Symmetry protected exceptional points of interacting fermions

Non-Hermitian quantum systems can exhibit spectral degeneracies known as exceptional points, where two or more eigenvectors coalesce, leading to a nondiagonalizable Jordan block. It is known that symmetries can enhance the abundance of exceptional points in noninteracting systems. Here we investigate the fate of such symmetry protected exceptional points in the presence of a symmetry preserving interaction between fermions and find that (i) exceptional points are stable in the presence of the interaction. Their propagation through the parameter space leads to the formation of characteristic exceptional “fans.” In addition, (ii) we identify a new source for exceptional points which are only present due to the interaction. These points emerge from diagonalizable degeneracies in the noninteracting case. Beyond their creation and stability, (iii) we also find that exceptional points can annihilate each other if they meet in parameter space with compatible many-body states forming a third order exceptional point at the endpoint. These phenomena are well captured by an “exceptional perturbation theory” starting from a noninteracting Hamiltonian.

Figure 1

(a) Non-Hermitian Hamiltonian with hopping $t=e^{i3/4\pi }/\sqrt{2}$ and interaction $U\in \mathbb{R}$. The hopping $j\to j+1$ picks up a phase defined by the twist angle $\phi \in [0,2\pi )$. (b) and (c) Propagation of EPs of two interacting fermions (dark lines) as a function of $\phi$ ($\mathrm{\Delta }=0.02$) and $U$ for sizes $L=6$ (b) and $L=18$ (c). EPs forming the fan feature (i) emanate from the same symmetry protected EP for $m=0.7$ at ${\phi }_{\text{e}}$ (cf. Fig. 2). EPs can emerge from diagonalizable degeneracies (ii), and annihilate each other (iii). The color scale indicates the minimal angle ${\alpha }_{ij}=arccos\left(\right|\langle {\mathrm{\Psi }}_i^R|{\mathrm{\Psi }}_j^R\rangle \left|\right)$ between two right eigenvectors.

Figure 2

Complex eigenenergies $E_{(k,\pm )}$, Eq. (3), of the single particle Bloch Hamiltonian in Eq. (1) for $m=0.7$. For simplicity we show here continuous momenta of an infinite chain, so that we do not need to rely on the twist angle $\phi$ here. $k_{\text{e}}=2arctan[(10+\sqrt{151})/17]$ indicates the single particle EP shown in Fig. 1.

Figure 3

Comparison of our analytic predictions (dashed lines) of EPs to the numerical simulation for $L=6$ and $m=0.7$. We identify three different twist angles where (i) EPs are inherited from the single particle spectrum at ${\phi }_{\text{e}}$, (ii) emerge from a diagonalizable degeneracy at ${\phi }_{\text{d}}$, and (iii) annihilate each other at ${\phi }_{\text{a}}$. Panel (a) and (b) show the analytical predictions from Eqs. (8) and (12) and identify the states forming the exceptional lines.

Figure 4

EPs generated by two interacting fermions in a system of length $L=3$ (left) and $L=6$ (right). The color code in (a) and (b) indicates the minimal angle enclosed by two eigenvectors for each point in the parameter space spanned by the interaction strength $U$ and twist angle $\phi$. Dark blue lines indicate the path of EPs which are inherited from the single particle spectrum (i) ${\phi }_{\text{e}}$, or emerge from a diagonalizable degeneracy at $U=0$ (ii) ${\phi }_{\text{d}}$, and form an endpoint at (iii) ${\phi }_{\text{a}}$ and $U_{\text{a}}\ne 0$. We evaluate the minimal angle on the circles (parametrized by $\theta$) drawn in (a) [(b)] and show the results on a semilogarithmic scale in (c) [(d)] for $m=0.7$. The point $P$ refers to Fig. 8.

Figure 5

Spreading of exceptional lines for $L=7$ and $m=0.6$ with a Hermitian interaction ($U\in \mathbb{R}$) in (a) and an anti-Hermitian interaction ($U\in i\mathbb{R}$) in (b). The EP located in the single particle spectrum is obtained for $k_{\text{e}}=6$ and ${\phi }_{\text{e}}=2\{\pi +arctan[(5-\sqrt{41})/8]\}L-2\pi k_{\text{e}}$, $\mathrm{\Delta }=0.05$. The predicted trajectories of EPs are plotted as dashed lines according to the interaction strength in Eq. (B11).

