Parametric Instabilities of Correlated Quantum Matter

TL;DR

This work develops a general framework to parametrize the dynamic driving of low-energy collective bosonic modes in correlated quantum materials by periodic modulation of microscopic parameters. The drive enters through two-boson terms that arise only when the modulation alters the fluctuation vacuum, linking parametric instabilities to fidelity susceptibility and furnishing a diagnostic tool for quantum fluctuations and phase transitions. By deriving the collective-mode Hamiltonian and its Bogoliubov structure, the paper shows how the parametric term is governed by off-diagonal Bogoliubov couplings and how the resulting instabilities can melt or reconfigure orders, sometimes stabilizing novel steady states with subharmonic responses. The authors illustrate the framework with quantum Hall bilayers and a flat-band toy model, uncovering how quantum geometry and proximity to quantum phase boundaries amplify parametric susceptibility and proposing concrete experimental probes such as interlayer tunneling spectroscopy and time-resolved measurements, thereby providing a practical toolbox for control and exploration of non-equilibrium states in correlated materials.

Abstract

Strongly correlated quantum materials exhibit a rich landscape of ordered phases with highly tunable properties, making them an intriguing platform for exploring non-equilibrium phenomena. A key to many of these phases is collective bosonic excitations, encoding fluctuations of the underlying order. In this work, we develop a general theoretical framework for parametric driving of such modes, whereby periodic modulation of microscopic parameters generates resonant two-boson processes. We show that the feasibility and strength of this drive depend sensitively on whether the targeted parameter alters the properties of the bosonic excitations vacuum, linking potential parametric instabilities directly to the fidelity susceptibility of the ground state. The driving facilitates nonthermal melting of the parent orders, as well as stabilization of novel steady states with experimentally distinct signatures. Through microscopic case-studies of correlated electronic systems, we identify promising driving knobs, highlight the role of quantum geometry in the collective modes susceptibility, and propose realistic experimental probes. Collective excitations are a powerful resource for steering correlated phases out of equilibrium, and will likely have several applications in quantum science. Our work provides the toolbox for controlling these excitations.

Figures (5)

