Characterization of degenerate supersymmetric ground states of the Nicolai supersymmetric fermion lattice model by symmetry breakdown

A supersymmetric fermion lattice model defined by Nicolai [Ni] is a pioneering work on (non-relativistic) supersymmetric quantum mechanics. This model, which we call Nicolai model, even predates Witten's supersymmetric quantum mechanical model [Wi], see [J1] [J2] for some historical remarks. It has been shown that the Nicolai model has highly degenerate supersymmetric ground states [M1] [LScSh] which give rise to interesting dynamical properties. The aim of this paper is to discuss the degeneracy of supersymmetric ground states of the Nicolai model from the viewpoint of symmetry breakdown. Based on our previous findings [M1] we will classify all classical supersymmetric ground states in terms of breakdown of local fermionic symmetries (supersymmetries) hidden in the model.

1.1. Supersymmetric fermion lattice model by Nicolai

We introduce a spinless fermion lattice model on one-dimensional integer lattice $\mathbb{Z}$ given by Nicolai [Ni]. For each site $i\in \mathbb{Z}$ let c_i and ${c}_{i}^{\enspace {\ast}}$ denote the annihilation and the creation of a spinless fermion at i. They obey the canonical anticommutation relations: for all $i,j\in \mathbb{Z}$

$$\begin{align}\hfill \left\{{c}_{i}^{\enspace {\ast}},{c}_{j}\right\}& ={\delta }_{i,j}\enspace 1,\hfill \\ \hfill \left\{{c}_{i}^{\enspace {\ast}},{c}_{j}^{\enspace {\ast}}\right\}& =\left\{{c}_{i},{c}_{j}\right\}=0.\hfill \end{align} \tag{ 1.1 }$$

For each site $i\in \mathbb{Z}$ the fermion number operator is defined by

$$\begin{equation}{n}_{i}{:=}{c}_{i}^{\enspace {\ast}}{c}_{i}.\end{equation} \tag{ 1.2 }$$

A formal infinite sum $N{:=}{\sum }_{i\in \mathbb{Z}}{n}_{i}$ will denote the total fermion number operator. Let

$$\begin{equation}Q=\sum _{i\in \mathbb{Z}}{q}_{2i},\quad \text{where}\;\enspace {q}_{2i}{:=}{c}_{2i+1}{c}_{2i}^{{\ast}}{c}_{2i-1}.\end{equation} \tag{ 1.3 }$$

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

$$\begin{equation}\left\{{\left(-1\right)}^{N},\enspace Q\right\}=\left\{{\left(-1\right)}^{N},\enspace {Q}^{{\ast}}\right\}=0.\end{equation} \tag{ 1.4 }$$

It is essential that the nilpotent property is satisfied:

$$\begin{equation}{Q}^{2}=0={{Q}^{{\ast}}}^{2}.\end{equation} \tag{ 1.5 }$$

The supersymmetric Hamiltonian is given by

$$\begin{equation}H{:=}\left\{Q,\enspace {Q}^{{\ast}}\right\}.\end{equation} \tag{ 1.6 }$$

The pair of supercharges Q, Q* and the supersymmetric Hamiltonian H satisfy the $\mathcal{N}=2$ supersymmetry relation [We], although there is no boson in the model.

The explicit form of H can be easily computed as

$$\begin{align}\hfill H& =\sum _{i\in \mathbb{Z}}\left\{{c}_{2i}^{{\ast}}{c}_{2i-1}{c}_{2i+2}{c}_{2i+3}^{{\ast}}+{c}_{2i-1}^{{\ast}}{c}_{2i}{c}_{2i+3}{c}_{2i+2}^{{\ast}}\right.\hfill \\ \hfill & \enspace \left.+{c}_{2i}^{{\ast}}{c}_{2i}{c}_{2i+1}{c}_{2i+1}^{{\ast}}+{c}_{2i-1}^{{\ast}}{c}_{2i-1}{c}_{2i}{c}_{2i}^{{\ast}}-{c}_{2i-1}^{{\ast}}{c}_{2i-1}{c}_{2i+1}{c}_{2i+1}^{{\ast}}\right\}.\hfill \end{align} \tag{ 1.7 }$$

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

$$\begin{equation}{\gamma }_{\theta }\left({c}_{i}\right)={\mathrm{e}}^{-\mathrm{i}\theta }{c}_{i},\quad {\gamma }_{\theta }\left({c}_{i}^{{\ast}}\right)={\mathrm{e}}^{\mathrm{i}\theta }{c}_{i}^{{\ast}},\quad \forall i\in \mathbb{Z}.\end{equation} \tag{ 1.8 }$$

The particle–hole transformation is given by the ${\mathbb{Z}}_{2}$ action:

$$\begin{equation}\rho \left({c}_{i}\right)={c}_{i}^{{\ast}},\quad \rho \left({c}_{i}^{{\ast}}\right)={c}_{i},\quad \forall i\in \mathbb{Z}.\end{equation} \tag{ 1.9 }$$

Let σ denote the shift-translation automorphism group defined by

$$\begin{equation}{\sigma }_{k}\left({c}_{i}\right)={c}_{i+k},\quad {\sigma }_{k}\left({c}_{i}^{{\ast}}\right)={c}_{i+k}^{{\ast}}\quad \forall i\in \mathbb{Z},\enspace \;\text{for}\;\text{each}\;\enspace k\in \mathbb{Z}.\end{equation} \tag{ 1.10 }$$

The Hamiltonian H (1.7) is invariant under γ_θ (θ ∈ [0, 2π]), so it has the global U(1)-symmetry. It has particle–hole symmetry as ρ(H) = H, which follows from ρ(Q) = −Q* and ρ(Q*) = −Q. Finally, H is invariant under translation by two sites, as σ_2k (H) = H for any $k\in \mathbb{Z}$, whereas the full translation symmetry is explicitly broken as σ_2k+1(H) ≠ H. We will see in section 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] that gives a general framework of supersymmetric fermion lattice systems. Although it is not absolutely necessary, the C*-algebraic formulation is helpful to formulate our pertinent problem and gives a clue to solve it.

For each finite $\mathrm{I}{\Subset}\mathbb{Z}$, $\mathcal{A}\left(\mathrm{I}\right)$ denotes the finite-dimensional algebra generated by $\left\{{c}_{i}^{\enspace {\ast}},\enspace {c}_{i}\enspace ;\enspace i\in \mathrm{I}\right\}$, where the notation '$\mathrm{I}{\Subset}\mathbb{Z}$' means that $\mathrm{I}\subset \mathbb{Z}$ and the number of sites |I| in I is finite. The union of all these $\mathcal{A}\left(\mathrm{I}\right)$ defines the local algebra:

$$\begin{equation}{\mathcal{A}}_{{\circ}}{:=}\bigcup _{\mathrm{I}{\Subset}\mathbb{Z}}\mathcal{A}\left(\mathrm{I}\right).\end{equation} \tag{ 1.11 }$$

The norm completion of the local algebra ${\mathcal{A}}_{{\circ}}$ gives a C*-system $\mathcal{A}$ called the CAR algebra.

Let Θ denote the fermion grading automorphism on $\mathcal{A}$ given as:

$$\begin{equation}{\Theta}\left({c}_{i}\right)=-{c}_{i},\quad {\Theta}\left({c}_{i}^{\enspace {\ast}}\right)=-{c}_{i}^{\enspace {\ast}},\quad \forall i\in \mathbb{Z}.\end{equation} \tag{ 1.12 }$$

The fermion system $\mathcal{A}$ is decomposed into the even part ${\mathcal{A}}_{+}$ and the odd part ${\mathcal{A}}_{-}$ as

$$\begin{equation}\mathcal{A}={\mathcal{A}}_{+}\oplus {\mathcal{A}}_{-},\quad {\mathcal{A}}_{+}=\left\{A\in \mathcal{A}\vert \enspace {\Theta}\left(A\right)=A\right\},\quad {\mathcal{A}}_{-}=\left\{A\in \mathcal{A}\vert \enspace {\Theta}\left(A\right)=-A\right\}.\end{equation} \tag{ 1.13 }$$

Any element of ${\mathcal{A}}_{+}$ is a linear sum of even monomials of fermion field operators, while that of ${\mathcal{A}}_{-}$ is a linear sum of odd monomials of fermion field operators. Similarly, for each $\mathrm{I}{\Subset}\mathbb{Z}$,

$$\begin{equation}\mathcal{A}\left(\mathrm{I}\right)=\mathcal{A}{\left(\mathrm{I}\right)}_{+}\oplus \mathcal{A}{\left(\mathrm{I}\right)}_{-},\quad \mathcal{A}{\left(\mathrm{I}\right)}_{+}{:=}\mathcal{A}\left(\mathrm{I}\right)\cap {\mathcal{A}}_{+},\quad \mathcal{A}{\left(\mathrm{I}\right)}_{-}{:=}\mathcal{A}\left(\mathrm{I}\right)\cap {\mathcal{A}}_{-},\end{equation} \tag{ 1.14 }$$

and for the local algebra

$$\begin{equation}{\mathcal{A}}_{{\circ}}={{\mathcal{A}}_{{\circ}}}_{+}\oplus {{\mathcal{A}}_{{\circ}}}_{-},\quad {{\mathcal{A}}_{{\circ}}}_{+}{:=}{\mathcal{A}}_{{\circ}}\cap {\mathcal{A}}_{+},\quad {{\mathcal{A}}_{{\circ}}}_{-}{:=}{\mathcal{A}}_{{\circ}}\cap {\mathcal{A}}_{-}.\end{equation} \tag{ 1.15 }$$

