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Exactly solvable higher-order Liouvillian exceptional points in dissipative fermionic systems

Mingtao Xu, Wei Yi

TL;DR

This work shows that quadratic open-fermion systems can host higher-order Liouvillian exceptional points (EPs) whose order scales with system size, realized in the quasisteady state of a hybrid, partially post-selected Liouvillian. By employing the third-quantization framework, the Liouvillian is analyzed through a shape matrix that, when non-diagonalizable, produces EPs with orders up to 1+K2, potentially reaching the system size n+1. A concrete solvable model of a dissipative fermion chain demonstrates EP5 for n=4, and the perturbation by many-body quantum jumps lifts the degeneracy with fractional power-law scalings in the Liouvillian gap, Delta ∝ z^{1/(1+d)}. These fractional scalings manifest in both the spectrum and the approach to the quasisteady state, offering observable signatures for higher-order Liouvillian EPs and advancing non-Hermitian physics in quantum many-body open systems.

Abstract

We propose a general class of open fermionic models where quadratic Liouvillians governing the dissipative dynamics feature exactly solvable higher-order exceptional points (EPs). Invoking the formalism of third quantization, we show that, among the multiple EPs of Liouvillian, an EP with its order approaching the system size arises as the quasisteady state of the system, leading to a gapless Liouvillian spectrum. By introducing perturbations, in the form of many-body quantum-jump processes, these higher-order EPs break down, leading to finite Liouvillian gaps with fractional power-law scalings. While the power-law scaling is a signature of the higher-order EP, its explicit form is sensitively dependent on the many-body perturbation. Finally, we discuss the steady-state approaching dynamic which can serve as detectable signals for the higher-order Liouvillian EPs.

Exactly solvable higher-order Liouvillian exceptional points in dissipative fermionic systems

TL;DR

This work shows that quadratic open-fermion systems can host higher-order Liouvillian exceptional points (EPs) whose order scales with system size, realized in the quasisteady state of a hybrid, partially post-selected Liouvillian. By employing the third-quantization framework, the Liouvillian is analyzed through a shape matrix that, when non-diagonalizable, produces EPs with orders up to 1+K2, potentially reaching the system size n+1. A concrete solvable model of a dissipative fermion chain demonstrates EP5 for n=4, and the perturbation by many-body quantum jumps lifts the degeneracy with fractional power-law scalings in the Liouvillian gap, Delta ∝ z^{1/(1+d)}. These fractional scalings manifest in both the spectrum and the approach to the quasisteady state, offering observable signatures for higher-order Liouvillian EPs and advancing non-Hermitian physics in quantum many-body open systems.

Abstract

We propose a general class of open fermionic models where quadratic Liouvillians governing the dissipative dynamics feature exactly solvable higher-order exceptional points (EPs). Invoking the formalism of third quantization, we show that, among the multiple EPs of Liouvillian, an EP with its order approaching the system size arises as the quasisteady state of the system, leading to a gapless Liouvillian spectrum. By introducing perturbations, in the form of many-body quantum-jump processes, these higher-order EPs break down, leading to finite Liouvillian gaps with fractional power-law scalings. While the power-law scaling is a signature of the higher-order EP, its explicit form is sensitively dependent on the many-body perturbation. Finally, we discuss the steady-state approaching dynamic which can serve as detectable signals for the higher-order Liouvillian EPs.
Paper Structure (11 sections, 2 theorems, 58 equations, 2 figures)

This paper contains 11 sections, 2 theorems, 58 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Dissipative dynamics under a hybrid Liouvillian
  3. Hybrid master equation under the third quantization
  4. Nondiagonalizable shape matrix: higher-order Liouvillian EPs
  5. A solvable example
  6. Higher-order Liovillian EP
  7. Perturbation near the higher-order EP
  8. Conclusion
  9. A proof that the matrices $T_\pm$ are nondiagonalizable.
  10. Properties of eigenmatrices
  11. Exact result for $d=1$

Key Result

Lemma 1

Given a diagonalizble matrix $A$, the block matrix $M=$ is diagonalizable if there exists $X$, such that $B=[X,A]$.

Figures (2)

  • Figure 1: Liouvillian eigenspectrum and particle number dynamics for the dissipative fermion chain in Eqs. (\ref{['example_1']}) and (\ref{['example_2']}). We set the system parameters as $t=1$, $\gamma=2$, and $n=4$. (a)(b) The rapidities and hybrid-Liouvillian spectrum, respectively, the hybrid-Liouvillian spectrum is purely imaginary and highly degenerate. (c)(d) Dynamics of $\exp(n\gamma t)\braket{N}_t$ and $1/\braket{\tilde{N}}_t$ for different initial states $\rho(0,n_0)$ characterized by $n_0$, respectively.
  • Figure 2: Liouvillian spectrum and dynamics subject to perturbations of the form $z\mathcal{L}_d$. We set the system parameters as $t=1,\gamma=2,n=4$. (a)(b) The perturbed spectra of the hybrid Liouvillian, $\mathcal{L}_H'=\mathcal{L}_H+z\mathcal{L}_d$ with $z=10^{-5}$, in the vicinity of (a) $\lambda=-8$ and (b) $-8+2i$, respectively, under different perturbations characterized by $d$. As an illustration, in panel (a), we mark the Liouvillian gap $\Delta$ for $d=4$. (c) The Liouvillian spectral gap $\Delta$ of the perturbed hybrid-Liouvillian spectra as functions of $z^{1/(1+d)}$. (d) Dynamics of $1/\braket{\tilde{N}}_t$ under perturbations with different $d$ but a fixed $z=10^{-5}$. The system is initialized in $\rho(0,n_0=1)=c_1^\dagger\ket{0}\bra{0}c_1$.

Theorems & Definitions (4)

  • Lemma 1
  • Proof 1
  • Theorem 1
  • Proof 2