Real-Space Dynamical Mean-Field Theory (R-DMFT) for non-Hermitian strongly correlated systems

1. Background and Motivation

Non-Hermitian quantum systems have recently garnered significant interest due to their ability to describe open quantum systems, systems with gain and loss, or effective models for complex physical scenarios. Unlike traditional Hermitian systems, non-Hermitian Hamiltonians can exhibit complex eigenvalues and non-orthogonal eigenvectors, giving rise to fascinating phenomena such as the non-Hermitian skin effect (NHSE). The NHSE is characterized by the localization of a macroscopic number of eigenstates at the boundaries under open boundary conditions, challenging the conventional understanding of bulk-boundary correspondence.

The Hubbard model, a cornerstone of condensed matter physics, provides a powerful framework for studying strongly correlated electron systems, capturing phenomena such as Mott transitions, magnetism, and superconductivity. While traditionally Hermitian with symmetric hopping, the introduction of asymmetric hopping extends the Hubbard model into the non-Hermitian domain. This modification can represent biased particle flow, dissipation, or coupling to external environments, making it a versatile tool for exploring new physics.

We recently studied the non-Hermitian Hubbard model with asymmetric hopping, focusing on the interplay between the NHSE and strong electronic correlations. We employed the real-space dynamical mean-field theory (RDMFT) (also called inhomogeneous DMFT), a non-perturbative method renowned for its ability to accurately capture local correlation effects in strongly interacting systems. By analyzing the end-to-end Green’s function as probes of directional amplification, we demonstrate that strong correlations can suppress the skin effect, leading to a crossover from boundary-dominated to correlation-driven dynamics. Our systematic study reveals that correlations dominate at small to intermediate strength of asymmetric hopping, inducing exponential decay in the end-to-end Green’s function, but higher strength of asymmetric hopping can restore amplification even at strong interaction. These results illuminate the tunable interplay between correlations and non-Hermitian physics, suggesting avenues for engineering non-reciprocal transport in correlated open quantum systems.

However, here we highlight the R-DMFT algorithm for anyone interested!

2. Model and Methods

2.1 Non-Hermitian Hubbard Model

The Hamiltonian for the non-Hermitian Hubbard model on a 1D lattice is:

$$H = - \sum_{j,\sigma} \left( t_R c_{j+1,\sigma}^\dagger c_{j,\sigma} + t_L c_{j,\sigma}^\dagger c_{j+1,\sigma} \right) + U \sum_j n_{j\uparrow} n_{j\downarrow}$$

Where:

• $c_{j,\sigma}^\dagger, c_{j,\sigma}$: Fermionic creation and annihilation operators at site $j$.
• $\sigma = \uparrow, \downarrow$: Spin of the electron.
• $t_R = t + \gamma$ and $t_L = t - \gamma$: Asymmetric hopping amplitudes, where $\gamma \neq 0$ denotes non-Hermiticity strength.
• $U$: On-site Hubbard interaction strength.
• $n_{j\sigma} = c_{j,\sigma}^\dagger c_{j,\sigma}$: Number operator.

The lattice has $L$ sites with open boundary conditions (OBC) to study the skin effect, where states localize at boundaries.

2.2 Real-Space DMFT Algorithm

This algorithm uses real-space DMFT to account for site-dependent properties, focusing on the retarded Green’s function $G^R(\omega)$ to compute the local density of states (LDOS). The self-energy is assumed local but varies across sites: $\Sigma_{ij}^R(\omega) = \delta_{ij} \Sigma_i^R(\omega)$.

• Step 1: Initialize Parameters and Self-Energy

  • Lattice size: Set $L$.
  • Hopping parameters: Define $t_R = t + \gamma, t_L = t - \gamma$.
  • Interaction: Set $U$.
  • Frequency grid: Choose real frequencies $\omega \in [-\omega_{\text{max}}, \omega_{\text{max}}]$.
  • Broadening factor: Set $\eta > 0$ (e.g., $\eta = 0.01$).
  • Initial self-energy: For each site $i$, initialize $\Sigma_i^R(\omega)$ (e.g., $\Sigma_i^R(\omega) = 0$ or Hartree term $U n_{\text{opp}}$).

