Ferromagnetically ordered states in the Hubbard model on the $H_{00}$ hexagonal golden-mean tiling

Quasiperiodic systems have attracted considerable interest since the discovery of the Al-Mn quasicrystal Shechtman et al. (1984). Their properties are of equal interest, in part driven by the observation of behaviour traditionally observed in periodic systems. For example, electron correlations in quasicrystals have been actively studied after quantum critical behavior was observed in the Au-Al-Yb quasicrystal Deguchi et al. (2012). Similarly, long-range correlative states have been reported despite the lack of periodicity inherent in these materials: such as superconductivity in the Al-Zn-Mg quasicrystal Kamiya et al. (2018), and ferromagnetically ordered states in the Au-Ga-X (X = Gd, Tb, Dy) quasicrystals Tamura et al. (2021); Takeuchi et al. (2023). These studies have stimulated, and continue to motivate theoretical investigations on electron correlations and the spontaneously symmetry breaking states in quasicrystals Okabe and Niizeki (1988); Wessel et al. (2003); Jagannathan et al. (2007); Watanabe and Miyake (2013); Takemori and Koga (2015); Takemura et al. (2015); Andrade et al. (2015); Fulga et al. (2016); Otsuki and Kusunose (2016); Sakai et al. (2017); Araújo and Andrade (2019); Sakai and Arita (2019); Varjas et al. (2019); Duncan et al. (2020); Cao et al. (2020); Takemori et al. (2020); Hauck et al. (2021); Ghadimi et al. (2021); Sakai and Koga (2021). For example, magnetically ordered states in the Hubbard model on quasiperiodic bipartite tilings have been studied, including the Penrose Jagannathan et al. (2007); Koga and Tsunetsugu (2017), Ammann-Beenker Wessel et al. (2003); Jagannathan and Schulz (1997); Koga (2020); Oktel (2021), and Socolar dodecagonal Koga (2021). One of the common properties among the majority of these studies is the existence of strictly localized states with $E=0$ (i.e., confined states) in the non-interacting case Kohmoto and Sutherland (1986); Arai et al. (1988); Koga and Tsunetsugu (2017); Koga (2020); Oktel (2021); Koga (2021); Keskiner and Oktel (2022); Matsubara et al. (2023). This leads to interesting magnetic properties in the weak coupling limit.

Recently, we introduced a family of golden–mean hexagonal and trigonal aperiodic tilings produced using a generalization of de Bruijn’s grid method  Coates et al. (2023). In this work, we showcased the structural properties and substitution rules of two ‘special’ cases of this family. These are the $H_{00}$ and $H_{\frac{1}{2}\frac{1}{2}}$ tilings, where the subscript refers to the tunable grid-shift parameters used in their construction (for more details, see  Coates et al. (2023)). These tilings hold distinct structural properties compared to the Penrose, Ammann-Beenker, and Socolar tilings – not only do they share rotational symmetries associated with periodic systems, but, they also possess a sublattice imbalance due to their vertex structure. However, they are still rooted in the ‘physical’ world of experimentally observed trigonal and hexagonal quasiperiodic systems  Woods et al. (2014); Uri et al. (2023); Oka and Koshino (2021); Coates et al. (2020).

It is therefore desirable to study magnetic properties on quasiperiodic systems with sublattice imbalances, in order to systematically understand and compare correlated electron behavior across the widest range of relevant quasiperiodic tilings. In fact, we have already shown the effect which an imbalance has on the magnetic states on one of the special cases from the hexagonal family; the $H_{\frac{1}{2}\frac{1}{2}}$ hexagonal golden-mean tiling realizes a ferrimagnetically ordered state in the ground state Koga and Coates (2022), which is in contrast to that in the Penrose Koga and Tsunetsugu (2017), Ammann-Beenker Koga (2020), and Socolar dodecagonal tilings Koga (2021) where antiferromagnetically ordered states are realized without a uniform magnetization.

In this paper, we discuss the relevant properties of the $H_{00}$ tiling structure and then study the macroscopically degenerate states with $E=0$ in the tight-binding model, which should play an important role for finding magnetic properties in the weak coupling limit. We clarify that two extended states appear in one of the sublattices, while confined states appear in the other. Furthermore, we obtain the exact fraction of the confined states in terms of Lieb’s theorem Lieb (1989), considering magnetism in the weak coupling limit. We also discuss how magnetic properties are affected by electron correlations in the half-filled Hubbard model.

