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0 Characterization of degenerate supersymmetric

ground states of the Nicolai supersymmetric

fermion lattice model by symmetry breakdown

Hosho Katsura,∗,†,‡ Hajime Moriya,§ Yu Nakayama¶

August 21 2020

keywords: Supersymmetric fermion lattice model. Ground states. Symme-try breakdown.

Abstract

We study a supersymmetric fermion lattice model defined by Her-

mann Nicolai. We show that its infinitely many classical supersym-

metric ground states are associated to breakdown of hidden local su-

persymmetries.

1 Introduction

A supersymmetric fermion lattice model defined by Nicolai [Ni] is a pio-neering work on (non-relativistic) supersymmetric quantum mechanics. Thismodel, which we call Nicolai model, even predates Witten’s supersymmetricquantum mechanical model [Wi], see [J1] [J2] for some historical remarks.

∗Department of Physics, Graduate School of Science, The University of Tokyo, 7-3-1,

Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan†Institute for Physics of Intelligence, The University of Tokyo, 7-3-1, Hongo, Bunkyo-

ku, Tokyo, 113-0033, Japan‡Trans-scale Quantum Science Institute, The University of Tokyo, 7-3-1, Hongo,

Bunkyo-ku, Tokyo, 113-0033, Japan§Institute of Science and Engineering, Kanazawa University, Kakuma-machi, Kanazawa

920-1192, Japan. hmoriya4@se.kanazawa-u.ac.jp¶Department of Physics, Rikkyo University, Toshima, Tokyo 171-8501, Japan

1

It has been shown that the Nicolai model has highly degenerate supersym-metric ground states [M1] [LScSh] which give rise to interesting dynamicalproperties. The aim of this paper is to discuss the degeneracy of supersym-metric ground states of the Nicolai model from the viewpoint of symmetrybreakdown. Based on our previous findings [M1] we will classify all classi-cal supersymmetric ground states in terms of breakdown of local fermionicsymmetries (supersymmetries) hidden in the model.

1.1 Supersymmetric fermion lattice model by Nicolai

We introduce a spinless fermion lattice model on one-dimensional integerlattice Z given by Nicolai [Ni]. For each site i ∈ Z let ci and c ∗i denotethe annihilation and the creation of a spinless fermion at i. They obey thecanonical anticommutation relations: For all i, j ∈ Z

c ∗i , cj = δi,j 1,

c ∗i , c∗j = ci, cj = 0. (1.1)

For each site i ∈ Z the fermion number operator is defined by

ni := c ∗i ci. (1.2)

A formal infinite sum N :=∑

i∈Z ni will denote the total fermion numberoperator. Let

Q =∑

i∈Z

q2i, where q2i := c2i+1c∗2ic2i−1. (1.3)

We see that Q and its adjoint Q∗ are fermion operators in the sense that

(−1)N , Q = (−1)N , Q∗ = 0. (1.4)

It is essential that the nilpotent property is satisfied:

Q2 = 0 = Q∗2. (1.5)

The supersymmetric Hamiltonian is given by

H := Q, Q∗. (1.6)

2

The pair of supercharges Q, Q∗ and the supersymmetric Hamiltonian H

satisfy the N = 2 supersymmetry relation [We], although there is no bosonin the model.

The explicit form of H can be easily computed as

H =∑

i∈Z

c∗2ic2i−1c2i+2c

∗2i+3 + c∗2i−1c2ic2i+3c

∗2i+2

+ c∗2ic2ic2i+1c∗2i+1 + c∗2i−1c2i−1c2ic

∗2i − c∗2i−1c2i−1c2i+1c

∗2i+1

. (1.7)

The Nicolai model has some obvious symmetries. The global U(1)-symmetrygroup γθ (θ ∈ [0, 2π]) is defined by

γθ(ci) = e−iθci, γθ(c∗i ) = eiθc∗i , ∀i ∈ Z. (1.8)

The particle-hole transformation is given by the Z2 action:

ρ(ci) = c∗i , ρ(c∗i ) = ci, ∀i ∈ Z. (1.9)

Let σ denote the shift-translation automorphism group defined by

σk(ci) = ci+k, σk(c∗i ) = c∗i+k ∀i ∈ Z, for each k ∈ Z. (1.10)

The Hamiltonian H (1.7) is invariant under γθ (θ ∈ [0, 2π]), so it has theglobal U(1)-symmetry. It has particle-hole symmetry as ρ(H) = H , whichfollows from ρ(Q) = −Q∗ and ρ(Q∗) = −Q. Finally, H is invariant undertranslation by two sites, as σ2k(H) = H for any k ∈ Z, whereas the fulltranslation symmetry is explicitly broken as σ2k+1(H) 6= H . We will see in§2 that the Nicolai model has other local symmetries.

1.2 Mathematical preliminary

In this subsection, we introduce some basic notations. We refer to [M2] thatgives a general framework of supersymmetric fermion lattice systems. Al-though it is not absolutely necessary, the C∗-algebraic formulation is helpfulto formulate our pertinent problem and gives a clue to solve it.

For each finite I ⋐ Z, A(I) denotes the finite-dimensional algebra gener-ated by c ∗i , ci ; i ∈ I, where the notation ‘I ⋐ Z’ means that I ⊂ Z and thenumber of sites |I| in I is finite. The union of all these A(I) defines the localalgebra:

A :=⋃

I⋐Z

A(I). (1.11)

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The norm completion of the local algebra A gives a C∗-system A called the

CAR algebra.Let Θ denote the fermion grading automorphism on A given as:

Θ(ci) = −ci, Θ(c ∗i ) = −c ∗i , ∀i ∈ Z. (1.12)

The fermion system A is decomposed into the even part A+ and the oddpart A− as

A = A+ ⊕A−, A+ = A ∈ A| Θ(A) = A, A− = A ∈ A| Θ(A) = −A.(1.13)

Any element of A+ is a linear sum of even monomials of fermion field op-erators, while that of A− is a linear sum of odd monomials of fermion fieldoperators. Similarly, for each I ⋐ Z,

A(I) = A(I)+ ⊕A(I)−, A(I)+ := A(I)∩A+, A(I)− := A(I)∩A−, (1.14)

and for the local algebra

A = A+ ⊕A−, A+ := A ∩ A+, A− := A ∩ A−. (1.15)

Define the graded commutator [ , ]Θ on A by the mixture of the com-mutator [ , ] and the anti-commutator , as