Define the graded commutator [ , ]_Θ on $\mathcal{A}$ by the mixture of the commutator [ , ] and the anti-commutator { , } as

$$\begin{align}\hfill {\left[{A}_{+},\enspace B\right]}_{{\Theta}}& =\left[{A}_{+},\enspace B\right]\equiv {A}_{+}B-B{A}_{+}\quad \text{for}\;\;{A}_{+}\in {\mathcal{A}}_{+},\enspace B\in \mathcal{A},\hfill \\ \hfill {\left[A,\enspace {B}_{+}\right]}_{{\Theta}}& =\left[A,\enspace {B}_{+}\right]=A{B}_{+}-{B}_{+}A\quad \text{for}\;\;A\in \mathcal{A},\enspace {B}_{+}\in {\mathcal{A}}_{+},\hfill \\ \hfill {\left[{A}_{-},\enspace {B}_{-}\right]}_{{\Theta}}& =\left\{{A}_{-},\enspace {B}_{-}\right\}\equiv {A}_{-}{B}_{-}+{B}_{-}{A}_{-}\quad \text{for}\;\;{A}_{-},{B}_{-}\in {\mathcal{A}}_{-}.\hfill \end{align} \tag{ 1.16 }$$

Consider the superderivation generated by the nilpotent supercharge Q:

$$\begin{equation}{\delta }_{Q}\left(A\right){:=}{\left[Q,\enspace A\right]}_{{\Theta}}\quad \text{for}\;\;A\in {\mathcal{A}}_{{\circ}}.\end{equation} \tag{ 1.17 }$$

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

$$\begin{equation}{\delta }_{Q}\cdot {\Theta}=-{\Theta}\cdot {\delta }_{Q},\end{equation} \tag{ 1.18 }$$

and that the graded Leibniz rule holds:

$$\begin{equation}{\delta }_{Q}\left(AB\right)={\delta }_{Q}\left(A\right)B+{\Theta}\left(A\right){\delta }_{Q}\left(B\right)\quad \text{for}\enspace \enspace A,B\in \mathcal{A}.\end{equation} \tag{ 1.19 }$$

A state (i.e. normalized positive linear functional on $\mathcal{A}$) is called a supersymmetric (ground) state if and only if it is invariant under the superderivation δ_Q , 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 associated with normalized vectors. For a supersymmetric model, if there is a supersymmetric state, then the supersymmetry is unbroken. If there exists no supersymmetric state, then the supersymmetry is spontaneously broken. The Nicolai model has many supersymmetric states as we will see later. Hence its supersymmetry is unbroken.

1.3. Classical supersymmetric ground states of the Nicolai model

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

Let |1⟩_i and |0⟩_i denote the occupied and empty vectors of the spinless fermion at site i, respectively. For each $i\in \mathbb{Z}$

$$\begin{equation}{c}_{i}\vert {1\rangle }_{i}=\vert {0\rangle }_{i},\;\;{c}_{i}^{\enspace {\ast}}\vert {1\rangle }_{i}=0,\;\;{c}_{i}^{\enspace {\ast}}\vert {0\rangle }_{i}=\vert {1\rangle }_{i},\;\;{c}_{i}\vert {0\rangle }_{i}=0.\end{equation} \tag{ 1.20 }$$

When there is no fear of confusion, we will omit the subscript and write simply |1⟩ and |0⟩.

We identify general (not necessarily supersymmetric) classical states on the fermion lattice system by classical configurations on $\mathbb{Z}$.

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

$$\begin{equation}\vert g{\left(n\right)}_{n\in \mathbb{Z}}\rangle {:=}\cdots \otimes \vert g{\left(i-1\right)\rangle }_{i-1}\otimes \vert g{\left(i\right)\rangle }_{i}\otimes \vert g{\left(i+1\right)\rangle }_{i+1}\otimes \cdots \enspace .\end{equation} \tag{ 1.21 }$$

This infinite product vector determines a state ψ_g(n) on the fermion system $\mathcal{A}$ which will be called the classical state associated to the configuration g(n) over $\mathbb{Z}$. Let ${\iota }_{0}\left(n\right){:=}0\quad \forall n\in \mathbb{Z}$. Then

$$\begin{equation}{{\Omega}}_{0}{:=}\vert {\iota }_{0}{\left(n\right)}_{n\in \mathbb{Z}}\rangle =\cdots \otimes \vert 0\rangle \otimes \vert 0\rangle \otimes \vert 0\rangle \otimes \vert 0\rangle \otimes \vert 0\rangle \otimes \vert 0\rangle \cdots \enspace .\end{equation} \tag{ 1.22 }$$

The above Ω₀ is called the Fock vector, and its associated translation-invariant state ψ₀ on $\mathcal{A}$ is called the Fock state. Similarly let ${\iota }_{1}\left(n\right){:=}1\forall n\in \mathbb{Z}$. Then

$$\begin{equation}{{\Omega}}_{1}{:=}\vert {\iota }_{1}{\left(n\right)}_{n\in \mathbb{Z}}\rangle =\cdots \otimes \vert 1\rangle \otimes \vert 1\rangle \otimes \vert 1\rangle \otimes \vert 1\rangle \otimes \vert 1\rangle \otimes \vert 1\rangle \cdots \enspace .\end{equation} \tag{ 1.23 }$$

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

To each classical configuration over $\mathbb{Z}$ we assign an operator by the following rule.

Definition 1.2. For each $i\in \mathbb{Z}$ let ${\hat{\kappa }}_{i}$ denote the map from {0, 1} into $\mathcal{A}\left(\left\{i\right\}\right)$ given as

$$\begin{equation}{\hat{\kappa }}_{i}\left(0\right){:=}\mathbf{1},\quad {\hat{\kappa }}_{i}\left(1\right){:=}{c}_{i}^{\enspace {\ast}}.\end{equation} \tag{ 1.24 }$$

For each classical configuration g(n) over $\mathbb{Z}$ define the infinite-product of fermion field operators:

$$\begin{equation}\hat{\mathcal{O}}\left(g\right){:=}\prod _{i\in \mathbb{Z}}{\hat{\kappa }}_{i}\left(g\left(i\right)\right)=\cdots {\hat{\kappa }}_{i-1}\left(g\left(i-1\right)\right){\hat{\kappa }}_{i}\left(g\left(i\right)\right){\hat{\kappa }}_{i+1}\left(g\left(i+1\right)\right)\cdots \enspace ,\end{equation} \tag{ 1.25 }$$

where the multiplication is taken in the increasing order of $i\in \mathbb{Z}$. If g(n) has a compact support, then

$$\begin{equation}\hat{\mathcal{O}}\left(g\right)\in {\mathcal{A}}_{{\circ}}.\end{equation} \tag{ 1.26 }$$

Otherwise $\hat{\mathcal{O}}\left(g\right)$ denotes a formal operator which does not belong to $\mathcal{A}$.

We have the following obvious correspondence between product vectors given in definition 1.1 and product operators given in definition 1.2 via the Fock representation.

Proposition 1.3. Let Ω₀ denote the Fock vector given in (1.22). For any classical configuration g(n) over $\mathbb{Z}$, the following identity holds:

$$\begin{equation}\hat{\mathcal{O}}\left(g\right){{\Omega}}_{0}=\vert g{\left(n\right)}_{n\in \mathbb{Z}}\rangle .\end{equation} \tag{ 1.27 }$$

Proof. The desired identity directly follows from definitions 1.1 and 1.2.□

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 to give all classical supersymmetric ground states of the Nicolai model. For this purpose, we introduce the following class of classical configurations.

Definition 1.4. Consider three consecutive sites {2i − 1, 2i, 2i + 1} centered at an even site 2i ($i\in \mathbb{Z}$). There are 2³ configurations (i.e. eight {0, 1}-valued functions) on {2i − 1, 2i, 2i + 1}. Let '0, 1, 0' and '1, 0, 1' be called forbidden triplets. If a classical configuration $g\left(n\right)\enspace \left(n\in \mathbb{Z}\right)$ does not include any of such forbidden triplets over $\mathbb{Z}$, then it is called a ground-state configuration over $\mathbb{Z}$ (for the Nicolai model). The set of all ground-state configurations over $\mathbb{Z}$ is denoted by ϒ. The set of all ground-state configurations whose supports are included in some finite region is denoted by ϒ_◦. The set of all ground-state configurations whose supports are included in a finite region $\mathrm{I}{\Subset}\mathbb{Z}$ is denoted by ϒ_I.

The following proposition classifies all the classical supersymmetric ground states in terms of classical configurations justifying our nomenclature 'ground-state configurations' of definition 1.4. It is based on the following fact that can 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 q_2i and ${q}_{2i}^{{\ast}}$ unless those are |0⟩_2i−1 ⊗ |1⟩_2i ⊗ |0⟩_2i+1 or |1⟩_2i−1 ⊗ |0⟩_2i ⊗ |1⟩_2i+1 which correspond to the forbidden triplets, {g(2i − 1) = 0, g(2i) = 1, g(2i + 1) = 0} and {g(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 $g\left(n\right)\enspace \left(n\in \mathbb{Z}\right)$ at all, then the corresponding product vector $\hat{\mathcal{O}}\left(g\right){{\Omega}}_{0}$ is annihilated by both Q and Q*, hence it is a supersymmetric ground state, whereas if there is at least one forbidden triplet in the sequence of $g\left(n\right)\enspace \left(n\in \mathbb{Z}\right)$, then either Q or Q*, or both do not annihilate $\hat{\mathcal{O}}\left(g\right){{\Omega}}_{0}$ and so it is not supersymmetric. See [M1] for the detail.

Proposition 1.5. A classical state on the fermion lattice system $\mathcal{A}$ is supersymmetric for the Nicolai model if and only if its associated configuration g(n) over $\mathbb{Z}$ is a ground-state configuration for the Nicolai model as stated in definition 1.4, namely 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 meaning of 'supersymmetric ground states' upon finite regions, as it is not so obvious due to the boundary.

