,

*w*) ∉

*E*and

*v*belongs to at most Δ(

*G*) edges. Hence, $gap(H\u0303\Lambda (n))\u22651\u2212\Delta (G)\u03f5G(n)$, and combining these bounds yields

*ϵ*

_{G}(

*n*) < 1/Δ(

*G*) if

*n*≥

*n*(Δ(

*G*)). The remainder of this paper is focused on proving this inequality.

### C. Transfer operator estimates

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 *Y*_{v} ∪ *Y*_{w}. 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

*v*

_{L}and

*v*

_{R}as the “left” and “right” vertex, respectively, associated with an edge (

*v*

_{L},

*v*

_{R}) ∈

*E*, and decompose

*n*sites decorating the edge (

*v*

_{L},

*v*

_{R}), and $X#=Yv#\Cn$ for # ∈ {

*L*,

*R*} (see Fig. 2). The desired transfer operators are those associated with these three regions.

*C*

_{n}is the

*n*-fold composition $En$, where $E:M2(C)\u2192M2(C)$ is the (well known) single vertex spin-1 AKLT transfer operator,

*S*

^{U}=

*σ*

^{U}/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,

*d*

_{#}= deg(

*v*

_{#}) for # ∈ {

*L*,

*R*}, and denote by

*X*

_{L}and

*X*

_{R}, respectively. These are the composition of the undecorated transfer operator and

*n*copies of $E\u2297d#\u22121$,

*d*

_{L}=

*d*, by definition

*v*

_{L}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}

*d*spin-1/2 particles, and let $\varphi kd\u2208(C2)\u2297d$ be the normalized symmetric vector with

*k*up spins. These satisfy the recursive formula

*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.

*v*

_{R}. This yields $W\u0303kd\u2254K\u2297d\u22121(Vkd)*=(\u22121)k\u22121(Wd\u2212kd)*$, which when combined with (9)–(12) implies

#### 2. Bounding *ϵ*_{G}(*n*) via transfer operator quantities

*ϵ*

_{G}(

*n*) for

*G*= (

*V*,

*E*) is a consequence of bounding

*a*(

*n*) from (10), the minimal eigenvalues $qL(d,n)$ and $qR(d,n)$, respectively, of

*∞*norm of $EL(d,n)$ and $ER(d,n)$. Here, we recall that, with respect to the Hilbert–Schmidt norm ∥·∥

_{2},

*Let*(

*v*

_{L},

*v*

_{R})

*be an edge of a simple graph*

*G*

*between vertices with degrees*

*d*

_{L}

*and*

*d*

_{R}

*, respectively, and define*

*If*$maxbL(dL,n),bR(dR,n),bL(dL,n)bR(dR,n)4a(n)<1$

*, then*$\u03f5(vL,vR)(n)\u2264\delta dL,dR(n)$,

*where*

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 $YvL\u222aYvR$ are all injective, see, e.g., Ref. 10, Corollary 3.4, which is necessary for their approach.

*b*(

*d*,

*n*) so that by (7) and Proposition 3,

*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.

*∞*norm in (18) rather than the norm induced by the operator norm, i.e.,

*a*(

*n*) is invariant under this change of norms, as an elementary, but tedious, calculation shows

*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.

## III. MATCHINGS OPERATORS

*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.

### A. Basic notions of matchings and matching operators

*m*≥ 2 and recall that the spectral decomposition of the Casimir operator $C(m)=(\u2211i=1mSi)2$ acting on $(C2)\u2297m$ is

*j*, and

*j*

_{0}= 0 if

*m*is even and

*j*

_{0}= 1/2 otherwise. Moreover, recall that $Q(m,j)=Psym(m)$ is the projection onto the symmetric subspace when $j=\u230am2\u230b+j0$.

*matching*.

#### 1. Definition and properties of matchings

*r*

*-matching*of [

*m*] ≔ {1, …,

*m*} is a collection of

*r*unordered pairs [meaning that (

*a,b*) is the same as (

*b,a*)]

*p*≔ {

*a*

_{i},

*b*

_{i}: 1 ≤

*i*≤

*r*} has 2

*r*elements. The set of all

*r*-matchings is denoted by $Mrm$. For consistency, $M0m$ is the set consisting of the

*empty matching*,

*p*= {}.