[A+, B]Θ = [A+, B] ≡ A+B − BA+ for A+ ∈ A+, B ∈ A,

[A, B+]Θ = [A, B+] = AB+ −B+A for A ∈ A, B+ ∈ A+,

[A−, B−]Θ = A−, B− ≡ A−B− +B−A− for A−, B− ∈ A−. (1.16)

Consider the superderivation generated by the nilpotent supercharge Q:

δQ(A) := [Q, A]Θ for A ∈ A. (1.17)

We see that δQ is a linear map that anticommutes with the grading:

δQ ·Θ = −Θ · δQ, (1.18)

and that the graded Leibniz rule holds:

δQ(AB) = δQ(A)B +Θ(A)δQ(B) for A,B ∈ A. (1.19)

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A state (i.e. normalized positive linear functional on A) is called a super-symmetric (ground) state if and only if it is invariant under the superderiva-tion δQ, equivalently its state vector (determined by the GNS representation)is annihilated by both the supercharge Q and its adjoint Q∗. In this paper,we deal with only pure states on finite systems that are always associatedwith normalized vectors. For a supersymmetric model, if there is a super-symmetric state, then the supersymmetry is unbroken. If there exists no su-persymmetric state, then the supersymmetry is spontaneously broken. TheNicolai model has many supersymmetric states as we will see later. Henceits supersymmetry is unbroken.

1.3 Classical supersymmetric ground states of the Nico-

lai model

In this paper, we focus on classical supersymmetric ground states which willbe stated below. This subsection is indebted to [M1].

Let |1〉i and |0〉i denote the occupied and empty vectors of the spinlessfermion at site i, respectively. For each i ∈ Z

ci|1〉i = |0〉i, c∗i |1〉i = 0, c ∗i |0〉i = |1〉i, ci|0〉i = 0. (1.20)

When there is no fear of confusion, we will omit the subscript and writesimply |1〉 and |0〉.

We identify general (not necessarily supersymmetric) classical states onthe fermion lattice system by classical configurations on Z.

Definition 1.1. Let g(n) denote an arbitrary 0, 1-valued function over Z.It is called a classical configuration over Z. For any classical configurationg(n) define

|g(n)n∈Z〉 := · · · ⊗ |g(i− 1)〉i−1 ⊗ |g(i)〉i ⊗ |g(i+ 1)〉i+1 ⊗ · · · . (1.21)

This infinite product vector determines a state ψg(n) on the fermion systemA which will be called the classical state associated to the configuration g(n)over Z. Let ι0(n) := 0 ∀n ∈ Z. Then

Ω0 := |ι0(n)n∈Z〉 = · · · ⊗ |0〉 ⊗ |0〉 ⊗ |0〉 ⊗ |0〉 ⊗ |0〉 ⊗ |0〉 · · · . (1.22)

The above Ω0 is called the Fock vector, and its associated translation-invariantstate ψ0 on A is called the Fock state. Similarly let ι1(n) := 1 ∀n ∈ Z. Then

Ω1 := |ι1(n)n∈Z〉 = · · · ⊗ |1〉 ⊗ |1〉 ⊗ |1〉 ⊗ |1〉 ⊗ |1〉 ⊗ |1〉 · · · . (1.23)

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The above Ω1 is called the fully-occupied vector, and its associated translation-invariant state ψ1 on A is called the fully-occupied state.

To each classical configuration over Z we assign an operator by the fol-lowing rule.

Definition 1.2. For each i ∈ Z let κi denote the map from 0, 1 into A(i)given as

κi(0) := 1, κi(1) := c ∗i . (1.24)

For each classical configuration g(n) over Z define the infinite-product offermion field operators:

O(g) :=∏

i∈Z

κi (g(i)) = · · · κi−1 (g(i− 1)) κi (g(i)) κi+1 (g(i+ 1)) · · · , (1.25)

where the multiplication is taken in the increasing order of i ∈ Z. If g(n) hasa compact support, then

O(g) ∈ A. (1.26)

Otherwise O(g) denotes a formal operator which does not belong to A.

We have the following obvious correspondence between product vectorsgiven in Definition 1.1 and product operators given in Definition 1.2 via theFock representation.

Proposition 1.3. Let Ω0 denote the Fock vector given in (1.22). For anyclassical configuration g(n) over Z, the following identity holds:

O(g)Ω0 = |g(n)n∈Z〉. (1.27)

Proof. The desired identity directly follows from Definition 1.1 and Definition1.2.

It is easy to see that the Fock state ψ0 and the fully-occupied state ψ1

are supersymmetric ground state for the Nicolai model. We would like togive all classical supersymmetric ground states of the Nicolai model. For thispurpose, we introduce the following class of classical configurations.

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Definition 1.4. Consider three consecutive sites 2i−1, 2i, 2i+1 centeredat an even site 2i (i ∈ Z). There are 23 configurations (i.e. eight 0, 1-valuedfunctions) on 2i−1, 2i, 2i+1. Let “0, 1, 0” and “1, 0, 1” be called forbiddentriplets. If a classical configuration g(n) (n ∈ Z) does not include any of suchforbidden triplets over Z, then it is called a ground-state configuration overZ (for the Nicolai model). The set of all ground-state configurations over Z isdenoted by Υ. The set of all ground-state configurations whose supports areincluded in some finite region is denoted by Υ. The set of all ground-stateconfigurations whose supports are included in a finite region I ⋐ Z is denotedby ΥI.

The following proposition classifies all the classical supersymmetric groundstates in terms of classical configurations justifying our nomenclature “ground-state configurations” of Definition 1.4. It is based on the following fact thatcan be easily checked by using (1.20): The product vector |g(2i− 1)〉2i−1 ⊗|g(2i)〉2i ⊗ |g(2i + 1)〉2i+1 is annihilated by both q2i and q∗2i unless thoseare |0〉2i−1 ⊗ |1〉2i ⊗ |0〉2i+1 or |1〉2i−1 ⊗ |0〉2i ⊗ |1〉2i+1 which correspond tothe forbidden triplets, g(2i − 1) = 0, g(2i) = 1, g(2i + 1) = 0 andg(2i − 1) = 1, g(2i) = 0, g(2i + 1) = 1, respectively. Theorem 2 [M1]established that if there appears no forbidden triplet in the sequence of g(n)(n ∈ Z) at all, then the corresponding product vector O(g)Ω0 is annihilatedby bothQ andQ∗, hence it is a supersymmetric ground state, whereas if thereis at least one forbidden triplet in the sequence of g(n) (n ∈ Z), then eitherQ or Q∗, or both do not annihilate O(g)Ω0 and so it is not supersymmetric.See [M1] for the detail.