To deal with the Nicolai model which has period-2 translational symmetry not full translation symmetry, rather than, it is convenient to consider the special finite intervals of $\mathbb{Z}$ whose edges are both even, see proposition 4.1 given later. Namely for $k,l\in \mathbb{Z}$ (k < l) we take

$$\begin{equation}{\text{I}}_{k,l}\equiv \left[2k,\enspace 2k+1,\enspace 2\left(k+1\right),\dots ,2\left(l-1\right),\enspace 2l-1,\enspace ,2l\right].\end{equation} \tag{ 1.28 }$$

We see that |I_k,l | is 2(l − k) + 1.

Now we give the precise definition of supersymmetric ground states on the finite interval I_k,l .

Definition 1.6. Consider any finite interval ${\text{I}}_{k,l}\equiv \left[2k,\enspace 2k+1,\enspace 2\left(k+1\right),\dots ,2\left(l-1\right),\enspace 2l-1,\enspace 2l\right]\quad \left(k,l\in \mathbb{Z}\enspace k{< }l\right)$. Let

$$\begin{equation}Q\left[k,l\right]\equiv \sum _{i=k}^{l}{q}_{2i}\in \mathcal{A}{\left(\left\{2k-1\right\}\cup {\text{I}}_{k,l}\cup \left\{2l+1\right\}\right)}_{-},\end{equation} \tag{ 1.29 }$$

where ${q}_{2i}\equiv -{c}_{2i-1}{c}_{2i}^{{\ast}}{c}_{2i+1}$ as defined in (1.3). A state on $\mathcal{A}\left({\text{I}}_{k,l}\right)$ is called a free-boundary supersymmetric ground state if its arbitrary state-extension to $\mathcal{A}\left(\left\{2k-1\right\}\cup {\text{I}}_{k,l}\cup \left\{2l+1\right\}\right)$ is invariant under the superderivation δ_Q[k,l] associated to the local supercharge Q[k, l], equivalently its associated vector is annihilated by both Q[k, l] and Q[k,l]*.

First note that

$$\begin{equation}{\delta }_{Q\left[k,l\right]}={\delta }_{Q}\;\;\text{on}\;\;\mathcal{A}\left({\text{I}}_{k,l}\right),\end{equation} \tag{ 1.30 }$$

where $Q={\sum }_{i\in \mathbb{Z}}{q}_{2i}$ as in (1.3). Namely, upon the subsystem $\mathcal{A}\left({\text{I}}_{k,l}\right)$, finite supercharge Q[k, l] sitting on a slightly larger region {2k − 1} ∪ I_k,l ∪ {2l + 1} gives the same action as of the total supercharge Q. Second, we address what 'its arbitrary state-extension' exactly means. According to [ArM], for every classical state on the given local system, any state-extension of it to a larger system is also a classical state (or mixture of such). In the present case it is described as follows. By proposition 1.3 any classical state of $\mathcal{A}\left({\text{I}}_{k,l}\right)$ is determined by a {0, 1}-valued function g(n) on I_k,l . Any state-extension of it to $\mathcal{A}\left(\left\{2k-1\right\}\cup {\text{I}}_{k,l}\cup \left\{2l+1\right\}\right)$ is determined by $\tilde {g}\left(n\right)$ on {2k − 1} ∪ I_k,l ∪ {2l + 1} satisfying that

$$\begin{equation}\tilde {g}\left(n\right)=g\left(n\right)\quad \text{for}\enspace \forall n\in {\text{I}}_{k,l},\quad \tilde {g}\left(2k-1\right)=0\enspace \;\text{or}\;1,\quad \tilde {g}\left(2l+1\right)=0\enspace \;\text{or}\;1.\end{equation} \tag{ 1.31 }$$

Due to the choice of the marginal points {2k − 1, 2l + 1} there are four possibilities.

To find configurations associated to definition 1.6, we introduce a subclass of ϒ_◦ given in definition 1.4 requiring certain boundary conditions as follows.

Definition 1.7. Let I_k,l ≡ [2k, 2k + 1, 2(k + 1), ..., 2(l − 1), 2l − 1, 2l] ($k,l\in \mathbb{Z}$ s.t. k < l) as before. The set of all $g\left(n\right)\in {{\Upsilon}}_{k,l}\equiv {{\Upsilon}}_{{\text{I}}_{k,l}}$ satisfying the following boundary conditions

$$\begin{equation}g\left(2k\right)=g\left(2k+1\right)\,\,\text{and}\,\,g\left(2l-1\right)=g\left(2l\right)\end{equation} \tag{ 1.32 }$$

will be denoted by ${\hat{{\Upsilon}}}_{k,l}$.

In proposition 1.5 we gave one-to-one correspondence between classical supersymmetric ground states and ground-state configurations over $\mathbb{Z}$. We can see analogous correspondence on finite regions I_k,l as follows.

Proposition 1.8. A classical state on the finite system $\mathcal{A}\left({\text{I}}_{k,l}\right)$ is free-boundary supersymmetric (definition 1.6) if and only if its associated configuration g(n) on I_k,l belongs to ${\hat{{\Upsilon}}}_{k,l}$ (definition 1.7).

Proof. We will see the if part as follows. For all i ∈ {k + 1, k + 2, ..., l − 1} both q_2i and ${q}_{2i}^{{\ast}}$ annihilate any product vector corresponding to g(n) ∈ ϒ_k,l . So we only have to see the marginal points k and l. For any given $g\left(n\right)\in {\hat{{\Upsilon}}}_{k,l}$ its extension to {2k − 1} ∪ I_k,l ∪ {2l + 1} will be denoted as $\tilde {g}\left(n\right)$. We see that $\tilde {g}\left(2k-1\right)$ is arbitrary, $\tilde {g}\left(2k\right)=\tilde {g}\left(2k+1\right)$, $\tilde {g}\left(2l-1\right)=\tilde {g}\left(2l\right)$, and $\tilde {g}\left(2l+1\right)$ is arbitrary. So there is no forbidden sequence on {2k − 1, 2k, 2k + 1}. Thus both q_2k and ${q}_{2k}^{{\ast}}$ annihilate any product vector corresponding to $\tilde {g}\left(n\right)$. Similarly, both q_2l and ${q}_{2l}^{{\ast}}$ annihilate the product vector corresponding to $\tilde {g}\left(n\right)$. The only if part can be shown as in theorem 2 [M1].□

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

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

$$\begin{equation}f\left(2i-1\right)=-1,\quad f\left(2i\right)=+1,\quad f\left(2i+1\right)=-1\end{equation} \tag{ 2.1 }$$

nor

$$\begin{equation}f\left(2i-1\right)=+1,\quad f\left(2i\right)=-1,\quad f\left(2i+1\right)=+1\end{equation} \tag{ 2.2 }$$

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

$$\begin{equation}f\left(2k\right)=f\left(2k+1\right)=+1\quad \text{or}\quad f\left(2k\right)=f\left(2k+1\right)=-1\end{equation} \tag{ 2.3 }$$

and

$$\begin{equation}f\left(2l-1\right)=f\left(2l\right)=+1\quad \text{or}\quad f\left(2l-1\right)=f\left(2l\right)=-1.\end{equation} \tag{ 2.4 }$$

The set of all {−1, +1}-valued sequences on I_k,l satisfying the above conditions is denoted by ${\hat{{\Xi}}}_{k,l}$. The union of ${\hat{{\Xi}}}_{k,l}$ over all $k,l\in \mathbb{Z}$ (k < l) is denoted by $\hat{{\Xi}}$:

$$\begin{equation}\hat{{\Xi}}{:=}\bigcup _{k,l\in \mathbb{Z}\enspace \left(k{< }l\right)}{\hat{{\Xi}}}_{k,l}.\end{equation} \tag{ 2.5 }$$

Take any $p,q\in \mathbb{Z}\enspace \left(p{< }q\right)$. Let

$$\begin{equation}\hat{{\Xi}}\left(p,q\right){:=}\bigcup _{k,l\in \mathbb{Z};\enspace p{\leqslant}k{< }l{\leqslant}q}{\hat{{\Xi}}}_{k,\enspace l}.\end{equation} \tag{ 2.6 }$$

Each $f\in \hat{{\Xi}}$ is called a local {−1, +1}-sequence of conservation for the Nicolai model.

Remark 2.2. The requirements (2.3) and (2.4) on the edges of I_k,l are essential to make 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 ${\hat{{\Xi}}}_{k,l}$ is roughly ${\left(\frac{{2}^{3}-2}{2}\right)}^{\left(l-k\right)}={3}^{\left(l-k\right)}={3}^{m/2}$, where m = 2(l − k) denotes approximately the size of the system (i.e. the number of sites in I_k,l ).

It is convenient to consider the following subclasses of $\hat{{\Xi}}$.

Definition 2.4. For each $k,l\in \mathbb{Z}$ (k < l) let ${r}_{\left[2k,2l\right]}^{+}\in {\hat{{\Xi}}}_{k,l}$ and ${r}_{\left[2k,2l\right]}^{-}\in {\hat{{\Xi}}}_{k,l}$ denote the constants over I_k,l as

$$\begin{equation}{r}_{\left[2k,2l\right]}^{+}\left(i\right)=+1\,\,\forall i\in {\text{I}}_{k,l},\quad {r}_{\left[2k,2l\right]}^{-}\left(i\right)=-1\,\,\forall i\in {\text{I}}_{k,l}.\end{equation} \tag{ 2.7 }$$

The set $\left\{{r}_{\left[2k,2l\right]}^{+}\right\}$ over all $k,l\in \mathbb{Z}$ (k < l) will be denoted as ${\hat{{\Xi}}}_{\enspace +1\text{const.}}$, and the set $\left\{{r}_{\left[2k,2l\right]}^{-}\right\}$ over all $k,l\in \mathbb{Z}$ (k < l) will be denoted as ${\hat{{\Xi}}}_{\enspace -1\text{const.}}$. Let ${\hat{{\Xi}}}_{\text{const.}}{:=}{\hat{{\Xi}}}_{\enspace +1\text{const.}}\cup {\hat{{\Xi}}}_{\enspace -1\text{const.}}$.

