Correlation-induced sensitivity and non-Hermitian skin effect of quasiparticles

Non-Hermitian (NH) Hamiltonians have been shown to exhibit unique signatures, including the NH skin effect and an exponential spectral sensitivity with respect to boundary conditions. Here, we investigate as to what extent these remarkable phenomena, recently predicted and observed in a broad range of settings, may also occur in closed correlated fermionic systems that are governed by a Hermitian many-body Hamiltonian. There, an effectively NH quasiparticle description naturally arises in the Green's function formalism due to interparticle scattering that represents an inherent source of dissipation. As a concrete platform we construct an extended interacting Su-Schrieffer-Heeger (SSH) model subject to varying boundary conditions, which we analyze using exact diagonalization and nonequilibrium Green's function methods. That way, we clearly identify the presence of the aforementioned NH phenomena in the quasiparticle properties of this Hermitian model system.

Figure 1

(a) Illustration of the tight-binding Hamiltonian ${\mathcal{H}}_0$ [Eq. (5)] with parameters $t_1,t_2$ and two-body interaction $\mathcal{V}$ [Eq. (7)] with sublattice-specific interaction strengths $U_{\mathrm{AA}},U_{\mathrm{BB}}$ in a two-leg ladder geometry, with sublattice A (B) as the red upper (blue lower) leg. (b) Illustration of the broken ring geometry with generalized boundary condition parameter $\mathrm{\Gamma }\in [0,1]$, where $\mathrm{\Gamma }=0$ ($\mathrm{\Gamma }=1$) corresponds to open (periodic) boundary conditions. (c) Schematic illustration of our present program, where the path of solid arrows is followed to demonstrate its (qualitative) commutation with the beaten dashed path.

Figure 2

(a) Exact diagonalization data ($L=14$ sites) and (b) second Born approximation data ($L=400$ sites) for effective NH Hamiltonians under periodic boundary conditions (PBC). Left panels: complex spectra with PBC showing a point gap, reference point $E_{\mathrm{b}}=1-0.2\mathrm{i}$ denoted as a red star; the red shaded area contains all the reference points, where formally $\nu =1$ holds. Right panels: corresponding nontrivial winding of the determinant around the origin, following Eq. (9), with reference point $E_{\mathrm{b}}=1-0.2\mathrm{i}$. Data obtained for $t_2=-t_1=1, U_{\mathrm{AA}}=1.5, U_{\mathrm{BB}}=0.1, T=5$. The arrows and the colors in the right panels show in both cases a counterclockwise winding corresponding to $\nu =1$.

Figure 3

(a) Spectral collapse under open boundary conditions (OBC) for SBA data (red, $L=100$ sites) and ED data (blue, $L=14$ sites), respectively, shown on top of the corresponding PBC data (grey). The spectra drastically change to an arc shape inside the shaded area associated with $\nu =1$. (b) Exponential localization of the total spectral weight of right eigenstates as a function of the site index, exemplifying the NH skin effect for SBA data (red, $L=100$ sites) and ED data (blue, $L=14$ sites). Data obtained for $t_2=-t_1=1, U_{\mathrm{AA}}=1.5, U_{\mathrm{BB}}=0.1, T=5$.

Figure 4

Energy shift $\mathrm{\Delta }E$ as a function of the number of sites $L$ in response to switching on a small end-to-end hopping $\mathrm{\Gamma }={10}^{-5}$. The red points show the shift $|\mathrm{\Delta }{E}_0|$ of the spectrally isolated in-gap mode [see Eq. (13)], and an exponential fit (black dashed line) closely matching our numerical data is plotted as a guide to the eye. The fit excludes the last three points at $L=91,95,101$, where the exponential behavior crosses over to a saturated regime. The blue line shows the overall energy shift $|\mathrm{\Delta }{E}_{\mathrm{tot}}|$ of the spectrum per unit length [see Eq. (15)]. The inset shows the spectra of the effective Hamiltonian ${\mathcal{H}}_{\mathrm{eff}}\left(\mathrm{\Gamma }\right)$ for $\mathrm{\Gamma }=0$ (crosses) and $\mathrm{\Gamma }={10}^{-5}$ (circles) for $L=81$ sites, where the in-gap mode is highlighted in red. Data obtained via SBA for $t_2=-t_1=1, U_{\mathrm{AA}}=1.5, U_{\mathrm{BB}}=0.1, T=5$.

Figure 5

The Keldysh contour used in our calculations, running from $t_0$ to the maximal time $t_{\max }$ back to $t_0$ and ends at the imaginary time $t_0-i\beta$. The complete contour is defined as sum over all time branches: forward ${\mathcal{C}}_{-}$, backward ${\mathcal{C}}_{+}$, and imaginary (Matsubara) branch ${\mathcal{C}}_M$. Depending on the time arguments, one recovers the ordinary GF like Matsubara or time-ordered component.

Figure 6

Illustration of the self-energy in the second Born approximation ${\mathrm{\Sigma }}_{SBA}$, which includes all diagrams up to second order (up to two interaction lines). The sum contains the Fock diagram, Hartree diagram, exchange diagram, and the direct diagram (from upper left to lower right).

Figure 7

Entries of the anti-Hermitian imaginary part of the self-energy $\mathrm{\Sigma }$ for different values of the interaction parameters $U_{\mathrm{AA}}$ and $U_{\mathrm{BB}}$ (indicated above each panel). A diagonal block corresponding to a bulk unit cell is highlighted in red to guide the eye. It is possible to observe how the staggered pattern of interaction (first and second panels) is mirrored in the diagonal entries of the self-energy, while the absence of any staggering (third panel) is also inherited by the self-energy. Decomposing the diagonal 2-by-2 blocks on the basis of Pauli matrices one notices that factors of the form of Eq. (6) are by far dominant, with ${\gamma }_z\simeq 0.1836,-0.1836$ and ${\gamma }_0\simeq 0.1855,0.1855$ on average for the first two panels, while for the third one ${\gamma }_z=0$ within machine error and ${\gamma }_0=0.4219$. This figure also shows that the self-energy has a more complex structure than the simple target model of Eq. (6), since also other (off-diagonal) entries are nonvanishing (though smaller than the dominant diagonal terms).