Linear and Non-Linear Response of Quadratic Lindbladians

TL;DR

This work develops a Lindblad-Keldysh formalism for quadratic Lindbladians to compute finite-frequency linear and non-linear spectroscopic responses of open quantum systems. By exploiting a third-quantized, non-Hermitian single-particle representation, the authors express response functions in terms of biorthogonal eigenmodes of the effective operator $\Xi_{\mathbf{k}}$, enabling exact treatment of non-equilibrium steady states for fermionic and bosonic systems. The framework is applied to a boundary-driven XY spin chain, dissipatively coupled Bernal bilayer graphene, and a bosonic optical lattice, revealing gapless dispersive spin modes, dissipation-induced non-centrosymmetric non-linear optics, and momentum-dependent occupations, respectively. The results illustrate how dissipation and non-Hermiticity reshape spectral functions, optical conductivities, and nonlinear responses, providing a robust toolkit for interpreting experiments on driven-dissipative quantum materials. Overall, this approach opens avenues for probing dissipative topology, exceptional points, and interacting extensions in open quantum systems with experimentally accessible spectroscopic probes.

Abstract

Quadratic Lindbladians encompass a rich class of dissipative electronic and bosonic quantum systems, which have been predicted to host new and exotic physics. In this study, we develop a Lindblad-Keldysh spectroscopic response formalism for open quantum systems that elucidates their steady-state response properties and dissipative phase transitions via finite-frequency linear and non-linear probes. As illustrative examples, we utilize this formalism to calculate the (1) density and dynamic spin susceptibilities of a boundary driven XY model at and near criticality, (2) linear and non-linear optical responses in Bernal bilayer graphene coupled to dissipative leads, and (3) steady state susceptibilities in a bosonic optical lattice. We find that the XY model spin density wavelength diverges with critical exponent 1/2, and there are gapless dispersive modes in the dynamic spin response that originate from the underlying spin density wave order; additionally the dispersing modes of the weak and ultra-strong dissipation limits exhibit a striking correspondence since the boundary dissipators couple only weakly to the bulk in both cases. In the optical response of the Bernal bilayer, we find that the diamagnetic response can decrease with increasing occupation, as opposed to in closed systems where the response increases monotonically with occupation; we study the effect of second harmonic generation and shift current and find that these responses, forbidden in centrosymmetric closed systems, can manifest in these open systems as a result of dissipation. We compare this formalism to its equilibrium counterpart and draw analogies between these non-interacting open systems and strongly interacting closed systems.

Figures (8)

• Figure 1: Schematic illustration of the Keldysh contour over which the evolution of the density matrix is conducted. Right fermions $r$ live on the right-moving contour. Left fermions $\ell$ live on the left-moving contour.
• Figure 2: Feynman diagrams for dissipative linear response. (a) The density/diamagnetic response at frequency $\Omega$ is given by a trace over the Keldysh Green's function and vertex $O$ here taken as a current vertex for electromagnetic response $O=j^{\mu\nu}$. (b) The paramagnetic response is composed two diagrams involving Keldysh, retarded, and advanced Green's functions, illustrated for the case where $O=j^\mu$ and $O'=j^\nu$ to describe electromagnetic response.
• Figure 3: In second order response, a set of triangle diagrams are relevant. There are four diagrams, corresponding to the lines of Eq. (\ref{['eq:triangle']}), proceeding counterclockwise from the lower right. Here we illustrate the case of current response to light where $O=j^\mu$, $O'=j^\nu$ and $O"=j^\rho$.
• Figure 4: The transverse-field XY model with boundary dissipation exhibits spin density wave (SDW) and paramagnetic (PM) phases where the wavelength of the spin density waves diverges at the critical point; the SDW amplitude depends on the dissipation rate and vanishes at the critical point. (a) Schematic of an XY chain coupled to ferromagnetic reservoirs on the boundaries. (b) SDWs near criticality in the XY model with $J_x=1$ and $J_y=1/3$ for $\Gamma=1$ and $N=1000$. (c) Phase diagram; at large fields the phase is paramagnetic except at $J_y=-J_x$ (purple line) where there is only a SDW phase because of a "seesaw" mechanism. At the Ising point, $J_y=0$ (dotted line), the oscillatory phase vanishes. Additionally, precisely at $J_x=J_y$ the system is the XX critical model which does not exhibit an oscillatory phase; here, $J_y$ is measured in units of $J_x$, i.e. $J_x=1$. (d-e) SDW amplitude depends on the dissipation rate and peaks when $\Gamma\sim J$ so that transitions from the reservoir to the second site is maximized as understood using second order perturbation theory, illustrated for a $N=30$ chain with $J_x=1$, $J_y=1/3$(f) SDW wavelength diverges at the critical point with critical exponent $-1/2$, illustrated for $N=1000$, $J_x=\Gamma=1$, $J_y=1/3$ although the exponent is the same for different parameter choices. (g) The SDW amplitude vanishes at the critical point, independent of $\Gamma$, and this dominates the spin-spin correlations.
• Figure 5: Dynamic spin susceptibility, $|\mathcal{S}(q,\Omega)|$ of the boundary driven XY model with $J_x=1$ and $J_y=1/3$ with $N=301$ sites. The critical point for spin-density wave (SDW) order is at $h_\perp=1$ independent of $\Gamma$. Sub-panels (i) and (ii) correspond to small transverse fields where SDW wave order exists. Sub-panels (iii) are at the critical point, and Sub-panels (iv) are at strong fields in the paramagnetic regime. Additionally, we compare $\Gamma=0.01$, $1$ and $100$ in panels (a), (b), and (c) respectively. In panels (b), $\Gamma=1\sim J$ maximizes the magnitude of the spin-spin correlation and for the $\Gamma=1$ panels we divide $\mathcal{S}$ by 10 so that we can use a uniform color scale across all panels. In this case inversion symmetry is broken by the bath as effects of the boundary dissipation are felt deep in the bulk of the system. Choosing $\Gamma$ much smaller (a) (or larger (c)) than $J$ leads to an approximate restoration of inversion symmetry as the effects of dissipation are primarily localized to the edges of the system. This can be understood in terms of Fig. 4(e) where second-order perturbation theory dictates that the amplitude of tunneling from the first site to the second site (and further into the bulk) is maximized for $\Gamma\sim J$. In contrast to the gapped excitations of the closed system, the dissipative system hosts gapless dispersive excitations as the Lindbladian steady state exhibits long-wavelength SDW fluctuations on top of a paramagnetic background. Wavevectors $\pm q_\mathrm{SDW}$ are illustrated with dashed red dashed vertical lines. Wavevectors $\pm2q_\mathrm{SDW} ~\textrm{mod}~ 2\pi$ are illustrated with black dotted vertical lines correspond to gapless modes dispersing from zero frequency.