Proposition 1.5. A classical state on the fermion lattice system A is su-persymmetric for the Nicolai model if and only if its associated configurationg(n) over Z is a ground-state configuration for the Nicolai model as stated inDefinition 1.4, namely, if and only if g(n) ∈ Υ.

1.4 Supersymmetric ground states on subsystems

We shall discuss supersymmetric ground states on finite subsystems. First,we specify finite regions that we will consider. Second, we specify the meaningof “supersymmetric ground states” upon finite regions, as it is not so obviousdue to the boundary.

To deal with the Nicolai model which has period-2 translational symmetrynot full translation symmetry it is convenient to consider the special finite

7

intervals of Z whose edges are both even, see Proposition 4.1 given later.Namely for k, l ∈ Z (k < l) we take

Ik,l ≡ [2k, 2k + 1, 2(k + 1), · · · · · · , 2(l− 1), 2l − 1, , 2l]. (1.28)

We see that |Ik,l| is 2(l − k) + 1.Now we give the precise definition of supersymmetric ground states on

the finite interval Ik,l.

Definition 1.6. Consider any finite interval Ik,l ≡ [2k, 2k + 1, 2(k +1), · · · , 2(l− 1), 2l − 1, 2l] (k, l ∈ Z k < l). Let

Q[k, l] ≡l∑

i=k

q2i ∈ A (2k − 1 ∪ Ik,l ∪ 2l + 1)− , (1.29)

where q2i ≡ −c2i−1c∗2ic2i+1 as defined in (1.3). A state on A(Ik,l) is called

a free-boundary supersymmetric ground state if its arbitrary state-extensionto A (2k − 1 ∪ Ik,l ∪ 2l + 1) is invariant under the superderivation δQ[k,l]

associated to the local supercharge Q[k, l], equivalently its associated vectoris annihilated by both Q[k, l] and Q[k, l]∗.

First note that

δQ[k,l] = δQ on A(Ik,l), (1.30)

where Q =∑

i∈Z q2i as in (1.3). Namely, upon the subsystem A(Ik,l), finitesupercharge Q[k, l] sitting on a slightly larger region 2k−1∪ Ik,l∪2l+1gives the same action as of the total supercharge Q. Second, we address what“its arbitrary state-extension” exactly means. According to [ArM], for everyclassical state on the given local system, any state-extension of it to a largersystem is also a classical state (or mixture of such). In the present case itis described as follows. By Proposition 1.3 any classical state of A(Ik,l) isdetermined by a 0, 1-valued function g(n) on Ik,l. Any state-extension ofit to A (2k − 1 ∪ Ik,l ∪ 2l + 1) is determined by g(n) on 2k− 1 ∪ Ik,l ∪2l + 1 satisfying that

g(n) = g(n) for ∀n ∈ Ik,l, g(2k − 1) = 0 or 1, g(2l + 1) = 0 or 1. (1.31)

Due to the choice of the marginal points 2k − 1, 2l + 1 there are fourpossibilities.

To find configurations associated to Definition 1.6, we introduce a subclassof Υ given in Definition 1.4 requiring certain boundary conditions as follows.

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Definition 1.7. Let Ik,l ≡ [2k, 2k + 1, 2(k + 1), · · · , 2(l − 1), 2l − 1, 2l](k, l ∈ Z s.t. k < l) as before. The set of all g(n) ∈ Υk,l ≡ ΥIk,l satisfyingthe following boundary conditions

g(2k) = g(2k + 1) and g(2l− 1) = g(2l) (1.32)

will be denoted by Υk,l.

In Proposition 1.5 we gave one-to-one correspondence between classicalsupersymmetric ground states and ground-state configurations over Z. Wecan see analogous correspondence on finite regions Ik,l as follows.

Proposition 1.8. A classical state on the finite system A(Ik,l) is free-boundarysupersymmetric (Definition 1.6) if and only if its associated configuration

g(n) on Ik,l belongs to Υk,l (Definition 1.7).

Proof. We will see the if part as follows. For all i ∈ k + 1, k + 2, · · · , l− 1both q2i and q

∗2i annihilate any product vector corresponding to g(n) ∈ Υk,l.

So we only have to see the marginal points k and l. For any given g(n) ∈ Υk,l

its extension to 2k−1∪ Ik,l∪2l+1 will be denoted as g(n). We see thatg(2k − 1) is arbitary, g(2k) = g(2k + 1), g(2l − 1) = g(2l), and g(2l + 1) isarbitrary. So there is no forbidden sequence on 2k−1, 2k, 2k+1. Thus bothq2k and q∗2k annihilate any product vector corresponding to g(n). Similarly,both q2l and q∗2l annihilate the product vector corresponding to g(n). Theonly if part can be shown as in Theorem 2 [M1].

2 Hidden local fermionic symmetries

We will show that there are infinitely many local fermionic symmetries hiddenin the Nicolai model. To this end, we need some preparation.

Definition 2.1. Take any finite interval Ik,l defined in (1.28). Let f be a−1,+1-valued sequence on Ik,l. For any consecutive triplet 2i−1, 2i, 2i+1 ⊂ Ik,l (i ∈ Z) assume that neither

f(2i− 1) = −1, f(2i) = +1, f(2i+ 1) = −1 (2.1)

nor

f(2i− 1) = +1, f(2i) = −1, f(2i+ 1) = +1 (2.2)

9

holds. Furthermore assume that f is constant on the left-end pair sites2k, 2k + 1 and on the right-end pair sites 2l − 1, 2l:

f(2k) = f(2k + 1) = +1 or f(2k) = f(2k + 1) = −1 (2.3)

and

f(2l− 1) = f(2l) = +1 or f(2l − 1) = f(2l) = −1. (2.4)

The set of all −1,+1-valued sequences on Ik,l satisfying the above condi-

tions is denoted by Ξk,l. The union of Ξk,l over all k, l ∈ Z (k < l) is denoted

by Ξ:

Ξ :=⋃

k,l∈Z (k<l)

Ξk,l. (2.5)

Take any p, q ∈ Z (p < q). Let

Ξ(p, q) :=⋃

k,l∈Z; p≤k<l≤q

Ξk, l. (2.6)

Each f ∈ Ξ is called a local −1,+1-sequence of conservation for the Nicolaimodel.

