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Real-time edge dynamics of non-Hermitian lattices

Tian-Hua Yang, Chen Fang

Abstract

We derive the asymptotic forms of the Green's function at the open edges of general non-Hermitian band systems in all dimensions in the long-time limit, using a modified saddle-point approximation and the analytic continuation of the momentum. The edge dynamics is determined by the "dominant saddle point", a complex momentum, which, contrary to previous conjectures, may lie outside the generalized Brillouin zone. From this result, we obtain the effective edge Hamiltonians that evidently, as demonstrated by extensive numerical simulations, characterize the dynamics on the edges, and can be probed in real-time experiments or spectroscopies.

Real-time edge dynamics of non-Hermitian lattices

Abstract

We derive the asymptotic forms of the Green's function at the open edges of general non-Hermitian band systems in all dimensions in the long-time limit, using a modified saddle-point approximation and the analytic continuation of the momentum. The edge dynamics is determined by the "dominant saddle point", a complex momentum, which, contrary to previous conjectures, may lie outside the generalized Brillouin zone. From this result, we obtain the effective edge Hamiltonians that evidently, as demonstrated by extensive numerical simulations, characterize the dynamics on the edges, and can be probed in real-time experiments or spectroscopies.

Paper Structure

This paper contains 41 sections, 19 theorems, 163 equations, 28 figures, 1 table.

Table of Contents

  1. Introduction.
  2. The saddle point method.
  3. Edge dynamics.
  4. Notes added.
  5. Data availability.
  6. Acknowledgements.
  7. From Laplace's approximation to the saddle point method
  8. Laplace's approximation for real integrals
  9. The saddle point approximation for contour integrals
  10. Higher-dimensional lattices and multi-band Hamiltonians
  11. The gradient flow method for saddle point selection
  12. Motivation and outline
  13. Formulation of the gradient descent
  14. The algorithm for determining relevant saddle points
  15. World-line Green's functions
  16. ...and 26 more sections

Key Result

Theorem A.1

For an integral $I(\lambda)=\int_{a}^{b}h(x)e^{\lambda f(x)}\mathrm{d}x$, given that $h(x)$ and $f(x)$ are sufficiently well-behaved (for example, are analytic), if $f(x)$ attains the only maximum on the interval at a point $c\in(a,b)$, then as $\lambda\to+\infty$, provided that $f^{\prime\prime}(c)$ and $h(c)$ are non-vanishing.

Figures (28)

  • Figure 1: Illustration of the BZGD process for finding the DSP. The model used is $H(z)=z + (2 + 0.3i)z^{-1} + 0.5i z^2 - 0.8iz^{-2}$. (a) The BZ in the complex $z$ plane. The background color represents $\mathrm{Re}H(z)$, corresponding to the phase of $e^{-i H t}$. The vector field indicates $-\nabla \mathrm{Im}H(z)$, pointing in the direction where $\mathrm{Im} H(z)$ decreases. The color on the BZ represents $\mathrm{Im}H(z)$. No point on the BZ satisfies the two conditions for the SP approximation. (b) The BZ is deformed along the vector field to progressively reduce $\mathrm{Im} H$ on the contour. (c) As the deformation continues, we encounter SPs (red dots) at which $\nabla \mathrm{Im} H(z)=0$. Upon reaching an SP, the contour branches along the SP's descending manifold, forming part of a Lefschetz thimble. (d) The final deformed contour consists of a combination of several Lefschetz thimbles associated with SPs. The SPs contributing to this contour are the RSPs (red dots with colored thimbles), and those who do not contribute are irrelevant ones (orange dot with gray thimble). The RSP with the largest $\mathrm{Im} H(z)$ is identified as the DSP.
  • Figure 2: Numerical results for a 1D one-band model, "Model 1C", which has up to next-nearest neighbor hopping with complex coefficients (see End Matter for the expression). All calculations are performed on an open chain of length $L=500$. The sites on the chain are indexed starting from $x=0$. (a) The PBC and OBC energy spectra along with the SPs. The shaded region is the permissible range for RSP energies. The DSP is shown to be outside the point gap. (b, c) Comparison of $G(0,0;t)$ against the theoretical prediction Eq. \ref{['eq:Gxxt-1D-OBC']}. (d) The amplitude of $\tilde{\psi}(x;t) = G(x,0;t)/G(0,0;t)$, compared to the theoretical prediction $\tilde{\psi}(x)=\langle x|\dot z_s\rangle /\langle 0|\dot z_s\rangle$.
  • Figure 3: Numerical results for a one-band 2D model, "Model 2A", which has up to next-nearest neighbor hoppings in the $x$ direction, nearest neighbor hopping in the $y$ direction, and nearest diagonal hoppings, all with randomly drawn complex coefficients (see End Matter for the expression). (a, b) Comparison of the Green's function $G(\mathbf x_1,\mathbf x_1;t)$ to theoretical prediction. The Hamiltonian is placed on a $120\times 120$ lattice, and $\mathbf x_1=(0,0)$ is the corner. (c) For the same model, placed on an $L\times W$ lattice, with $L=200$ and $W=70$. A wave packet is placed on the edge at $\mathbf x_2=(L/2,W-1)$, and the edge evolution $G((x,W-1), \mathbf {x_2};t)$ is compared to the theoretical prediction, Eq. \ref{['eq:eff-Ham-time']}, given by the effective Hamiltonian.
  • Figure 4: Simulated local spectroscopy for model 1C, same as in Fig. \ref{['fig:1D-results']}. The LDOS $\mathrm{Re} G(x,x;\omega)$ is calculated for (a) $x$ in the bulk and (b) $x$ on the edge, for systems of length $L=1000$ and $L=500$ respectively, by Fourier transforming a time evolution up to $T=60$. The peaks of the LDOS coincide with the real parts of the energies of some of the SPs.
  • Figure 5: Illustration of the RSP-finding algorithm. (a) The Lefschetz thimbles and the ascending flows of the SPs. The model is the same as in Fig. \ref{['fig:sublevel']}. Red dots are the SPs, the lines with purple ends are the descending manifolds (Lefschetz thimbles), and lines with red ends are ascending manifolds. For the three RSPs as indicated in Fig. \ref{['fig:sublevel']}(d), their ascending manifolds cross the BZ, while the irrelevant SP's does not. (b) The ascending gradient flow of an RSP in a 2D system. The model used is $H(z_1,z_2)=z_1z_2 - iz_1^{-1}z_2^{-1} + (1 + 0.5i) z_1 z_2^{-1} + (1 - 0.3i) z_1^{-1} z_2 + (0.7 + 0.3i) z_1 + (0.1 - 0.3i) z_2^{-1}$. Each closed contour indicated the $\boldsymbol{\mu}(s;\boldsymbol{\xi})$ curve as a parametric curve with respect to $\boldsymbol{\xi}$ for a given $s$. The curve touches the origin in the $\boldsymbol{\mu}$ plane at $s\approx 0.05$, and develops a winding with respect to the origin afterwards.
  • ...and 23 more figures

Theorems & Definitions (19)

  • Theorem A.1: Laplace's Approximation
  • Theorem A.2
  • Theorem A.3: Laplace's Approximation in Higher Dimensions
  • Theorem A.4: Method of Steepest Descent / Saddle Point Approximation
  • Theorem A.5: More General Form of Saddle Point Approximation
  • Theorem B.1: Stokes's Theorem
  • Proposition C.1
  • Proposition C.2
  • Proposition C.3
  • Proposition C.4
  • ...and 9 more