Boundary dissipative spin chains with partial solvability inherited from system Hamiltonians

The notion of the spectrum generating algebra (SGA), or also called the dynamical symmetry, has been introduced in various contexts [7, 25, 6, 87, 29, 40]. In this paper, we shall say that the model has the SGA if there exists a spectrum generating operator $Q^{{\dagger}}$ that satisfies an algebraic relation,

$\displaystyle[H,\,Q^{{\dagger}}]-\mathcal{E}Q^{{\dagger}}=0,$

(3)

for a real constant $\mathcal{E}$. If a model has the SGA and some of its energy eigenstates are known, one can construct towers of eigenstates by applying the operator $Q^{{\dagger}}$ to those known states repeatedly. The observed energy spectrum is then equally spaced with the interval $\mathcal{E}$ due to the algebraic relation (3).

One of the simplest examples that show the SGA is a free-fermion model. The Hamiltonian (denoted by $H_{\rm FF}$) is diagonalized in the momentum space,

$\displaystyle H_{\rm FF}=\sum_{k}\Lambda_{k}\widetilde{\eta}_{k}^{{\dagger}}\widetilde{\eta}_{k},$

(4)

with real eigenvalues $\Lambda_{k}$ and a fermion creation operator $\widetilde{\eta}_{k}^{{\dagger}}$, and thus satisfies the SGA,

$\displaystyle[H_{\rm FF},\,\widetilde{\eta}_{k}^{{\dagger}}]=\Lambda_{k}\widetilde{\eta}_{k}^{{\dagger}},$

(5)

due to the anti-commutation relations for $\widetilde{\eta}_{k}^{{\dagger}}$ and $\widetilde{\eta}_{k}$. Then all the energy eigenstates are created by applying the fermion creation operators $\widetilde{\eta}_{k}^{{\dagger}}$ to the obvious vacuum state $|0\rangle$, i.e., the zero-energy eigenstate,

$\displaystyle H_{\rm FF}\,\widetilde{\eta}_{k_{1}}^{{\dagger}}\cdots\widetilde{\eta}_{k_{n}}^{{\dagger}}|0\rangle=(\Lambda_{k_{1}}+\cdots+\Lambda_{k_{n}})\,\widetilde{\eta}_{k_{1}}^{{\dagger}}\cdots\widetilde{\eta}_{k_{n}}^{{\dagger}}|0\rangle.$

(6)

In this example, the fermion creation operator plays a role of the spectrum generating operator for each mode $k$.

Sometimes there appears the SGA only in a subspace $W$ of the entire Hilbert space $\mathcal{H}$,

$\displaystyle[H,\,Q^{{\dagger}}]-\mathcal{E}Q^{{\dagger}}\big{|}_{W}=0,\quad W\subset\mathcal{H}.$

(7)

We call this type of the SGA structure that emerges only in the subspace $W$ “the restricted spectrum generating algebra (rSGA)”. As in the case of the SGA that holds in the entire Hilbert space, one can construct a tower of energy eigenstates in the subspace $W$ by repeatedly applying the operator $Q^{{\dagger}}$ to an obvious energy eigenstate $|\Psi_{0}\rangle$ (if it may exist).

One of the simplest examples equipped with the rSGA is the perturbed spin-$1$ $XY$ model [76], whose local Hamiltonian is given by

$\displaystyle h_{j,j+1}^{XY}=\frac{J}{2}(S_{j}^{+}S_{j+1}^{-}+S_{j}^{-}S_{j+1}^{+})+\frac{m}{2}(S_{j}^{z}+S_{j+1}^{z}),$

(8)

where $S_{j}^{\pm}=S_{j}^{x}\pm iS_{j}^{y}$ and $S_{j}^{z}$ are spin-1 operators, and $J$ and $m$ are real coefficients. This model is considered to be non-integrable, and therefore, the energy eigenstates are in general not exactly solvable. However, one can easily find that the fully-polarized state $|22\dots 2\rangle$ (and $|00\dots 0\rangle$) is obviously an energy eigenstate with the eigenenergy $-mN$ ($mN$). Accordingly, one can derive some of the excited states exactly in the form of $(Q^{{\dagger}})^{n}|22\dots 2\rangle$ (that allows a quasiparticle picture), where $Q^{\dagger}$ creates quasiparticles of spin-2 magnons carrying momentum $k=\pi$