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 $i\in \mathbb{Z}$ let ζ_i denote the assignment from {−1, +1} into the fermion field at i given as

$$\begin{equation}{\zeta }_{i}\left(-1\right){:=}{c}_{i},\quad {\zeta }_{i}\left(+1\right){:=}{c}_{i}^{\enspace {\ast}}.\end{equation} \tag{ 2.8 }$$

Take any pair of integers $k,l\in \mathbb{Z}$ such that k < l. For each $f\in {\hat{{\Xi}}}_{k,l}$, set

$$\begin{align}\hfill \mathcal{Q}\left(f\right)& {:=}\prod _{i=2k}^{2l}{\zeta }_{i}\left(f\left(i\right)\right)\hfill \\ \hfill & \equiv {\zeta }_{2k}\left(f\left(2k\right)\right){\zeta }_{2k+1}\left(f\left(2k+1\right)\right)\cdots {\zeta }_{2l-1}\left(f\left(2l-1\right)\right){\zeta }_{2l}\left(f\left(2l\right)\right)\in \mathcal{A}{\left({\text{I}}_{k,l}\right)}_{-},\hfill \end{align} \tag{ 2.9 }$$

where the multiplication is taken in the increasing order as above. The formulas (2.9) for all $k,l\in \mathbb{Z}$ (k < l) yield a unique assignment $\mathcal{Q}$ from $\hat{{\Xi}}$ to ${{\mathcal{A}}_{{\circ}}}_{-}$.

By definition 2.5 the following local fermion operators are assigned to ±-characters supported on the segment I_k,l of definition 2.4. For $k,l\in \mathbb{Z}$ (k < l)

$$\begin{align}\hfill \mathcal{Q}\left({r}_{\left[2k,2l\right]}^{+}\right)& {:=}{c}_{2k}^{\enspace {\ast}}{c}_{2k+1}^{\enspace {\ast}}\cdots {c}_{2l-1}^{\enspace {\ast}}{c}_{2l}^{\enspace {\ast}}\in \mathcal{A}{\left({\text{I}}_{k,l}\right)}_{-},\hfill \\ \hfill \mathcal{Q}\left({r}_{\left[2k,2l\right]}^{-}\right)& {:=}{c}_{2k}{c}_{2k+1}\cdots {c}_{2l-1}{c}_{2l}\in \mathcal{A}{\left({\text{I}}_{k,l}\right)}_{-}.\hfill \end{align} \tag{ 2.10 }$$

The following is the main result of this section.

Theorem 2.6. For every $f\in \hat{{\Xi}}$

$$\begin{equation}\left[H,\enspace \mathcal{Q}\left(f\right)\right]=0=\left[H,\enspace \mathcal{Q}{\left(f\right)}^{{\ast}}\right],\end{equation} \tag{ 2.11 }$$

where H denotes the Hamiltonian of the Nicolai model over $\mathbb{Z}$.

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

$$\begin{equation}\left\{Q,\enspace \mathcal{Q}\left(f\right)\right\}=\left\{{Q}^{{\ast}},\enspace \mathcal{Q}\left(f\right)\right\}=0,\end{equation} \tag{ 2.12 }$$

and that

$$\begin{equation}\left\{Q,\enspace \mathcal{Q}{\left(f\right)}^{{\ast}}\right\}=\left\{{Q}^{{\ast}},\enspace \mathcal{Q}{\left(f\right)}^{{\ast}}\right\}=0,\end{equation} \tag{ 2.13 }$$

as the former implies $\left[H,\enspace \mathcal{Q}\left(f\right)\right]=0$ and the latter implies $\left[H,\enspace \mathcal{Q}{\left(f\right)}^{{\ast}}\right]=0$ by the graded Leibniz rule of superderivations (1.19). Recall $Q={\sum }_{i\in \mathbb{Z}}{q}_{2i}$ and ${q}_{2i}\equiv {c}_{2i+1}{c}_{2i}^{{\ast}}{c}_{2i-1}$ defined in (1.3). By definitions 2.1 and 2.5, we have

$$\begin{equation}\mathcal{Q}\left(f\right){q}_{2i}=0={q}_{2i}\mathcal{Q}\left(f\right),\quad \mathcal{Q}\left(f\right){q}_{2i}^{{\ast}}=0={q}_{2i}^{{\ast}}\mathcal{Q}\left(f\right)\quad \text{for}\enspace \text{all}\quad i\in \mathbb{Z}.\end{equation} \tag{ 2.14 }$$

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

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

Definition 2.7. For each local {−1, +1}-sequence of conservation $f\in \hat{{\Xi}}$, $\mathcal{Q}\left(f\right)$ is called the local fermionic constant of motion associated to f, and the pair $\left\{\mathcal{Q}\left(f\right),\mathcal{Q}{\left(f\right)}^{{\ast}}\right\}$ is called the local fermionic charge associated to f.

Remark 2.8. Fermionic symmetry satisfying the supersymmetry relation other than the dynamical supersymmetry is sometimes called kinematical supersymmetry. See e.g. [NSakY]. Hence the local fermionic charge $\left\{\mathcal{Q}\left(f\right),\mathcal{Q}{\left(f\right)}^{{\ast}}\right\}$ for any $f\in \hat{{\Xi}}$ gives a local kinematical supersymmetry.

Remark 2.9. Any operator of the algebra generated by $\left\{\mathcal{Q}\left(f\right)\in {\mathcal{A}}_{{\circ}}\vert \enspace f\in \hat{{\Xi}}\right\}$ is a local constant of motion. There exist many such self-adjoint bosonic operators that generate (bosonic) symmetries for the Nicolai model. For example, we obtain a bosonic constant ${n}_{2k}{n}_{2k+1}\cdots {n}_{2l-1}{n}_{2l}\in \mathcal{A}{\left({\text{I}}_{k,l}\right)}_{+}$ from the product $\mathcal{Q}\left({r}_{\left[2k,2l\right]}^{+}\right)\enspace \enspace \text{and}\enspace \enspace \mathcal{Q}\left({r}_{\left[2k,2l\right]}^{-}\right)$.

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

Theorem 3.1. Take any segment I_k,l indexed by $k,l\in \mathbb{Z}$ (k < l) as in (1.28). Any classical free-boundary supersymmetric ground state on $\mathcal{A}\left({\text{I}}_{k,l}\right)$ (definition 1.6) can be constructed by some finitely many applications of operators $\mathcal{Q}\left(f\right)$ (and $\mathcal{Q}{\left(f\right)}^{{\ast}}$) with $f\in \hat{{\Xi}}\left(k,l\right)$ (definition 2.1) to the Fock vector Ω₀ (1.22), similarly, to the fully-occupied vector Ω₁ (1.23).

By definition 1.1 we can identify every classical supersymmetric ground state ψ_g(n) on $\mathcal{A}$ with its corresponding classical configuration g(n) over $\mathbb{Z}$, and vice versa. By proposition 1.8 we can identify the set of all classical free-boundary supersymmetric ground states on $\mathcal{A}\left({\text{I}}_{k,l}\right)$ (definition 1.6) with ${\hat{{\Upsilon}}}_{k,l}$ (definition 1.7). We will frequently use those identifications in what follows.

The following lemma implies that the latter part (using Ω₁) of theorem 3.1 holds once the former part (using Ω₀) is proved. Also it will be used frequently in the proof.

Lemma 3.2. For any $n\in \mathbb{N}$, if $g\in {\hat{{\Upsilon}}}_{0,n}$ can be constructed by some finitely many applications of local fermionic charges within I_0,n to the Fock vector Ω₀. Then it can be constructed by some applications of local fermionic charges within I_0,n to the fully-occupied state Ω₁.

Proof. For any $f\in \hat{{\Xi}}\left(0,n\right)$, $-f\in \hat{{\Xi}}\left(0,n\right)$ by definition. From (2.9) in definition 2.5

$$\begin{equation}\mathcal{Q}\left(-f\right)=\rho \left(\mathcal{Q}\left(f\right)\right),\end{equation} \tag{ 3.1 }$$

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

Obviously it is enough to show theorem 3.1 by setting k = 0 and $l=\forall n\in \mathbb{N}$ by shift-translations. Thus we will prove the following.

Proposition 3.3. For any $n\in \mathbb{N}$, every $g\in {\hat{{\Upsilon}}}_{0,n}$ can be constructed by some applications of local fermionic charges within I_0,n :

$$\begin{equation}\left\{\mathcal{Q}\left(f\right);\enspace f\in \hat{{\Xi}}\left(0,n\right)\equiv \bigcup _{m\in \left\{1,2,\dots ,n\right\}}{\hat{{\Xi}}}_{0,m}\right\}\end{equation} \tag{ 3.2 }$$

to the Fock vector Ω₀ (1.22).

Proof. We need concrete forms of elements in $\hat{{\Xi}}$ which are listed in appendix A.1.

First, let us consider the case n = 1. ${\hat{{\Upsilon}}}_{0,1}$ consists of the following two sequences on I_0,1:

${\hat{{\Upsilon}}}_{0,1}$	0	1	2
${g}_{\left[0,2\right]}^{{\circ}}$	0	0	0
${g}_{\left[0,2\right]}^{{\bullet}}$	1	1	1

The classical configuration ${g}_{\left[0,2\right]}^{{\circ}}$ corresponds to the Fock vector Ω₀ (restricted to the local region [0, 1, 2]). We shall write simply ${g}_{\left[0,2\right]}^{{\circ}}={{\Omega}}_{0}$, and this identification will be used hereafter. On the other hand, ${g}_{\left[0,2\right]}^{{\bullet}}$ corresponds to $\mathcal{Q}\left({r}_{\left[0,2\right]}^{+}\right){{\Omega}}_{0}$ which is the fully-occupied state on I_0,1, where ${r}_{\left[0,2\right]}^{+}\in {\hat{{\Xi}}}_{0,1}$. Thus we obtain ${g}_{\left[0,2\right]}^{{\bullet}}={r}_{\left[0,2\right]}^{+}{{\Omega}}_{0}$ which is the desired formula.

