Phase transitions in a non-Hermitian Su-Schrieffer-Heeger model via Krylov spread complexity

TL;DR

The paper investigates phase transitions in a $\\mathcal{PT}$-symmetric non-Hermitian SSH model with an imaginary chemical potential by using Krylov spread complexity and Krylov fidelity. It shows that the derivatives of the Krylov spread density $\mathcal{C}_\Omega$ for the unitary path from the Hermitian ground state to the non-Hermitian vacuum reveal non-analytic behavior at boundaries where the spectrum changes from real to complex or from complex to purely imaginary. In the $\\mathcal{PT}$-broken (imaginary) regime, the time-dependent Krylov spread approaches a stationary value with two distinct dynamical phases, identified via the Krylov fidelity and controlled by the slowest dissipation modes. Together, these results establish Krylov spread measures as sensitive probes of spectral and dynamical phase transitions in non-Hermitian, $\\mathcal{PT}$-symmetric systems and uncover hidden dynamical structures beyond conventional analyses.

Abstract

We investigate phase transitions in a non-Hermitian Su-Schrieffer-Heeger (SSH) model with an imaginary chemical potential via Krylov spread complexity and Krylov fidelity. The spread witnesses the $\mathcal{PT}$-transition for the non-Hermitian Bogoliubov vacuum of the SSH Hamiltonian, where the spectrum goes from purely real to complex (oscillatory dynamics to damped oscillations). In addition, it also witnesses the transition occurring in the $\mathcal{PT}$-broken phase, where the spectrum goes from complex to purely imaginary (damped oscillations to sheer decay). For a purely imaginary spectrum, the Krylov spread fidelity, which measures how the time-dependent spread reaches its stationary state value, serves as a probe of previously undetected dynamical phase transitions.

Figures (4)

• Figure 1: Negative branch $-\Lambda(k)$ of the spectrum \ref{['eq:7']} in terms of the parameters $(h,\gamma)$ for $J=1$. Panel (a) displays the $h$-$\gamma$ plane with the six subsets I-VI identified in the main text. Note that regions II and IV are, respectively, all the black and all the red lines separating the two-dimensional regions (I, III, V, and VI). Panels (b)-(d) display the real (solid lines) and imaginary (dotted lines) parts of $-\Lambda(k)$ for several values of $(h,\gamma)$. Specifically, in Panel (b), $h=1$ and $\gamma \in \{1/2,1, 3/2, 2,3\}$; in panel (c), $h=2$ and $\gamma \in \{3/2, 2, 3\}$; in panel (d), $h=3$ and $\gamma \in \{1,2,5/2, 3,4 \}$. For each of the panels (b), (c), and (d), the values of $\gamma$ go from bottom to top in the vertical cross-sections of diagram (a) indicated by arrows b, c, and d, referring to the corresponding panels. The line colors in (b)-(d) correspond to the position of the ordered pair $(h,\gamma)$ in the diagram (a), where each pair is located in one of the distinct colored regions.
• Figure 2: (a) Krylov spread complexity density $\mathcal{C}_\Omega$ [Eq. \ref{['eq:15']}] of the unitary evolution taking $\ket{\text{GS}}$ [Eq. \ref{['eq:10']}] to $\ket{\Omega}$ [Eq. \ref{['eq:12']}] via the Hamiltonian $H_\Omega$, Eq. \ref{['eq:13']}, at $J= 1$. (b) $1-\bar{\mathcal{C}}$, where $\bar{\mathcal{C}}$ is the time-averaged Krylov spread complexity density \ref{['eq:av1']} of a non-Hermitian evolution of $\ket{\text{GS}}$ induced by Hamiltonian \ref{['eq:1']} of the non-Hermitian SSH model. In regions IV and V (see Fig. \ref{['fig:1']}), the spectrum is purely imaginary, and it can have maximally two gapless points in region IV. Thus, the state $\ket{\text{GS}}$ reaches $\ket{\Omega}$ in the infinite-time limit in the non-Hermitian evolution, and the two spreads coincide in such regions, i.e., $\mathcal{C}_\Omega = \bar{\mathcal{C}} = 1/2$.
• Figure 3: Transverse cross-sections of the spread $\mathcal{C}_\Omega$, Eq. \ref{['eq:15.1']}. (a) Spread as a function of $\gamma$ for $h\in \{-2,0,2,3 \}$. (b) Spread as a function of $h$ for $\gamma \in \{0,3/2,2,3\}$. In all cases, $J=1$.
• Figure 4: Schematic phase diagram defined via the Krylov fidelity \ref{['eq:b5.1']} with $J=1$. Region 1 (in gray) corresponds to the time \ref{['eq:b7.1']}, and Region 2 (in light blue) corresponds to the time \ref{['eq:b7.2']}. The points in red correspond to Region IV (see Fig. \ref{['fig:1']}), where $\Lambda(k) = i\Gamma(k) = 0$, so we do not associate any $t^*$ with those points. The white regions correspond to the points where the spectrum is either purely or partially real, so there is no $t^*$ for these regions.