The paper is organized as follows. In Sec. II, we briefly describe the properties of the $H_{00}$ hexagonal golden-mean tiling needed for our work. In Sec. III, we introduce the half-filled Hubbard model on the $H_{00}$ hexagonal golden-mean tiling. Then, we study the macroscopically degenerate states with $E=0$ in Sec. IV. By means of the real-space Hartree approximations, we clarify how a magnetically ordered state is realized in the Hubbard model in Sec. V. Finally, crossover behavior in the ordered state is addressed by mapping the spatial distribution of the magnetization to perpendicular space. A summary is given in the last section.

Here, we will give an overview and describe the relevant properties of the $H_{00}$ hexagonal golden-mean tiling which we need for our calculations. The tiling is composed of large hexagons (LH), parallelograms (P), and small hexagons (SH). A section of the tiling, and the schematics of its proto-tiles are shown in Figure 1. The vertex system of the tiling is bipartite, since it is composed of polygons with even edges (hexagons and parallelograms). For our work, we require the exact fractions of tile and vertex frequencies across the tiling, which we take directly from  Coates et al. (2023), in which we explicitly explain our methods of derivation.

In the thermodynamic limit, the fractions of the LH, P, and SH tiles are given as:

where $\tau$ is the golden-mean $(1+\sqrt{5})/2$. Similarly, there are seven types of vertices: A, B, C, D, E, F, and G vertices, which are explicitly shown in Figure 2. Their fractions across the tiling are given as:

As the tiling is bipartite, trivially, we have two distinct sublattices of vertices. These sublattices can be distinguished either by their occupation of distinct sub-planes in perpendicular space  Coates et al. (2023), or by grouping by their coordination number. For example, one sublattice consists of A, B, and C vertices (coordination number of 3), while the other consists of D, E, F, and G vertices (coordination numbers of 4, 5, 6, and 4, respectively). From here on, the sublattice including A, B, and C vertices is denoted as the $b$ sublattice and the other is denoted as the $w$ sublattice.

As we previously mentioned, this sublattice structure is in contrast to that of the well-known bipartite tilings such as the Penrose and Ammann-Beenker tilings, where half of the vertices for each type exist in both sublattices. The sublattice structure inherent in the $H_{00}$ tiling, however, leads to the sublattice imbalance $\Delta$, such that Coates et al. (2023):

where $f_{b}$ and $f_{w}$ are the fractions of the $b$ and $w$ sublattices, respectively.

We note the following property which is convenient for reducing the computational cost of mean-field calculations. In the $H_{00}$ hexagonal tiling, certain tiles or vertices have a local threefold rotational symmetry, e.g. the LH and SH tiles, and the A and F vertices, as seen in Figure 2. Following the substitution rules in Coates et al. (2023), this threefold ‘group’ is changed in a cyclical manner as: LH tile $\rightarrow$ SH tile $\rightarrow$ A vertex $\rightarrow$ F vertex $\rightarrow$ LH tile $\rightarrow$ $\cdots$. Therefore, the system belongs to the point group $C_{3v}$ when one generates the tiling by iteratively applying the deflation rule to an LH or SH tile as its seed, allowing us to save computational time by applying symmetry operations. However, in the thermodynamic limit, the entire system has sixfold rotational symmetry, which is seen in Fourier space Coates et al. (2023).

We study the Hubbard model on the $H_{00}$ hexagonal golden-mean tiling, which is given by the following Hamiltonian

where $c_{i\sigma}$ $(c^{\dagger}_{i\sigma})$ annihilates (creates) an electron with spin $\sigma\,(=\uparrow,\,\downarrow)$ at the $i$th site and $n_{i\sigma}=c^{\dagger}_{i\sigma}c_{i\sigma}$. $t$ is the nearest-neighbor transfer integral and $U$ is the on-site Coulomb interaction. For simplicity, we have assumed that the magnitude of the hopping integral is uniform in the system. The chemical potential is always $\mu=U/2$ when the electron density is fixed to be half filling.

To discuss magnetic properties in the Hubbard model, we make use of the real-space mean-field theory. This method has an advantage in treating large clusters, which is crucial to clarify magnetic properties inherent in the quasiperiodic systems. Here, we introduce the site-dependent mean-field $\braket{n_{i\sigma}}$ and the mean-field Hamiltonian is then given as

For given values of $\braket{n_{i\sigma}}$, we numerically diagonalize the Hamiltonian $H^{\rm MF}$, update $\braket{n_{i\sigma}}$, and repeat this procedure until the result converges. The uniform and staggered magnetizations $m^{\pm}$ are given as

where $N_{\alpha}$ ($m_{\alpha}$) is the number of the sites (the average of the magnetization) in the $\alpha$ sublattice and $m_{i}$ is the local magnetization at the $i$th site.

