Discriminant indicator with generalized rotational symmetry

Consider an $N\times N$-Hamiltonian $H(\bm{k})$ for two-dimensional systems with generalized $C_{n}$ symmetry under the periodic boundary condition [70, 98, 72, 99, 97, 101, 51, 100, 61]. In this case, the Hamiltonian satisfies

with

and a unitary operator $U_{C_{n}}$ satisfying $U_{C_{n}}^{n}=\mbox{1}\mbox{l}_{N\times N}$.

The above non-Hermitian Hamiltonian may host exceptional points which are characterized by the discriminant number [59, 60, 78]

The integral is taken along a closed path in the BZ. The discriminant $\Delta(\bm{k})$ of the characteristic polynomial $P(\bm{k},E)=\det[H(\bm{k})-E\mbox{1}\mbox{l}_{N\times N}]=\sum_{j=0}^{N}a_{j}E^{j}$ is defined as

Here, $\epsilon_{n}$ denote the eigenvalues of $H(\bm{k})$. $\Delta(\bm{k})$ can be rewritten as

with $b_{j}=ja_{j}$ [102, 103]. Clearly, $\Delta(\bm{k})$ is determined by the coefficients of the characteristic polynomial. If $\nu$ takes a non-zero value, $\Delta(\bm{k})$ vanishes at a point inside of the loop. Such a point corresponds to an exceptional point because Eq. (4) indicates the band touching on this point.

In the presence of the generalized $C_{n}$ symmetry, the discriminant $\Delta(\bm{k})$ satisfies

which is proven in Sec. II.2.

By making use of Eq. (6), we obtain discriminant indicators for two-dimensional systems with generalized $C_{n}$ symmetry ($n=4$, $6$),

Here, $\nu_{C_{4}}$ and $\nu_{C_{6}}$ denote the discriminant number computed along the closed path illustrated in Fig. 1(a) [Fig. 1(b)]. Symbols $\Gamma$ and $M$ denote the high-symmetry points in the BZ [see Fig. 1]. Because of the constraint written in Eq. (6), the discriminant becomes real at $\Gamma$ and $M$ points.

We note that for $n=2$, the problem is reduced to the case for the generalized inversion symmetry, which is discussed in Appendix A. For systems with generalized $C_{3}$ symmetry, the Hamiltonian becomes Hermitian ¹¹1 This can be seen as follows, $H(\bm{k})=U_{C_{3}}^{3}H(\bm{k})U_{C_{3}}^{-3}=H^{\dagger}(R_{3}^{3}\bm{k})=H^{\dagger}(\bm{k})$.. In Secs. II.3 and II.4, we derive these indicators. These indicators can be computed without the input of the reference energy, and predict the presence of exceptional points.

For the systems with generalized $C_{n}$ symmetry, the polynomial $P(\bm{k},E)$ satisfies

Here, we have used $\det A^{T}=\det A$ and $\det(AB)=\det A\det B=\det B\det A$. We note that the discriminant can be computed only from coefficients of the polynomial. It results in Eq. (6). It is straight forward to extend the above arguments to three-dimensional cases. Specifically, for three-dimensional systems with generalized $C_{n}$ symmetry about the $z$axis, the discriminant satisfies

with $\bm{k}_{\|}^{T}=(k_{x},k_{y})$.

In order to derive Eq. (7a), let us consider the discriminant number evaluated along the closed path illustrated in Fig. 1(a). This integral can be decomposed into integrals along the path $\Gamma_{i}$ ($i=1,\cdots,4$),

Here, $\Gamma_{i}$ ($i=1,\cdots,4$) are the paths between the high-symmetry points [see Fig. 1(a)]. First, we note that $p_{2}+p_{3}$ vanishes because applying the generalized $C_{4}$ operator twice maps the path $\Gamma_{3}$ to the path $\Gamma_{3}^{\prime}$ which is equivalent to the path $\Gamma_{2}$ [see Fig. 1(a)] due to the periodicity of the BZ.

Second, let us evaluate $p_{1}+p_{4}$. The integration of $\Delta(\bm{k})$ along the path $\Gamma_{4}$ is mapped to that of $\Delta^{\ast}(\bm{k})$ along the opposite direction of $\Gamma_{1}$ by the generalized $C_{4}$ symmetry. As a result, the integration $p_{1}+p_{4}$ is simplified as

with some integer $N_{0}$. From the second to the third lines, we have used Eq. (6). We note that this discussion can be extended to the cases of generalized $C_{6}$ symmetry.

From these results, we obtain the relation

Here, we used the fact that $\Delta(\bm{k})$ is real at $\Gamma$ and $M$ points because of Eq. (6). Therefore, we obtain the discriminant indicator with generalized $C_{4}$ symmetry in Eq. (7a). We note that even if the exceptional points exist on the red line in Fig. 1(a), we can avoid by the continuous deformation of the integration path [24]. We also note that if $\Delta=0$, exceptional points emerge at these points.

In order to derive Eq. (7b), let us consider the discriminant number composed along the closed path illustrated in Fig. 1(b). We decompose the path for the integral into four paths $p_{i}$ ($i=1,\cdots,4$) in a similar way to the previous section [see Eq. (10)].

