Exceptional points and spectral cusps from density-wave fluctuation

Abstract

We report two types of singularities that arise from fluctuations during the formation of charge- or spin-density waves. The first is the exceptional point (EP), corresponding to a higher-order pole of the retarded Green's function. Such EPs lead to algebraic corrections in the decay of quasiparticle occupations and are observable through time-resolved angle-resolved photoemission spectroscopy (Tr-ARPES). The second is a spectral cusp, defined by the coalescence of three extrema in the real-frequency spectral function $A(\mathbf{k}, ω)$. This cusp enforces the formation of Fermi arcs and induces a "threading" structure in the nearby band structure, both of which are directly observable in ARPES.

Figures (6)

• Figure 1: (a) Schematic Fermi surface consisting of one electron pocket (black) and one hole pocket (red), coupled by a density-wave order with ordering vector $\mathbf{Q}$. (b),(c) Real-space distributions of the density-wave order parameter $\Delta(\mathbf r)$. From (b) to (c), lowering temperature toward the transition enhances the correlation length $\xi_T$, leading to stronger spatial correlations.
• Figure 2: (a) Conditions for emergence of EPs. The black line corresponds to Eq.\ref{['Eq:DpLine']}, while the colored lines correspond to Eq.\ref{['Eq:DeltaEqgamma']}; the red curve represents a lower temperature (larger $\xi_T$) than the blue one. (b) and (c) Dynamical signatures of a simple pole and a second-order pole. Panel (c) shows the doubly logarithmic plot after applying the exponential correction $\delta \tilde{n}_{\mathbf k} = e^{2\mathrm{Im}\omega_{\mathbf k}t}\delta n_{\mathbf k}$, highlighting the algebraic behavior.
• Figure 3: (a) Origin of the cusp. Green and blue lines indicate degeneracies $\omega_1=\omega_2$ and $\omega_2=\omega_3$; their intersection (red dot) marks the cusp. In the gray region, $A(\mathbf{k},\omega)$ exhibits two peaks (three extrema), while in the white region only a single peak remains. (a1)–(a6) Representative $A(\mathbf{k},\omega)$ at selected $\mathbf{k}$ points in (a). Panels (a1), (a3), and (a5) correspond to momenta on the green line, blue line, and at the cusp. Panel (a2) shows the two-peak regime; (a4) and (a6) the single-peak regime. (b),(c) Trajectories of spectral extrema $\omega_{i\mathbf{k}}$ along loops that either enclose (solid) or avoid (dashed) the cusp, as marked in (a). The colored surface denotes the real-frequency plane. Black curves lie within it and correspond to observable extrema in $A(\mathbf{k},\omega)$, while orange curves depart from it, indicating annihilated extrema beyond the real-frequency plane.
• Figure 4: Simulation of the model in Eq. \ref{['Eq:LatticeModel']} with parameters $(t_1, t_2, t_3, t_4, \mu, \xi\Delta_0)=(-0.9, 1.4, -0.85, -0.85, 1.45, 0.92)$. (a) Fermi surface: blue lines denote electron-like pockets and red lines denote hole-like pockets, with nesting occurring between those around $\Gamma=(0,0)$ and $M=(\pi,0)$. (b) Conditions for EP (red dots) and cusp (blue dots) formation. (c) Dynamical response with $\Delta_0=0.1$ at EP and a generic momentum ($\mathbf{k}=(0.2,0.5)$) away from EP. Panel (c2) shows the exponential correction $\delta \tilde{n}_{\mathbf k} = e^{2\mathrm{Im}\omega_{\mathbf k}t}\delta n_{\mathbf k}$, highlighting the algebraic behavior. (d) and (e) Simulation of the spectral function on a $120 \times 120$ lattice with $\Delta_0=0.5$. (d) Evolution of $A(\mathbf{k},\omega)$ along a closed loop around the cusp (black dotted path in panel (b)). (e) Intensity map of $-\partial_\omega^2A(\mathbf k,E_\mathrm{cp})$. (e1), (e2) Spectral functions at points indicated by the orange and red arrows in (e); dashed lines mark $\omega = E_\mathrm{cp}$.
• Figure A1: Illustration based on the model in Eq. \ref{['Eq:LatticeModel']} with parameters $(t_1, t_2, t_3, t_4, \mu, \Delta_0, \xi)=(-0.9,,1.4,,-0.85,,-0.85,,1.45,,0.5,,1.845)$. (a) Schematic of the cusp: the blue curve denotes the possible Fermi surface (with Fermi arcs forming its subsets), and the black line indicates the condition where a dip and a peak coincide. (b),(c) Spectral functions at the black and red points in panels (a) and (d). The dashed line indicates $\omega = E_\mathrm{cp}$. (d) Intensity plot of the second derivative of the spectral function with respect to frequency.