*r*-matchings is easily calculated with the multinomial coefficient,

*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 $p\u2208Mrm$ 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 $Mrm$.

*m*≥ 2. For any $0\u2264r\u2264\u230am2\u230b$ and distinct pair

*i*,

*j*∈ [

*m*], set

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 *A*_{r}(*i*, *j*), and similarly for the others.

*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

*:*

For each set *X*_{r}(*i*, *j*) from Definition 2, the result follows from constructing an *n*-to-1 mapping between *X*_{r}(*i*, *j*) and the appropriate set of matchings.

- A bijection between
*A*_{r}(*i*,*j*) and*D*_{r−1}(*i*,*j*) is given bywhile a bijection between$p\u2208Ar(i,j)\u21a6p\{(i,j)}\u2208Dr\u22121(i,j),$*D*_{r−1}(*i*,*j*) and $Mr\u22121m\u22122$ results from recognizing*D*_{r−1}(*i*,*j*) as the set of all*r*− 1 pairings on the set [*m*]\{*i*,*j*}. - For
*p*∈*B*_{r}(*i*,*j*) define the mapping*p*↦*p*′ by decomposingfor some$p={(i,x)}\u222ap\u2032orp={(j,x)}\u222ap\u2032,$*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. - For
*p*∈*C*_{r}(*i*,*j*), the mapping*p*↦*p*′ is defined by writingwhere$p={(x,i),(y,j)}\u222ap\u2032,$*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.*

*m*≥ 2. Then, the

*r*

*-matching operator*$Mr(m)\u2208B((C2)\u2297m)$ is

*S*

_{p}does not matter. Moreover, each matching operator is clearly nonzero since

$E\u2297m(Mr(m))=3\u22122rMr(m)$ since $E(1)=1$ and $E\u2297E(S\u22c5S)=19(S\u22c5S)$.

- $M1(m)$ is related to the Casimir operator
*C*^{(m)}via(31)$C(m)=3m41+2M1(m).$ - Every matching operator $Mr(m)$ is Hermitian as well as
*SU*(2) and permutation symmetric (since $Mrm$ is permutation invariant). Therefore, $Mr(m)\u2208Z(m)$ by Schur–Weyl duality. However, we provide an alternative way of verifying this inclusion. Lemma 7 establishes thatfor appropriate coefficients$M1(m)\u22c5Mr(m)=crMr\u22121(m)+arMr(m)+brMr+1(m)\u2200r\u22650,$*a*_{r},*b*_{r},*c*_{r}. 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).

*The set of matching operators* ${Mr(m)}r=0\u230am2\u230b$ *forms a Hilbert–Schmidt orthogonal basis of* $Z(m)$*.*

Since each $Mr(m)\u2208Z(m)$ 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 $Z(m)$.

*r*

_{1}≠

*r*

_{2}. Since each matching operator is Hermitian, it trivially follows that

*S*

_{p}

*S*

_{q}) = 0 unless ∪

*p*= ∪

*q*. As $\u222ap=2r1\u22602r2=\u222aq$, this cannot occur. This proves orthogonality.□

*ϵ*

_{i,j,k}denote the Levi-Civita symbol, and define

*τ*of the indices, i.e.,

*E*

_{τ(a),τ(b),τ(c)}= −

*E*

_{a,b,c}. Using $SiSj=14\delta i,j1+i2\u03f5i,j,kSk$, the following relations hold:

*For all*$0\u2264r\u2264\u230am2\u230b$

*, the Hilbert–Schmidt norm of the*

*r*

*-matching operator satisfies*

*E*

_{a,b,c}over the

*b*-th index is zero, it follows from (34) that

*S*

_{p}

*S*

_{p}) = 0 unless ∪

*p*= ∪

*q*. Since tr(

*S*

_{p}

*S*

_{q}) is invariant under permutations of the factors, without loss of generality, assume that ∪

*p*= ∪

*q*= {1, …, 2

*r*}. Therefore, the norm calculation reduces to

To calculate $\Vert Mr(2r)\Vert 22$, consider first an arbitrary pair $p,q\u2208Mr2r$, which we refer to as *perfect matchings* over 2*r* elements. As we now show, pairs (*p*, *q*) of perfect matchings of 2*r* elements are in bijection with permutations $\pi \u2208S2r$ of 2*r* elements that have no odd length cycles, which we denote by $S2re$.