Second, let us consider the case n = 2. ${\hat{{\Upsilon}}}_{0,2}$ consists of the following six sequences on I_0,2:

${\hat{{\Upsilon}}}_{0,2}$	0	1	2	3	4
${g}_{\left[0,4\right]}^{{\circ}}$	0	0	0	0	0
${g}_{\left[0,4\right]}^{1}$	0	0	0	1	1
${g}_{\left[0,4\right]}^{2}$	0	0	1	1	1
${g}_{\left[0,4\right]}^{3}$	1	1	1	0	0
${g}_{\left[0,4\right]}^{4}$	1	1	0	0	0
${g}_{\left[0,4\right]}^{{\bullet}}$	1	1	1	1	1

We have ${g}_{\left[0,4\right]}^{{\circ}}={{\Omega}}_{0}$ and ${g}_{\left[0,4\right]}^{{\bullet}}={r}_{\left[0,4\right]}^{+}{{\Omega}}_{0}={c}_{0}^{{\ast}}{c}_{1}^{{\ast}}{c}_{2}^{{\ast}}{c}_{3}^{{\ast}}{c}_{4}^{{\ast}}{{\Omega}}_{0}$ (the fully-occupied state on I_0,2) according to (A.3). We have

$$\begin{align*}\hfill {g}_{\left[0,4\right]}^{3}={r}_{\left[0,2\right]}^{+}{{\Omega}}_{0}={c}_{0}^{{\ast}}{c}_{1}^{{\ast}}{c}_{2}^{{\ast}}{{\Omega}}_{0},\quad {r}_{\left[0,2\right]}^{+}\in {\hat{{\Xi}}}_{0,1}\\ \hfill {g}_{\left[0,4\right]}^{2}={r}_{\left[2,4\right]}^{+}{{\Omega}}_{0}={c}_{2}^{{\ast}}{c}_{3}^{{\ast}}{c}_{4}^{{\ast}}{{\Omega}}_{0},\quad {r}_{\left[2,4\right]}^{+}\in {\hat{{\Xi}}}_{1,2}.\end{align*}$$

To get ${g}_{\left[0,4\right]}^{1}$ and ${g}_{\left[0,4\right]}^{4}$ we use 'double' actions as

$$\begin{equation}{g}_{\left[0,4\right]}^{1}={r}_{\left[0,2\right]}^{-}{g}_{\left[0,4\right]}^{{\bullet}}={r}_{\left[0,2\right]}^{-}{r}_{\left[0,4\right]}^{+}{{\Omega}}_{0}={c}_{3}^{{\ast}}{c}_{4}^{{\ast}}{{\Omega}}_{0},\quad {r}_{\left[0,2\right]}^{-}\in {\hat{{\Xi}}}_{0,1},\enspace {r}_{\left[0,4\right]}^{+}\in {\hat{{\Xi}}}_{0,2},\end{equation} \tag{ 3.3 }$$

and similarly

$$\begin{equation}{g}_{\left[0,4\right]}^{4}={r}_{\left[2,4\right]}^{-}{g}_{\left[0,4\right]}^{{\bullet}}={r}_{\left[2,4\right]}^{-}{r}_{\left[0,4\right]}^{+}{{\Omega}}_{0}={c}_{0}^{{\ast}}{c}_{1}^{{\ast}}{{\Omega}}_{0},\quad {r}_{\left[2,4\right]}^{-}\in {\hat{{\Xi}}}_{1,2},\enspace {r}_{\left[0,4\right]}^{+}\in {\hat{{\Xi}}}_{0,2}.\end{equation} \tag{ 3.4 }$$

We have derived all the elements of ${\hat{{\Upsilon}}}_{0,2}$ and accordingly all the classical free-boundary supersymmetric ground states on $\mathcal{A}\left({\text{I}}_{0,2}\right)$ from the Fock vector Ω₀. Let us note that ${g}_{\left[0,4\right]}^{1}$ and ${g}_{\left[0,4\right]}^{3}$ are mapped to each other by the particle–hole transformation, and so are ${g}_{\left[0,4\right]}^{2}$ and ${g}_{\left[0,4\right]}^{4}$. 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 ${\hat{{\Upsilon}}}_{0,2}$ (all the classical free-boundary supersymmetric ground states on $\mathcal{A}\left({\text{I}}_{0,2}\right)$) from the fully-occupied vector Ω₁ in place of Ω₀. This fact is important.

Let us consider the case n = 3. ${\hat{{\Upsilon}}}_{0,3}$ consists of the following 18 sequences on I_0,3:

${\hat{{\Upsilon}}}_{0,3}$	0	1	2	3	4	5	6
${g}_{\left[0,6\right]}^{{\circ}}$	0	0	0	0	0	0	0
${g}_{\left[0,6\right]}^{1}$	0	0	0	1	1	0	0
${g}_{\left[0,6\right]}^{2}$	0	0	1	1	0	0	0
${g}_{\left[0,6\right]}^{3}$	0	0	0	1	0	0	0
${g}_{\left[0,6\right]}^{4}$	0	0	1	1	1	0	0
${g}_{\left[0,6\right]}^{5}$	0	0	0	0	0	1	1
${g}_{\left[0,6\right]}^{6}$	0	0	0	0	1	1	1
${g}_{\left[0,6\right]}^{7}$	0	0	0	1	1	1	1
${g}_{\left[0,6\right]}^{8}$	0	0	1	1	1	1	1
${g}_{\left[0,6\right]}^{9}$	1	1	1	1	1	0	0
${g}_{\left[0,6\right]}^{10}$	1	1	1	1	0	0	0
${g}_{\left[0,6\right]}^{11}$	1	1	1	0	0	0	0
${g}_{\left[0,6\right]}^{12}$	1	1	0	0	0	0	0
${g}_{\left[0,6\right]}^{{\bullet}}$	1	1	1	1	1	1	1
${g}_{\left[0,6\right]}^{13}$	1	1	1	0	0	1	1
${g}_{\left[0,6\right]}^{14}$	1	1	0	0	1	1	1
${g}_{\left[0,6\right]}^{15}$	1	1	1	0	1	1	1
${g}_{\left[0,6\right]}^{16}$	1	1	0	0	0	1	1

We will generate all the above ${g}_{\left[0,6\right]}^{{\ast}}$. Obviously ${g}_{\left[0,6\right]}^{{\circ}}={{\Omega}}_{0}$, and ${g}_{\left[0,6\right]}^{{\bullet}}={r}_{\left[0,6\right]}^{+}{{\Omega}}_{0}={c}_{0}^{{\ast}}{c}_{1}^{{\ast}}{c}_{2}^{{\ast}}{c}_{3}^{{\ast}}{c}_{4}^{{\ast}}{c}_{5}^{{\ast}}{c}_{6}^{{\ast}}{{\Omega}}_{0}$ which is the fully-occupied vector Ω₁ restricted to I_0,3.

The restriction of ${g}_{\left[0,6\right]}^{1}$ to [0, 4] is ${g}_{\left[0,4\right]}^{1}$, and the restriction of ${g}_{\left[0,6\right]}^{4}$ to [0, 4] is ${g}_{\left[0,4\right]}^{2}$. The restriction of ${g}_{\left[0,6\right]}^{2}$ to [2, 6] is ${g}_{\left[2,6\right]}^{4}$, which is the translation of ${g}_{\left[0,4\right]}^{4}$ used before. Thus each of ${g}_{\left[0,6\right]}^{1}\enspace {g}_{\left[0,6\right]}^{4}$ and ${g}_{\left[0,6\right]}^{2}$ can be given as in the case n = 2.

The restriction of ${g}_{\left[0,6\right]}^{13}$ to [0, 4] is ${g}_{\left[0,4\right]}^{3}$, the restriction of ${g}_{\left[0,6\right]}^{14}$ to [2, 6] is ${g}_{\left[2,6\right]}^{2}$, the restriction of ${g}_{\left[0,6\right]}^{16}$ to [0, 4] is ${g}_{\left[0,4\right]}^{4}$ (also the restriction of ${g}_{\left[0,6\right]}^{16}$ to [2, 6] is ${g}_{\left[2,6\right]}^{4}$). Hence each of ${g}_{\left[0,6\right]}^{13}$, ${g}_{\left[0,6\right]}^{14}$ and ${g}_{\left[0,6\right]}^{16}$ can be given as in the case n = 2. Note that ${{\Omega}}_{1}{\vert }_{\left[0,6\right]}={r}_{\left[0,6\right]}^{+}{{\Omega}}_{0}{\vert }_{\left[0,6\right]}$ with ${r}_{\left[0,6\right]}^{+}\in \mathcal{A}\left({\text{I}}_{0,3}\right)$. Therefore each of ${g}_{\left[0,6\right]}^{13}\enspace {g}_{\left[0,6\right]}^{14}$ and ${g}_{\left[0,6\right]}^{16}$ can be generated by local supercharges in [0, 6] applied to Ω₀.