Firstly, we discuss the integration of $p_{2}+p_{3}$ in a similar way to the previous case [see Eq. (11)]. In the presence of the generalized $C_{6}$ symmetry, the integration of $\Delta(\bm{k})$ along $\Gamma_{2}$ is mapped to the one of $\Delta^{\ast}(\bm{k})$ along $\Gamma_{2}^{\prime}$. Additionally, from the periodicity in the BZ, this integration is mapped to one along the opposite direction of $\Gamma_{3}$. Thus, in a similar way to the previous case, $p_{2}+p_{3}$ is simplified as

with some integer $N_{0}$.

Secondly, let us simplify $p_{1}+p_{4}$. The integration of $\Delta(\bm{k})$ along $\Gamma_{1}$ is mapped to the integration of $\Delta^{\ast}(\bm{k})$ along the opposite direction of $\Gamma_{4}$ by the generalized $C_{6}$ operator. Hence, the integration $p_{1}+p_{4}$ is

with some integer $N_{0}^{\prime}$.

Putting Eqs. (13) and (14) together, we obtain Eq. (7b). Here, we used the fact that $\Delta(\bm{k})$ is real at the $\Gamma$ and $M$ points because of Eq. (6).

Let us see the discriminant number in the three-dimensional BZ and exceptional points from loops. In the presence of generalized $C_{4}$ symmetry about the $z$axis, the following indicator captures the exceptional loops crossing one of the planes at $k_{z}=0$ and $\pi$;

Here, $\Gamma$, $M$, $Z$, and $A$ denote high-symmetry points as shown in Fig. 1(c). We note that $\Delta(\Gamma)$, $\Delta(M)$, $\Delta(Z)$, and $\Delta(A)$ are real because of Eq. (9). All possible configurations of $\text{sgn}\Delta(\bm{k})$ giving $\xi_{C_{n},3D}=-1$ are shown in Appendix B. The above indicator is obtained by applying the argument in Sec. II.3 to the two-dimensional subsystems at $k_{z}=0$ and $\pi$ where generalized $C_{4}$ symmetry is preserved. In a similar way, we can introduce the indicator for systems with generalized $C_{6}$ symmetry,

where $\Gamma$, $M$, $A$, and $L$ are high-symmetry points in Fig. 1(d). We note that $\Delta(\Gamma)$, $\Delta(M)$, $\Delta(A)$, and $\Delta(L)$ are real because of Eq. (9).

Let us consider the two-dimensional toy model with generalized $C_{4}$ symmetry in Fig. 2(a), whose Hamiltonian is defined in Appendix C. In Fig. 2(a), the red (blue) dots denote the on-site potentials $i\delta$ ($-i\delta$). The red (blue) lines denote the hoppings $1+\delta t$ ($1-\delta t$).

The above model respects generalized $C_{4}$ symmetry whose rotation axis is illustrated by a cross in Fig. 2(a). The explicit form of $U_{C_{4}}$ is written in Appendix C.

The numerical results for discriminant indicators are displayed in Fig. 2(b). The discriminant indicator takes vales of $\xi_{C_{4},2D}=1$ and $-1$ in yellow and blue regions, respectively. We note that $\xi_{C_{4},2D}$ vanishes for $\delta t=0$ or $\delta=0$ since $\Delta(\bm{k})$ vanishes at the high-symmetry point in the BZ. Additionally, at the boundary of the phases, $\xi_{C_{4},2D}$ vanishes for the same reason. For $\xi_{C_{4},2D}=-1$, the exceptional points can be observed by computing $\text{arg}\Delta(\bm{k})/\pi$. As an example, we show the map of $\text{arg}\Delta(\bm{k})/\pi$ in momentum space for $(\delta,\ \delta t)=(0.1,\ 0.5)$ [see Fig. 2(c)]. Green and black dots in Fig. 2(c) represent exceptional points characterized by $\nu=1$ and $-1$, respectively. This result indicates that the discriminant number along the path in Fig. 1(a) takes a value of $-1$.

Figures 2(e) and 2(f) display the band structure along the line $k_{x}=k_{y}=k$ illustrated in Fig. 2(d). Figs. 2(e) and 2(f), we can find exceptional points.

We consider the two-dimensional toy model with generalized $C_{6}$ symmetry in Fig. 3(a), whose Hamiltonian is defined in Appendix C. In this figure, the red (blue) dots denote the on-site potentials $i\delta$ ($-i\delta$). The red (blue) lines denote the non-reciprocal hoppings. Specifically, the hopping along the red (blue) arrows is $1+\gamma_{1}$ ($1+\gamma_{2}$). The hopping opposite to the red (blue) arrows is $1-\gamma_{1}$ ($1-\gamma_{2}$). For the blue lines without arrows, the hopping is $1$ for both directions.

The above model respects generalized $C_{6}$ symmetry whose rotation axis is illustrated by a cross in Fig. 3(a). The explicit form of $U_{C_{6}}$ is written in Appendix C.

