Superconfined Antiferromagnons on the Two-Dimensional Penrose Lattice

Abstract

We find novel confined states in the spin-S nearest-neighbor antiferromagnetic Heisenberg model on the two-dimensional Penrose lattice. Linear spin waves have massively degenerate eigenstates strictly confined to tricoordinated sites. They contrast with the well-known itinerant analogs in the tight-binding model, where electrons are confined but extended to both tricoordinated and pentacoordinated sites. It is the site potentials in the spin-wave Hamiltonian, originating from Coulomb interactions between electrons, that confine spin waves to minimally coordinated sites only. Confined states in the tight-binding Hamiltonian consist of six types of building blocks, whereas those in the spin-wave Hamiltonian consist of only four of them. Confined spin waves are robust against 1/S corrections. Emergent O(S⁰) interactions further confine—superconfine—spin waves into two separate groups within tricoordinated sites.

1. Introduction

Quasicrystals have aperiodic but long-range ordered atomic arrangements [1,2] to exhibit non-crystallographic rotational symmetry and self-similarity. Since the discovery of such a structure in an Al-Mn alloy [3], numerous transition-metal alloys have been tried and proved to be quasicrystalline [4,5,6,7]. In particular, magnetic long-range order was reported in rare-earth-based icosahedral quasicrystals, namely ferromagnetic order in Au-Ga-R (R = Gd, Tb, and Dy) [8,9] and antiferromagnetic order in Au-In-Eu [10], confirmed by neutron diffraction and magnetization measurements. These findings have provided a conclusive answer to the long-standing question of whether magnetic long-range order can emerge in quasicrystals [11,12,13,14,15]. Motivated by the observations of magnetic long-range order, as well as quantum criticality [16], superconductivity [17], and related phenomena, theoretical studies based on ideal models are essential for understanding how aperiodic geometry governs collective excitations [18,19,20,21,22,23,24].

The ${\mathbf{C}}_{5\mathrm{v}}$ Penrose tiling [25] is a two-dimensional realization of quasiperiodicity. It not only deserves investigation in itself but also serves as an ideal platform for exploring quasicrystals, especially of icosahedral point group symmetry. A striking feature of its electronic structure is the macroscopic degeneracy of the tight-binding Hamiltonian in the vertex model, where electrons hop between adjacent vertices connected by rhombus edges [26,27]. This Hamiltonian hosts a macroscopically degenerate zero-energy eigenlevel [28,29,30,31,32,33,34], which originates from a set of distinctive eigenstates known as confined states, whose wavefunctions are strictly confined to finite domains. These states are critical in nature—neither localized nor extended in the conventional sense—as they lack any characteristic length scale. Arai et al. proposed that the confined states are composed of six independent building blocks [29], which was numerically demonstrated indeed [34].

Macroscopic zero-energy degeneracy is found in various quasiperiodic tight-binding models as well, such as the generalized Penrose lattice [35], Ammann–Beenker lattice [36,37], Socolar dodecagonal lattice [38,39], hexagonal golden-mean tilings [40,41], quasiperiodic twisted bilayer systems [42], and the recently discovered aperiodic monotile [43]. Confined states do not necessarily occur at zero energy but can emerge at nonzero energies as well. In the center model on the Penrose lattice, with an s-like orbital placed at the center of each rhombus, non-interacting electrons exhibit confined states at energy $2t$ [44,45,46], where t is the hopping integral. In the $\pi$-flux model on the Ammann–Beenker lattice, confined states appear at energies $\pm \sqrt{2}t$, $\pm 2t$, and $\pm 2\sqrt{2}t$, as well as at zero energy, depending on flux configurations [47].

While such macroscopic degeneracy has been extensively studied in electron hopping models, analogous phenomena occur in spin systems as well. The half-filled single-band Hubbard model maps onto a spin $-{\displaystyle \frac{1}{2}}$ Heisenberg antiferromagnet in the strong correlation limit [48,49]. Quantum Monte Carlo (QMC) simulations reveal that the spatially averaged staggered magnetization for the Penrose-lattice Heisenberg antiferromagnet is larger than that of the square lattice, despite both having the same average coordination number of four [50,51,52]. This suggests that the structural inhomogeneity suppresses quantum fluctuations. The local magnetizations obtained from linear spin-wave (LSW) theory qualitatively agree with the QMC results in terms of their coordination-number dependence, and the ground-state energies from LSW and QMC calculations also agree to within $2\%$ [50,53]. Spin dynamics has also been investigated via inelastic photon [54,55] and neutron [56] scattering calculations. Within the harmonic-oscillator approximation for the Heisenberg antiferromagnet on the Penrose lattice, a prominent peak appears in the density of states at energy $3JS$ [53], where J is the exchange interaction, and S is the spin magnitude. At this energy, numerous degenerate states emerge, characterized by string-like wavefunctions that form closed loops without a characteristic length scale. These states manifest as nearly flat-band excitations in the dynamic structure factor, spanning almost the entire momentum space [56]. A perpendicular-space [57] analysis attributes their origin to antiferromagnons strongly confined to tricoordinated sites.

