Exceptional points near first- and second-order quantum phase transitions

We study the impact of quantum phase transitions (QPTs) on the distribution of exceptional points (EPs) of the Hamiltonian in the complex-extended parameter domain. Analyzing first- and second-order QPTs in the Lipkin-Meshkov-Glick model we find an exponentially and polynomially close approach of EPs to the respective critical point with increasing size of the system. If the critical Hamiltonian is subject to random perturbations of various kinds, the averaged distribution of EPs close to the critical point still carries decisive information on the QPT type. We therefore claim that properties of the EP distribution represent a parametrization-independent signature of criticality in quantum systems.

Figure 1

Energy spectrum and EPs for the first-order QPT LMG Hamiltonian (4) with $N=15$. (a) The $\lambda \leftrightarrow -\lambda$ symmetric energy spectrum with the avoided crossings of levels demarcated by dots. (b) The EP pattern, in which selected EPs are assigned to the corresponding avoided crossing in (a) by the dot types. (c) and (d) Evolution of real and imaginary parts of individual level energies $E_n\left(\mathbf{\lambda }\right)$ along a path $\mathbf{\lambda }=0+i\mu$. (c) The real parts merge and (d) the imaginary parts diverge as the path crosses individual EPs in (b) [mergers in (c) are invisible due to very small energy differences, while in (d) they are emphasized by $1/\mathrm{Im}\lambda$ scaling of $\mathrm{Im}E]$.

Figure 2

(a) Energy spectrum and (b) EPs for the second-order QPT LMG Hamiltonian (6) with $N=15$. Only the EPs closest to the real axis are assigned to the corresponding avoided crossings.

Figure 3

(a) Energy spectrum and (b) EPs for the first-order QPT LMG Hamiltonian (8) with $N=15$.

Figure 4

Evolution of the single EP located closest to the QPT critical point with increasing size $N$. (a) First-order QPT LMG Hamiltonians (4) and (8) showing an exponential decrease of $\mathrm{Im}{\mathbf{\lambda }}_1^{\mathrm{EP}}$. (b) Second-order QPT LMG Hamiltonian (6) with an algebraic decrease of $\mathrm{Im}{\mathbf{\lambda }}_1^{\mathrm{EP}}$. A linear fit of the lin-log dependences in (a) yields $\mathrm{Im}{\lambda }_1^{\mathrm{EP}}\propto N^{-\zeta }{e}^{-\eta N}$, where $(\zeta ,\eta )\doteq (0.52,1.49)$ for ${\widehat{H}}^{\mathrm{QPT}1}$ and $(0.56,1.12)$ for ${\widehat{H}}^{\mathrm{QPT}{1}^{\prime }}$. The log-log dependence in (b) is consistent with $\mathrm{Im}{\lambda }_1^{\mathrm{EP}}\propto N^{-\kappa \left(N\right)}$, the evolution of $\kappa \left(N\right)$ being shown in the inset (an estimated asymptotic value deduced from the calculation up to $N\approx 5400$ is $\kappa \doteq 0.666$; the tilted line in the main graph is a linear fit through the last three points).

Figure 5

Ensemble-averaged distributions of DPs (real crossings) along $\left|\lambda \right|=\left|\mathrm{Re}\mathbf{\lambda }\right|$ for Hamiltonian (1) with the diagonal random interaction $\widehat{V}={\widehat{V}}^{\mathrm{diag}}$ from Eq. (18) for $d=16$ ($N=15$). The calculation was done via formula (25). (a) Rectangular distribution and (b) normal distribution of diagonal matrix elements; the functions $F$ from Eq. (27) are shown in the respective insets. Individual curves show results for three unperturbed Hamiltonians $\widehat{H}\left(0\right)={\widehat{H}}^{\mathrm{c}1}$, ${\widehat{H}}^{\mathrm{c}2}$, and ${\widehat{H}}^{\mathrm{HO}}$.

Figure 6

Ensemble-averaged distributions of EPs for the Hamiltonian (1) with the GOE random interaction $\widehat{V}={\widehat{V}}^{\mathrm{full}}$ from Eq. (19) for $d=16$ ($N=15$). (a) The EP distribution as a function of the absolute value $\left|\mathbf{\lambda }\right|$; the three curves correspond to the three choices of the free Hamiltonians (13). Also shown are the EP distributions in the whole complex plane for (b) $\widehat{H}\left(0\right)={\widehat{H}}^{\mathrm{c}1}$, (c) $\widehat{H}\left(0\right)={\widehat{H}}^{\mathrm{c}2}$, and (d) $\widehat{H}\left(0\right)={\widehat{H}}^{\mathrm{HO}}$. The distribution was calculated from $\sim 8400$ random-matrix realizations ($\sim {10}^6$ complex-conjugate pairs of EPs).

Figure 7

Same as in Fig. 6, but for the off-diagonal random interaction ensemble $\widehat{V}={\widehat{V}}^{\text{off-diag}}$ from Eq. (20).

Figure 8

Behavior of EPs near $\mathbf{\lambda }=0$ for $\widehat{H}\left(0\right)={\widehat{H}}^{\mathrm{c}2}$, ${\widehat{H}}^{\mathrm{HO}}$, and $\widehat{V}={\widehat{V}}^{\mathrm{full}}$ with increasing dimension ($d=8,16,32,64$). The main panel shows the ensemble average $\langle \left|{\mathbf{\lambda }}_1^{\mathrm{EP}}\right|\rangle$ of the deviation of the closest EP from the origin. The inset shows the value ${\lambda }^{\mathrm{thr}}$ determined as the lowest value $|{\mathbf{\lambda }}_i^{\mathrm{EP}}|$ in a sample of approximately ${10}^6$ generated EPs. Linear fits (lines) indicate a faster convergence of both $\langle \left|{\mathbf{\lambda }}_1^{\mathrm{EP}}\right|\rangle$ and ${\lambda }^{\mathrm{thr}}$ to zero for ${\widehat{H}}^{\mathrm{c}2}$ than for ${\widehat{H}}^{\mathrm{HO}}$.