Here, we discuss electronic properties in the noninteracting case ($U=0$), where the model Hamiltonian is reduced to the tightbinding model. Diagonalizing the Hamiltonian for the system with $N=1\,767\,438$, we obtain the density of states as

where $N$ ($=\sum_{\alpha}N_{\alpha}$) is the number of the sites in the whole system and $\epsilon_{i}$ is the $i$th eigenenergy. The results are shown in Figure 3.

We find the delta-function peak at $E=0$, suggesting the existence of macroscopically degenerate states. In fact, the clear jump singularity appears at $E=0$ in the integrated density of states. These states should be important for magnetic properties in the weak coupling limit. In the next section, we discuss the macroscopically degenerate states with $E=0$.

Here, we focus on the degenerate states with $E=0$ in the tightbinding model. Since the $H_{00}$ hexagonal golden-mean tiling is bipartite, these states should exist in both sublattices. First, we focus on the $w$ sublattice composed of D, E, F, and G vertices. It is clarified that the number of the degenerate states in the sublattice is at most two, which will be proven in Appendix A. This proof is based on the fact that there exist no tiles with zero amplitudes. Since all tiles have finite amplitudes in either corner site, this should indicate the existence of extended states. To clarify this, we consider the detail of the $w$ sublattice. Figure 1(a) shows that the $w$ sublattice can be divided into three groups $(w_{1},w_{2},w_{3})$, which are shown as the numbers in the open circles. Each site in the $b$ sublattice connects to three nearest-neighbor sites belonging to each of the $w_{1}$, $w_{2}$, and $w_{3}$ sublattices. Again, this is proven in Appendix B. Therefore, two states $|\Psi_{\pm}\rangle$ are the exact eigenstates with $E=0$ in the $w$ sublattice, where the amplitudes are given as

where $i_{n}\;(n=1,2,3)$ is the site index in the $w_{n}$ sublattice and $\omega_{\pm}(=\exp[\pm 2\pi i/3])$ is a solution of the equation $x^{2}+x+1=0$. Since finite amplitudes appear in the whole system, these states can be regarded as the extended states. Therefore, we can say that there exist only two extended states with $E=0$ in the $w$ sublattice.

By contrast, there are the other macroscopically degenerate states in the $b$ sublattice. We construct a simple form, considering their linear combinations, as discussed in previous papers Koga and Tsunetsugu (2017); Kohmoto and Sutherland (1986); Arai et al. (1988). The states can be represented to be exactly localized in a certain region and can be regarded as confined states. These are in contrast to the extended states in the $w$ sublattice. Five simple examples of the confined states $\Psi_{1},\Psi_{2},\cdots,$ and $\Psi_{5}$ are explicitly shown in Figure 4.

According to Conway’s theorem, a certain diagram appears repeatedly in the quasiperiodic tiling, in general. This means that each confined state exists with a finite fraction in the tiling. The diagram for the site structures of $\Psi_{1}$ and $\Psi_{2}$, which is shown in Figure 4, always appears around the F vertex due to the matching rule of the tiles. Therefore, the fractions for these confined states are given by the fraction of the F vertex, $f_{1}=f_{2}=1/(4\tau^{7})$. On the other hand, the site structures for $\Psi_{3}$, $\Psi_{4}$, and $\Psi_{5}$ do not always appear around the LH and SH tiles, and A vertices, respectively. Taking the tiling structure into account, we obtain the fractions of $\Psi_{3}$, $\Psi_{4}$, and $\Psi_{5}$ as $f_{3}=(\tau^{-3}+\tau^{-8})/4$, $f_{4}=3/4\tau^{7}$, and $f_{5}=(\tau^{-6}-\tau^{-11})/4$, respectively. In the tightbinding model on the $H_{00}$ hexagonal golden-mean tiling, there are many kinds of confined states (not shown) and therefore the fraction of the confined states $f^{C}$ is bounded by $f^{C}\geq\sum_{i=1}^{5}f_{i}\sim 0.120$.

Next, we try to directly obtain the exact fraction of the confined states, making use of magnetic properties at half filling Koga and Coates (2022). According to Lieb’s theorem, the ground state of the half-filled Hubbard model has a total spin $S_{tot}=N\Delta/2=N/(4\tau^{2})$ for arbitrary $U$. In the weak coupling limit, the magnetically ordered state originates only from the macroscopically degenerate states with $E=0$. Two extended states in the $w$ sublattice should be negligible in the thermodynamic limit. Therefore, magnetic properties little depend on these states and mainly depend on the confined states in the $b$ sublattice. Thus, the uniform magnetization can be given as $m^{+}=f^{C}/2$, where $f^{C}$ is the fraction of the confined states. From these two equations, we obtain the exact fraction of the confined states as

This is consistent with the numerical results $f^{C}=336288/1767438\sim 0.190$ for the finite cluster with $N=1\,767\,438$.