We demonstrate that the macroscopically degenerate spin-wave excitations in the Penrose-lattice Heisenberg antiferromagnet are confined states of antiferromagnons—the magnetic analogs of the well-known confined states in the tight-binding model.

2. Model and Method

We consider the spin-S nearest-neighbor antiferromagnetic Heisenberg model on the two-dimensional Penrose lattice, which consists of bipartite sublattices A and B (Figure 1). The Hamiltonian reads

$$\begin{array}{c}\hfill \mathcal{H}=J\sum _{i\in \mathrm{A}}\sum _{j\in \mathrm{B}}{l}_{i,j}{\mathit{S}}_{{\mathit{r}}_i}·{\mathit{S}}_{{\mathit{r}}_j}(J>0).\end{array}$$

(1)

We denote the vector spin operators on the A and B sublattices by ${\mathit{S}}_{{\mathit{r}}_i}$ ($i=1,2,\dots ,L_{\mathrm{A}}$) and ${\mathbf{S}}_{{\mathit{r}}_j}$$(j=1,2,\dots ,L_{\mathrm{B}})$, respectively, where ${\mathit{r}}_l$ serves as a generic site label running over both sublattices. The total number of sites is $L=L_{\mathrm{A}}+L_{\mathrm{B}}$. The linkage identifier $l_{i,j}$ equals 1 if the vertices ${\mathit{r}}_i$ and ${\mathit{r}}_j$ are connected, and 0 otherwise. We express the Hamiltonian (1) in terms of the Holstein–Primakoff bosons [58],

$$\begin{array}{cc}\hfill & S_{{\mathit{r}}_i}^{+}={\left(2S-a_i^{†}{a}_i\right)}^{{\displaystyle \frac{1}{2}}}{a}_i,S_{{\mathit{r}}_i}^{-}=a_i^{†}{\left(2S-a_i^{†}{a}_i\right)}^{{\displaystyle \frac{1}{2}}},S_{{\mathit{r}}_i}^z=S-a_i^{†}{a}_i,\hfill \\ \hfill & S_{{\mathit{r}}_j}^{+}=b_j^{†}{\left(2S-b_j^{†}{b}_j\right)}^{{\displaystyle \frac{1}{2}}},S_{{\mathit{r}}_j}^{-}={\left(2S-b_j^{†}{b}_j\right)}^{{\displaystyle \frac{1}{2}}}{b}_j,S_{{\mathit{r}}_j}^z=b_j^{†}{b}_j-S,\hfill \end{array}$$

(2)

and expand it in descending powers of S [59],

$$\begin{array}{c}\hfill \mathcal{H}={\mathcal{H}}^{\left(2\right)}+{\mathcal{H}}^{\left(1\right)}+O\left(S^0\right),\end{array}$$

(3)

where ${\mathcal{H}}^{\left(m\right)}$, on the order of $S^m$, reads

$$\begin{array}{cc}\hfill & {\mathcal{H}}^{\left(2\right)}=-J{S}^2\sum _{i\in \mathrm{A}}\sum _{j\in \mathrm{B}}{l}_{i,j},{\mathcal{H}}^{\left(1\right)}=JS\sum _{i\in \mathrm{A}}\sum _{j\in \mathrm{B}}{l}_{i,j}\left(a_i^{†}{a}_i+b_j^{†}{b}_j+a_i{b}_j+a_i^{†}{b}_j^{†}\right).\hfill \end{array}$$

(4)

We define the up-to-$O\left(S^1\right)$ LSW Hamiltonian

$$\begin{array}{c}\hfill {\mathcal{H}}^{\left[1\right]}\equiv {\mathcal{H}}^{\left(2\right)}+{\mathcal{H}}^{\left(1\right)}.\end{array}$$

(5)

We introduce the row vectors ${\mathit{d}}^{†}$, ${\mathit{a}}^{†}$, and ${}^t\mathit{b}$, of dimension L, $L_{\mathrm{A}}$, and $L_{\mathrm{B}}$, respectively, as

$$\begin{array}{c}\hfill {\mathit{d}}^{†}=\left[a_1^{†},\dots ,a_{L_{\mathrm{A}}}^{†},b_1,\dots ,b_{L_{\mathrm{B}}}\right]\equiv \left[{\mathit{a}}^{†},{}^t\mathit{b}\right].\end{array}$$

(6)