We have

$$\begin{equation*}{g}_{\left[0,6\right]}^{3}={r}_{\left[0,2\right]}^{-}{r}_{\left[4,6\right]}^{-}{{\Omega}}_{1}={r}_{\left[0,2\right]}^{-}{r}_{\left[4,6\right]}^{-}{r}_{\left[0,6\right]}^{+}{{\Omega}}_{0}={c}_{3}^{{\ast}}{{\Omega}}_{0},\end{equation*}$$

and

$$\begin{equation*}{g}_{\left[0,6\right]}^{15}={r}_{\left[0,2\right]}^{+}{r}_{\left[4,6\right]}^{+}{{\Omega}}_{0}={c}_{0}^{{\ast}}{c}_{1}^{{\ast}}{c}_{2}^{{\ast}}{c}_{4}^{{\ast}}{c}_{5}^{{\ast}}{c}_{6}^{{\ast}}{{\Omega}}_{0}.\end{equation*}$$

We have

$$\begin{align*}\hfill {g}_{\left[0,6\right]}^{5}& ={r}_{\left[0,4\right]}^{-}{{\Omega}}_{1}={r}_{\left[0,4\right]}^{-}{r}_{\left[0,6\right]}^{+}{{\Omega}}_{0}={c}_{5}^{{\ast}}{c}_{6}^{{\ast}}{{\Omega}}_{0},\hfill \\ \hfill {g}_{\left[0,6\right]}^{6}& ={r}_{\left[4,6\right]}^{+}{{\Omega}}_{0}={c}_{4}^{{\ast}}{c}_{5}^{{\ast}}{c}_{6}^{{\ast}}{{\Omega}}_{0},\hfill \\ \hfill {g}_{\left[0,6\right]}^{7}& ={r}_{\left[0,2\right]}^{-}{{\Omega}}_{1}={r}_{\left[0,2\right]}^{-}{r}_{\left[0,6\right]}^{+}{{\Omega}}_{0}={c}_{3}^{{\ast}}{c}_{4}^{{\ast}}{c}_{5}^{{\ast}}{c}_{6}^{{\ast}}{{\Omega}}_{0},\hfill \\ \hfill {g}_{\left[0,6\right]}^{8}& ={r}_{\left[2,6\right]}^{+}{{\Omega}}_{0}={c}_{2}^{{\ast}}{c}_{3}^{{\ast}}{c}_{4}^{{\ast}}{c}_{5}^{{\ast}}{c}_{6}^{{\ast}}{{\Omega}}_{0},\hfill \end{align*}$$

and similarly

$$\begin{align*}\hfill {g}_{\left[0,6\right]}^{9}& ={r}_{\left[0,4\right]}^{+}{{\Omega}}_{0}={c}_{0}^{{\ast}}{c}_{1}^{{\ast}}{c}_{2}^{{\ast}}{c}_{3}^{{\ast}}{c}_{4}^{{\ast}}{{\Omega}}_{0},\hfill \\ \hfill {g}_{\left[0,6\right]}^{10}& ={r}_{\left[4,6\right]}^{-}{{\Omega}}_{1}={r}_{\left[4,6\right]}^{-}{r}_{\left[0,6\right]}^{+}{{\Omega}}_{0}={c}_{0}^{{\ast}}{c}_{1}^{{\ast}}{c}_{2}^{{\ast}}{c}_{3}^{{\ast}}{{\Omega}}_{0},\hfill \\ \hfill {g}_{\left[0,6\right]}^{11}& ={r}_{\left[0,2\right]}^{+}{{\Omega}}_{0}={c}_{0}^{{\ast}}{c}_{1}^{{\ast}}{c}_{2}^{{\ast}}{{\Omega}}_{0},\hfill \\ \hfill {g}_{\left[0,6\right]}^{12}& ={r}_{\left[2,6\right]}^{-}{{\Omega}}_{1}={r}_{\left[2,6\right]}^{-}{r}_{\left[0,6\right]}^{+}{{\Omega}}_{0}={c}_{0}^{{\ast}}{c}_{1}^{{\ast}}{{\Omega}}_{0}.\hfill \end{align*}$$

We have now derived all the sequences of ${\hat{{\Upsilon}}}_{0,3}$, i.e. all the classical free-boundary supersymmetric ground states on $\mathcal{A}\left({\text{I}}_{0,3}\right)$.

We will start the argument of induction. We have verified the statement for n = 1, 2, 3. Now let us assume that the statement holds for any integer from $1\in \mathbb{N}$ up to $n\in \mathbb{N}$. We are going to show that the statement holds for $n+1\in \mathbb{N}$. Concretely, we will construct ${\hat{{\Upsilon}}}_{0,n+1}$ from ${\hat{{\Upsilon}}}_{p,q}$ (0 ⩽ p < q ⩽ n + 1) where 0 < p or q < n + 1.

We divide ${\hat{{\Upsilon}}}_{0,n+1}$ into four cases (case I–IV) as below. We shall indicate how the induction argument can be applied to each of them.

Case I: We deal with all $g\in {\hat{{\Upsilon}}}_{0,n+1}$ whose left and right ends are

$$\begin{equation}g\left(0\right)=g\left(1\right)=0,\quad g\left(2n+1\right)=g\left(2\left(n+1\right)\right)=0.\end{equation} \tag{ 3.5 }$$

${\hat{{\Upsilon}}}_{0,n+1}$	0	1	2	3	⋯	⋯	⋯	2n − 1	2n	2n + 1	2(n + 1)
I-1	0	0	0	0	∗∗∗	∗∗∗	∗∗∗	0	0	0	0
I-2	0	0	0	0	∗∗∗	∗∗∗	∗∗∗	1	1	0	0
I-3	0	0	0	0	∗∗∗	∗∗∗	∗∗∗	1	0	0	0
I-4	0	0	1	1	∗∗∗	∗∗∗	∗∗∗	0	0	0	0
I-5	0	0	1	1	∗∗∗	∗∗∗	∗∗∗	1	1	0	0
I-6	0	0	1	1	∗∗∗	∗∗∗	∗∗∗	1	0	0	0
I-7	0	0	0	1	∗∗∗	∗∗∗	∗∗∗	0	0	0	0
I-8	0	0	0	1	∗∗∗	∗∗∗	∗∗∗	1	1	0	0
I-9	0	0	0	1	∗∗∗	∗∗∗	∗∗∗	1	0	0	0

Note that '∗∗∗'s in the middle mean some appropriate sequences of 0, 1 so that the sequence belongs to ${\hat{{\Upsilon}}}_{0,n+1}$, not being arbitrary.

All the above elements in ${\hat{{\Upsilon}}}_{0,n+1}$ except I-9 belong to ${\hat{{\Upsilon}}}_{1,n+1}$ or to ${\hat{{\Upsilon}}}_{0,n}$ when being restricted to [2, 2(n + 1)] or to [0, 2n], respectively. By applying ${r}_{\left[0,2\right]}^{+}$ to the vector of I-9, we get

${\hat{{\Upsilon}}}_{0,n+1}$	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 ${\hat{{\Upsilon}}}_{1,n+1}$ 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 ${{\Omega}}_{1}={r}_{\left[0,2\left(n+1\right)\right]}^{+}{{\Omega}}_{0}$ on the segment [0, 2(n + 1)] as noted in lemma 3.2. Hence we can construct I-9 by applying some local supercharges in [0, 2(n + 1)] to Ω₀. In this way we have made all the configurations of case I by the specified rule.

By applying ${r}_{\left[2n,2\left(n+1\right)\right]}^{+}$ to the vector of I-9, we get

${\hat{{\Upsilon}}}_{0,n+1}$	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 ${\hat{{\Upsilon}}}_{0,n}$ when being restricted to [0, 2n]. Therefore 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 applying some local supercharges in [0, 2(n + 1)] to Ω₀.

Case II:

We deal with all $g\in {\hat{{\Upsilon}}}_{0,n+1}$ whose left and right ends are

$$\begin{equation}g\left(0\right)=g\left(1\right)=1,\quad g\left(2n+1\right)=g\left(2\left(n+1\right)\right)=1.\end{equation} \tag{ 3.6 }$$

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

Case III:

We deal with all $g\in {\hat{{\Upsilon}}}_{0,n+1}$ whose left and right ends are

$$\begin{equation}g\left(0\right)=g\left(1\right)=0,\quad g\left(2n+1\right)=g\left(2\left(n+1\right)\right)=1.\end{equation} \tag{ 3.7 }$$

${\hat{{\Upsilon}}}_{0,n+1}$	0	1	2	3	⋯	⋯	⋯	2n − 1	2n	2n + 1	2(n + 1)
III-1	0	0	0	0	∗∗∗	∗∗∗	∗∗∗	0	0	1	1
III-2	0	0	0	0	∗∗∗	∗∗∗	∗∗∗	1	1	1	1
III-3	0	0	0	0	∗∗∗	∗∗∗	∗∗∗	0	1	1	1
III-4	0	0	1	1	∗∗∗	∗∗∗	∗∗∗	0	0	1	1
III-5	0	0	1	1	∗∗∗	∗∗∗	∗∗∗	1	1	1	1
III-6	0	0	1	1	∗∗∗	∗∗∗	∗∗∗	0	1	1	1
III-7	0	0	0	1	∗∗∗	∗∗∗	∗∗∗	0	0	1	1
III-8	0	0	0	1	∗∗∗	∗∗∗	∗∗∗	1	1	1	1
III-9	0	0	0	1	∗∗∗	∗∗∗	∗∗∗	0	1	1	1

All the above elements in ${\hat{{\Upsilon}}}_{0,n+1}$ except III-9 belong to ${\hat{{\Upsilon}}}_{1,n+1}$ or to ${\hat{{\Upsilon}}}_{0,n}$ when being restricted to [2, 2(n + 1)] or to [0, 2n], respectively. By applying ${r}_{\left[0,2\right]}^{+}$ to the vector of III-9, we get

${\hat{{\Upsilon}}}_{0,n+1}$	0	1	2	3	⋯	⋯	⋯	2n − 1	2n	2n + 1	2(n + 1)
New III-9	1	1	1	1	∗∗∗	∗∗∗	∗∗∗	0	1	1	1

'New III-9' above belongs to ${\hat{{\Upsilon}}}_{1,n+1}$ 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 can construct I-9 by applying some local supercharges in [0, 2(n + 1)] to Ω₀. We have completed the assertion for case III.