$\displaystyle Q^{{\dagger}}=\sum_{x=1}^{N}(-1)^{x}(S_{x}^{+})^{2}.$

(9)

The spin-$2$ magnon creation operator $Q^{{\dagger}}$ then satisfies the rSGA,

$\displaystyle[H_{XY},\,Q^{{\dagger}}]+2mQ^{{\dagger}}\big{|}_{W}=0,$

(10)

in the subspace $W$ spanned by the quasiparticle excitation states

$\displaystyle W={\rm span}\{(Q^{{\dagger}})^{n}|22\dots 2\rangle\}_{n\in\{0,1,\dots,N\}}.$

(11)

The rSGA produces the equally-spaced eigenvalue spectrum in the solvable subspace $W$, which can explain the persistent oscillation observed in the Loschmidt echo [76, 16]. This implies that the perturbed spin-$1$ $XY$ model never relaxes to any steady state, if the initial state has large enough overlap with the solvable subspace $W$.

A more involved example exhibiting the rSGA is a family of the spin-$1$ Affleck-Kennedy-Lieb-Tasaki (AKLT)-type model [1, 48]. The local Hamiltonian is given by

	$\displaystyle h_{j,j+1}^{\rm AKLT}=$	$\displaystyle\frac{1}{2}h^{00}_{00}(S_{j}^{x}S_{j+1}^{x}+S_{j}^{y}S_{j+1}^{y}+S_{j}^{z}S_{j+1}^{z})$		(12)
		$\displaystyle-\left(\frac{1}{2}h^{00}_{00}+\frac{a_{1}^{2}}{a_{0}a_{2}}h^{11}_{11}\right)(S_{j}^{x}S_{j+1}^{x}+S_{j}^{y}S_{j+1}^{y}+S_{j}^{z}S_{j+1}^{z})^{2}$
		$\displaystyle-\left(\frac{1}{2}h^{00}_{00}-\frac{a_{1}^{2}}{a_{0}a_{2}}\Big{(}\frac{a_{1}^{2}}{a_{0}a_{2}}-1\Big{)}h^{11}_{11}\right)(S_{j}^{x}S_{j+1}^{x}+S_{j}^{y}S_{j+1}^{y})^{2}$
		$\displaystyle-\left(h^{00}_{00}+\Big{(}\frac{a_{1}^{2}}{a_{0}a_{2}}-1\Big{)}h^{11}_{11}\right)(S_{j}^{z}S_{j+1}^{z})^{2}$
		$\displaystyle+\left(\frac{1}{2}h^{00}_{00}+\Big{(}\frac{a_{1}^{4}}{a_{0}^{2}a_{2}^{2}}-1\Big{)}h^{11}_{11}\right)((S_{j}^{z})^{2}+(S_{j+1}^{z})^{2})+\left(1-2\frac{a_{1}^{4}}{a_{0}^{2}a_{2}^{2}}\right)h^{11}_{11},$

in which $h^{00}_{00}$ and $h^{11}_{11}$ are free real parameters, while $a_{0}$, $a_{1}$, and $a_{2}$ are free complex parameters under the condition that $a_{1}^{2}/(a_{0}a_{2})\in\mathbb{R}$. Note that the original AKLT model is realized by choosing $h^{11}_{11}/h^{00}_{00}=2/3$ and $a_{0}=-\sqrt{2}a_{1}=-a_{2}=\sqrt{2/3}$. Although this Hamiltonian is non-integrable, it has been known for a long time that the ground state and some of the excited states are exactly solvable [1, 4], to which a new list of solvable excited states has been added recently in the context of the QMBS states [59].

The zero-energy state of the AKLT-type model is written in the form of the matrix product state,

$\displaystyle|\Psi_{0}\rangle=\sum_{m_{1},\dots m_{N}\in\{0,1,2\}}{\rm tr}_{a}(A_{m_{1}}\cdots A_{m_{N}})|m_{1}\dots m_{N}\rangle,$

(13)

with the frustration-free condition

$\displaystyle h_{j,j+1}^{\rm AKLT}\vec{A}_{j}\vec{A}_{j+1}=0,\quad\vec{A}_{j}=\begin{pmatrix}A_{0}\\ A_{1}\\ A_{2}\end{pmatrix}_{j},$

(14)

which holds for $j=1,2,\dots,N$. The matrix-valued elements $A_{0}$, $A_{1}$, and $A_{2}$ are given by the Pauli matrices

		$\displaystyle A_{0}=a_{0}\sigma^{+},\quad A_{1}=a_{1}\sigma^{Z},\quad A_{2}=a_{2}\sigma^{-},$		(15)
		$\displaystyle\sigma^{\pm}=\sigma^{x}\pm i\sigma^{y},$

in which the coefficients $a_{0}$, $a_{1}$, and $a_{2}$ are the same as in the Hamiltonian (12). Therefore, the bond dimension of this matrix product state is two.