Then, we can compactly express the LSW Hamiltonian in matrix form as

$$\begin{array}{cc}\hfill & {\mathcal{H}}^{\left[1\right]}={\mathcal{H}}^{\left(2\right)}+{\epsilon }^{\left(1\right)}+JS\left[{\mathit{a}}^{†},{}^t\mathit{b}\right]\left[\begin{array}{cc}{\mathbf{Z}}_{\mathrm{AA}}& {\mathbf{C}}_{\mathrm{AB}}\\ {\mathbf{C}}_{\mathrm{AB}}^{†}& {\mathbf{Z}}_{\mathrm{BB}}\end{array}\right]\left[\begin{array}{c}\mathit{a}\\ {}^t\mathit{b}^{†}\end{array}\right]\hfill \\ \hfill & \equiv {\mathcal{H}}^{\left(2\right)}+{\epsilon }^{\left(1\right)}+JS{\mathit{d}}^{†}\mathbf{M}\mathit{d};{\epsilon }^{\left(1\right)}=-JS\sum _{i\in \mathrm{A}}\sum _{j\in \mathrm{B}}{l}_{i,j},\hfill \end{array}$$

(7)

where ${\mathbf{Z}}_{\mathrm{AA}}$ and ${\mathbf{Z}}_{\mathrm{BB}}$ are diagonal matrices of dimension $L_{\mathrm{A}}\times L_{\mathrm{A}}$ and $L_{\mathrm{B}}\times L_{\mathrm{B}}$, respectively, and ${\mathbf{C}}_{\mathrm{AB}}$ is the biadjacency matrix of dimension $L_{\mathrm{A}}\times L_{\mathrm{B}}$, as

$$\begin{array}{c}\hfill {\left[{\mathbf{Z}}_{\mathrm{AA}}\right]}_{i,i^{\prime }}=z_i{\delta }_{i,i^{\prime }},{\left[{\mathbf{Z}}_{\mathrm{BB}}\right]}_{j,j^{\prime }}=z_j{\delta }_{j,j^{\prime }},{\left[{\mathbf{C}}_{\mathrm{AB}}\right]}_{i,j}=l_{i,j},\end{array}$$

(8)

where $z_l$ is the coordination number of the site at ${\mathit{r}}_l$. Denoting the diagonal and off-diagonal blocks of the LSW Hamiltonian matrix $\mathbf{M}$ as

$$\begin{array}{c}\hfill \mathbf{Z}\equiv \left[\begin{array}{cc}{\mathbf{Z}}_{\mathrm{AA}}& {\mathbf{O}}_{\mathrm{AB}}\\ {\mathbf{O}}_{\mathrm{AB}}^{†}& {\mathbf{Z}}_{\mathrm{BB}}\end{array}\right],\mathbf{T}\equiv \left[\begin{array}{cc}{\mathbf{O}}_{\mathrm{AA}}& {\mathbf{C}}_{\mathrm{AB}}\\ {\mathbf{C}}_{\mathrm{AB}}^{†}& {\mathbf{O}}_{\mathrm{BB}}\end{array}\right],\end{array}$$

(9)

where ${\mathbf{O}}_{\mathrm{AA}}$, ${\mathbf{O}}_{\mathrm{BB}}$, and ${\mathbf{O}}_{\mathrm{AB}}$ are zero matrices of dimension $L_{\mathrm{A}}\times L_{\mathrm{A}}$, $L_{\mathrm{B}}\times L_{\mathrm{B}}$, and $L_{\mathrm{A}}\times L_{\mathrm{B}}$, respectively, we obtain

$$\begin{array}{c}\hfill \mathbf{M}=\mathbf{Z}+\mathbf{T}.\end{array}$$

(10)

We carry out the Bogoliubov transformation [60,61]

$$\begin{array}{cc}\hfill & \mathit{d}=\left[\begin{array}{cc}\mathbf{V}& \mathbf{X}\\ \mathbf{Y}& \mathbf{W}\end{array}\right]\mathit{\alpha };{\left[\mathbf{V}\right]}_{i,k_{-}}=v_{i,k_{-}},{\left[\mathbf{W}\right]}_{j,k_{+}}=w_{j,k_{+}},{\left[\mathbf{X}\right]}_{i,k_{+}}=x_{i,k_{+}},{\left[\mathbf{Y}\right]}_{j,k_{-}}=y_{j,k_{-}}\hfill \end{array}$$

(11)

with the matrices $\mathbf{V}$, $\mathbf{W}$, $\mathbf{X}$, and $\mathbf{Y}$, of dimension $L_{\mathrm{A}}\times L_{-}$, $L_{\mathrm{B}}\times L_{+}$, $L_{\mathrm{A}}\times L_{+}$, and $L_{\mathrm{B}}\times L_{-}$, respectively, to obtain quasiparticle magnons

$$\begin{array}{c}\hfill \left[{\alpha }_1^{-†},\dots ,{\alpha }_{L_{-}}^{-†},{\alpha }_1^{+},\dots ,{\alpha }_{L_{+}}^{+}\right]\equiv {\mathit{\alpha }}^{†}\end{array}$$