Remark 2.2. The requirements (2.3) (2.4) on the edges of Ik,l are essential tomake conservation laws for the Nicolai model.

Remark 2.3. By crude estimate we can see that the number of local −1,+1-sequences of conservation in Ξk,l is roughly (2

3−22

)(l−k) = 3(l−k) = 3m/2, wherem = 2(l − k) denotes approximately the size of the system (i.e. the numberof sites in Ik,l).

It is convenient to consider the following subclasses of Ξ.

Definition 2.4. For each k, l ∈ Z (k < l) let r+[2k,2l] ∈ Ξk,l and r−[2k,2l] ∈ Ξk,l

denote the constants over Ik,l as

r+[2k,2l](i) = +1 ∀i ∈ Ik,l, r−[2k,2l](i) = −1 ∀i ∈ Ik,l. (2.7)

The setr+[2k,2l]

over all k, l ∈ Z (k < l) will be denoted as Ξ+1const., and

the setr−[2k,2l]

over all k, l ∈ Z (k < l) will be denoted as Ξ−1const.. Let

Ξconst. := Ξ+1const. ∪ Ξ−1const..

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We shall give a rule to assign a local fermion operator for every local−1,+1-sequence of conservation of Definition 2.1.

Definition 2.5. For each i ∈ Z let ζi denote the assignment from −1,+1into the fermion field at i given as

ζi(−1) := ci, ζi(+1) := c ∗i . (2.8)

Take any pair of integers k, l ∈ Z such that k < l. For each f ∈ Ξk,l, set

Q(f) :=

2l∏

i=2k

ζi (f(i))

≡ ζ2k (f(2k)) ζ2k+1 (f(2k + 1)) · · · · · · ζ2l−1 (f(2l − 1)) ζ2l (f(2l)) ∈ A(Ik,l)−,(2.9)

where the multiplication is taken in the increasing order as above. Theformulas (2.9) for all k, l ∈ Z (k < l) yield a unique assignment Q from Ξ toA−.

By Definition 2.5 the following local fermion operators are assigned to±-characters supported on the segment Ik,l of Definition 2.4. For k, l ∈ Z

(k < l)

Q(r+[2k,2l]) := c ∗2kc∗2k+1 · · · c

∗2l−1c

∗2l ∈ A(Ik,l)−,

Q(r−[2k,2l]) := c2kc2k+1 · · · c2l−1c2l ∈ A(Ik,l)−. (2.10)

The following is the main result of this section.

Theorem 2.6. For every f ∈ Ξ

[H, Q(f)] = 0 = [H, Q(f)∗], (2.11)

where H denotes the Hamiltonian of the Nicolai model over Z.

Proof. This theorem is established in [M1]. Because of its importance andthe reader’s convenience, we will provide its more formal derivation below.It suffices to show that

Q, Q(f) = Q∗, Q(f) = 0, (2.12)

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and thatQ, Q(f)∗ = Q∗, Q(f)∗ = 0, (2.13)

as the former implies [H, Q(f)] = 0 and the latter implies [H, Q(f)∗] = 0by the graded Leibniz rule of superderivations (1.19). Recall Q =

∑i∈Z q2i

and q2i ≡ c2i+1c∗2ic2i−1 defined in (1.3). By Definitions 2.1 2.5, we have

Q(f)q2i = 0 = q2iQ(f), Q(f)q∗2i = 0 = q∗2iQ(f) for all i ∈ Z. (2.14)

From the above we obtain (2.12) and (2.13).

Theorem 2.6 says that the Nicolai model has infinitely many local fermionicconstants. Those generate local fermionic symmetries.

Definition 2.7. For each local −1,+1-sequence of conservation f ∈ Ξ,Q(f) is called the local fermionic constant of motion associated to f , andthe pair Q(f),Q(f)∗ is called the local fermionic charge associated to f .

Remark 2.8. Fermionic symmetry satisfying the supersymmetry relation otherthan the dynamical supersymmetry is sometimes called kinematical super-symmetry. See e.g. [NSakY]. Hence the local fermionic charge Q(f),Q(f)∗for any f ∈ Ξ gives a local kinematical supersymmetry.

Remark 2.9. Any operator of the algebra generated by Q(f) ∈ A| f ∈ Ξis a local constant of motion. There exist many such self-adjoint bosonicoperators that generate (bosonic) symmetries for the Nicolai model. Forexample, we obtain a bosonic constant n2kn2k+1 · · ·n2l−1n2l ∈ A(Ik,l)+ fromthe multiple Q(r+[2k,2l])Q(r−[2k,2l]) where each of them is defined in (2.10).

3 Degenerate classical supersymmetric ground

states and broken local fermionic symme-

tries

In this section we will relate the high degeneracy of ground states shownin § 1.3 to the existence of many local fermionic symmetries shown in § 2.In particular, we will show that every classical supersymmetric ground statecan be constructed from (broken) local fermionic symmetries.

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Theorem 3.1. Take any segment Ik,l indexed by k, l ∈ Z (k < l) as in(1.28). Any classical free-boundary supersymmetric ground state on A(Ik,l)(Definition 1.6) can be constructed by some finitely many applications ofoperators Q(f) (and Q(f)∗) with f ∈ Ξ(k, l) (Definition 2.1) to the Fockvector Ω0 (1.22), similarly, to the fully-occupied vector Ω1 (1.23).

By Definition 1.1 we can identify every classical supersymmetric groundstate ψg(n) on A with its corresponding classical configuration g(n) over Z,and vice versa. By Proposition 1.8 we can identify the set of all classicalfree-boundary supersymmetric ground states on A(Ik,l) (Definition 1.6) with

Υk,l (Definition 1.7). We will frequently use those identifications in whatfollows.

The following lemma implies that the latter part (using Ω1) of Theorem3.1 holds once the former part (using Ω0) is proved. Also it will be usedfrequently in the proof.

Lemma 3.2. For any n ∈ N, if g ∈ Υ0,n can be constructed by some finitelymany applications of local fermionic charges within I0,n to the Fock vector Ω0.Then it can be constructed by some applications of local fermionic chargeswithin I0,n to the fully-occupied state Ω1.