We wish to note that in the $H_{00}$ hexagonal golden-mean tiling the extended states appear in addition to the confined states. The extended states are also found in the tight-binding model on the $H_{\frac{1}{2}\frac{1}{2}}$ hexagonal tiling, although this was not discussed in our previous work Koga and Coates (2022).

Here, we discuss magnetic properties in the half-filled Hubbard model on the $H_{00}$ hexagonal golden-mean tiling. We mainly treat the system with $N=256\,636$ by means of real-space mean-field approximations. When the system is non-interacting, the macroscopically degenerate states appear at the Fermi level, as shown in Figure 3. The introduction of interaction leads to a magnetically ordered state with finite magnetizations: the magnetization profile for the case with $U/t=1.0\times 10^{-7}$ is shown in Figure 5(a), where red circles indicate positive magnetizations, and its size is proportional to the magnitude.

We find finite magnetizations only in the $b$ sublattice, as discussed above. In particular, the magnetizations in the A vertices are smaller than those in the B and C vertices. This quantitative difference is clearly found in the distribution of the magnetization in Figure 6(a), where the magnetizations on the A vertices are $m\sim 0.1$, while those on the B and C vertices are $m\sim 0.16$ .

This behaviour can be explained by the spatial distribution of the confined states. When one considers the local tiling structure for the confined states $\Psi_{1},\Psi_{2},\cdots,\Psi_{5}$ (see Figure 4), the A vertices have an amplitude only in the wave function $\Psi_{5}$. On the other hand, multiple confined states have amplitudes at B and C vertices. This should lead to a difference in the magnetizations, namely, the confined states in the larger regions have amplitudes in many sites and therefore have a minimal effect on the magnetizations at the A vertices. By contrast, we find no magnetization in the $w$ sublattice, which is consistent with the fact that two extended states little affect magnetic properties in the weak coupling limit. From these results, we can say that, in the weak coupling limit, the ferromagnetically ordered state is realized with the total uniform moment $m^{+}=1/(4\tau^{2})$.

Increasing the interaction strength, the local magnetization in the $b$ sublattice monotonically increases and the magnetizations in the other sublattice are induced. The spatial structure in the magnetization for $U/t=1$ is shown in Figure 5(b). The magnetization $m\sim-0.05$ is induced in the $w$ sublattice, as shown in Figure 6(b). Further increasing the interaction strength $U$ changes the distribution of the local magnetizations: when $U/t=5$, the magnetization is almost $m\sim\pm 0.4$, as shown in Figure 6(c).

Figure 7 shows the change in the distribution of the local moments. When $U/t\lesssim 1$, the distribution is similar to that in the weak coupling limit $U/t\rightarrow+0$. Namely, a sharp peak appears at $m<0$ (the $w$ sublattice), while some peaks appear at $m>0$ (the $b$ sublattice). When $U/t\gtrsim 3$, distinct behavior appears in the magnetic distribution. In the strong coupling case, the local magnetization should be classified into some groups. In the $b$ sublattice with $m>0$, the magnetization at the A vertices is distinct from that at the B and C vertices, and this behavior appears in the whole parameter space. On the other hand, in the $w$ sublattice with $m<0$, the magnetization is classified into three groups characteristic of the coordination number $z$. Namely, $m\sim-0.37$ for D and G vertices with $z=4$, $m\sim-0.36$ for the E vertices, and $m\sim-0.35$ for the F vertices when $U/t=5$, as shown in Figure 6(c). This is distinct from the weak coupling case.

The crossover between the weak and strong coupling regimes occurs around $U/t\sim 2$. In the strong coupling limit $U/t\rightarrow\infty$, the Hubbard model is reduced to the antiferromagnetic Heisenberg model with nearest-neighbor couplings $J=4t^{2}/U$. The mean-field ground state is described by the staggered moment $m_{j}\rightarrow\pm 1/2$. This means that the mean-field approach cannot correctly describe the reduction of the magnetic moment due to quantum fluctuations. Therefore, an alternative method is necessary to clarify magnetic properties in this regime, which is beyond the scope of the present study. Nevertheless, interesting magnetic properties inherent in the $H_{00}$ hexagonal golden-mean tiling, eg. the ferromagnetically ordered state in the weak coupling limit, can be captured even in our simple mean-field method.