(12)

and the diagonalized LSW Hamiltonian

$$\begin{array}{c}\hfill {\mathcal{H}}^{\left[1\right]}={\mathcal{H}}^{\left(2\right)}+{\epsilon }^{\left(1\right)}+{\displaystyle \sum _{k_{+}=1}^{L_{+}}}{\epsilon }_{k_{+}}^{+}+\sum _{\sigma =\mp }{\displaystyle \sum _{k_{\sigma }=1}^{L_{\sigma }}}{\epsilon }_{k_{\sigma }}^{\sigma }{\alpha }_{k_{\sigma }}^{\sigma †}{\alpha }_{k_{\sigma }}^{\sigma },\end{array}$$

(13)

where ${\alpha }_{k_{\sigma }}^{\sigma †}$ creates a ferromagnetic ($\sigma =-$) or antiferromagnetic ($\sigma =+$) magnon of energy ${\epsilon }_{k_{\sigma }}^{\sigma }$.

We define the site-resolved density of states for the quasiparticle magnon excitation spectrum

$$\begin{array}{cc}\hfill & \rho \left(\omega \right)={\displaystyle \sum _{l=1}^L}{\rho }_l\left(\omega \right)={\displaystyle \sum _{z=3}^7}{\rho }_z\left(\omega \right)={\displaystyle \sum _{z=3}^7}{\displaystyle \frac{1}{L}}\left\{{\displaystyle \sum _{k_{-}=1}^{L_{-}}}{\displaystyle \frac{{\sum }_{i(z_i=z)}{\left|v_{i,k_{-}}\right|}^2+{\sum }_{j(z_j=z)}{\left|y_{j,k_{-}}\right|}^2}{{\sum }_{i\in \mathrm{A}}{\left|v_{i,k_{-}}\right|}^2+{\sum }_{j\in \mathrm{B}}{\left|y_{j,k_{-}}\right|}^2}}\delta \left(\hslash \omega -{\epsilon }_{k_{-}}^{-}\right)\right.\hfill \\ \hfill & \left.+{\displaystyle \sum _{k_{+}=1}^{L_{+}}}{\displaystyle \frac{{\sum }_{i(z_i=z)}{\left|x_{i,k_{+}}\right|}^2+{\sum }_{j(z_j=z)}{\left|w_{j,k_{+}}\right|}^2}{{\sum }_{i\in \mathrm{A}}{\left|x_{i,k_{+}}\right|}^2+{\sum }_{j\in \mathrm{B}}{\left|w_{j,k_{+}}\right|}^2}}\delta \left(\hslash \omega -{\epsilon }_{k_{+}}^{+}\right)\right\}.\hfill \end{array}$$

(14)

Figure 2 shows the coordination-number-resolved density of states ${\rho }_z\left(\omega \right)$, where z ranges from 3 to 7 (see Appendix A). The spectral weight shifts toward sites with higher coordination numbers as the energy increases, a feature attributable to the immobility of spin deviations in the Ising limit. In this limit, the excitation spectrum consists of discrete eigenlevels at $zJS$. The correlation between coordination number and excitation energy persists even when the degeneracy is lifted by spin fluctuations. Notably, the sharp peak at $\hslash \omega =3JS$ indicates macroscopically degenerate confined states.

3. Confined Antiferromagnons

Figure 3 demonstrates the spatial confinement of degenerate states at ${\epsilon }_k^{\mp }=3JS$ for the LSW Hamiltonian and at ${\epsilon }_k=0$ for the tight-binding Hamiltonian (see Appendix B) on the Penrose lattice. Here, ${\epsilon }_k$ denotes the eigenvalues of the tight-binding Hamiltonian. In both cases, these confined states are spatially separated by “forbidden ladders” [29,34], which are one-dimensional, unbranched closed loops with zero wavefunction amplitude. Within each domain enclosed by these forbidden ladders, the confined states exhibit nonzero amplitudes exclusively on either the A or B sublattice. The alternating pattern of confinement domains across the forbidden ladders matches perfectly between Figure 3a,b. A key distinction between the two types of confined states lies in their spatial distributions: in the LSW eigenstates (Figure 3a), antiferromagnons are confined only to tricoordinated sites, whereas in the tight-binding eigenstates (Figure 3b), electrons are confined to tricoordinated and pentacoordinated sites.