Besides the ground state, it has been found that several excited states are solvable, most of which admit the quasiparticle description [16]. Here we focus on the excited states expressed by the spin-$2$ magnons carrying momentum $k=\pi$. They are created by the operator given in (9), which also satisfies the rSGA for the AKLT-type Hamiltonian,

$\displaystyle[H_{\rm AKLT},\,Q^{{\dagger}}]-2h^{00}_{00}Q^{{\dagger}}\big{|}_{W}=0,$

(16)

in the subspace $W\subset\mathcal{H}$ spanned by the quasiparticle excited states

$\displaystyle W={\rm span}\{(Q^{{\dagger}})^{n}|\Psi_{0}\rangle\}_{n\in\{0,1,\dots,\lfloor N/2\rfloor\}}.$

(17)

Thus, the quasiparticle creation operator plays a role of the spectrum generating operator that creates a tower of solvable energy eigenstates on top of the matrix product zero-energy state (13).

Recent studies on partial solvability have provided more formal understanding about the emergence of the rSGA in terms of quasisymmetry [64, 63, 66, 67, 72] for the former example and its deformation for the latter example [73].

Hilbert space fragmentation (HSF) is characterized by exponentially many block-diagonal structures of the Hamiltonian $H$, which are caused by non-obvious symmetries of $H$. Due to the block-diagonal structure of the Hilbert space, each subsapce is never accessed from the other subspaces by time evolution. Among several mechanisms for HSF [65, 74, 37], we focus on the fragmentation observed for the Hilbert space of spin chains due to the presence of “frozen” spin configurations under the action of a Hamiltonian. Those “frozen” spin configurations are called “irreducible strings” (IS) [23, 5, 53], which are associated with a non-obvious symmetry of the Hamiltonian.

In order to explain the HSF induced by IS, let us consider spin chains with arbitrary spin-$s$ (instead of spin-$1$). Suppose that we have a spin-$s$ chain with nearest-neighbor interactions, whose local Hamiltonian is given by

	$\displaystyle h^{\rm HSF}$	$\displaystyle=\sum_{s,s^{\prime}\in A}h^{s,s^{\prime}}_{s,s^{\prime}}|ss^{\prime}\rangle\langle ss^{\prime}|+\sum_{t,t^{\prime}\in B}h^{t,t^{\prime}}_{t,t^{\prime}}|tt^{\prime}\rangle\langle tt^{\prime}|$
		$\displaystyle\quad+\sum_{s\in A,\,t\in B}\Big{(}h^{s,t}_{t,s}|ts\rangle\langle st|+h^{t,s}_{s,t}|st\rangle\langle ts|+h^{s,t}_{s,t}|st\rangle\langle st|+h^{t,s}_{t,s}|ts\rangle\langle ts|\Big{)}.$		(18)

Here $A$ and $B$ represent subsets of the labels for local states $A,B\subset\{0,1,\dots 2s\}$, which satisfy $A\cup B=\{0,1,\dots 2s\}$ and $A\cap B=\emptyset$. We also call the labels for local states “species”. Since the Hamiltonian given in the form of Eq. (2.2) never exchanges the species in each subset $A$ or $B$, the configuration (i.e., IS) in each subset $A$ and $B$ is “frozen” under the action of the Hamiltonian. The existence of such a frozen partial configuration causes fragmentation of the Hilbert space, i.e., the the Hilbert space is fragmented according to the partial configurations in the subsets $A$ and $B$.

In each of the fragmented subspaces, one can observe that a spin-$1/2$ model (with two local states) is embedded in the following way. The entire Hilbert space of the spin-$s$ chain is $\mathbb{C}^{(2s+1)N}$, which consists of the tensor product of $N$ local linear spaces $\mathbb{C}^{2s+1}$ spanned by the $(2s+1)$ basis vectors $|0\rangle,|1\rangle,\dots,|2s+1\rangle$ corresponding to the spin degrees of freedom. In this basis, a state in $\mathbb{C}^{(2s+1)N}$ is labeled by the spin configuration. Alternatively, one can use another basis with local states labeled by $A$ and $B$, and configurations realized within $A$ and $B$ (Fig. 2).