• Step 2: Construct Non-Interacting Hamiltonian Build the non-interacting Hamiltonian $H_0$ as an $L \times L$ tridiagonal matrix for OBC: $H_0(i, i+1) = -t_R$ and $H_0(i+1, i) = -t_L$. Example for $L = 4$:

  $$H_0 = \begin{pmatrix} 0 & -t_R & 0 & 0 \ -t_L & 0 & -t_R & 0 \ 0 & -t_L & 0 & -t_R \ 0 & 0 & -t_L & 0 \end{pmatrix}$$

• Step 3: Compute the Lattice Green’s Function Form the diagonal self-energy matrix $\Sigma^R(\omega)$ and compute: $$G^R(\omega) = \left[ (\omega + i\eta) I - H_0 - \Sigma^R(\omega) \right]^{-1}$$

• Step 4: Extract Local Green’s Functions For each site $i$, extract the diagonal element: $$G_{ii}^R(\omega) = \left[ G^R(\omega) \right]_{ii}$$

• Step 5: Compute the Weiss Field and Hybridization Calculate the inverse Weiss field:

  $$\mathcal{G}_{0,i}^R(\omega)^{-1} = G_{ii}^R(\omega)^{-1} + \Sigma_i^R(\omega)$$

  The hybridization function is:

  $$\Delta_i^R(\omega) = \omega - G_{ii}^R(\omega)^{-1} - \Sigma_i^R(\omega)$$

• Step 6: Solve the Impurity Problem Solve the Anderson impurity model for each site. Using Iterated Perturbation Theory (IPT) at half-filling: $$\Sigma_i^R(\omega) = -i U^2 \int_{0}^\infty dt e^{i\omega t}[\beta_i(t)^2\alpha_i(-t) + \alpha_i(t)^2\beta_i(-t)]$$ where: $$\begin{Bmatrix} \alpha_i(t) \ \beta_i(t) \end{Bmatrix} = \int_{-\infty}^{\infty} \mathrm{d}\omega, e^{-i \omega t} \rho_{0,i}(\omega) f(\pm \omega)$$

• Step 7: Update Self-Energy Mix the new and old self-energies for stability: $$\Sigma_i^R(\omega)^{\text{new}} = \alpha \Sigma_i^R(\omega)^{\text{computed}} + (1 - \alpha) \Sigma_i^R(\omega)^{\text{old}}$$

• Step 8: Check for Convergence $$\text{error} = \max_{i,\omega} \left| \Sigma_i^R(\omega)^{\text{new}} - \Sigma_i^R(\omega)^{\text{old}} \right|$$ If $\text{error} < \text{tolerance}$, stop; otherwise, return to Step 3.

• Step 9: Analyze the Skin Effect Compute the spectral function: $$A_i(\omega) = -\frac{1}{\pi} \Im G_{ii}^R(\omega)$$ The skin effect is observed if $A_i(\omega)$ shows enhanced values at boundary sites ($i=1$ or $i=L$).

3. Implementation of the Impurity Solver

As defined in the IPT approximation, the self-energy is computed through time-domain integrals. In numerical implementations, these Fourier transforms are carried out in discrete form using the Fast Fourier Transform (FFT), which scales as $\mathcal{O}(N\log N)$.

Discrete Fourier Transform (DFT):

$$\tilde{f}_k = \sum_{n=0}^{N-1}f_n e^{-i\frac{2\pi n k}{N}}$$

Inverse Discrete Fourier Transform (IDFT): $$f_n = \frac{1}{N}\sum_{k=0}^{N-1} \tilde{f}_ke^{i\frac{2\pi n k}{N}}$$