We derive the constraints on the wavefunction amplitudes of the confined states. The matrix form of the LSW Hamiltonian (7) reduces the problem to an eigenvalue problem for the matrix $\mathbf{M}=\mathbf{Z}+\mathbf{T}$. We first examine the off-diagonal connectivity matrix $\mathbf{T}$, which corresponds to the tight-binding Hamiltonian with the hopping integral set to unity. Using the argument from the tight-binding model (see Appendix B), we obtain the local constraints at each site ${\mathit{r}}_l$ for the LSW wavefunction (11) to be an eigenvector of $\mathbf{T}$ with zero eigenvalue, which read

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\displaystyle \sum _{j\in \mathrm{B}}}{l}_{i,j}{y}_{j,k_{-}}=0\hfill & (l=i\in \mathrm{A})\hfill \\ {\displaystyle \sum _{i\in \mathrm{A}}}{l}_{i,j}{v}_{i,k_{-}}=0\hfill & (l=j\in \mathrm{B})\hfill \end{array}\right.\end{array}$$

(15)

for the $\sigma =-$ mode and

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\displaystyle \sum _{j\in \mathrm{B}}}{l}_{i,j}{w}_{j,k_{+}}=0\hfill & (l=i\in \mathrm{A})\hfill \\ {\displaystyle \sum _{i\in \mathrm{A}}}{l}_{i,j}{x}_{i,k_{+}}=0\hfill & (l=j\in \mathrm{B})\hfill \end{array}\right.\end{array}$$

(16)

for the $\sigma =+$ mode. As established in Refs. [29,34], for the connectivity matrix $\mathbf{T}$ on the infinite Penrose lattice, the only local solutions satisfying Equations (15) and (16) are the six types of building blocks shown in Figure 4. We define the L-dimensional vector

$$\begin{array}{c}\hfill {\mathit{\psi }}_{n,m}=\left[\begin{array}{c}{\psi }_{n,m}\left({\mathit{r}}_1\right)\\ ⋮\\ {\psi }_{n,m}\left({\mathit{r}}_L\right)\end{array}\right],{\left[{\mathit{\psi }}_{n,m}\right]}_l={\psi }_{n,m}\left({\mathit{r}}_l\right),\end{array}$$

(17)

where $n=1,\dots ,6$ labels the type of building block, $m=1,\dots ,N_n$ labels each block of type-n, and $N_n$ is the total number of such blocks. Each ${\mathit{\psi }}_{n,m}$ is an unnormalized eigenvector of $\mathbf{T}$ satisfying $\mathbf{T}{\mathit{\psi }}_{n,m}=0$. The wavefunction amplitudes ${\psi }_{n,m}\left({\mathit{r}}_l\right)$ at each site take the values specified in Figure 4 for the corresponding tiles of the Penrose lattice and are zero elsewhere. The number of nonzero components for each type of building block is summarized in

Table 1. Building-block properties.

Type	1	2	3	4	5	6
Number of constituent sites	36	31	28	66	36	81
Number of finite-amplitude sites	10	10	12	28	19	41

.

Next, we examine the effect of the diagonal local potential matrix $\mathbf{Z}$ on the vectors ${\mathit{\psi }}_{n,m}$. Applying the matrix $\mathbf{M}$ to the vector ${\mathit{\psi }}_{n,m}$ yields

$$\begin{array}{c}\hfill \mathbf{M}{\mathit{\psi }}_{n,m}=\mathbf{Z}{\mathit{\psi }}_{n,m}+\mathbf{T}{\mathit{\psi }}_{n,m}=\mathbf{Z}{\mathit{\psi }}_{n,m}.\end{array}$$

(18)

Since $\mathbf{Z}$ is a diagonal matrix whose elements are the coordination numbers $z_l$ at each site ${\mathit{r}}_l$, we obtain

$$\begin{array}{c}\hfill {\left[\mathbf{Z}{\mathit{\psi }}_{n,m}\right]}_l=z_l{\psi }_{n,m}\left({\mathit{r}}_l\right).\end{array}$$

(19)

When the coordination number $z_l$ is uniform ($z_l=z$) across all sites with nonzero amplitudes, ${\mathit{\psi }}_{n,m}$ is a simultaneous eigenvector of $\mathbf{Z}$ and $\mathbf{M}$, such that $\mathbf{Z}{\mathit{\psi }}_{n,m}=z{\mathit{\psi }}_{n,m}$ and $\mathbf{M}{\mathit{\psi }}_{n,m}=z{\mathit{\psi }}_{n,m}$. Therefore, the type-1 to type-4 building blocks, in which all sites with nonzero wavefunction amplitude have coordination number $z=3$, and are eigenvectors of $\mathbf{M}$ with eigenvalue 3

$$\begin{array}{c}\hfill \mathbf{M}{\mathit{\psi }}_{n,m}=3{\mathit{\psi }}_{n,m}(n=1,2,3,4).\end{array}$$

(20)

In contrast, the type-5 and type-6 building blocks, which contain both $z=3$ and $z=5$ sites with nonzero wavefunction amplitudes are eigenvectors of $\mathbf{T}$ but not of $\mathbf{M}$. Thus, the confined states at ${\epsilon }_k^{\mp }=3JS$ in the LSW Hamiltonian are linear combinations of type-1 to type-4 building blocks, whereas those at ${\epsilon }_k=0$ in the tight-binding Hamiltonian are linear combinations of type-1 to type-6 building blocks. Examples of the coverings of the Penrose lattice by these building blocks are shown in Figure A5.