Finally, we wish to demonstrate the spatial profile of the magnetizations characteristic of the $H_{00}$ hexagonal golden-mean tiling. To this purpose, we map the tiling to perpendicular space ${\bm{r}}^{\perp}$, where the positions in perpendicular space have one-to-one correspondence with the positions in physical space. We have previously shown that there are four ${\bm{r}}^{\perp}$ windows, which can be labelled by pairs of integer heights  Coates et al. (2023), which we re-label here. These heights correspond to where each vertex of the tiling projects onto the body-diagonals of the two 3–dimensional cubes which can be formed by the 6–dimensional super-space basis vectors ${\bm{n}}=(n_{0},n_{1},\cdots,n_{5})$  Coates et al. (2023). Thus, the four ${\bm{r}}^{\perp}$ planes of the $H_{00}$ hexagonal golden-mean tiling are described by these heights as ${\bm{r}}^{\perp}=(x^{\perp},y^{\perp})$ where $x^{\perp}=0,1$ and $y^{\perp}=0,1$. The A, B, and C vertices uniquely occupy the (0, 0) and (1, 1) planes, with the remaining vertices occupying the remaining planes, which is schematically shown in Figure 8.

The magnetization profiles of the (0, 0) and (0, 1) planes in perpendicular space are shown in Figure 9, where we show the absolute values of the local magnetizations. As the (1, 0) and (1, 1) planes are equivalent, it is unnecessary to show them. When $U/t=1.0\times 10^{-7}$, the system is essentially the same as that with $U\rightarrow 0$, where no magnetization appears in the planes (0, 1) and (1, 0) for the $w$ sublattice. This is consistent with the fact that the extended states have little effect on the magnetic properties in the $w$ sublattice. By contrast, finite magnetization appears across the entirety of the (0, 0) and (1, 1) planes, implying that the spontaneous magnetizations appear in the $b$ sublattice. Therefore, we can say that the ferromagnetically ordered state is realized in the weak coupling limit.

We also find a spatial pattern in the B and C vertex regions in the (0, 0) and (1, 1) planes, and, a spatial pattern with a tiny difference appears in the magnetic moments in the A region, as shown in Figure 10. These suggest the existence of many kinds of confined states in relatively large regions. This is because the overlapping structure in the confined states should classify the vertices into hierarchical groups, which yields a detailed structure in perpendicular space, distinct from the simple pattern for the vertices (see Figure 8).

Upon increasing the interaction strength, all vertex sites have magnetizations, as shown in Figure 9(b). In the strong coupling case, the Coulomb interactions become crucial to stabilize the ferrimagnetically ordered states with staggered moments. When $U/t=5$, the local magnetization takes large values. In this case, the magnitude of local magnetizations can be classified into two groups in the $b$ sublattice and three groups in the $w$ sublattice, discussed above.

Before concluding, we would like to summarize and compare the magnetic properties in the Hubbard models on the Penrose, Ammann-Beenker, Socolar dodecagonal, $H_{\frac{1}{2}\frac{1}{2}}$ hexagonal golden-mean, and $H_{00}$ hexagonal golden-mean tilings. One of the common features is the existence of confined states at $E=0$ in the noninteracting case ($U=0$), which play a crucial role in stabilizing the magnetically ordered states in the weak coupling limit. Nevertheless, their confined state properties are distinct from each other. The number of types of confined states are six in the Penrose case Kohmoto and Sutherland (1986); Arai et al. (1988), while it should be infinite in the others. As for sublattice structures, the $H_{\frac{1}{2}\frac{1}{2}}$ hexagonal golden-mean tiling has a sublattice imbalance, leading to a ferrimagnetically ordered state even in the weak coupling limit Koga and Coates (2022). In our $H_{00}$ tiling, however, there exists a sublattice imbalance such that the confined states appear in one of the sublattices, leading to a ferromagnetically ordered state in the weak coupling limit. The other tilings have an equivalent sublattice structure, and the corresponding Hubbard model shows the antiferromagnetically ordered state without a uniform magnetization.

We have studied magnetic properties in the half-filled Hubbard model on the $H_{00}$ hexagonal golden-mean tiling by means of the real-space mean-field approach. We have found the delta-function peak in the density of states of the tight-binding model, implying the existence of macroscopically degenerate confined states at $E=0$. We have then clarified that two extended states exist in the $w$ sublattice and the confined states appear only in the $b$ sublattice. For the above properties, we have obtained the exact fraction of the confined states as $1/2\tau^{2}$. The introduction of the Coulomb interaction lifts the macroscopic degeneracy at the Fermi level and drives the system to a ferromagnetically ordered state. We have clarified how the spatial distribution of the magnetizations continuously changes with increasing interaction strength. Crossover behaviour in the magnetically ordered states has been discussed by applying perpendicular space analysis to the local magnetizations.