Proof. For any f ∈ Ξ(0, n), −f ∈ Ξ(0, n) by definition. From (2.9) in Defi-nition 2.5

Q(−f) = ρ(Q(f)), (3.1)

where ρ is the particle-hole transformation defined in (1.9). Thus by usingthe particle-hole transformation, we can use Ω0 and Ω1 interchangeably.

Obviously it is enough to show Theorem 3.1 by setting k = 0 and l =∀n ∈ N by shift-translations. Thus we will prove the following.

Proposition 3.3. For any n ∈ N, every g ∈ Υ0,n can be constructed by someapplications of local fermionic charges within I0,n:

Q(f); f ∈ Ξ(0, n) ≡

⋃

m∈1,2,··· ,n

Ξ0,m

(3.2)

to the Fock vector Ω0 (1.22).

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Proof. We need concrete forms of elements in Ξ which are listed in §A.1.First, let us consider the case n = 1. Υ0,1 consists of the following two

sequences on I0,1:

Υ0,1 0 1 2g[0,2] 0 0 0

g•[0,2] 1 1 1

The classical configuration g[0,2] corresponds to the Fock vector Ω0 (restricted

to the local region [0, 1, 2]). We shall write simply g[0,2] = Ω0, and thisidentification will be used hereafter. On the other hand, g•[0,2] corresponds to

Q(r+[0,2])Ω0 which is the fully-occupied state on I0,1, where r+[0,2] ∈ Ξ0,1. Thus

we obtain g•[0,2] = r+[0,2]Ω0 which is the desired formula.

Second, let us consider the case n = 2. Υ0,2 consists of the following 6sequences on I0,2:

Υ0,2 0 1 2 3 4g[0,4] 0 0 0 0 0

g1[0,4] 0 0 0 1 1

g2[0,4] 0 0 1 1 1

g3[0,4] 1 1 1 0 0

g4[0,4] 1 1 0 0 0

g•[0,4] 1 1 1 1 1

We have g[0,4] = Ω0 and g•[0,4] = r+[0,4]Ω0 = c∗0c∗1c

∗2c

∗3c

∗4Ω0 (the fully-occupied

state on I0,2) according to (A.3). We have

g3[0,4] = r+[0,2]Ω0 = c∗0c∗1c

∗2Ω0, r+[0,2] ∈ Ξ0,1

g2[0,4] = r+[2,4]Ω0 = c∗2c∗3c

∗4Ω0, r+[2,4] ∈ Ξ1,2.

To get g1[0,4] and g4[0,4] we use ‘double’ applications as

g1[0,4] = r−[0,2]g•[0,4] = r−[0,2]r

+[0,4]Ω0 = c∗3c

∗4Ω0, r−[0,2] ∈ Ξ0,1, r

+[0,4] ∈ Ξ0,2, (3.3)

and similarly

g4[0,4] = r−[2,4]g•[0,4] = r−[2,4]r

+[0,4]Ω0 = c∗0c

∗1Ω0, r−[2,4] ∈ Ξ1,2, r

+[0,4] ∈ Ξ0,2. (3.4)

14

We have derived all the elements of Υ0,2 and accordingly all the classical free-boundary supersymmetric ground states on A(I0,2) from the Fock vector Ω0.Let us note that g1[0,4] and g

3[0,4] are mapped to each other by the particle-hole

transformation, and so are g2[0,4] and g4[0,4]. However, as shown above, we do

not need to use the particle-hole transformation.In an analogous manner, we can get all the elements of Υ0,2 (all the

classical free-boundary supersymmetric ground states on A(I0,2)) from thefully-occupied vector Ω1 in place of Ω0. This fact is important.

Let us consider the case n = 3. Υ0,3 consists of the following 18 sequenceson I0,3:

Υ0,3 0 1 2 3 4 5 6g[0,6] 0 0 0 0 0 0 0

g1[0,6] 0 0 0 1 1 0 0

g2[0,6] 0 0 1 1 0 0 0

g3[0,6] 0 0 0 1 0 0 0

g4[0,6] 0 0 1 1 1 0 0

g5[0,6] 0 0 0 0 0 1 1

g6[0,6] 0 0 0 0 1 1 1

g7[0,6] 0 0 0 1 1 1 1

g8[0,6] 0 0 1 1 1 1 1

g9[0,6] 1 1 1 1 1 0 0

g10[0,6] 1 1 1 1 0 0 0

g11[0,6] 1 1 1 0 0 0 0

g12[0,6] 1 1 0 0 0 0 0

g•[0,6] 1 1 1 1 1 1 1

g13[0,6] 1 1 1 0 0 1 1

g14[0,6] 1 1 0 0 1 1 1

g15[0,6] 1 1 1 0 1 1 1

g16[0,6] 1 1 0 0 0 1 1

We will generate all the above g∗[0,6]. Obviously g[0,6] = Ω0, and g•[0,6] =

r+[0,6]Ω0 = c∗0c∗1c

∗2c

∗3c

∗4c

∗5c

∗6Ω0 which is the fully-occupied vector Ω1 restricted to

I0,3.The restriction of g1[0,6] to [0, 4] is g1[0,4], and the restriction of g4[0,6] to [0, 4]

is g2[0,4]. The restriction of g2[0,6] to [2, 6] is g4[2,6], which is the translation of

15

g4[0,4] used before. Thus each of g1[0,6] g4[0,6] and g

2[0,6] can be given as in the case

n = 2.The restriction of g13[0,6] to [0, 4] is g3[0,4], the restriction of g14[0,6] to [2, 6] is

g2[2,6], the restriction of g16[0,6] to [0, 4] is g4[0,4] (also the restriction of g16[0,6] to

[2, 6] is g4[2,6]). Hence each of g13[0,6], g14[0,6] and g

16[0,6] can be given as in the case

n = 2. Note that Ω1|[0,6] = r+[0,6]Ω0|[0,6] with r+[0,6] ∈ A(I0,3). Therefore each of

g13[0,6] g14[0,6] and g

16[0,6] can be generated by local supercharges in [0, 6] applied

to Ω0.We have

g3[0,6] = r−[0,2]r−[4,6]Ω1 = r−[0,2]r

−[4,6]r

+[0,6]Ω0 = c∗3Ω0,

and

g15[0,6] = r+[0,2]r+[4,6]Ω0 = c∗0c

∗1c

∗2c

∗4c

∗5c

∗6Ω0.

