Spontaneous Formation of Exceptional Points at the Onset of Magnetism

We reveal how symmetry-protected nodal points in topological semimetals may be promoted to pairs of generically stable exceptional points (EPs) by symmetry-breaking fluctuations at the onset of long-range order. This intriguing interplay between non-Hermitian (NH) topology and spontaneous symmetry breaking is exemplified by a magnetic NH Weyl phase spontaneously emerging at the surface of a strongly correlated three-dimensional topological insulator, when entering the ferromagnetic regime from a high-temperature paramagnetic phase. Here, electronic excitations with opposite spin acquire significantly different lifetimes, thus giving rise to an anti-Hermitian structure in spin that is incompatible with the chiral spin texture of the nodal surface states, and hence facilitate the spontaneous formation of EPs. We present numerical evidence of this phenomenon by solving a microscopic multiband Hubbard model nonperturbatively in the framework of dynamical mean-field theory.

Figure 1

Formation of surface EPs from ferromagnetic fluctuations in a topological insulator setup with slab geometry. In the noninteracting case, a paramagnetic surface semimetal with Dirac dispersion is found around high-symmetry points in the Brillouin zone (left). Adding correlations, in a range of intermediate temperatures, the onset of ferromagnetism with small $z$ magnetization and sizable magnetic fluctuations stabilizes a non-Hermitian Weyl phase (right). There, the quasiparticle dispersion exhibits a pair of exceptional points along the $k_z$ direction, from which a characteristic square root dispersion emanates and which are connected by a flat non-Hermitian Fermi arc.

Figure 2

Magnetization $m_z$ as a function of $T$ for fixed filling $n=0.39$, $U=13$, $U^{\prime }=7$, $J=6$. The yellow shaded region represents the area where the magnetization is low enough to allow for EPs to appear at the Fermi energy. At larger $T$ (gray shaded), the system is in a paramagnetic phase. At smaller $T$ (white region), the system is ferromagnetic, but the difference in the real part of the spin-dependent self-energies is so high as to surmount the topological band gap.

Figure 3

Site-resolved spectral function on the real frequency axis, for different orbitals (color) and spin (full and dashed lines). The parameters of the model are $(U,U^{\prime },J,T,\mu )=(13.0,7.0,6.0,0.13,2.00)$. Top: refers to the slab edge. Bottom: refers to the bulk. The auxiliary orbital 1 (green) does not contribute spectral weight at the Fermi level, i.e., the physics is dominated by the EP and Fermi arc in the protected surface-state region. Top inset: an averaged spectral weight for orbitals 2 and 3, resolved for spin and site, is plotted around the Fermi level, comparing the surface layer (solid lines) and a bulk layer (dashed lines).

Figure 4

Effective Hamiltonian and self-energies for parameters $(U,U^{\prime },J,T,\mu )=(13.0,7.0,6.0,0.13,2.00)$ (a) Real part of the eigenvalue dispersion of the effective Hamiltonian near the Fermi level. The bands are plotted along the $k_z$ direction for the $k_x$ value at which the EPs sit. The two EPs have positive (solid circle) and negative (dotted circle) vorticity $\nu$, respectively. Between the EPs, the real energy dispersion of the surface bands is zero along a line (non-Hermitian Fermi arc) whose shape is shown in the inset. (b),(c) Real and imaginary parts of the orbital-diagonal self-energy of orbital 2 on the Matsubara axis (orbital 3 is analogous). Color encodes spin, and the multiple lines refer to the different planes of the slab. (d) Imaginary part of the eigenvalue dispersion of the effective Hamiltonian. Between the EPs, the energy dispersion of the surface bands is purely imaginary.