Higher-order Topological Hyperbolic Lattices

Recently, hyperbolic lattices have been experimentally realized in circuit quantum electrodynamics Houck2019nat and electric circuits Zhang2022NC ; Lenggenhager2022NC ; Boettcher2022arxiv , igniting great interest in study of various properties of hyperbolic lattices Houck2019CMP ; Park2020PRL ; Gorshkov2020PRA ; Rayan2021scia ; Matsuki2021JOP ; Rouxinol2021arxiv ; Rayan2022PNAS ; Gorshkov2022PRL ; Boettcher2022PRL ; Thomale2022PRB ; Urwyler2022arxiv ; Zhou2022PRB ; Mao2022arxiv ; Bzdusek2022PRB ; Mosseri2022PRB , such as topological properties, hyperbolic band theory and flat band properties. Different from a hyperbolic lattice, it is a well-known fact that only twofold, threefold, fourfold or sixfold rotational symmetry is permitted in two-dimensional (2D) crystalline materials. In other words, one can only use regular $p$-sided polygons with $p=3$, $4$ or $6$ to tessellate the 2D Euclidean plane. However, for a hyperbolic lattice with constant negative curvature, such a restriction is lifted so that one can use regular $p$-gons for any integer $p>2$ to tessellate a hyperbolic plane [see Fig. 1(a)]. As a result, any $p$-fold rotational symmetry can be realized in a hyperbolic lattice. This leads to a natural question of whether the enriched rotational symmetry of a hyperbolic lattice will result in any new topological phases beyond crystalline systems.

Recent generalizations of topological phases to the higher-order case provide us with an opportunity to study the effects of the enriched rotational symmetry. Different from the conventional first-order topological system, such a topological phase supports $(n-m)$-dimensional ($1<m\leq n$) gapless boundary modes for an $n$-dimensional system Taylor2017Science ; Fritz2012PRL ; ZhangFan2013PRL ; Slager2015PRB ; Brouwer2017PRL ; FangChen2017PRL ; Schindler2018SA ; Taylor2018PRB(R) ; Brouwer2019PRX ; Roy2019PRB ; Fulga2019PRL ; Yang2019PRL ; Roy2020PPR ; Xu2020PRL ; Parameswaran2020PRL ; Xu2020PPR ; Xu2020PPB ; AYang2020PRL ; Xu2020NJP ; Seradjeh2019PRB ; Wang2021PRL . For instance, a two-dimensional (2D) second-order topological insulator may support two, four or six zero-energy corner modes Taylor2017Science ; Yang2019PRL ; AYang2020PRL . Since the number of corner modes is closely related to the crystalline symmetry of a system, one thus may wonder whether the new rotational symmetry in hyperbolic lattices can allow for the existence of new higher-order topological phases that cannot exist in a crystalline material.

In this work, we theoretically demonstrate the existence of higher-order topological phases in hyperbolic lattices with eight-fold, twelve-fold, sixteen-fold or twenty-fold rotational symmetry by constructing and exploring tight-binding models on the lattices. Such Hamiltonians respect the combination of time-reversal symmetry and $p$-fold ($p=$8, 12, 16 or 20) rotational symmetry, which is not allowed in a crystalline lattice. While a quasicrystal may allow for the eight-fold or twelve-fold rotational symmetry, to the best of our knowledge, it is unclear whether sixteen-fold or twenty-fold rotational symmetry can occur there Sunbook . For clarify of presentation, we mainly focus on a hyperbolic $\{8,3\}$ lattice where regular $8$-gons are used to tessellate a hyperbolic plane such that each lattice site is connected to three neighboring sites [see Fig. 1(b)]. Note that for the Euclidean plane, only $\{3,6\}$, $\{4,4\}$ and $\{6,3\}$ lattices can achieve the tessellation. For the hyperbolic $\{8,3\}$ lattice, we find a gapped and a gapless higher-order topological hyperbolic phase by numerically computing the quadrupole moment and energy band properties of a four-dimensional (4D) momentum-space Hamiltonian based on the hyperbolic band theory Rayan2021scia . We further study a Hamiltonian on a hyperbolic lattice without translational symmetry and find the existence of eight-fold degenerate zero-energy modes localized at eight corners in a phase with a finite edge energy gap. The topology of this phase is characterized by the corner charge. Interestingly, there also appears a gapless phase with vanishing edge energy gap where states near zero energy are mainly localized on the boundary of the hyperbolic lattice. In addition, the real space results also show a reentrant gapped topological phase with zero-energy corner modes, which may arise from finite-size effects. Finally, we show the existence of twelve, sixteen or twenty zero-energy corner modes in a hyperbolic lattice with the corresponding rotational symmetry.