Suppose that the subset $A$ consists of $N_{A}$ species and $B$ consists of $N_{B}$ ($=(2s+1)-N_{A}$) species. The total degree of freedom is then calculated as

$\displaystyle\sum_{n=0}^{N}\begin{pmatrix}N\\ n\end{pmatrix}\cdot N_{A}^{n}N_{B}^{N-n}=(N_{A}+N_{B})^{N},$

(19)

which matches the dimension of the Hilbert space in the first description (${\rm dim}\,\mathbb{C}^{(2s+1)N}$), indicating that the two different bases describe the same Hilbert space. For the Hamiltonian with HSF (2.2), the configurations within $A$ and $B$ (i.e., IS) are frozen, while the labels $A$ and $B$ are unconstrained. The ($A,B$) degrees of freedom form the $N$-fold tensor product of the two-dimensional local linear space $\mathbb{C}^{2}$. The projection onto such a $2^{N}$-dimensional subspace is defined by restricting to the configurations in $A$ and $B$ specified by IS (denoted by a projector $P_{\rm IS}$) and then by identifying all the local states in each of the subspaces $A$ and $B$ (Fig. 2). In this way, the Hamiltonian (2.2) is reduced to a spin-$1/2$ chain with the nearest-neighbor interactions.

The first example in which partial solvability is robust against boundary dissipators is given by an rSGA-induced partially solvable system. Let us consider a system in which a zero-energy state $|\Psi_{0}\rangle$ is analytically known. The rSGA solvability is characterized by the existence of a spectrum generating operator $Q^{{\dagger}}$ that satisfies the rSGA (7) with the Hamiltonian in a subspace $W$ spanned by states constructed by applying $Q^{{\dagger}}$ to the zero-energy state,

$\displaystyle W=\{|\Psi_{n}\rangle\}_{n},\quad|\Psi_{n}\rangle=(Q^{{\dagger}})^{n}|\Psi_{0}\rangle.$

(36)

Thus, the states $|\Psi_{n}\rangle$ are exactly solvable energy eigenstates of the Hamiltonian with the eigenenergies $\mathcal{E}n$. It often occurs that the solvable excited states admit the single-mode quasiparticle description,

$\displaystyle Q^{{\dagger}}=\sum_{x=1}^{N}e^{ikx}q^{{\dagger}}_{x},$

in which $q_{x}^{{\dagger}}$ is a local operator non-trivially acting on the $x$th site.

Based on these, it is natural to ask whether the solvable states can survive even when quasiparticles are injected from both of the edges of the system. Such a situation is realized by taking the boundary quantum jump operators as

$\displaystyle A_{\rm L}=q_{1}^{{\dagger}},\quad A_{\rm R}=q_{N}^{{\dagger}}.$

(37)

The density matrix $\rho_{nn}$ for the solvable energy eigenstate $|\Psi_{n}\rangle$, as it commutes with the Hamiltonian by definition (i.e., $[\rho_{nn},H]=0$), becomes the steady state of the GKSL equation (35) if

		$\displaystyle\mathcal{D}_{\alpha}(\rho_{nn})=0,\quad\forall\alpha,$		(38)
		$\displaystyle\rho_{nn}=|\Psi_{n}\rangle\langle\Psi_{n}|.$

The pure state $\rho_{nn}$ that satisfies this condition together with the commutativity with the Hamiltonian is known as “the dark state”, which has been introduced in the context of atomic physics and optics [24, 38].

The AKLT-type model (12) is one of the examples whose solvable energy eigenstates satisfy the dark-state condition (38) in the presence of the boundary quasiparticle dissipators. The zero-energy eigenstate is given by the matrix product state, as has been explained in Sec. 2.1, but with the boundary deformation,

$\displaystyle|\Psi^{(v_{\rm L},v_{\rm R})}_{0}\rangle=\sum_{m_{1},\dots,m_{N}\in\{0,1,2\}}{{}_{a}}\langle v_{\rm L}|A_{m_{1}}\cdots A_{m_{N}}|v_{\rm R}\rangle_{a}\cdot|m_{1}\dots m_{N}\rangle,$