4. Boundary-Condition Effects

The fractions of type-n building blocks in the thermodynamic limit, $f_n\equiv {lim}_{L\to \infty }{N}_n/L$, can be determined exactly using either the inflation-deflation rule [29,62] or the perpendicular-space accounting method [32]. These two independent methods conclude the same results. The values of $f_n$ are summarized in

Table 2. The fraction of each building block.

Building Blocks	Their Fraction
Type-1	$f_1={\displaystyle \frac{1}{{\tau }^8(1+{\tau }^2)}}\simeq 5.883\times {10}^{-3}$
Type-2	$f_2={\displaystyle \frac{1}{{\tau }^6(1+{\tau }^2)}}\simeq 1.540\times {10}^{-2}$
Type-3	$f_3={\displaystyle \frac{2}{{\tau }^8}}\simeq 4.257\times {10}^{-2}$
Type-4	$f_4={\displaystyle \frac{2}{{\tau }^{10}}}\simeq 1.626\times {10}^{-2}$
Type-5	$f_5={\displaystyle \frac{1}{{\tau }^9}}\simeq 1.316\times {10}^{-2}$
Type-6	$f_6={\displaystyle \frac{1}{{\tau }^{11}}}\simeq 5.025\times {10}^{-3}$
Total (LSW)	${\displaystyle \sum _{n=1}^4}{f}_n=280-173\tau \simeq 8.012\times {10}^{-2}$
Total (tight-binding)	${\displaystyle \sum _{n=1}^6}{f}_n=81-50\tau \simeq 9.830\times {10}^{-2}$

. The total fraction of confined states is given by ${\sum }_{n=1}^4{f}_n=280-173\tau \simeq 8.012\times {10}^{-2}$ for the LSW confined states at ${\epsilon }_k^{\mp }=3JS$, and by ${\sum }_{n=1}^6{f}_n=81-50\tau \simeq 9.830\times {10}^{-2}$ for the tight-binding confined states at ${\epsilon }_k=0$, where $\tau \equiv {\displaystyle \frac{1+\sqrt{5}}{2}}$ is the golden number.

Let $N(\hslash \omega )$ denote the number of states with energy $\hslash \omega$. Figure 5 shows the finite-size scaling of the fractions $N\left(3JS\right)/L$ and $N\left(0\right)/L$ obtained numerically, corresponding to the ${\epsilon }_k^{\mp }=3JS$ eigenstates of the LSW Hamiltonian and ${\epsilon }_k=0$ eigenstates of the tight-binding Hamiltonian, respectively. For finite L, the fraction of confined states deviates from its thermodynamic-limit value. Since the wavefunctions of the confined states are strictly localized within the finite-extent building blocks shown in Figure 4, the Penrose lattice must be sufficiently large to accommodate a proper tiling by these building blocks. In the thermodynamic limit, the fraction of LSW ${\epsilon }_k^{\mp }=3JS$ confined states $N\left(3JS\right)/L$ converges to the estimated value $280-173\tau$ in boundary-free periodic approximants, but remains approximately $6\%$ lower even at L = 65,751 in an open-boundary cluster, as is shown in Figure 5a. In contrast, as shown in Figure 5b, the fraction of tight-binding ${\epsilon }_k=0$ confined states $N\left(0\right)/L$ converges to the exact value $81-50\tau$ regardless of the boundary conditions.

Figure 6 illustrates the effects of open boundaries. For the LSW confined states, blank domains without local solutions emerge near the boundaries, as shown in Figure 6a. This arises because the coordination number $z_l$ at boundary sites is reduced compared to the bulk, making it virtually impossible to simultaneously satisfy both the constraint that the wavefunction amplitude is nonzero only on tricoordinated sites and either Equation (15) or (16). As a result, ribbon-like blank domains, approximately one building block wide, form along the boundary and lead to an underestimation of $N\left(3JS\right)/L$ relative to the exact value. The fraction of such blank domains is estimated to be $O(1/\sqrt{L})$ in an L-site system, which is consistent in order of magnitude with the observed discrepancy in $N\left(3JS\right)/L$. In contrast, in boundary-free periodic approximants, the coordination numbers remain unaffected, thus preventing the formation of such blank domains. Consequently, the fraction $N\left(3JS\right)/L$ converges to the exact value $280-173\tau$, demonstrating that only type-1 to type-4 building blocks constitute the LSW confined states. In the case of the tight-binding Hamiltonian, i.e., the matrix $\mathbf{T}$, some of the ${\epsilon }_k=0$ eigenstates include building blocks other than types 1 to 6 as is shown in Figure 6b. As a result, the entire Penrose lattice can be covered, excluding the one-dimensional forbidden ladders, and $N\left(0\right)/L$ converges to the exact value $81-50\tau$ under both boundary conditions.