Given (*p*, *q*), the corresponding permutation *π* is built as follows: let *a*_{1} ∈ [2*r*] be arbitrary, and inductively define *a*_{2i} to be the element connected to *a*_{2i−1} in *p*, and *a*_{2i+1} to be the element connected to *a*_{2i} in *q*. Then, by properties of the matching, it must be that there is some finite *l* such that *a*_{2l+1} = *a*_{1}. Therefore, (*a*_{1}, …, *a*_{2l}) is a cycle of length 2*l* 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 $\pi \u2208S2re$, define (*p*, *q*) as follows: for each cycle (*a*_{1}, …, *a*_{2l}) in *π*, add (*a*_{2i−1}, *a*_{2i}) to *p* and (*a*_{2i}, *a*_{2i+1}) to *q*, for each *i* = 0, …, *l* (where addition is taken modulo 2*l*). Then *p* and *q* are perfect matchings as they both have *r* disjoint pairs.

*N*(

*π*) the number of cycles of

*π*, and

*ℓ*

_{1}, …,

*ℓ*

_{N(π)}be the length of the cycles. Then by (36),

*ℓ*

_{1}+ ⋯ +

*ℓ*

_{N(π)}= 2

*r*. Therefore, we have proved that

*h*(2

*r*,

*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*(2

*r*,

*k*). This is computed in Lemma 16, which shows

### B. A recursion relation for the matching operators

*Fix*

*m*≥ 2

*, and for all*$0\u2264r\u2264s\u2254\u230am2\u230b$

*define*

*Then, the matchings operators satisfy*

*where, for consistency, one takes*$M\u22121(m)=Ms+1(m)=0$

*.*

We first prove the main technical result needed to prove Lemma 7, which makes use of the partition elements *X*_{r}(*i*, *j*) of $Mrm$ from Definition 2.

*Fix*

*m*≥ 2

*. Then the following relations hold for any*$0\u2264r\u2264\u230am2\u230b$

*:*

*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.

By the previous remark, we need only consider the values of *r* such that *X*_{r}(*i*, *j*) is nonempty. This is independent of (*i*, *j*).

The identity (42) follows from noting that any matching $p\u2208Mrm$ belongs to precisely *r* different sets *A*_{r}(*i*, *j*), namely, those that are associated with the elements (*i*, *j*) ∈ *p*.

For (43), every matching $p\u2208Mrm$ belongs to precisely 2*r*(*m* − 2*r*) different sets *B*_{r}(*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* ∈ *C*_{r}(*i*, *j*) if and only if *i*, *j* ∈ ∪*p* but (*i*, *j*)∉*p*. Hence, given any fixed $p\u2208Mrm$ there are exactly $2r(2r\u22122)2=2r(r\u22121)$ distinct choices for the set {*i*, *j*} such that *p* ∈ *C*_{r}(*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* ∈ *D*_{r}(*i*, *j*) if and only if *i*, *j* ∈ [*m*]\∪*p*. For any *r*-matching *p*, there are $m\u22122r2$ distinct choices for (*i*, *j*).□

We drop the superscript *m* and set *S*_{i,j} = *S*_{i} · *S*_{j} to simplify notation.

*r*= 1, the result is a consequence of breaking up the summation,

*i*,

*j*} ∩ {

*k*,

*l*}|, and then applying the identities from (34). Since

*E*

_{i,j,k}= −

*E*

_{k,j,i}, this yields

*r*> 1. We write the product

*M*

_{1}

*M*

_{r}as

*X*∈ {

*A*,

*B*,

*C*,

*D*} separately.