We show the numerical result for the discriminant indicator $\xi_{C_{6},2D}$ in Eq. (7b) for $\delta=0.1$ [see Fig. 3(b)]. In Fig. 3(b), there exist two phases where the discriminant indicator takes values of $\xi_{C_{6},2D}=1$ (yellow area) and $-1$ (blue area). We note that $\xi_{C_{6},2D}$ vanishes for $\gamma_{1}=0$ and $\gamma_{2}=0$ since $\Delta(\bm{k})$ vanishes at the high-symmetry point in the BZ. Additionally, at the boundary between phases, $\xi_{C_{6},2D}$ vanishes for the same reason. In the blue-colored region, the indicator predicts the presence of exceptional points. To be concrete, we compute $\text{arg}\Delta(\bm{k})/\pi$ in the momentum space for $(\delta,\ \gamma_{1},\ \gamma_{2})=(0.1,\ 0.3,\ 0.3)$ [see Figs. 3(c) and 3(d)]. Green and black dots in Fig. 3(d) represent exceptional points characterized by $\nu=1$ and $-1$, respectively. Figure 3(d) indicates that the discriminant number computed along the path in Fig. 1(b) takes a value of $-1$.

Figures 3(e) and 3(f) plot the band structure for $(\delta,\gamma_{1},\gamma_{2})=(0.1,\ 0.3,\ 0.3)$, which indicates the emergence of the exceptional point at the point denoted by the red dots in Figs. 3(e) and 3(f).

Consider the three-dimensional toy model with generalized fourfold rotational symmetry whose Hamiltonian reads

with

Here, $\sigma_{i}$ ($i=1$, $2$, $3$) denote the Pauli matrices. The above model respects generalized $C_{4}$ symmetry about the $z$axis with $U_{C_{4}}=\sigma_{3}$.

Figure 4(a) displays the numerical result for the discriminant indicator $\xi_{C_{4},3D}$. In Fig. 4(a), there exist two phases where the discriminant indicator takes values of $\xi_{C_{4},3D}=1$ (yellow area) and $\xi_{C_{4},3D}=-1$ (blue area). For $\xi_{C_{4},3D}=-1$, the exceptional loops crossing the plane for $k_{z}=0$ or $\pi$ can be obtained by computing the map of $\text{arg}\Delta(\bm{k})/\pi$. As an example, we show the maps of $\text{arg}\Delta(k_{x},k_{y},0)$ and $\text{arg}\Delta(k_{x},k_{y},\pi)$ in momentum space for $(d_{1},d_{2})=(1,1)$ [see Figs. 4(b) and 4(c)]. In Figs. 4(b) and 4(c), green and black dots represent exceptional loops characterized by $\nu=1$ and $-1$, respectively. By comparing these figures, we obtain the emergence of the exceptional loops crossing the plane for $k_{z}=\pi$.

Figure 4(d) displays the momentum points $\bm{k}$ satisfying $\text{abs}\Delta(\bm{k})<0.05$, indicating the emergence of exceptional loops. We note that exceptional loops predicted by the nontrivial value of $\xi_{C_{4},3D}$ merge at the line specified by $(k_{x},k_{y})=(0,0)$ or $(\pi,\pi)$, originating from the symmetry constraint on the exceptional points for fixed $k_{z}$ in three dimensions (for more details see Appendix D).

Consider the three-dimensional toy model with generalized sixfold rotational symmetry whose Hamiltonian reads

with

Here, $C(k_{x},k_{y})$ and $S(k_{x},k_{y})$ are defined as

The above model respects generalized $C_{6}$ symmetry about the $z$axis with $U_{C_{6}}=\sigma_{3}$.

Figure 5(a) displays the numerical result of the discriminant $\xi_{C_{6},3D}$. In Fig. 5(a), there exist two phases where the discriminant indicator takes values of $\xi_{C_{6},3D}=1$ (yellow area) and $\xi_{C_{6},3D}=-1$ (blue area). For $\xi_{C_{6},3D}=-1$, the exceptional loops crossing the plane for $k_{z}=0$ or $\pi$ can be obtained by computing the map of $\text{arg}\Delta(\bm{k})/\pi$. As an example, we show the maps of $\text{arg}\Delta(k_{x},k_{y},0)$ and $\text{arg}\Delta(k_{x},k_{y},\pi)$ in the momentum space for $(d_{1},d_{2})=(1,1)$ [see Figs. 5(b) and 5(c)]. In Fig. 5(c), green and black dots represent exceptional loops characterized by $\nu=1$ and $-1$, respectively. By comparing Fig. 5(b) with 5(c), we obtain the emergence of the exceptional loops crossing the plane for $k_{z}=\pi$.

Figure 5(d) displays the momentum points $\bm{k}$ satisfying $\text{abs}\Delta(\bm{k})<0.05$, indicating the emergence of exceptional loops. We note that, in contrast to the case of generalized $C_{4}$ symmetry in Sec. III.3, generalized $C_{6}$ symmetry does not predict the merging of the exceptional loops since the exceptional loops can cross the boundary of the BZ (for more details see Appendix D).