We have

g5[0,6] = r−[0,4]Ω1 = r−[0,4]r+[0,6]Ω0 = c∗5c

∗6Ω0,

g6[0,6] = r+[4,6]Ω0 = c∗4c∗5c

∗6Ω0,

g7[0,6] = r−[0,2]Ω1 = r−[0,2]r+[0,6]Ω0 = c∗3c

∗4c

∗5c

∗6Ω0,

g8[0,6] = r+[2,6]Ω0 = c∗2c∗3c

∗4c

∗5c

∗6Ω0,

and similarly

g9[0,6] = r+[0,4]Ω0 = c∗0c∗1c

∗2c

∗3c

∗4Ω0,

g10[0,6] = r−[4,6]Ω1 = r−[4,6]r+[0,6]Ω0 = c∗0c

∗1c

∗2c

∗3Ω0,

g11[0,6] = r+[0,2]Ω0 = c∗0c∗1c

∗2Ω0,

g12[0,6] = r−[2,6]Ω1 = r−[2,6]r+[0,6]Ω0 = c∗0c

∗1Ω0.

We have now derived all the sequences of Υ0,3, i.e. all the classical free-boundary supersymmetric ground states on A(I0,3).

We will start the argument of induction. We have verified the statementfor n = 1, 2, 3. Now let us assume that the statement holds for any integer

16

from 1 ∈ N up to n ∈ N. We are going to show that the statement holds forn+1 ∈ N. Concretely, we will construct Υ0,n+1 from Υp,q (0 ≤ p < q ≤ n+1)where 0 < p or q < n + 1.

We divide Υ0,n+1 into four cases (Case I-IV) as below. We shall indicatehow the induction argument can be applied to each of them.

Case I:We deal with all g ∈ Υ0,n+1 whose left and right ends are

g(0) = g(1) = 0, g(2n+ 1) = g(2(n+ 1)) = 0. (3.5)

Υ0,n+1 0 1 2 3 · · · · · · · · · 2n− 1 2n 2n+ 1 2(n+ 1)I-1 0 0 0 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 0 0 0I-2 0 0 0 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 1 1 0 0I-3 0 0 0 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 1 0 0 0I-4 0 0 1 1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 0 0 0I-5 0 0 1 1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 1 1 0 0I-6 0 0 1 1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 1 0 0 0I-7 0 0 0 1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 0 0 0I-8 0 0 0 1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 1 1 0 0I-9 0 0 0 1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 1 0 0 0

Note that ‘∗ ∗ ∗’s in the middle mean some appropriate sequences of 0, 1so that the sequence belongs to Υ0,n+1, not being arbitrary.

All the above elements in Υ0,n+1 except I-9 belong to Υ1,n+1 or to Υ0,n

when being restricted to [2, 2(n+ 1)] or to [0, 2n], respectively. By applyingr+[0,2] to the vector of I-9, we get

Υ0,n+1 0 1 2 3 · · · · · · · · · 2n− 1 2n 2n+ 1 2(n+ 1)New I-9 1 1 1 1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 1 0 0 0

“New I-9” above belongs to Υ1,n+1 when being restricted to [2, 2(n + 1)].Therefore we can obtain New I-9 by applying some local supercharges in[2, 2(n+1)] to Ω1 (not Ω0 here). Note that Ω1 = r+[0,2(n+1)]Ω0 on the segment

[0, 2(n+1)] as noted in Lemma 3.2. Hence we can construct I-9 by applyingsome local supercharges in [0, 2(n+ 1)] to Ω0. In this way we have made allthe configurations of Case I by the specified rule.

By applying r+[2n,2(n+1)] to the vector of I-9, we get

17

Υ0,n+1 0 1 2 3 · · · · · · · · · 2n− 1 2n 2n+ 1 2(n+ 1)New I-9(2) 0 0 0 1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 1 1 1 1

“New I-9(2)” above belongs to Υ0,n when being restricted to [0, 2n]. There-fore we can obtain New I-9(2) by applying some local supercharges in [0, 2n]to Ω1 (not Ω0 here). By noting Lemma 3.2 we can construct I-9 by applyingsome local supercharges in [0, 2(n+ 1)] to Ω0.

Case II:We deal with all g ∈ Υ0,n+1 whose left and right ends are

g(0) = g(1) = 1, g(2n+ 1) = g(2(n+ 1)) = 1. (3.6)

The proof for Case II can be done in the same way as done for Case I.

Case III:We deal with all g ∈ Υ0,n+1 whose left and right ends are

g(0) = g(1) = 0, g(2n+ 1) = g(2(n+ 1)) = 1. (3.7)

Υ0,n+1 0 1 2 3 · · · · · · · · · 2n− 1 2n 2n+ 1 2(n+ 1)III-1 0 0 0 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 0 1 1III-2 0 0 0 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 1 1 1 1III-3 0 0 0 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 1 1 1III-4 0 0 1 1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 0 1 1III-5 0 0 1 1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 1 1 1 1III-6 0 0 1 1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 1 1 1III-7 0 0 0 1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 0 1 1III-8 0 0 0 1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 1 1 1 1III-9 0 0 0 1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 1 1 1

All the above elements in Υ0,n+1 except III-9 belong to Υ1,n+1 or to Υ0,n

when being restricted to [2, 2(n+ 1)] or to [0, 2n], respectively. By applyingr+[0,2] to the vector of III-9, we get

Υ0,n+1 0 1 2 3 · · · · · · · · · 2n− 1 2n 2n+ 1 2(n+ 1)New III-9 1 1 1 1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 1 1 1

18

“New III-9” above belongs to Υ1,n+1 when being restricted to [2, 2(n+1)].Therefore we can obtain New III-9 by applying some local supercharges in[2, 2(n + 1)] to Ω1 (not Ω0 here). Note that Ω1 can be constructed from Ω0

on [0, 2(n + 1)] by using local supercharges on [0, 2(n + 1)]. Thus we canconstruct I-9 by applying some local supercharges in [0, 2(n+ 1)] to Ω0. Wehave completed the assertion for Case III.

Case IV:We deal with all g ∈ Υ0,n+1 whose left and right ends are

g(0) = g(1) = 1, g(2n+ 1) = g(2(n+ 1)) = 0. (3.8)

The proof for Case IV is similar to that for Case III given above.

In conclusion, for all the cases (Case I-IV) we have generated all the

elements of Υ0,n+1 from Υ0,n and Υ1,n+1. Hence by the induction, we haveshown the statement.