*i*<

*j*and consider the simplest case,

*D*

_{r}(

*i*,

*j*). Since

*D*

_{r}(

*i*,

*j*) ∋

*p*↦

*p*∪ (

*i*,

*j*) ∈

*A*

_{r+1}(

*i*,

*j*) is a bijection,

*i*<

*j*and applying Lemma 8 yields

*p*∈

*A*

_{r}(

*i*,

*j*), one can write

*S*

_{p}=

*S*

_{i,j}

*S*

_{p\(i,j)}, which by (34) implies

*A*

_{r}(

*i*,

*j*) and

*D*

_{r−1}(

*i*,

*j*) and then Lemma 8 gives

*B*

_{r}(

*i*,

*j*), for which one can write

*r*− 1-matchings on [

*m*]\{

*x*,

*i*,

*j*}. Then, applying (34) and using

*E*

_{i,j,x}= −

*E*

_{j,i,x}one finds

*x*,

*p*′ and then all

*i*<

*j*produces the final identity,

*C*

_{r}(

*i*,

*j*), we begin by expanding

*r*− 2 pairings on [

*m*]\{

*i*,

*j*,

*x*,

*y*}. Then, by (34),

*E*

_{j,i,x}and using $SaSb=14\delta a,b1+i2\u03f5a,b,cSc$, the last term above can be calculated as

_{a}

*ϵ*

_{a,b,c}

*ϵ*

_{a,d,e}=

*δ*

_{b,d}

*δ*

_{c,e}−

*δ*

_{b,e}

*δ*

_{c,d}.

*S*

_{i,j}

*S*

_{j,x}

*S*

_{i,y}by exchanging

*i*and

*j*in the above formulas. Putting all of this together and using again the anti-symmetric property of

*E*

_{a,b,c}, one finds

*x*,

*y*and

*p*′ produces

*p*′ ∪ (

*x*,

*y*) ∈

*D*

_{r−1}(

*i*,

*j*), that each

*p*= {(

*x*

_{1},

*y*

_{1}), …, (

*x*

_{r−1},

*y*

_{r−1})} ∈

*D*

_{r}(

*i*,

*j*) appears exactly

*r*− 1 times on the LHS, and the bijection

*i*<

*j*and applying Lemma 8 shows

Inserting (47)–(49) and (53) into (46) produces the desired expression for *M*_{1}*M*_{r}.□

*M*^{(m)}the vector of

*s*+ 1 operators given by

*Fix*

*m*≥ 3

*, and let*

*B*

*be the*(

*s*+ 1) × (

*s*+ 1)

*tridiagonal matrix,*

*whose entries*

*a*

_{r}

*,*

*b*

_{r}

*,*and

*c*

_{r}

*are defined in Lemma 7. Then for each*$v\u2208Cs+1$

*,*

*a*

_{0}= 0, this is equivalent to

*v*′ =

*Bv*.□

The following are two immediate consequences of Lemma 9:

*Fix* *m* ≥ 3 *and define* $e0=(1,0,\u2026,0)\u2208Cs+1$*. The matrix* *B* *from Lemma 9 satisfies the following two properties:*

$q(M1(m))=(q(B)e0)\u22c5M(m)$

*for any polynomial*$q\u2208C[t]$*.**The set*{*e*_{0}, …,*e*_{s}}*is a basis for*$Cs+1$,*where**e*_{r}=*B*^{r}*e*_{0}*.*

*q*by linearity. Moreover, for any 0 ≤

*r*≤

*s*, by the definition of

*B*in Lemma 9,

*e*

_{r}= (

*e*

_{r}(0), …,

*e*

_{r}(

*s*)) where

*p*

_{j}such that

*p*

_{j}(

*λ*

_{j}) = 1 and

*p*

_{j}(

*λ*

_{k}) = 0 for all

*k*≠

*j*, one has

*p*

_{j}is easy to find, e.g., $pj(t)=\u220fi\u2260jt\u2212\lambda i\u220fi\u2260j\lambda j\u2212\lambda i$. However, now one needs to evaluate

*p*

_{j}(

*B*), and so we turn to considering the spectrum and eigenvectors of

*B*.

*The spectrum of*

*B*

*and of*$M1(m)$

*are the same up to multiplicities. If*$w\u2208(C2)\u2297m$

*is an eigenvector of*$M1(m)$

*, then*$y\u2208Cs+1$

*defined by*

*is a left eigenvector of*

*B*

*with the same eigenvalue.*

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.