Model with hyperbolic translational symmetry.—To demonstrate the existence of higher-order topological phases in hyperbolic lattices, we will construct two types of tight-binding models in a hyperbolic lattice described by the Poincaré disk model [see Fig. 1(b)]. We start by constructing the first tight-binding model in a hyperbolic ${\{8,3\}}$ lattice with hyperbolic translational symmetry by following Ref. Urwyler2022arxiv . We first construct the onsite and hopping term inside the first unit cell [the region enclosed by the green curve in Fig. 1(b)], ${H}_{0}=\sum_{\alpha,\beta}[\sum_{i}m|1i\alpha\rangle[\tau_{z}\sigma_{0}]_{\alpha\beta}\langle 1i\beta|+\sum_{\left\langle i,j\right\rangle}|1i\alpha\rangle[T(\theta_{ij})]_{\alpha\beta}\langle 1j\beta|]$, where $|ri\alpha\rangle$ denotes the state of the $\alpha$th degree of freedom at the site $i$ in the $r$th unit cell in the Poincaré disk. At each site, there are four degrees of freedom, and $\left\{\tau_{\nu}\right\}$ and $\left\{\sigma_{\nu}\right\}$ with $\nu=x,y,z$ are two sets of Pauli matrices that act on these degrees of freedom. In $H_{0}$, the first term $m\tau_{z}\sigma_{0}$ describes the on-site energy, and the second term depicts the hopping between two neighboring connected sites inside the unit cell with the hopping matrix $T(\theta_{ij})=[t_{0}{\tau_{z}}{\sigma_{0}}-{\text{i}}{t_{1}}(\cos\theta_{ij}\tau_{x}\sigma_{x}+\sin\theta_{ij}\tau_{x}\sigma_{y})+g\cos(p\theta_{ij}/2)\tau_{y}\sigma_{0}]/2$ with $p=8$, where $\theta_{ij}$ is the polar angle of the vector from the site $j$ to site $i$ in the first unit cell. In the following, we will set the system parameters $t_{0}=t_{1}=1$ as the units of energy.

For the hopping term between the sites in the first unit cell and the sites in the neighboring four unit cells described by the set $S_{1}$, we define it as $H_{1}=\sum_{r\in S_{1}}\sum_{\alpha,\beta}\sum_{\left\langle i,j\right\rangle}|ri\alpha\rangle[T(\tilde{\theta}_{(ri),(1j)})]_{\alpha\beta}\langle 1j\beta|+\text{H.c.}$. To ensure that the system has $C_{p}T$ symmetry (here $p=8$) where $C_{p}$ is the $p$-fold rotational operator and $T$ is the time-reversal operator, we have to consider a modified $\tilde{\theta}_{(ri),(1j)}$ (see Supplemental Material Sec. S-2). The entire Hamiltonian can be generated by applying translational operations (generated by generators $\gamma_{1},\dots,\gamma_{4}$).

When $g=0$, the system respects in-plane mirror symmetry $M_{1}=\tau_{z}\sigma_{z}$, time-reversal symmetry ${T}=i\sigma_{y}\kappa$ where $\kappa$ is the complex conjugate operator, chiral symmetry $\Gamma=\tau_{x}\sigma_{z}$, and thus particle-hole symmetry $\Xi=\tau_{x}\sigma_{x}\kappa$. Owing to the eight-fold rotational symmetry about the $z$ axis preserved by the hyperbolic lattice, the Hamiltonian also respects the eight-fold rotational symmetry ${C}_{p}=\tau_{0}e^{-i\frac{\pi}{p}\sigma_{z}}R_{p}$ where $R_{p}|i\alpha\rangle\equiv|g_{p}(i)\alpha\rangle$ with $g_{p}(i)$ rotating the lattice site $i$ by an angle $2\pi/p$ about the $z$ axis (here $p=8$). With these internal symmetry, the system belongs to the DIII class corresponding to a $\mathbb{Z}_{2}$ topological insulator whose nontrivial phase exhibits helical edge modes. To generate a higher-order phase with corner modes, we add the term $g\cos(4\theta_{ij})\tau_{y}\sigma_{0}$ to break the time-reversal symmetry so as to open the gap of the helical modes at a boundary; this term thus acts as an edge mass term. As this term changes its sign once $\theta_{ij}$ increases by $\pi/4$, a corner state may arise at the location where the mass flips its sign. Since the change of sign occurs eight times, total eight corner modes may appear. While the term also breaks the $C_{8}$ symmetry, the combination of $T$ and $C_{8}$ symmetry is still conserved. The symmetry ensures that the number of corner modes must be an integer multiple of eight given the fact that if there is a zero-energy corner mode $|\psi_{c}\rangle$ mainly localized at ${\bf r}$, then $C_{8}T|\psi_{c}\rangle$ is also a zero-energy corner mode mainly localized at $g_{8}({\bf r})$.