Case IV:

We deal with all $g\in {\hat{{\Upsilon}}}_{0,n+1}$ whose left and right ends are

$$\begin{equation}g\left(0\right)=g\left(1\right)=1,\quad g\left(2n+1\right)=g\left(2\left(n+1\right)\right)=0.\end{equation} \tag{ 3.8 }$$

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 ${\hat{{\Upsilon}}}_{0,n+1}$ from ${\hat{{\Upsilon}}}_{0,n}$ and ${\hat{{\Upsilon}}}_{1,n+1}$. Hence by the induction, we have shown the statement.□

The number of classical supersymmetric ground states can be computed explicitly.

Proposition 3.4. The number of classical free-boundary supersymmetric ground states on I_0,n ($n\in \mathbb{N}$) is 2 ⋅ 3ⁿ⁻¹.

Proof. This computation is given by the transfer-matrix method. We first divide I_0,n into n-sequential pairs as

$$\begin{equation*}{\text{I}}_{0,n}=\left\{0,1,2\right\}\cup \left\{3,4\right\}\cdots \cup \left\{2k-1,2k\right\}\cup \cdots \cup \left\{2n-1,2n\right\},\end{equation*}$$

where the first group exceptionally consists of three sites {0, 1, 2}. On each {2k − 1, 2k} all classical configurations are possible. However, to connect {2k − 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

$$\begin{equation}T{:=}\left[\begin{matrix}\hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \end{matrix}\right].\end{equation} \tag{ 3.9 }$$

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

$$\begin{align*}\hfill 0,0,0,\dots ,0,0,\\ \hfill 0,0,0,\dots ,1,1,\\ \hfill 0,0,1,\dots ,0,0,\\ \hfill 0,0,1,\dots ,1,1,\\ \hfill 1,1,0,\dots ,0,0,\\ \hfill 1,1,0,\dots ,1,1,\\ \hfill 1,1,1,\dots ,0,0,\\ \hfill 1,1,1,\dots ,1,1,\end{align*}$$

which correspond to (1, 1), (1, 4), (2, 1), (2, 4), (3, 1), (3, 4), (4, 1) and (4, 4) elements of Tⁿ⁻¹, respectively. Those amount to 2 ⋅ 3ⁿ⁻¹.□

We determined all classical supersymmetric ground states of the Nicolai supersymmetric fermion lattice model, and explained the high degeneracy of ground states by breakdown of its infinitely many local fermionic symmetries. The above finding may recall other supersymmetric models with many ground 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 the Nicolai model is shown, but the precise form of these states is not specified. In this paper, we consider only classical ground states. To determine all the ground states (furthermore all eigenstates) we need more detailed spectral property of the Hamiltonian and its symmetries (including bosonic ones).

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

The final comment is concerned with some technical point. We have chosen special subregions (I_k,l ) and the boundary conditions (the free-boundary supersymmetric condition) which seem artificial. However, as long as we consider classical states only, there is no loss of generality with this choice as follows.

Proposition 4.1. Given any finite subset Λ of $\mathbb{Z}$. Any classical supersymmetric state on Λ can be given by restriction of some classical free-boundary supersymmetric ground states on some larger I_k,l that includes Λ.

Proof. First recall the one-to-one correspondence between the set of classical supersymmetric ground states on Λ and ϒ_Λ by proposition 1.5. Recall the one-to-one correspondence between the set of classical free-boundary supersymmetric ground states on I_k,l and ${\hat{{\Upsilon}}}_{k,l}$ by proposition 1.8. Thus any classical supersymmetric ground state on Λ can be extended to at least one classical free-boundary supersymmetric ground state on I_k,l that includes Λ.□

HK was supported in part by JSPS Grant-in-Aid for Scientic Research on Innovative Areas No. JP18H04478 and JP20H04630, and JSPS KAKENHI Grant No. 18K03445. HM would like to thank Prof Arai and Dr Huijse for helpful discussion. HM acknowledges Riyu-1 group of Kanazawa University for encouragement.

A.1. Forms of $\hat{{\Xi}}$

We will give concrete examples for local {−1, +1}-sequences of conservation of definition 2.1 and their associated local fermion operators of definition 2.5. First we see that ${\hat{{\Xi}}}_{0,1}$ on I_0,1 ≡ [0, 1, 2] consists of two ±-characters only.

${\hat{{\Xi}}}_{0,1}$	0	1	2
${r}_{\left[0,2\right]}^{-}$	−1	−1	−1
${r}_{\left[0,2\right]}^{+}$	+1	+1	+1

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

$$\begin{align}\hfill \mathcal{Q}\left({r}_{\left[0,2\right]}^{-}\right)& ={c}_{0}{c}_{1}{c}_{2}\in \mathcal{A}{\left({\text{I}}_{0,1}\right)}_{-},\hfill \\ \hfill \mathcal{Q}\left({r}_{\left[0,2\right]}^{+}\right)& ={c}_{0}^{{\ast}}{c}_{1}^{{\ast}}{c}_{2}^{{\ast}}\in \mathcal{A}{\left({\text{I}}_{0,1}\right)}_{-}.\hfill \end{align} \tag{ A.1 }$$

We consider a next smallest segment I_0,2 ≡ [0, 1, 2, 3, 4] by setting k = 0 and l = 2. The space ${\hat{{\Xi}}}_{0,2}$ on I_0,2 consists of the following five {−1, +1}-sequences:

${\hat{{\Xi}}}_{0,2}$	0	1	2	3	4
${r}_{\left[0,4\right]}^{-}$	−1	−1	−1	−1	−1
${u}_{\left[0,4\right]}^{\mathrm{i}}$	−1	−1	−1	+1	+1
${u}_{\left[0,4\right]}^{\text{ii}}$	−1	−1	+1	+1	+1
${v}_{\left[0,4\right]}^{\mathrm{i}}$	+1	+1	+1	−1	−1
${v}_{\left[0,4\right]}^{\text{ii}}$	+1	+1	−1	−1	−1
${r}_{\left[0,4\right]}^{+}$	+1	+1	+1	+1	+1

Note that

$$\begin{equation}{r}_{\left[0,4\right]}^{-}=-{r}_{\left[0,4\right]}^{+},\,\,{u}_{\left[0,4\right]}^{\mathrm{i}}=-{v}_{\left[0,4\right]}^{\mathrm{i}},\,\,{u}_{\left[0,4\right]}^{\text{ii}}=-{v}_{\left[0,4\right]}^{\text{ii}}.\end{equation} \tag{ A.2 }$$

By (2.9) of definition 2.5 we have

$$\begin{align}\hfill \mathcal{Q}\left({r}_{\left[0,4\right]}^{-}\right)& ={c}_{0}{c}_{1}{c}_{2}{c}_{3}{c}_{4}\in \mathcal{A}{\left({\text{I}}_{0,2}\right)}_{-},\hfill \\ \hfill \mathcal{Q}\left({u}_{\left[0,4\right]}^{\mathrm{i}}\right)& ={c}_{0}{c}_{1}{c}_{2}{c}_{3}^{{\ast}}{c}_{4}^{{\ast}}\in \mathcal{A}{\left({\text{I}}_{0,2}\right)}_{-},\hfill \\ \hfill \mathcal{Q}\left({u}_{\left[0,4\right]}^{\text{ii}}\right)& ={c}_{0}{c}_{1}{c}_{2}^{{\ast}}{c}_{3}^{{\ast}}{c}_{4}^{{\ast}}\in \mathcal{A}{\left({\text{I}}_{0,2}\right)}_{-},\hfill \\ \hfill \mathcal{Q}\left({v}_{\left[0,4\right]}^{\mathrm{i}}\right)& ={c}_{0}^{{\ast}}{c}_{1}^{{\ast}}{c}_{2}^{{\ast}}{c}_{3}{c}_{4}\in \mathcal{A}{\left({\text{I}}_{0,2}\right)}_{-},\hfill \\ \hfill \mathcal{Q}\left({v}_{\left[0,4\right]}^{\text{ii}}\right)& ={c}_{0}^{{\ast}}{c}_{1}^{{\ast}}{c}_{2}{c}_{3}{c}_{4}\in \mathcal{A}{\left({\text{I}}_{0,2}\right)}_{-},\hfill \\ \hfill \mathcal{Q}\left({r}_{\left[0,4\right]}^{+}\right)& ={c}_{0}^{{\ast}}{c}_{1}^{{\ast}}{c}_{2}^{{\ast}}{c}_{3}^{{\ast}}{c}_{4}^{{\ast}}\in \mathcal{A}{\left({\text{I}}_{0,2}\right)}_{-}.\hfill \end{align} \tag{ A.3 }$$

We then consider the segment I_0,3 ≡ [0, 1, 2, 3, 4, 5, 6] taking k = 0 and l = 3. By definition it consists of 5 + 4 + 4 + 5 = 18 {−1, +1}-sequences:

${\hat{{\Xi}}}_{0,3}$	0	1	2	3	4	5	6
${s}_{\left[0,6\right]}^{{\circ}}$	−1	−1	−1	−1	−1	−1	−1
${s}_{\left[0,6\right]}^{\mathrm{i}}$	−1	−1	−1	+1	+1	−1	−1
${s}_{\left[0,6\right]}^{\text{ii}}$	−1	−1	+1	+1	−1	−1	−1
${s}_{\left[0,6\right]}^{\text{iii}}$	−1	−1	−1	+1	−1	−1	−1
${s}_{\left[0,6\right]}^{\text{iv}}$	−1	−1	+1	+1	+1	−1	−1
${u}_{\left[0,6\right]}^{\mathrm{i}}$	−1	−1	−1	−1	−1	+1	+1
${u}_{\left[0,6\right]}^{\text{ii}}$	−1	−1	−1	−1	+1	+1	+1
${u}_{\left[0,6\right]}^{\text{iii}}$	−1	−1	−1	+1	+1	+1	+1
${u}_{\left[0,6\right]}^{\text{iv}}$	−1	−1	+1	+1	+1	+1	+1
${v}_{\left[0,6\right]}^{\mathrm{i}}$	+1	+1	+1	+1	+1	−1	−1
${v}_{\left[0,6\right]}^{\text{ii}}$	+1	+1	+1	+1	−1	−1	−1
${v}_{\left[0,6\right]}^{\text{iii}}$	+1	+1	+1	−1	−1	−1	−1
${v}_{\left[0,6\right]}^{\text{iv}}$	+1	+1	−1	−1	−1	−1	−1
${t}_{\left[0,6\right]}^{{\bullet}}$	+1	+1	+1	+1	+1	+1	+1
${t}_{\left[0,6\right]}^{\mathrm{i}}$	+1	+1	+1	−1	−1	+1	+1
${t}_{\left[0,6\right]}^{\text{ii}}$	+1	+1	−1	−1	+1	+1	+1
${t}_{\left[0,6\right]}^{\text{iii}}$	+1	+1	+1	−1	+1	+1	+1
${t}_{\left[0,6\right]}^{\text{iv}}$	+1	+1	−1	−1	−1	+1	+1

Note that ${s}_{\left[0,6\right]}^{{\circ}}\equiv {r}_{\left[0,6\right]}^{-}$ and ${t}_{\left[0,6\right]}^{{\bullet}}\equiv {r}_{\left[0,6\right]}^{+}$ and that

$$\begin{align}\hfill {s}_{\left[0,6\right]}^{{\circ}}=-{t}_{\left[0,6\right]}^{{\bullet}},\enspace {s}_{\left[0,6\right]}^{k}=-{t}_{\left[0,6\right]}^{k},\,\,\forall k\in \left\{\mathrm{i},\mathrm{i}\mathrm{i},\mathrm{i}\mathrm{i}\mathrm{i},\mathrm{i}\mathrm{v}\right\}\\ \hfill {u}_{\left[0,6\right]}^{k}=-{v}_{\left[0,6\right]}^{k},\,\,\forall k\in \left\{\mathrm{i},\mathrm{i}\mathrm{i},\mathrm{i}\mathrm{i}\mathrm{i},\mathrm{i}\mathrm{v}\right\}.\end{align} \tag{ A.4 }$$

Using the rule we obtain the following list of 18 fermion operators associated to ${\hat{{\Xi}}}_{0,3}$:

$$\begin{align}\hfill \mathcal{Q}\left({s}_{\left[0,6\right]}^{{\circ}}\right)& ={c}_{0}{c}_{1}{c}_{2}{c}_{3}{c}_{4}{c}_{5}{c}_{6}\in \mathcal{A}{\left({\text{I}}_{0,3}\right)}_{-},\hfill \\ \hfill \mathcal{Q}\left({s}_{\left[0,6\right]}^{\mathrm{i}}\right)& ={c}_{0}{c}_{1}{c}_{2}{c}_{3}^{{\ast}}{c}_{4}^{{\ast}}{c}_{5}{c}_{6}\in \mathcal{A}{\left({\text{I}}_{0,3}\right)}_{-},\hfill \\ \hfill \mathcal{Q}\left({s}_{\left[0,6\right]}^{\text{ii}}\right)& ={c}_{0}{c}_{1}{c}_{2}^{{\ast}}{c}_{3}^{{\ast}}{c}_{4}{c}_{5}{c}_{6}\in \mathcal{A}{\left({\text{I}}_{0,3}\right)}_{-},\hfill \\ \hfill \mathcal{Q}\left({s}_{\left[0,6\right]}^{\text{iii}}\right)& ={c}_{0}{c}_{1}{c}_{2}{c}_{3}^{{\ast}}{c}_{4}{c}_{5}{c}_{6}\in \mathcal{A}{\left({\text{I}}_{0,3}\right)}_{-},\hfill \\ \hfill \mathcal{Q}\left({s}_{\left[0,6\right]}^{\text{iv}}\right)& ={c}_{0}{c}_{1}{c}_{2}^{{\ast}}{c}_{3}^{{\ast}}{c}_{4}^{{\ast}}{c}_{5}{c}_{6}\in \mathcal{A}{\left({\text{I}}_{0,3}\right)}_{-},\hfill \\ \hfill \mathcal{Q}\left({u}_{\left[0,6\right]}^{\mathrm{i}}\right)& ={c}_{0}{c}_{1}{c}_{2}{c}_{3}{c}_{4}{c}_{5}^{{\ast}}{c}_{6}^{{\ast}}\in \mathcal{A}{\left({\text{I}}_{0,3}\right)}_{-},\hfill \\ \hfill \mathcal{Q}\left({u}_{\left[0,6\right]}^{\text{ii}}\right)& ={c}_{0}{c}_{1}{c}_{2}{c}_{3}{c}_{4}^{{\ast}}{c}_{5}^{{\ast}}{c}_{6}^{{\ast}}\in \mathcal{A}{\left({\text{I}}_{0,3}\right)}_{-},\hfill \\ \hfill \mathcal{Q}\left({u}_{\left[0,6\right]}^{\text{iii}}\right)& ={c}_{0}{c}_{1}{c}_{2}{c}_{3}^{{\ast}}{c}_{4}^{{\ast}}{c}_{5}^{{\ast}}{c}_{6}^{{\ast}}\in \mathcal{A}{\left({\text{I}}_{0,3}\right)}_{-},\hfill \\ \hfill \mathcal{Q}\left({u}_{\left[0,6\right]}^{\text{iv}}\right)& ={c}_{0}{c}_{1}{c}_{2}^{{\ast}}{c}_{3}^{{\ast}}{c}_{4}^{{\ast}}{c}_{5}^{{\ast}}{c}_{6}^{{\ast}}\in \mathcal{A}{\left({\text{I}}_{0,3}\right)}_{-},\hfill \\ \hfill \mathcal{Q}\left({v}_{\left[0,6\right]}^{\mathrm{i}}\right)& ={c}_{0}^{{\ast}}{c}_{1}^{{\ast}}{c}_{2}^{{\ast}}{c}_{3}^{{\ast}}{c}_{4}^{{\ast}}{c}_{5}{c}_{6}\in \mathcal{A}{\left({\text{I}}_{0,3}\right)}_{-},\hfill \\ \hfill \mathcal{Q}\left({v}_{\left[0,6\right]}^{\text{ii}}\right)& ={c}_{0}^{{\ast}}{c}_{1}^{{\ast}}{c}_{2}^{{\ast}}{c}_{3}^{{\ast}}{c}_{4}{c}_{5}{c}_{6}\in \mathcal{A}{\left({\text{I}}_{0,3}\right)}_{-},\hfill \\ \hfill \mathcal{Q}\left({v}_{\left[0,6\right]}^{\text{iii}}\right)& ={c}_{0}^{{\ast}}{c}_{1}^{{\ast}}{c}_{2}^{{\ast}}{c}_{3}{c}_{4}{c}_{5}{c}_{6}\in \mathcal{A}{\left({\text{I}}_{0,3}\right)}_{-},\hfill \\ \hfill \mathcal{Q}\left({v}_{\left[0,6\right]}^{\text{iv}}\right)& ={c}_{0}^{{\ast}}{c}_{1}^{{\ast}}{c}_{2}{c}_{3}{c}_{4}{c}_{5}{c}_{6}\in \mathcal{A}{\left({\text{I}}_{0,3}\right)}_{-},\hfill \\ \hfill \mathcal{Q}\left({t}_{\left[0,6\right]}^{{\bullet}}\right)& ={c}_{0}^{{\ast}}{c}_{1}^{{\ast}}{c}_{2}^{{\ast}}{c}_{3}^{{\ast}}{c}_{4}^{{\ast}}{c}_{5}^{{\ast}}{c}_{6}^{{\ast}}\in \mathcal{A}{\left({\text{I}}_{0,3}\right)}_{-},\hfill \\ \hfill \mathcal{Q}\left({t}_{\left[0,6\right]}^{\mathrm{i}}\right)& ={c}_{0}^{{\ast}}{c}_{1}^{{\ast}}{c}_{2}^{{\ast}}{c}_{3}{c}_{4}{c}_{5}^{{\ast}}{c}_{6}^{{\ast}}\in \mathcal{A}{\left({\text{I}}_{0,3}\right)}_{-},\hfill \\ \hfill \mathcal{Q}\left({t}_{\left[0,6\right]}^{\text{ii}}\right)& ={c}_{0}^{{\ast}}{c}_{1}^{{\ast}}{c}_{2}{c}_{3}{c}_{4}^{{\ast}}{c}_{5}^{{\ast}}{c}_{6}^{{\ast}}\in \mathcal{A}{\left({\text{I}}_{0,3}\right)}_{-},\hfill \\ \hfill \mathcal{Q}\left({t}_{\left[0,6\right]}^{\text{iii}}\right)& ={c}_{0}^{{\ast}}{c}_{1}^{{\ast}}{c}_{2}^{{\ast}}{c}_{3}{c}_{4}^{{\ast}}{c}_{5}^{{\ast}}{c}_{6}^{{\ast}}\in \mathcal{A}{\left({\text{I}}_{0,3}\right)}_{-},\hfill \\ \hfill \mathcal{Q}\left({t}_{\left[0,6\right]}^{\text{iv}}\right)& ={c}_{0}^{{\ast}}{c}_{1}^{{\ast}}{c}_{2}{c}_{3}{c}_{4}{c}_{5}^{{\ast}}{c}_{6}^{{\ast}}\in \mathcal{A}{\left({\text{I}}_{0,3}\right)}_{-}.\hfill \end{align} \tag{ A.5 }$$