Figure 6

We are analyzing the nontrivial eigenvalues of the effective Hamiltonian $H_{3\times 3}^{\text{(ii)}}$ (left panels) and $H_{4\times 4}^{\text{(ii)}}$ (right panels). The lines of emergent EPs (dashed blue/red) are predicted via Eqs. (B19) and (B20). The left (right) panels are showing a system with $L=3$ ($L=6$) sites for $m=0.7$ and are modeled by an effective Hamiltonian of size $3\times 3$ ($4\times 4$). We can identify the analytical eigenvalues [Eq. (B17)] with ${\lambda }_{\text{a}}=0$, ${\lambda }_{+}=\delta$, and ${\lambda }_{-}=-\delta$ and the corresponding eigenstates $|{\mathrm{\Psi }}_{\text{a}}\rangle$, $|{\mathrm{\Psi }}_{+}\rangle$, and $|{\mathrm{\Psi }}_{-}\rangle$. Starting from $U=0$, we can track the eigenvalues continuously using Riemann surfaces as shown in the lower panels. We track the eigenvalues from $P_0$ at ${\phi }_{\text{p}}$ and $U=0$ to a finite $U$ at $P_1$. The initial eigenvalues are marked by crosses and the color shading from light to dark indicates the transition from $P_0\to P_1$ in (c) and (d). We find that the EP (red dashed line) is formed by states which are initially associated with ${\lambda }_{\text{a}}=0$ and ${\lambda }_{-}=-\delta$. While the left panel reveals only one intersection of the eigenvalues, the right panel shows two intersections. The second solution of Eq. (B19) is imaginary and therefore, only one is present in the case of a Hermitian interaction. The gray dots in the lower panels are showing the eigenvalues of the full Hamiltonian evaluated from $P_0\to P_1$ which agree remarkable well with the effective description.

Figure 7

Comparison of our analytic predictions (dashed lines) of EPs to the numerical simulation for $L=3$ and $m=0.7$. We identify three different twist angles where (i) EPs are inherited from the single particle spectrum at ${\phi }_{\text{e}}$, (ii) emerge from a diagonalizable degeneracy at ${\phi }_{\text{d}}$; and (iii) annihilate each other at ${\phi }_{\text{a}}$. Panel (a) and (b) show the analytical predictions from Eqs. (B11) and (B19) and identify the states forming the exceptional lines.

Figure 8

Robustness of the eigenvector associated with the lines of EPs for a system of $L=6$ sites and $m=0.7$ (cf. Fig. 4). Each reference point ($P$, $A_0$, $A_1$, and $B_0$) refers to an EP which can be found in Fig. 4. We determine the eigenvector $|{\mathrm{\Psi }}_{\text{EP}}^R\rangle$ for the reference point and calculate the minimal angle between all eigenstates for each point in the parameter space spanned by the twist angle $\phi$ and interaction strength $U$. The quantifier is given by ${\text{min}}_i{\beta }_i={\text{min}}_{i}arccos\left(\right|\langle {\mathrm{\Psi }}_i^R|{\mathrm{\Psi }}_{\text{EP}}^R\rangle \left|\right)$.

Figure 9

Numerical evaluation of the effective Hamiltonian $H_{\text{even}}^{\text{(iii)}}$ in Eq. (B22) where we determine the minimal angle between its eigenstates. The perturbative description captures all phenomena including the heredity (i), emergence (ii), and annihilation (iii) of EPs in a system with $L=6$ sites and $m=0.7$. The corresponding analytical predictions $U_{(q,\pm )}^m$ in Eq. (B11) derived from $H^{\text{(i)}}$ and $U_{\{{\mathrm{\Psi }}_{\text{a}},{\mathrm{\Psi }}_{-}\}}$ in Eq. (B20) derived from $H_{4\times 4}^{\text{(ii)}}$ agree with the numerical calculation.

Figure 10

(a) ($L=3$) and (b) $(L=6)$ are showing the minimal angle enclosed by all eigenvectors of a system parametrized on a sphere around the endpoints (iii) in Fig. 4 which is extended to a non-Hermitian interaction, $U\in \mathbb{C}$. The three-dimensional sphere is described using spherical coordinates ($\nu$ and $\eta$) via $[\phi ,\text{Re}\left(U\right),\text{Im}\left(U\right)]=[r_{\phi }cos\left(\nu \right)sin\left(\eta \right)+{\phi }_{\text{a}},r_{U}sin\left(\nu \right)sin\left(\eta \right)+U_{\text{a}},r_{U}cos\left(\eta \right)]$ where the endpoint (iii) is located at ${\phi }_{\text{a}}$ and $U_{\text{a}}$. We highlighted the incoming EPs in black which correspond to $A_1$ and $B_1$ in Fig. 4 and the outgoing points in red. We used a different radius compared to Fig. 4. $\eta =\pi /2$ corresponds to the Hermitian interaction $U\in \mathbb{R}$. We illustrated the two incoming (black) and the two outgoing (red) EPs on the three-dimensional sphere in (c) ($L=3$) and (d) ($L=6$).

Figure 11

The figure is showing the minimal angle for a system of $L=3$ sites with $m=0.7$ with three fermions. While panel (a) evaluates the full Hamiltonian, panels (b), (c), and (d) is restricted to the total conserved momentum $k_{\text{tot}}=0,1,2$ respectively. EPs are inherited from the single particle spectrum at $k_{\text{e}}=0$ and ${\phi }_{\text{e}}$. Panel (b) includes the predicted paths of EPs using Eq. (C7) for a total momentum of $k_{\text{tot}}=0$. Panels (c) and (d) reveal vertical lines of EPs which originate from $|{\mathrm{\Phi }}_2\rangle$ and $|{\mathrm{\Phi }}_1\rangle$. As in Fig. 7, EPs emerge from diagonalizable degeneracies at ${\phi }_{\text{d}}$ for $U=0$.

Figure 12

The figure is showing different disorder realization for $L=6$ sites and $m=0.7$. The disorder strength for panels (a), (b), (c), and (d) is $\delta =0.0,0.01,0.025,0.05$ respectively.