The number of classical supersymmetric ground states can be computedexplicitly.

Proposition 3.4. The number of classical free-boundary supersymmetricground states on I0,n (n ∈ N) is 2 · 3n−1.

Proof. This computation is given by the transfer-matrix method. We firstdivide I0,n into n-sequential pairs as

I0,n = 0, 1, 2 ∪ 3, 4 · · · ∪ 2k − 1, 2k ∪ · · · ∪ 2n− 1, 2n,

where the first group exceptionally consists of three sites 0, 1, 2. On each2k − 1, 2k all classical configurations are possible. However, to connect2k − 1, 2k and 2k + 1, 2(k + 1) we have to avoid the forbidden triplets:0, 1, 0 1, 0, 1 on 2k − 1, 2k, 2k + 1. So the transfer matrix should be

T :=

1 1 1 10 0 1 11 1 0 01 1 1 1

. (3.9)

19

By taking the edge condition (1.32) into account, the possible configurationsare any of

0, 0, 0, . . . , 0, 0,

0, 0, 0, . . . , 1, 1,

0, 0, 1, . . . , 0, 0,

0, 0, 1, . . . , 1, 1,

1, 1, 0, . . . , 0, 0,

1, 1, 0, . . . , 1, 1,

1, 1, 1, . . . , 0, 0,

1, 1, 1, . . . , 1, 1,

which correspond to (1, 1), (1, 4), (2, 1), (2, 4), (3, 1), (3, 4), (4, 1) and (4, 4)elements of T n−1, respectively. Those amount to 2 · 3n−1.

4 Discussion

We determined all classical supersymmetric ground states of the Nicolai su-persymmetric fermion lattice model, and explained the high degeneracy ofground states by breakdown of its infinitely many local fermionic symme-tries. The above finding may recall other supersymmetric models with manyground states such as the supersymmetric fermion lattice model by Fendley-Schoutens-de Boer-Nienhuis [FScdB] [FScNi] on two-dimensional lattice [vE],and some Wess-Zumino supersymmetry quantum mechanical model [A].

In [LScSh] the exact number of ground states on finite systems of theNicolai model is shown, but the precise form of these states is not specified.To determine all the ground states (furthermore all eigenstates) beyond theclassical ground states discussed in this paper we need more detailed spectralproperty of the Hamiltonian and its symmetries (including bosonic ones).

In [SanKN] [M3] an extended version of Nicolai model that breaks itsdynamical supersymmetry is studied. As we have seen, the (original) Nicolaimodel does not break its dynamical supersymmetry. However, it will breakits hidden supersymmetries for some ground states. It would be interestingto discuss breakdown of these hidden fermionic symmetries.

The final comment is concerned with some technical point. We have cho-sen special subregions (Ik,l) and the boundary conditions (the free-boundary

20

supersymmetric condition) which seem artificial. However, as long as weconsider classical states only, there is no loss of generality with this choice asfollows.

Proposition 4.1. Given any finite subset Λ of Z. Any classical supersym-metric state on Λ can be given by restriction of some classical free-boundarysupersymmetric ground state on some larger Ik,l that includes Λ.

Proof. First recall the one-to-one correspondence between the set of classi-cal supersymmetric ground states on Λ and ΥΛ by Proposition 1.5. Recallthe one-to-one correspondence between the set of classical free-boundary su-persymmetric ground states on Ik,l and Υk,l by Proposition 1.8. Thus anyclassical supersymmetric ground state on Λ can be extended to at least oneclassical free-boundary supersymmetric ground state on Ik,l that includesΛ.

Acknowledgments

H. K. was supported in part by JSPS Grant-in-Aid for Scientic Research onInnovative Areas No. JP18H04478 and JP20H04630, and JSPS KAKENHIGrant No. 18K03445. H. M. would like to thank Prof. Arai and Dr. Hui-jse for helpful discussion. H. M. acknowledges Riyu-1 group of KanazawaUniversity for encouragement.

References

[Ni] Nicolai H 1976 Supersymmetry and spin systems J. Phys. A: Math. Gen.9 1497-1505

[Wi] Witten E 1981 Dynamical breaking of supersymmetry Nucl. Phys.B185 513-554

[J1] Junker G 1996 Supersymmetric methods in quantum and statisticalphysics (Berlin-Heidelberg: Springer-Verlag)

[J2] Junker G 2017 40 years of supersymmetric quantum mechanics J. Phys.A: Math. Theor. 50 021001

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[M1] Moriya H 2018 Ergodicity breaking and Localization of the Nicolaisupersymmetric fermion lattice model J. Stat. Phys. 172 1270-1290

[LScSh] La R, Schoutens K and Shadrin S 2018 Ground states of Nicolai andZ2 Nicolai models J. Phys. A: Math. Theor. 52 02LT01

[We] Weinberg S 2000 The quantum theory of fields, Volume 3: Supersym-metry (Cambridge: Cambridge University Press)

[M2] Moriya H 2016 On supersymmetric fermion lattice systems. Ann. Inst.Henri. Poincare 17 2199-2236

[ArM] Araki H and Moriya H 2003 Joint extension of states of subsystemsfor a CAR system Comm. Math. Phys. 237 105-122

[NSakY] Nakayama Y, Sakaguchi M and Yoshida K 2009 Interacting SUSY-singlet matter in non-relativistic Chern-Simons theory J. Phys. A: Math.Theor. 42 195402

[FScdB] Fendley P, Schoutens K and de Boer J 2003 Lattice models withN=2 supersymmetry J. Phys. A: Math. Theor. 90 120402

[FScNi] Fendley P, Schoutens K and Nienhuis B 2003 Lattice fermion modelswith supersymmetry J. Phys. A: Math. Gen. 50 12399-12424

[vE] van Eerten H 2005 Extensive ground state entropy in supersymmetriclattice models J. Math. Phys. 46 123302

[A] Arai A 1989 Existence of infinitely many zero-energy states in a modelof supersymmetric quantum mechanics J. Math. Phys. 30 1164

[SanKN] Sannomiya N, Katsura H and Nakayama Y 2016 Supersymmetrybreaking and Nambu-Goldstone fermions in an extended Nicolai modelPhys. Rev. D 94 045014

[M3] Moriya H 2018 Supersymmetry breakdown for an extended version ofthe Nicolai supersymmetric fermion lattice model Phys. Rev. D 98 015018

22

A Appendix

A.1 Forms of Ξ

We will give concrete examples for local −1,+1-sequences of conservationof Definition 2.1 and their associated local fermion operators of Definition2.5. First we see that Ξ0,1 on I0,1 ≡ [0, 1, 2] consists of two ±-characters only.