Boundary-induced reduction of $N\left(3JS\right)$ may likewise occur in other quasiperiodic systems with confined states constructed from particular local building blocks, each consisting of same-coordinated sites. We evaluate the fraction of confined states for several open clusters of various sizes, which effectively correspond to different boundary terminations. The calculated fraction converges to the analytically derived value for the infinite-size lattice. This suggests that changing the boundary termination does not qualitatively alter the formation of finite-width blank domains near the boundaries. In addition, matching-rule-violating defect sites (see Appendix A.2 and Figure A3) emerge in periodic approximants of the Penrose lattice and locally obstruct the placement of the building blocks. As a result, point-like blank domains are formed around such defect sites, whereas boundary effects produce one-dimensional blank domains. The affected fractions therefore scale as $O(1/L)$ and $O(1/\sqrt{L})$, respectively. Owing to the local building-block nature of the LSW confined states on the Penrose lattice, the effects of boundaries, defects, and local disorder on the bulk confined-state fraction are expected to vanish in the thermodynamic limit.

5. Superconfined Antiferromagnons

We incorporate the $O\left(S^0\right)$ quantum corrections (see Appendix C) into the LSW Hamiltonian via the Wick decomposition [63,64]. Figure 7 compares the site-resolved density of states for the LSW and up-to-$O\left(S^0\right)$ interacting spin-wave (ISW) excitation spectra at $S={\displaystyle \frac{1}{2}}$, together with their mappings in the perpendicular space. In the LSW spectrum, a prominent peak at $\hslash \omega =3J/2$ corresponds to massively degenerate confined states (Figure 7a). The $O\left(S^0\right)$ magnon-magnon interactions introduce environment-dependent corrections into the ISW Hamiltonian (A19). In the absence of translational invariance, interaction-induced on-site potentials vary even among sites with the same coordination number [56]. As a result, the $O\left(S^0\right)$ quantum corrections not only shift the excitation spectrum upward but also lift the degeneracy at $\hslash \omega =3J/2$. The corresponding peaks remain prominent in the ISW spectrum, but split into two peaks around $1.6300J$ and $1.6725J$ (Figure 7a′).

The one-to-one correspondence between positions in the perpendicular space and local vertices in the physical space (Figure A2) confirms that the LSW spectral weight at $\hslash \omega =3J/2$ (Figure 7b,b′) is localized within the $z=3$ domains. In these domains, certain regions exhibit negligible spectral intensity, consistent with the perpendicular-space analysis of forbidden regions [32] associated with the type-1 to type-4 building blocks. The perpendicular-space mappings of the ISW spectrum reveal that the spectral weight remains concentrated in the $z=3$ domains for both the lower branch at $\hslash \omega =1.6300J$ (Figure 7c,c′) and the upper branch at $\hslash \omega =1.6725J$ (Figure 7d,d′). Compared to the LSW case, however, the intensity exhibits a more fragmented distribution, with each branch occupying distinct subdomains. The subdivision of the perpendicular-space domains reflects a finer classification of local environments in the physical space, arising from the inclusion of more distant surrounding geometry [56,65]. The ISW spectral weight distributions of the two branches are complementary within the $z=3$ domains. This indicates that the $O\left(S^0\right)$ quantum fluctuations distinguish these subtle geometric variations and resolve subsets of tricoordinated sites into distinct energy branches. The magnons forming these two branches remain well confined to tricoordinated sites and are thus “superconfined” to selected subsets of these sites.

6. Summary and Discussion

In the vertex model on the Penrose lattice, itinerant electrons exhibit macroscopically degenerate confined states at ${\epsilon }_k=0$. They consist of type-1 to type-6 building blocks. In contrast, in the antiferromagnetic Heisenberg model on the Penrose lattice, localized spins exhibit macroscopically degenerate confined states at ${\epsilon }_k^{\mp }=3JS$ within the harmonic-oscillator approximation. They consist only of type-1 to type-4 building blocks. The spin-wave Hamiltonian in the site representation has coordination-number-dependent positive on-site potentials, which more stabilize lower-coordinated sites. Type-1 to type-4 building blocks consist of tricoordinated sites only, while type-5 and type-6 building blocks consist of pentacoordinated as well as tricoordinated sites. Confined spin waves consist purely of tricoordinated sites.

