Correlations at higher-order exceptional points in non-Hermitian models

We investigate the decay of spatial correlations of $\mathcal{PT}$-symmetric non-Hermitian one-dimensional models that host higher-order exceptional points. Beyond a certain correlation length, they develop anomalous power-law behavior that indicates strong suppression of correlations in the non-Hermitian setups as compared to the Hermitian ones. The correlation length is also reflected in the entanglement entropy where it marks a change from logarithmic growth at short distance to a constant value at large distance, characteristic of an insulator, despite the spectrum being gapless. Two different families of models are investigated, both having a similar spectrum constrained by particle-hole symmetry. The first model offers an experimentally attractive way to generate arbitrary higher-order exceptional points and represents a non-Hermitian extension of the Dirac Hamiltonian for general spin. At the critical point, it displays a decay of the correlations $\sim 1/x^2$ and $1/x^3$ irrespective of the order of the exceptional point. The second model is constructed using unidirectional hopping and displays enhanced suppression of correlations $\sim 1/x^a, a\ge 2$ with a power law that depends on the order of the exceptional point.

Figure 1

(a) Sketch of energy dispersion (blue lines) in $\mathcal{PT}$-symmetric models near an ${\mathrm{EP}}_N$. Flat bands occur only for odd $N$. (b),(c) The exponents $a$ that describe the algebraic decay $\sim 1/x^a$ for all $N(N+1)/2$ possible combinations of real-space correlators as a function of the order $N$ of the EP. Exponents are extracted numerically from fitting the large distance $x$ behavior of the corresponding Green's functions in the two classes of models studied. (b) Suppression of correlations is limited to $1/x^2$ and $1/x^3$ decay in the model implementing the non-Hermitian general spin Dirac Hamiltonian irrespective of the order of the EP. (c) Correlations display a general $1/x^a$ decay, with $a\ge 2$ that depends on the order of the EP in the unidirectional model.

Figure 2

The Dirac Hamiltonian with spin $S$ is realized on a diamond lattice that hosts ${\mathrm{EP}}_N$, where $N$ is equal to $2S+1$. Losses and gains in hoppings are denoted by blue and red arrows, respectively, with the coefficients of the hoppings ${\alpha }_j$ being proportional to $(t-i\gamma )$ for losses and ${\alpha }_j^{*}$ to $(t+i\gamma )$ for gains. The unit cell, which is highlighted by a dotted line, contains $N$ atoms, and the sites have chemical potentials ${\mu }_j$. The amplitudes of the chemical potentials and hoppings ${\alpha }_j$ in each row $j$ are tuned according to Eq. (14).

Figure 3

Correlation functions in the diamond lattice at EPs, $\mu =\gamma$. Correlations for $N=3, \gamma =0.2t$ at filling (a) $\nu =1/3$ and (b) $\nu =2/3$ from numerics (colored symbols) follow closely the analytical lattice results (colored dashed lines) at large $n$ from Eqs. (18) and (19). (c) Short-distance behavior for $N=3, \nu =1/3$, at $\gamma =0.005t$ is qualitatively given by the continuum results in Eqs. (21) and (22). The matching color of a symbol from numerics and the dashed line from analytics identify the same correlation function. (d) For $N=4, \nu =1/2$, and $\gamma =0.2t$, correlations still decay at large distance as $1/n^2$ or $1/n^3$ (the dashed black lines are guidelines). Panels (a), (b), and (c) share the legend.

Figure 4

Charge-density imbalance with increasing rates $\gamma$ in the diamond lattice with ${\mathrm{EP}}_3$ at (a) $\nu =1/3$, (b) $\nu =2/3$, and (c) with ${\mathrm{EP}}_4, \nu =1/2$. The charge density is shown on each site in the unit cell numbered according to the convention shown in Fig. 2.

Figure 5

Bond current densities in the ground state as a function of dissipation rates $\gamma$ for (a) $N=3, \nu =1/3$; (b) $N=3, \nu =2/3$; and (c) $N=4, \nu =1/2$. Currents $j$ and rates $\gamma$ are in units of $t$.

Figure 6

(a) Entanglement entropy as a function of the subsystem size in unit cells $n$ at $\mu =\gamma =0.1t$. The lattice models host an ${\mathrm{EP}}_N$, where $N$ ranges from 2 to 7. (b) Evolution of the entanglement entropy for $N=4$ and for different rates $\gamma$. (c) Entanglement entropy for $n\ll t/\gamma$ (symbols) is fitted with Eq. (33) (dashed lines) at $\mu =\gamma ={10}^{-3}t$. (d) Saturation entropy (symbols) as a function of dissipation rates $\gamma$ for $n\gg t/\gamma$ is fitted with Eq. (34) (dashed lines). The legend is shared for all panels, and $\gamma$ is in units of $t$.

Figure 7

Tight-binding lattice model (37) hosting exceptional points ${\mathrm{EP}}_N$. Reciprocal hoppings are drawn as bidirectional red arrows, while blue arrows denote unidirectional couplings of uniform amplitude $t_0$. The sites encircled by a dotted line form a unit cell. The sites in the unit cell are counted from the top.

Figure 8

(a) Bands of the tight-binding model realizing two ${\mathrm{EP}}_2$. (b) The spatial variation, in unit cells $n$, for the three correlation functions $C_{i,j}\left(n\right)$ in the $N=2$ model (37). The symbols represent the numerical results, while the dashed lines denote the analytical results for the corresponding $C_{i,j}\left(n\right)$ in Eqs. (39).

Figure 9

The spatial variation, in unit cells $n$, of the six correlation functions $C_{i,j}\left(n\right)$ in the $N=3$ model (37) at filling (a) $\nu =1/3$ and (b) $\nu =2/3$ for $t_0=t$. Symbols represent the numerical results, while the dashed lines are the analytical results for the corresponding $C_{i,j}\left(n\right)$ in Eqs. (46), (47), and (20). Correlations at $\nu =1/3$ for $t_0={10}^{-3}t$, in the short-distance limit, are fitted with the corresponding continuum model results Eqs. (41) and (42). The legend is shared among panels (a), (b), and (c). The matching color of a symbol from numerics and the dashed line from analytics identify the same correlation function. (d) All 15 distinct correlation functions for $N=5$ exhibit stronger suppression in the large-distance limit, from $1/n^2$ up to $1/n^7$, as indicated by the dashed line.

Figure 10

The dependence of the charge density on the amplitude $t_0$ of the unidirectional hopping. Here, $t_0$ is measured in units of the maximal reciprocal hopping amplitude $t$. Panels (a) and (b) present the case $N=3$ at filling $\nu =1/3$ and $2/3$, respectively, while (c) shows the case $N=4, \nu =1/2$.

Figure 11

(a) Entanglement entropy as a function of the subsystem size in unit cells $n$ at $t_0=0.5t$. The lattice models host an ${\mathrm{EP}}_N$, with $N$ in between 2 and 7. (b) Typical evolution of the entropy when $N=4$ for different $t_0$ values. (c) Entanglement entropy for $n\ll t/t_0$ (symbols) is fitted with Eq. (51) (dashed lines) at $t_0={10}^{-3}t$. (d) Saturation entropy (symbols) as a function of $t_0$ for $n\gg t/t_0$ is fitted with Eq. (52) (dashed lines). The legend is shared for all panels, and $t_0$ is in units of $t$.