*p*≤

*s*,

*v*

_{p}≔ (

*B*−

*λ*)

*e*

_{p}, it follows from Corollary 10 that

*p*, and the vectors

*v*

_{p}and

*y*have real entries, the above equality implies

*y*is nonzero since

*y*(0) = ‖

*w*‖

^{2}≠ 0. Hence, (60) is equivalent to

*e*

_{0}, …,

*e*

_{s}} is a basis for $Cs+1$, it must be that (

*B*

^{t}−

*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.

*Fix*

*m*≥ 2

*and let*$D=diag(d0,\u2026,ds)\u2208Ms+1(C)$

*be the diagonal matrix whose entries are given by*

*If*

*y*

_{r}

*is a left eigenvector of*

*B*

*with eigenvalue*

*λ*

_{r}

*, then*

*Dy*

_{r}

*is a right eigenvector of*

*B*

*with the same eigenvalue. As a consequence,*

*and*$yrDyr\u2032=\delta r,r\u2032\u27e8yr|D|yr\u27e9$

*for every*

*r*,

*r*′ = 0,

*…*,

*s*

*.*

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

*y*be a left eigenvector of

*B*with eigenvalue

*λ*. By definition, it satisfies

*r*= 0 and

*r*=

*s*are interpreted by setting

*y*(−1) =

*y*(

*r*+ 1) = 0. Now consider

*y*′ =

*Dy*. Then,

*d*

_{r−1}/

*d*

_{r}=

*c*

_{r}/

*b*

_{r−1}for 1 ≤

*r*≤

*s*which, considering (63), implies

*y*′ =

*Dy*is a right eigenvector of

*B*as then

*B*) = {

*λ*

_{r}: 0 ≤

*r*≤

*s*} and let {

*y*

_{r}: 0 ≤

*r*≤

*s*} be the corresponding set of left eigenvectors. It is easy to verify that

*B*′ =

*D*

^{−1/2}

*BD*

^{1/2}is symmetric, and hence has an orthonormal basis of eigenvectors. Since each eigenvalue of

*B*is simple by Lemma 11, (64) implies that

*B*′. Hence, $yrDyr\u2032=\delta r,r\u2032\u27e8yr|D|yr\u27e9$, and by the spectral theorem

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).

*For any*

*m*≥ 2

*, the projection onto the symmetric subspace*$Psym(m)$

*is*

*p*is such that

*p*(

*λ*

_{s}) = 1 and

*p*(

*λ*) = 0 for all $\lambda \u2208spec(M1(m))\{\lambda s}$. Therefore,

*c*

^{(m)}(

*r*) = (

*p*(

*B*)

*e*

_{0}) (

*r*), and so by Lemma 12,

*y*

_{s}, $ys\u2032\u2254Dys$, and their scalar product.

*λ*

_{s}, Lemma 11 implies that the corresponding left eigenvector of

*B*is defined by

*y*

_{s}(0) = 1.

*D*, $ys\u2032(0)=1$ and $ys\u2032(r)=drys(r)$ for 1 ≤

*r*≤

*s*, where

## IV. THE UNIFORM SPECTRAL GAP

### A. Bounding $\Vert EL(d,n)\Vert \u221e$ and $qL(d,n)$

*U*∈ {

*X*,

*Y*,

*Z*} and the notation $(Mr(d\u22122))[d\u22121]\{j}$ denotes the matching operator associated with the

*d*− 2 indices that remain after removing

*j*from [

*d*− 1].

*SVD of*$EL(d,n)$).

*Let*

*d*≥ 3

*,*

*n*≥ 0

*, and set*

*α*

_{n}= 2(−1/3)

^{n}

*. Then,*

*The set*${VA/2(d\u22121)(\alpha n):A\u2208B}$

*is orthogonal, and the singular values of*$EL(d,n)$

*are*

*n*= 0, as $E(SU)=\u221213SU$ for each

*U*=

*X*,

*Y*,

*Z*implies

*τ*corresponding to $EL$ as a simple, but tedious, calculation shows

*τ*in the latter form from (74).

*r*≤ ⌊

*d*/2⌋. For each matching $p\u2208Mrd$ either (

*j*,

*d*) ∈

*p*for some 1 ≤

*j*≤

*d*− 1, or

*d*∉ ∪

*p*(in which case $p\u2208Mrd\u22121$). Therefore, $Mrd$ can be partitioned as

*d*− 1]\{

*j*}. The corresponding matching operator then factorizes as

*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., $M\u22121(d\u22122)=0$.