(39)

since the open boundary condition is imposed on the system. The boundary vectors $|v_{\rm L,R}\rangle\in V_{a}={\rm span}\{|\uparrow\rangle,|\downarrow\rangle\}$ in the auxiliary space must be properly chosen in order for (39) to be the zero-energy eigenstate. Especially when the model is frustration-free, as we consider in this paper, there are four degenerate zero-energy eigenstates, as no constraint is imposed on the boundary vectors. A tower of the solvable energy eigenstates are then independently constructed on top of each of the four degenerate zero-energy states by applying the spin-$2$ magnon creation operator $Q^{{\dagger}}$ (9). That is, the solvable subspace $W$ under the open boundary condition is composed of four separate subspaces specified by the boundary vectors,

		$\displaystyle W=W^{(\uparrow,\uparrow)}\oplus W^{(\uparrow,\downarrow)}\oplus W^{(\downarrow,\uparrow)}\oplus W^{(\downarrow,\downarrow)},$		(40)
		$\displaystyle W^{(v_{\rm L},v_{\rm R})}={\rm span}\{|\Psi_{n}^{(v_{\rm L},v_{\rm R})}\rangle\}_{n},\quad|\Psi_{n}^{(v_{\rm L},v_{\rm R})}\rangle=(Q^{{\dagger}})^{n}|\Psi_{0}^{(v_{\rm L},v_{\rm R})}\rangle.$

We set the boundary quasiparticle dissipators in such a way that the quasiparticles are coming into the system from both of the ends,

$\displaystyle A_{\rm L}=(S_{1}^{+})^{2},\quad A_{\rm R}=(S_{N}^{+})^{2}.$

(41)

With these dissipators, one of the solvable subspaces $W^{(\uparrow,\downarrow)}$ satisfies the dark state conditions (38),

		$\displaystyle A_{\alpha}|\psi^{(\uparrow,\downarrow)}\rangle=0,\quad\alpha\in\{{\rm L},{\rm R}\},$		(42)
		$\displaystyle|\psi^{(\uparrow,\downarrow)}\rangle\in W^{(\uparrow,\downarrow)}.$

(See Appendix A for the proof.) That is, by denoting the Liouvillian with the AKLT-type Hamiltonian and the dissipators (41) by $\mathcal{L}_{\rm AKLT}$, any diagonal density matrix in the subspace $W^{(\uparrow,\downarrow)}$ becomes a steady state of the GKSL equation,

	$\displaystyle\mathcal{L}_{\rm AKLT}(\rho_{\rm diag}^{(\uparrow,\downarrow)})=0,$		(43)
	$\displaystyle\rho_{\rm diag}^{(\uparrow,\downarrow)}=\sum_{n}p_{n}|\Psi_{n}^{(\uparrow,\downarrow)}\rangle\langle\Psi_{n}^{(\uparrow,\downarrow)}|,\quad\sum_{n}p_{n}=1,\,p_{n}\geq 0,\,\forall n.$

Note that, in the case of the perturbed spin-$1$ $XY$ model, which is another model with the hidden rSGA, there never exist such a solvable energy eigenstate that satisfies the dark-state condition (42).

The fact that any state in the subspace $W^{(\uparrow,\downarrow)}$ is the dark state (42) indicates that any density matrix in $W^{(\uparrow,\downarrow)}\otimes(W^{(\uparrow,\downarrow)})^{{\dagger}}$, even if it contains off-diagonal elements, becomes an eigenmode of the GKSL equation. Suppose that we have an off-diagonal element $\rho_{m,n}^{(\uparrow,\downarrow)}=|\Psi_{m}^{(\uparrow,\downarrow)}\rangle\langle\Psi_{n}^{(\uparrow,\downarrow)}|$ in a given density matrix. This is a dark state from the previous statement, and moreover, its commutator with the Hamiltonian is proportional to itself,

$\displaystyle[H,\,\rho_{m,n}^{(\uparrow,\downarrow)}]=2(m-n)h^{00}_{00}\rho_{m,n}^{(\uparrow,\downarrow)},$

(44)

which indicates that $\rho_{m,n}^{(\uparrow,\downarrow)}$ is an eigenmode of the GKSL equation. The relation (44) together with the dark-state condition (42) leads to the restricted spectrum generating algebra for the Liouvillian in the subspace $W^{(\uparrow,\downarrow)}\otimes(W^{(\uparrow,\downarrow)})^{{\dagger}}\subset\mathcal{H}\otimes\mathcal{H}^{{\dagger}}$,

$\displaystyle\mathcal{L}_{\rm AKLT}(\rho^{(\uparrow,\downarrow)}_{m,n})=-2i(m-n)h^{00}_{00}\rho^{(\uparrow,\downarrow)}_{m,n},$

(45)

giving an equally-spaced spectrum along the imaginary axis with the interval $2h^{00}_{00}$ embedded in the full spectrum of $\mathcal{L}_{\rm AKLT}$.