Ξ0,1 0 1 2r−[0,2] −1 −1 −1

r+[0,2] +1 +1 +1

By (2.9) of Definition 2.5 the corresponding local fermion operators are

Q(r−[0,2]) = c0c1c2 ∈ A(I0,1)−,

Q(r+[0,2]) = c∗0c∗1c

∗2 ∈ A(I0,1)−. (A.1)

We consider a next smallest segment I0,2 ≡ [0, 1, 2, 3, 4] by setting k = 0

and l = 2. The space Ξ0,2 on I0,2 consists of the following five −1,+1-sequences:

Ξ0,2 0 1 2 3 4r−[0,4] −1 −1 −1 −1 −1

ui[0,4] −1 −1 −1 +1 +1

uii[0,4] −1 −1 +1 +1 +1

vi[0,4] +1 +1 +1 −1 −1

vii[0,4] +1 +1 −1 −1 −1

r+[0,4] +1 +1 +1 +1 +1

Note that

r−[0,4] = −r+[0,4], ui[0,4] = −vi[0,4], u

ii[0,4] = −vii[0,4]. (A.2)

23

By (2.9) of Definition 2.5 we have

Q(r−[0,4]) = c0c1c2c3c4 ∈ A(I0,2)−,

Q(ui[0,4]) = c0c1c2c∗3c

∗4 ∈ A(I0,2)−,

Q(uii[0,4]) = c0c1c∗2c

∗3c

∗4 ∈ A(I0,2)−,

Q(vi[0,4]) = c∗0c∗1c

∗2c3c4 ∈ A(I0,2)−,

Q(vii[0,4]) = c∗0c∗1c2c3c4 ∈ A(I0,2)−,

Q(r+[0,4]) = c∗0c∗1c

∗2c

∗3c

∗4 ∈ A(I0,2)−. (A.3)

We then consider the segment I0,3 ≡ [0, 1, 2, 3, 4, 5, 6] taking k = 0 andl = 3. By definition it consists of 5 + 4 + 4 + 5 = 18 −1,+1-sequences:

Ξ0,3 0 1 2 3 4 5 6s[0,6] −1 −1 −1 −1 −1 −1 −1

si[0,6] −1 −1 −1 +1 +1 −1 −1

sii[0,6] −1 −1 +1 +1 −1 −1 −1

siii[0,6] −1 −1 −1 +1 −1 −1 −1

siv[0,6] −1 −1 +1 +1 +1 −1 −1

ui[0,6] −1 −1 −1 −1 −1 +1 +1

uii[0,6] −1 −1 −1 −1 +1 +1 +1

uiii[0,6] −1 −1 −1 +1 +1 +1 +1

uiv[0,6] −1 −1 +1 +1 +1 +1 +1

vi[0,6] +1 +1 +1 +1 +1 −1 −1

vii[0,6] +1 +1 +1 +1 −1 −1 −1

viii[0,6] +1 +1 +1 −1 −1 −1 −1

viv[0,6] +1 +1 −1 −1 −1 −1 −1

t•[0,6] +1 +1 +1 +1 +1 +1 +1

ti[0,6] +1 +1 +1 −1 −1 +1 +1

tii[0,6] +1 +1 −1 −1 +1 +1 +1

tiii[0,6] +1 +1 +1 −1 +1 +1 +1

tiv[0,6] +1 +1 −1 −1 −1 +1 +1

Note that s[0,6] ≡ r−[0,6] and t•[0,6] ≡ r+[0,6] and that

s[0,6] = −t•[0,6], sk[0,6] = −tk[0,6], ∀k ∈ i, ii, iii, iv

uk[0,6] = −vk[0,6], ∀k ∈ i, ii, iii, iv. (A.4)

24

Using the rule we obtain the following list of 18 fermion operators associatedto Ξ0,3:

Q(s[0,6]) = c0c1c2c3c4c5c6 ∈ A(I0,3)−,

Q(si[0,6]) = c0c1c2c∗3c

∗4c5c6 ∈ A(I0,3)−,

Q(sii[0,6]) = c0c1c∗2c

∗3c4c5c6 ∈ A(I0,3)−,

Q(siii[0,6]) = c0c1c2c∗3c4c5c6 ∈ A(I0,3)−,

Q(siv[0,6]) = c0c1c∗2c

∗3c

∗4c5c6 ∈ A(I0,3)−,

Q(ui[0,6]) = c0c1c2c3c4c∗5c

∗6 ∈ A(I0,3)−,

Q(uii[0,6]) = c0c1c2c3c∗4c

∗5c

∗6 ∈ A(I0,3)−,

Q(uiii[0,6]) = c0c1c2c∗3c

∗4c

∗5c

∗6 ∈ A(I0,3)−,

Q(uiv[0,6]) = c0c1c∗2c

∗3c

∗4c

∗5c

∗6 ∈ A(I0,3)−,

Q(vi[0,6]) = c∗0c∗1c

∗2c

∗3c

∗4c5c6 ∈ A(I0,3)−,

Q(vii[0,6]) = c∗0c∗1c

∗2c

∗3c4c5c6 ∈ A(I0,3)−,

Q(viii[0,6]) = c∗0c∗1c

∗2c3c4c5c6 ∈ A(I0,3)−,

Q(viv[0,6]) = c∗0c∗1c2c3c4c5c6 ∈ A(I0,3)−,

Q(t•[0,6]) = c∗0c∗1c

∗2c

∗3c

∗4c

∗5c

∗6 ∈ A(I0,3)−,

Q(ti[0,6]) = c∗0c∗1c

∗2c3c4c

∗5c

∗6 ∈ A(I0,3)−,

Q(tii[0,6]) = c∗0c∗1c2c3c

∗4c

∗5c

∗6 ∈ A(I0,3)−,

Q(tiii[0,6]) = c∗0c∗1c

∗2c3c

∗4c

∗5c

∗6 ∈ A(I0,3)−,

Q(tiv[0,6]) = c∗0c∗1c2c3c4c

∗5c

∗6 ∈ A(I0,3)−. (A.5)

25