Emergent $\mathcal{PT}$-symmetry breaking of collective modes with topological critical phenomena

Abstract

The spontaneous breaking of parity-time ($\mathcal{PT}$) symmetry yields rich critical behavior in non-Hermitian systems, and has stimulated much interest, albeit most previous studies were performed within the single-particle or mean-field framework. Here, by studying the collective excitations of a Fermi superfluid with $\mathcal{PT}$-symmetric spin-orbit coupling, we uncover an emergent $\mathcal{PT}$-symmetry breaking in the Anderson-Bogoliubov (AB) collective modes, even as the superfluid ground state retains an unbroken $\mathcal{PT}$ symmetry. {The critical point of the transition is marked by a non-analytic kink in the speed of sound, which derives from the coalescence and annihilation of the AB mode and its hole partner, reminiscent of the particle-antiparticle annihilation. The system consequently becomes immune to low-frequency external perturbations at the critical point, a phenomenon associated with the spectral topology of the complex quasiparticle dispersion. This critical phenomenon offers a fascinating route toward perturbation-free quantum states.

Figures (5)

• Figure 1: Pairing in a non-Hermitian superfluid.a Phase diagram on the occupation fraction and interaction coefficient ($\nu$--$U$) plane, where the background colors indicate the order parameter $\Delta$. The self-consistent ground state of the system undergoes a first-order phase transition from the normal phase (N) to the superfluid phase (SF) when increasing $U$. b and c: Bogoliubov quasiparticle spectra in the limit $\Delta=0$ (the normal phase) b and SF phase c in the complex plane. As shown in b, fermions in a potential Cooper pair are separated by a finite imaginary energy gap (vertical distance) on the Fermi surface, which underlies the first-order phase transition. The quasiparticle spectra are gapped in the SF phase [see c], which implies the emergence of quantized spectral winding number $W$ for the occupied bands. Different shades of grey in c mark regions for different choice of reference energy. Here we take the hopping strength $t_{\text{s}}/\varepsilon=1$ and the SOC strength $t_{\text{so}}/\varepsilon=0.5$ with the unit of energy as the recoil energy $\varepsilon=\hbar^2 a^{-2}/(2m)$, where $\hbar$ is the Plank constant, $a$ is the lattice constant (unit of length) and $m$ is the mass of fermions.
• Figure 2: Response function and collective modes.a Typical distribution of response function $\chi(q,\omega)$ on the complex frequency plane with a fixed perturbation momentum $q/\uppi=0.5$. The sharp peaks reflect the presence of collective modes. The surface colors are only employed to increase visibility. b Dispersions of collective modes, obtained from the divergent sharp peaks plotted in panel a. Note that the imaginary parts of the spectra are zero here, as we set the spin-orbit coupling (SOC) strength $t_{\text{so}}$ smaller than the hopping coefficient $t_{\text{s}}$. The red lower (blue higher) branch locates outside (inside) the spectral loop $\omega=E_{k+q/2}+E_{k-q/2}$ with quasi-energy $E_{k}$, which is marked by the dash-dotted curve in a. Here we take the parameters: hopping coefficient $t_{\text{s}}/\varepsilon=1$, SOC strength $t_{\text{so}}/\varepsilon=0.5$, occupation fraction $\nu=1/4$ and interaction coefficient $U/\varepsilon=4$ with the recoil energy $\varepsilon$, in our calculations.
• Figure 3: PT transition in the collective modes. Response function $\chi(q,\omega)$ in the low-frequency regime for spin-orbit coupling (SOC) strength $t_{\text{so}}/\varepsilon=0.8$ ($t_{\text{so}}/\varepsilon=1.2$) in the column aj ( bj), with $j=I,II,III$. Here aI and bI correspond to the perturbation momentum $q/\uppi=0.1$; aII and bII: $q/\uppi=0.2$; aIII and bIII: $q/\uppi=0.3$. c and d: Spectra of AB mode $\omega_{AB}$ (solid) and its hole partner $-\omega_{AB}$ (dashed) coalesce at the critical point (indicated by arrow), which corresponds to the real-to-complex transition point illustrated in panels as and bs. Locations of the critical point in c and d are indicated by gray lines, which are not poles of $\chi(q,\omega)$ and do not correspond to the spectra of collective modes. Inset of c: numerically calculated speed of sound (blue dots) near the critical point, which is well-fitted by a power-law function $\propto|t_{\text{s}}-t_{\text{so}}|^{\gamma}$ (red curves), with a critical exponent $\gamma\approx 1/2$, where $t_{\text{s}}$ is the hopping coefficient. Here we take $t_{\text{s}}/\varepsilon=1$, interaction strength $U/\varepsilon=4$ and occupation fraction $\nu=1/4$, with the recoil energy $\varepsilon$.
• Figure 4: Critical response.a Typical distribution of response function $|\chi|$ on the complex frequency plane at the critical point (take perturbation momentum $q/\uppi=0.5$ for example). Inset: the sectional view of $\chi$ along the real axis, where the red solid, blue dotted and black dashed curves denote the total, the part arising from the simple Bardeen-Cooper-Schrieffer (BCS) theory, and the part arising from the order-parameter fluctuations of $\chi$, respectively. We see that $\chi$ vanishes outside the gapped spectral loops of $\omega=E_{k+q/2}+E_{k-q/2}$ with quasi-energy $E_{k}$ (see also the shaded regions in the inset of a), whose shapes for different $q$ are shown in b. As a result, $\chi$ always vanishes for $\omega$ outside all such spectral loops for any $q$. Here we set the interaction strength $U/\varepsilon=4$, hopping coefficient $t_{\text{s}}/\varepsilon=t_{\text{so}}/\varepsilon=1$ and occupation coefficient $\nu=1/4$, where $t_{\text{so}}$ is the spin-orbit-coupling (SOC) strength.
• Figure 5: Softening of phonon velocity at the critical point. Numerically evaluated a expansion coefficients $J_{a,b,c}$ and b phonon velocity $\upsilon_{p}$ as functions of the spin-orbit coupling (SOC) strength $t_{\text{so}}$. Other parameters are the same as those in Fig. \ref{['fig:response_EP']}. Here we fix the interaction strength $U/\varepsilon=4$, hopping coefficient $t_{\text{s}}/\varepsilon=1$ and occupation coefficient $\nu=1/4$.