We now employ the twisted boundary conditions to construct the momentum-space Bloch Hamiltonian based on the hyperbolic band theory Rayan2021scia ; Rayan2022PNAS ; Mao2022arxiv . In the Hamiltonian, the hopping between two sites in two different unit cells which are connected by a translation operator $\gamma_{j}$ should carry an extra phase term $e^{-\text{i}k_{j}}$. We thus obtain a $64\times 64$ Hamiltonian $H({\bf k})=H(k_{1},k_{2},k_{3},k_{4})$ in a four-dimensional Brillouin zone with $k_{j}\in[0,2\pi]$ for $j=1,\dots,4$ (see Supplemental Material Sec. S-2) .

While time-reversal symmetry is broken in the Hamiltonian $H$, chiral symmetry is still preserved so that $H({\bf k})$ respects chiral symmetry. In light of the fact that the quadrupole moment Cho2019PRB ; Hughes2019PRB is protected to be quantized by chiral symmetry Xu2021PRB ; Shen2020PRL , we can utilize the quadrupole moment to characterize the topological property of $H({\bf k})$ spanned by two of the four momenta. Specifically, one can regard $H({\bf k})$ spanned by $k_{i}$ and $k_{j}$ ($i,j\in{1,\dots,4}$ and $i\neq j$) with the other two momenta $k_{\bar{i}}$ and $k_{\bar{j}}$ fixed as the momentum-space version of a Hamiltonian $H_{s}$ in an $L\times L$ square lattice. The quadrupole moment for the occupied states is defined as Cho2019PRB ; Hughes2019PRB ; Xu2021PRB ; Shen2020PRL

where $U_{o}=\left(|\psi_{1}\rangle,|\psi_{2}\rangle,\cdots,|\psi_{n_{c}}\rangle\right)$ with $|\psi_{j}\rangle$ being the $j$th occupied eigenstate of $H_{s}$ (one of $n_{c}=32L^{2}$ occupied states), and $\hat{D}=\mathrm{diag}\left\{e^{2\pi\text{i}x_{j}y_{j}/L^{2}}\right\}_{j=1}^{64L^{2}}$ with $(x_{j},y_{j})$ denoting the square position of the $j$th lattice site. Here, $Q_{0}$ is the contribution from the background positive charge distribution.

To distinguish between an insulating phase and a semimetal phase, we calculate the average quadrupole moment over all fixed momenta $(k_{\bar{i}},k_{\bar{j}})$

The system respects $C_{8}M_{1}$ symmetry, that is, $U_{C_{8}M_{1}}H(k_{1},k_{2},k_{3},k_{4})(U_{C_{8}M_{1}})^{-1}=H(-k_{4},k_{1},k_{2},k_{3})$. It follows that $\overline{Q}_{ij}$ should satisfy the following relations (see Supplemental Material Sec. S-3 for proof):

Note that a similar relation for the Chern number has been derived in Ref. Urwyler2022arxiv . We find that $\overline{Q}_{13}$ is always equal to zero and thus use $\overline{Q}_{12}$ to characterize the topological property.

The average quadrupole moment illustrates a sharp decline from a quantized value of $0.5$ to a nonzero fractional value as $g$ increases as shown in Fig. 2(a), indicating the presence of two distinct phases. One phase is a higher-order topological hyperbolic insulator with $\overline{Q}_{12}=0.5$. The other one is a higher-order topological hyperbolic semimetal with vanishing energy gap in the 4D momentum space footnote . In fact, there are several degenerate nodes in momentum space. In the Brillouin zone spanned by $(k_{3},k_{4})$, a part has the quadrupole moment $Q_{12}$ of $0.5$ and the other part has zero $Q_{12}$, leading to a fractional value of the average quadrupole moment (see Supplemental Material Sec. S-4). The existence of the gapless phase is in stark contrast to the higher-order phase on Euclidean square $\{4,4\}$ lattices where the system is always gapped as we increase the term that breaks the time-reversal symmetry Taylor2017Science ; Schindler2018SA .