• Figure 1: Parametric driving of collective bosonic excitations. (a) A generic two-parameter ($p_1,\,p_2$) phase diagram, with distinct phases A, B, and C. The system is perturbed without hitting a phase boundary (e.g., gray circle in B, purple circle in C) by periodically modulating $p_1$ and $p_2$ at a frequency $\omega_d$. Low-lying collective excitations within a given phase, could exhibit a parametric instability and amplification, if the modulation's driving frequency coincides with twice the energy of a collective mode. The severity of this instability relates to the quantum fluctuations in a given phase, and thus may depend on its proximity to various phase boundaries. (b) The low-lying collective excitations for the flat-band model we study in Sec. \ref{['sec:TBGproxy']} (black). The model has a quantum-geometry-related parameter $\zeta$, set to the value $\zeta=\zeta_0$. By slightly modifying this parameter, either increasing (red traces) or reducing $\zeta$ (blue traces) by 10%, the collective mode spectrum is altered. (c) Left: For the $n=2$, ${\bf Q=0}$ mode, marked by a purple star in (b), we plot $\delta z$, which is the strength of the induced two-boson term (blue) as a function of deviation from $\zeta_0$. The light green line has the slope $\frac{\partial\delta_z}{\partial\zeta}|_{\zeta\to\zeta_0}$, representing the parametric susceptibility with respect to $\zeta$. Right: Matrix elements of the perturbed Hamiltonian, see Eq. \ref{['eq:bosonicparametric']}. For perturbing $\zeta_0\to1.2\zeta_0$ we present the hybridization of this mode with other modes (top), and the respective two-boson term (bottom). The peaked contribution at $n=2$ is the extracted $\delta_z$ for this value of $\zeta$.
• Figure 2: Squeezing of vacuum fluctuations for generic ordered states. (a) Schematic depiction of the Bloch sphere for an SU(2) symmetric or easy-axis (along $\hat{z}$) ferromagnet. The ordering direction is indicated by the black arrow. Small fluctuations about the ordered state are represented by the red cloud. A change in the parameters of the parent Hamiltonian of this phase cannot change the nature of the fluctuations without explicitly breaking the remaining U(1) symmetry. (b) Bloch sphere of an easy-plane ferromagnet. The out-of-plane direction is inherently harder for the vacuum fluctuations as compared to in-plane. The fluctuation vacuum is thus squeezed (red oval cloud). Modification of Hamiltonian parameters will generically alter the amount of the aforementioned squeezing, depicted by the elongated blue cloud. The vacuum of fluctuations is susceptible to such parameter changes. (c) Generalized Bloch sphere for an antiferromagnet. The north pole represents the ordered state. The two fluctuation quadratures correspond to either a rotation of the Néel vector, (an easy direction) or to ferromagnetic alignment (hard direction). Squeezing, sensitive to microscopic Hamiltonian parameters, is thus an unavoidable consequence.
• Figure 3: Driving and probing quantum Hall double layer collective excitations (Sec. \ref{['sec:QHbilayer']}). (a) Schematic Bloch sphere of the ordered phase with zero (green) and finite (purple) displacement field. Since the equator hosting the order parameter is pushed upward on the sphere when $\langle\ell_z\rangle$ is finite (i.e., the system develops layer polarization), the quadrature squeezing is different. Hence, parametric instability is enabled. (b) When the system is parametrically driven, an oscillatory perpendicular electric field is induced in the system. Interlayer tunneling spectroscopy is thus expected to display coherent peaks at half the driving frequency (alongside the interlayer-coherence-associated zero-bias peak). (c) In a Josephson-analogous geometry (weak links denoted by dashed black lines), the relative phase of the condensate between the driven part (driving represented by fuzzy orange oval) and the static part would result in an ac counterflow current $I\left(t\right)$ dominated by a $\omega_d/2$ harmonic.
• Figure 4: Driving in the T-order phase of the flat band model (Sec. \ref{['sec:Torderflat']}). (a) Energy hierarchy within the low-energy manifold of the possible ordered ground states. The energy $u_{\zeta}$ separates the T/K orders from the rest of the manifold, and $\Delta_{\rm TK}$ further favors T-order over K. (b) The generalized Bloch sphere for the T-order case, positioned the north pole. The two fluctuation quadratures correspond to the easy K-order, and the hard sublattice ($\sigma$) order. The anisotropy of the fluctuation cloud corresponds to the squeezing in the fluctuation vacuum. Next to each order we schematically draw real-space spectroscopic signatures of the analogous TBG phase: T-order admits a Kekulé distortion of the graphene unit cell, $\sigma$ corresponds to finite sub-lattice polarization, and K-order is not discernible in such experiments IVCspectroscopycalugaru. Modulating the parameter $\Delta_{\rm TK}$ affects the squeezing, and thus enables parametric resonance. In the corresponding driven state one observes substantial sublattice oscillations at half the driving frequency.
• Figure 5: Pseudospin (sublattice) polarized order and its parametric driving (Sec. \ref{['sec:sigmaorderflat']}). (a) Energy hierarchy with a finite external sublattice potential [$\Delta_\sigma$ in Eq. \ref{['eq:pseudotbgsingleparticle']}]. The energy $u_{\zeta}$ brings the T/K orders down in energy, and $\Delta_{\rm TK}$ once again favors T-order over K. (b) The generalized Bloch sphere for the $\sigma$-order case, positioned the north pole. The two fluctuation quadratures correspond to the easy T-order, and the hard K-order, distinguished by the interaction energy $\Delta_{\rm TK}$. The anisotropy of the fluctuation cloud corresponds again to the vacuum. (c) Left: Parametric efficiency with respect to the quantum geometric parameter $\zeta$, Eq. \ref{['eq:parametricsuszeta']}. This parameter roughly corresponds to the Wannier spread in real-space of the wavefunctions corresponding to the active flat-bands (right schematic). The dashed line marks the phase boundary between the $\sigma$ phase below and the T-order stabilized above (hatched region), in accordance with the hierarchy in (a). Red line traces where $\eta_{{\rm par}.}^{\left(\zeta\right)}=1$. The parametric response is maximal when the interaction energy strongly depends on quantum geometry. The susceptibility diverges as one approaches the phase boundary, even at $\zeta$ much larger than its optimal value. Right: A vertical cut of the left colormap at $\zeta=0.65$, showing the divergence as the phase boundary is approached. Here, $\Delta_\sigma=u$.