*c*

^{(d)}(

*r*) from Theorem 13, the first summation can be rewritten as

*c*

^{(d)}(

*r*) shows

*S*

^{U}for all

*U*=

*X*,

*Y*,

*Z*. As the

*r*-th matching operator is the sum of all such

*S*

_{p}, by (70), $V\rho (d\u22121)(2)$ is also a sum of simple tensors, each of which has an even number of the spin operator

*S*

^{U}for all

*U*. The same argument shows that $VSU(d\u22121)(2)$ is a sum of simple tensors with an odd number of

*S*

^{U}, and an even number of

*S*

^{V}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 $\Vert EL(d,n)\Vert \u221e$ to prove Theorem 1.

*Fix*

*d*≥ 3

*. The minimal eigenvalue*$qL(d,n)$

*of*$QL(d,n)=EL(d,n)(1)$

*satisfies*

*In particular,*$QL(d,n)$

*is invertible for any*$n\u2265ln(d\u22121)ln(3)\u2212ln(ln(3))ln(3)+12$

*. For such*

*n*

*, one also has*

By Lemma 14, it is simple to calculate the spectrum of $QL(d,n)$ and $\Vert EL(d,n)\Vert \u221e$ 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 $Mr(d\u22121)$ as a polynomial in $M1(d\u22121)$ and again invoking the relationship to the Casimir operator.

d
. | $qL(d,n)$ . | $\Vert EL(d,n)\Vert \u221e$ . |
---|---|---|

1 | 2 | $2$ |

2 | $32$ | $32$ |

3 | 1 − 3^{−2n} | $2(1+3\u22124n\u22121)1/2$ |

4 | $58(1\u22123\u22122n)$ | $54(1+3\u22124n)1/2$ |

d
. | $qL(d,n)$ . | $\Vert EL(d,n)\Vert \u221e$ . |
---|---|---|

1 | 2 | $2$ |

2 | $32$ | $32$ |

3 | 1 − 3^{−2n} | $2(1+3\u22124n\u22121)1/2$ |

4 | $58(1\u22123\u22122n)$ | $54(1+3\u22124n)1/2$ |

*j*

_{0}is

*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*)).

*R*

_{L}‖, first use $\Vert S\u22c5S\Vert =34$ and (25) to bound the operator norm,

*R*

_{L}is bounded by

*U*,

*α*

_{n}= 2 · (−3)

^{−n}.

*r*-times and again applying Lemma 6, one deduces

*α*

_{n}and used

### B. Proof of Theorem 1

*G*= (

*V*,

*E*) be any simple graph such that $\Delta (G)=supv\u2208Vdeg(v)\u22653$. As discussed in Sec. II, Theorem 1 follows immediately from proving,

*d*

_{#}= deg(

*v*

_{#}) for # ∈ {

*L*,

*R*}. Then Proposition 3 shows that, as long as the maximum assumption is satisfied, $\u03f5G(n)\u2264sup(vL,vR)\u2208E\delta dL,dR(n)$, where

*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

*G*) =

*D*. Two key bounds follow.

*n*. As

*n*> ln(2)

*D*/ln(3) for all

*D*≥ 3, this is bounded by

*v*

_{L},

*v*

_{R}) ∈

*E*when

*n*≥

*n*(Δ(

*G*)).

*c*(

*D*,

*n*) < 2/3

^{n/2}by (91). The final expression is decreasing in

*n*. Since

*n*> ln(2)

*D*/ln(3), one finds

*ϵ*

_{n}< 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

*ϵ*

_{G}(

*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 $n:E\u2192N$. Let

*n*= min

_{e∈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 (*v*_{L}, *v*_{R}) of adjacent sites in *G* with degrees *d*_{L} and *d*_{R}. We show that $\u03f5(vL,vR)(n)\u2254\u2225GvLGvR\u2212GvL\u2227GvR\u2225\u2264\delta dL,dR(n)$, where $\delta dL,dR(n)$ is again the function from Proposition 3.