Model without hyperbolic translational symmetry.—For the model with translational symmetry, while the hyperbolic band theory predicts the existence of higher-order topological phases, we find that its energy spectrum in real space changes dramatically for different system sizes possibly due to large boundary effects (see Supplemental Material Sec. S-5). We therefore construct another tight-binding model Hamiltonian in a hyperbolic ${\{p,q\}}$ lattice as

where $|i\alpha\rangle$ denotes the state of the $\alpha$th degree of freedom at the site $i$ in the disk, and $\theta_{ij}$ is the polar angle of the vector from the site $j$ to the site $i$, rather than a modified one. This model respects the $C_{p}T$ symmetry. Note that $p$ should be an integer multiple of 4 to ensure that the Hamiltonian is Hermitian.

To illustrate the existence of zero-energy corner modes, we calculate the eigenenergies and eigenstates of the Hamiltonian in Eq. (4) in real space under open boundary conditions. We further compute the local density of states (DOS) defined as $\rho(E,\textbf{r})=\sum_{i,j}\delta(E-E_{i})|\Psi_{i,j}(\textbf{r})|^{2}$, where $\Psi_{i,j}(\textbf{r})$ is the $j$th component of the $i$th eigenstate at site r with the eigenenergy $E_{i}$.

The energy spectrum in Fig. 2(b) shows the existence of eight-fold degenerate zero-energy states when $0<g<1.15$. These modes are mainly localized near the corner positions on the boundary at the polar angle $\theta_{c}=\pi/8+n\pi/4$ with $n=0,1,\dots,7$ as shown by the local DOS in Fig. 3(a). Such a feature is also revealed by the average local DOS near a corner and the center of an edge (e.g., $\theta=0$) (here an edge refers to the collection of the boundary sites between two nearest neighboring corners). When $g=0$, the system is in the first-order topological phase, and thus the gapless edge states are almost equally distributed on the edge and corner sites. As we increase $g$, the energy gap at the edge is opened, leading to the appearance of corner modes, reflected by the increase (decrease) of the average local DOS at the corner (the center of an edge) [Fig. 2(c)]. In this regime, the average local DOS in the bulk almost vanishes.

To further characterize its topology, we calculate the corner charge $C_{S}$ (see Supplemental Material S-6 for its definition) and find that it approaches the quantized value of $0.5$ as shown in the inset of Fig. 2(c). In the Supplemental Material S-7, we show that with increasing the system size, while the mini-gap (determined by the first nonzero eigenenergy) declines, the edge energy gap (determined by the energy where the degeneracy suddenly drops from the eight-fold to the double one) remains almost unchanged. More interestingly, for a larger system, the corner charge becomes closer to $0.5$, indicating a better topological behavior, which arises due to a significant drop of the energy splitting of the zero-energy states. These results strongly suggest the existence of the topological phase in the thermodynamic limit. See Supplemental Material S-7 for finite-size analysis of the energy gap and local DOS.

As we further increase $g$, the energy spectrum in Fig. 2(b) becomes continuous near zero ($1.15<g<1.44$), leading to a gapless phase (vanishing edge energy gap) with finite local DOS in the bulk [Fig. 2(c)]. The local DOS at zero energy in Fig. 3(b) exhibits the distribution mainly localized on boundaries, implying that the gapless modes are mainly comprised of boundary modes including the states localized at corner positions and other positions at the boundary. Such a fact is also revealed by a significant rise of the average local DOS near a corner and the center of an edge when we enter into this regime [see Fig. 2(c)].

To show how the states near zero energy affects the local DOS when the system enters into the gapless phase, we approximate the density of an eigenstate by $n_{i}(\theta)=\sum_{j}|\Psi_{i,j}|^{2}\approx a+b\cos(8\theta)$, which is the expansion of the density in a Fourier series up to the first order. Here, we write the density as a function of the polar angle $\theta$ in the Poincaré disk for each site on the boundary, and $n_{i}(\theta+\pi/4)=n_{i}(\theta)$ due to the $C_{8}T$ symmetry. For this approximate density, if $b<0$, then it takes a maximum value at the corner position, i.e., $\theta=\theta_{c}$; if $b>0$, then the maximum occurs at the center of two neighboring corners, i.e., $\theta=\theta_{e}=n\pi/4$ with $n$ being an integer. For each state, we calculate the Fourier coefficient $b$ and then use the states with $b<0$ ($b>0$) to calculate the corresponding local DOS $\rho_{-}$ ($\rho_{+}$). We then expand $\rho_{\pm}$ as $c_{\pm}+d_{\pm}\cos(8\theta)$ and plot the Fourier coefficient $d_{\pm}$ in Fig. 2(d). In the region $0<g<1.15$, $d_{-}$ ($d_{+}$) is finite and negative (vanishes), consistent with our previous results that the eight zero-energy modes are mainly localized at the position near $\theta_{c}$. When the system enters into the gapless regime, we see a sharp rise of $d_{+}$, suggesting the appearance of states near zero energy with large occupation close to the center of an edge (see Supplemental Material Sec. S-8 for more discussion).