The hidden rSGA structure (45) of the Liouvillian evokes us the persistent oscillations observed for rSGA-induced partially solvable isolated quantum systems [90]. Indeed, if we choose an initial state in the subspace $W^{(\uparrow,\downarrow)}$,

$\displaystyle|\psi(0)\rangle=\sum_{n}a_{n}|\Psi_{n}^{(\uparrow,\downarrow)}\rangle\in W^{(\uparrow,\downarrow)},$

(46)

it is easy to show that persistent oscillations are observed for an observable $O$ in a long-time scale,

$\displaystyle\langle O(t)\rangle\sim\sum_{n\leq m}2\cos(2(m-n)h^{00}_{00}t)a_{m}a_{n}\,{\rm Re}\,O_{nm}\quad(t\to\infty).$

(47)

This indicates that the system prepared in the solvable subspace never relaxes to any steady state. Even if the initial state is generic, the long-lived oscillations survive as long as the initial state has a large enough overlap with the subspace $W^{(\uparrow,\downarrow)}$.

To see how the presence of the rSGA-induced solvable eigenmodes affects observables in partially solvable open spin chains, here we perform numerical simulations for the GKSL equation (35). We take the generalized AKLT model $H_{\rm AKLT}=\sum_{j=1}^{N-1}h_{j,j+1}^{\rm AKLT}$ (12) as the bulk Hamiltonian with a finite number of lattice sites $N$. We specifically focus on the case of the original AKLT model, i.e., by choosing $h_{11}^{11}/h_{00}^{00}=2/3,a_{0}=-\sqrt{2}a_{1}=-a_{2}=\sqrt{2/3}$. The dissipators are given by spin-2 creation operators (41) acting on each end of the chain. The explicit form of the GKSL equation that we solve in this section is:

	$\displaystyle\frac{d}{dt}\rho$	$\displaystyle=-i[H_{\rm AKLT},\rho]+\sum_{\alpha={\rm L,R}}\gamma_{\alpha}\left(A_{\alpha}\rho A_{\alpha}^{\dagger}-\frac{1}{2}\{A_{\alpha}^{\dagger}A_{\alpha},\rho\}\right),$		(48)
	$\displaystyle H_{\rm AKLT}$	$\displaystyle=\sum_{j=1}^{N-1}\left[\bm{S}_{j}\cdot\bm{S}_{j+1}+\frac{1}{3}(\bm{S}_{j}\cdot\bm{S}_{j+1})^{2}\right],$		(49)
	$\displaystyle A_{\rm L}$	$\displaystyle=(S_{1}^{+})^{2},\quad A_{\rm R}=(S_{N}^{+})^{2},$		(50)

which is numerically simulated by the quantum trajectory method [19, 15, 18] together with the exact diagonalization.

The initial state is chosen to be a product state of the spin-1 chain, whose spin configuration is nearly a Néel state $|N,S^{z}(0)\rangle$, depending on the values of $N$ and the initial total $S^{z}(0)$:

	$\displaystyle|0202\cdots 0202\rangle:$	$N$ is even,  $S^{z}(0)=0$		(51)
	$\displaystyle|0202\cdots 0201\rangle:$	$N$ is even,  $S^{z}(0)=1$		(52)
	$\displaystyle|0202\cdots 02021\rangle:$	$N$ is odd,  $S^{z}(0)=0$		(53)
	$\displaystyle|0202\cdots 02020\rangle:$	$N$ is odd,  $S^{z}(0)=1$		(54)

We identify the state labels $0,1,2$ with the spin configuration $\uparrow,0,\downarrow$, respectively. Among the initial states $|N,S^{z}(0)\rangle$ considered here, those with $S^{z}(0)=1$ have an overlap with the states in the subspace $W^{(\uparrow,\downarrow)}$, since they have $\uparrow$ spins at both ends of the chain after removing $S_{j}^{z}=0$ spins. Hence, we expect that the rSGA-induced solvable eigenmodes appear in those cases with $S^{z}(0)=1$.