What will happen to confined antiferromagnons when they are brought into interaction? The $O\left(S^0\right)$ quartic interactions lift the macroscopic degeneracy at ${\epsilon }_k^{\mp }=3JS$ to yield two branches, as is shown in Figure 7. Their spectral weights are distributed complementarily in the perpendicular space. In order to formulate these distributions, we define a distance R between two vertices such that the shortest route from one to the other consists of R bonds. With increasing R, vertices are more and more classified into subgroups and subdomains in the physical and perpendicular spaces, respectively. The eight fundamental domains are further subdivided into 15, 27, and 40 types for $R=1$, 2, and 3, respectively. Overlaying these subdivisions onto the perpendicular-space distributions for each branch shows that the spectral-weight patterns are well described at $R=3$ (Figure 8). This can be understood as follows. As an example, consider an arbitrary bond $\langle i,j\rangle$. In the LSW Hamiltonian, the matrix elements consist of on-site potentials determined by the coordination number of each site and uniform nearest-neighbor bonds. Thus, the LSW description is strictly local to each bond. In the ISW Hamiltonian, the $O\left(S^0\right)$ corrections to the quadratic terms contain site expectation values $\langle a_i^{†}{a}_i\rangle$ and $\langle b_j^{†}{b}_j\rangle$, and bond expectation values $\langle a_i{b}_j\rangle$ and $\langle a_i^{†}{b}_j^{†}\rangle$, as is shown in Equation (A16) (see Appendix C). These quantities depend on their local environments and therefore vary from bond to bond. They are self-consistently determined via matrix elements depending on the surrounding environments of each bond. Thus, the ISW description includes contributions beyond the bond in issue. Specifically, for the sites ${\mathit{r}}_i$ and ${\mathit{r}}_j(={\mathit{r}}_i+\mathit{\delta })$, the neighboring sites ${\mathit{r}}_{j^{\prime }}={\mathit{r}}_i+{\mathit{\delta }}_i$ ($\ne {\mathit{r}}_j$) and ${\mathit{r}}_{i^{\prime }}={\mathit{r}}_j+{\mathit{\delta }}_j$ ($\ne {\mathit{r}}_i$) also explicitly enter the ISW Hamiltonian. These sites are related by ${\mathit{r}}_{i^{\prime }}={\mathit{r}}_{j^{\prime }}-{\mathit{\delta }}_i+\mathit{\delta }+{\mathit{\delta }}_j$, implying that contributions up to a distance $R=3$ are relevant at the $O\left(S^0\right)$ level. This estimate is consistent with the spectral-weight distributions of the superconfined antiferromagnons.

What will happen to superconfined antiferromagnons when they are brought into further interactions higher than $O\left(S^0\right)$? The $O\left(S^{-1}\right)$ sextic interactions and still higher-order terms introduce additional corrections to the eigenvalues. Since the nearest-neighbor exchange interaction conserves magnetization, the expectation values in the quadratic terms obtained via Wick decomposition remain limited to the same types as in the $O\left(S^0\right)$ case: site expectation values $\langle a_i^{†}{a}_i\rangle$ and $\langle b_j^{†}{b}_j\rangle$, and bond expectation values $\langle a_i{b}_j\rangle$ and $\langle a_i^{†}{b}_j^{†}\rangle$. These expectation values are defined only on sites and nearest-neighbor bonds. The spatial range of the corrections remains unchanged from the $O\left(S^0\right)$ case. Thus, such higher-order corrections are unlikely to incorporate more distant environments to induce further branching. Superconfinement is not fractionalization up to infinity but dividing tricoordinated sites into essentially two groups.

Resonant inelastic x-ray scattering (RIXS) [66,67,68] may be a functional light probe to capture confined antiferromagnons of our interest. While spin-orbit-coupling-assisted magnetic Raman scattering [69,70] serves to reveal single magnon eigenlevels, the Raman operator of this type reads a total uniform magnetization in the polarization-dependent particular direction, and therefore acts only on rotation-invariant and mirror-symmetric magnons. It is thus useless to detect confined antiferromagnons consisting of mirror-antisymmetric building blocks. The single-magnon Raman scattering is further disappointing because it cannot have any access to the $\mathrm{SU}\left(2\right)$ symmetric Heisenberg Hamiltonian commutable with the total uniform magnetization in every direction. Inelastic neutron scattering [56,71] is indeed available in this context, but it requires relatively large bulk samples. RIXS works even with small samples and thin films. We can tune the momentum, polarization, and resonance energy of incoming and outgoing x-rays. Interestingly enough, photons can distinguish between sublattices. Circularly polarized light interacts with either a magnetization-enhancing or magnetization-reducing magnon. Each confined state resides in either the A or B sublattice, because its building blocks belong to either the A or B sublattice. There is a possibility of selectively accessing confined antiferromagnons according to their constituent building blocks. RIXS spectra possibly contain a nearly flat band at ${\epsilon }_k^{\mp }\simeq 3JS$.