In Fig. 2(b), we also see that with the further increase of $g$ until 2, the energy spectrum becomes gapped again with eight zero-energy states separated from the other states. The phase arises because of the overlap between edge wave functions, leading to the energy gap opening of the edge states. Such an overlap is reflected by the sudden drop of the proportion of edge states on the boundary with respect to $g$ [see Fig. S9(a) in Supplemental Material]. In other words, the finite-size effect opens the gap of the edge wave functions, leaving the corner modes at zero energy [also evidenced by the zero-energy local DOS in Fig. 3(c)]. The finite-size effect will be reduced by increasing the system size so that the gapless regime becomes smaller (see Supplemental Material Sec. S-7). In addition, when $g>2$, zero-energy corner modes bifurcate into four branches away from zero energy, and no corner states are observed in this phase [Fig. 3(d)] (see Supplemental Material Sec. S-7).

Higher-order topological phases with twelve, sixteen or twenty corner modes.—We now proceed to study the higher-order topological phase in a hyperbolic lattice respecting twelve-fold, sixteen-fold or twenty-fold rotational symmetry by constructing the Hamiltonian in hyperbolic $\{12,3\}$, $\{16,3\}$, or $\{20,3\}$ lattices. Such Hamiltonians respect chiral symmetry and the corresponding $C_{p}T$ symmetry with $p=12$, $16$, or $20$. Figure 4 illustrates the local DOS at zero energy in the Poincaré disk, indicating the existence of twelve, sixteen and twenty corner modes, respectively (see the energy spectrum in Supplemental Material S-9). All these phases arise from the allowed rotational symmetry of a hyperbolic lattice, and thus cannot exist in a crystalline material. Even in a quasicrystal, it is unclear whether sixteen-fold or twenty-fold rotational symmetry can occur, to the best of our knowledge Sunbook .

In summary, we have theoretically predicted higher-order topological phases in hyperbolic lattices with eight-fold, twelve-fold, sixteen-fold or twenty-fold rotational symmetry, which have eight, twelve, sixteen and twenty corner modes, respectively. Such phases are not allowed in a crystalline material. For the hyperbolic $\{8,3\}$ lattices, we identify a higher-order topological phase with a finite edge energy gap and a gapless phase. Given that hyperbolic lattices have been experimentally realized in circuit quantum electrodynamics Houck2019nat and electric circuits Zhang2022NC ; Lenggenhager2022NC ; Boettcher2022arxiv , higher-order topological phases in hyperbolic lattices may be observed in these systems. In fact, higher-order topological phases in square lattices have been observed in phononic Huber2018Nat , microwave Bahl2018Nat , electric circuit Thomale2018NP , and photonic systems Hafezi2019NP . Our work may also inspire the interest of studying higher-order topological phases in hyperbolic lattices in quantum simulators, such as cold atom systems Browaeys2016Science ; Lukin2016Science .

In the Supplemental Material, we will give a pedagogical introduction to how the hyperbolic lattices on the Poincaré disk are generated based on circular inversion in Section S-1, show how the modified angle is defined and present the momentum space Hamiltonian in Section S-2, prove that $\overline{Q}_{12}=\overline{Q}_{23}=\overline{Q}_{34}=\overline{Q}_{41}$ in Section S-3, provide more detailed discussion on the band and topological properties of the higher-order topological hyperbolic semimetal phase in the 4D momentum space in Section S-4, provide the energy spectrum and local DOS for the model with translational symmetry calculated in a geometry with open boundaries in Section S-5, utilize the corner charge to characterize the gapped topological phases in Section S-6, show that the higher-order topological phase continues to exist in a much larger system even though the mini-gap becomes very small and how the reentrant gapped phase and the branching arise in Section S-7, provide the density profile of the states in the vicinity of zero energy in the gapless phase in Section S-8, present the energy spectra of the hyperbolic {12,3}, {16,3} and {20,3} lattices with respect to the system parameter $g$ in Section S-9, and finally present the effect of weak disorder on topological phases in Section S-10.