*v*

_{L}but not to

*v*

_{R}as ${eL1,\u2026,eLdL\u22121}$, and similarly the ones that are incident to

*v*

_{R}but not to

*v*

_{L}as ${eR1,\u2026,eRdR\u22121}$. Then the transfer operators corresponding to the regions

*X*

_{L}and

*X*

_{R}are, respectively,

_{∞}is sub-multiplicative and $\u2225E\u2225\u221e=1$, it immediately follows that

*e*= (

*v*

_{L},

*v*

_{R}). 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,

*b*

_{L}(

*d*

_{L},

*n*) and

*b*

_{R}(

*d*

_{R},

*n*). From this is follows that $\u03f5(vL,vR)(n)$ is upper bounded by $\delta dL,dR(n)$.

## ACKNOWLEDGMENTS

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.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

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 AVAILABILITY

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

### APPENDIX A: GENERATING FUNCTION CALCULATIONS

We now state and prove the result used to calculate $\u2211k=1rh(2r,k)3k$ in Lemma 6.

*Fix*

*r*≥ 1

*and let*

*h*(

*2r*,

*k*)

*denote the number of permutations in*$S2re$

*with exactly*

*k*

*cycles. Then*

*where*$yr\u0304\u2254y(y+1)\cdots (y+r\u22121)$

*denotes the raising factorial.*

^{22}Let

*d*

_{r}= (

*r*− 1)! be the number of cyclic permutations of [

*r*], and set

*deck enumerator function*of the exponential family of even length cycles. The corresponding

*hand enumerator*is then

*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

### APPENDIX B: MODIFICATIONS TO SPECTRAL GAP ESTIMATES

Let us recall the bound on *ϵ*_{n} from Ref. 10, Proposition 3.6 applied to our setting:

*(*

*Ref. 10*

*, Proposition 3.6)*.

*For any edge*(

*v*

_{L},

*v*

_{R})

*of a simple graph*

*G*

*with degrees*

*d*

_{L}

*and*

*d*

_{R}

*, repsectively, let*

*If*$maxbLop(n),bRop(n),bLRop(n)<1$

*, then*$\u03f5n(vL,vR)\u2264\delta op(n)$,

*where*

#### 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.

_{Λ}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)].

_{Λ}, via

_{Λ}are injective. They also satisfy the following approximation bound, which is a variant of Ref. 10, Lemma 3.3.

*Let*$\Lambda \u2208{YvL\u222aYvR,YvL,YvR}$

*. Then for any*$B,C\u2208K\Lambda $

*,*

*where the constants are defined by*

*D*= 2 for the decorated AKLT models, the constant prefactor in the bound from Ref. 10, Lemma 3.3 is obtained from replacing

*C*

_{Λ}with

_{op}denotes the norm induced from the operator norm from Ref. 21.

*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,

*ρ*

_{min}≔ 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 $C\Lambda \u2032$ with *C*_{Λ}. This results in Proposition 3 from Sec. II C 1.

Similar to the proof of Ref. 10, Lemma 3.3, we prove the result for $\Lambda =YvL\u222aYvR$ as the other two cases follow from simple modifications of this case.

_{p}the Schatten

*p*-norm, and introduced the map $SB,C:M2dL\u22121(C)\u2192M2dR\u22121(C)$ defined by

*S*

_{B,C}(

*A*) ≔

*B**

*AC*.

*RT*‖

_{2}≤ ‖

*R*‖

_{∞}‖

*T*‖

_{2}to (B11) produces

#### 2. Comparing the two bounds for decorated AKLT models

*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

*δ*

^{∞}(

*n*) =

*δ*

_{d,d}(

*n*), and

*Suppose that*

*G*= (

*V*,

*E*)

*is a regular, simple graph such that*deg(

*v*) =

*d*

*for all*

*v*∈

*V*

*. Then, for any*$n\u2265ln(d\u22121)ln(3)\u2212ln(ln(3))ln(3)+12$

*, one has*$bLR\u221e(n)<bLRop(n)$

*and*

*Said differently,*

*δ*

^{∞}(

*n*) <

*δ*

^{op}(

*n*)

*.*

*x*≥ 0,

*α*

_{n}= 2(−3)

^{−n}. Then by (15),

*k*≤

*n*− 1 to bound

## REFERENCES

*Analytic Trends in Mathematical Physics, Contemporary Mathematics*