Steady-states and response functions of the periodically driven O(N) scalar field theory

TL;DR

This work analyzes a parametrically driven $O(N)$ scalar theory coupled to a Markovian bath, using large-$N$ techniques and Floquet theory to map a rich non-equilibrium phase diagram that includes uniform and spatially modulated symmetry-broken states, some with order-parameter dynamics at half the drive frequency. It develops both analytical (Mathieu–Hill/period-doubling) and numerical approaches to obtain steady states and their stability, and it couples the system to an electromagnetic field to reveal Meissner-like responses and the novel Meissner polariton arising when time-translation symmetry is broken. The study also uncovers how finite-temperature fluctuations shift the transition points and how the optical conductivity can exhibit superconducting-like features without a true Meissner effect, thereby offering a unified framework for light-induced orders and their electromagnetic signatures. The results provide theoretical guidance for experiments on light-driven superconductivity and related orders, suggesting observable fingerprints such as period-doubled superconducting states, PDWs, and Meissner-polariton modes in pump–probe setups.

Abstract

We investigate the phase diagram of a relativistic, parametrically driven O($N$)-symmetric theory coupled to a Markovian thermal bath. Our analysis reveals a rich variety of phases, including both uniform and spatially modulated symmetry-broken states, some of which feature an order parameter oscillating at half the drive frequency. When coupled to a background electromagnetic potential, these phases exhibit a Meissner effect, in the sense that the photon acquires a mass term. However, if the order parameter oscillates around a sufficiently small value, a fraction of an externally applied magnetic field can penetrate the sample in the form of a standing wave. We dub this property a \textit{Meissner polariton}, that is, a collective mode resulting from the hybridization of light with order parameter oscillations. Furthermore, near the onset of symmetry breaking, strong fluctuations give rise to a superconducting-like response even in the absence of a Meissner effect or of a Meissner polariton. Our results are relevant to experiments on light-induced orders, particularly superconductivity.

Figures (6)

• Figure 1: Panel (a): Phase diagram obtained by solving Eqs. \ref{['eq:EoM']} with parameters $u=\Omega^2$, and $\gamma=0.03\,\Omega$. Six phases appear: normal (gray), superconducting (SC,blue), period-doubling SC (red), pair density wave (PDW, blue hatches), PDW with period doubling (red hatches) and a quasi-periodic PDW (yellow). The symbols mark the points whose order-parameter time traces over the last ten drive periods are shown in panel (b). As detailed in App. \ref{['app: finite-q']}, we also performed a dynamical stability analysis. The hatched regions indicate an instability of the normal state towards a PDW, oscillating at either half the drive frequency (red hatched regions), labeled as PDW with period doubling (PDW w PD), or a at the drive frequency, labeled as PDW withoud period doubling (PDW w/o PD). The yellow region indicates a portion of the phase diagram where the steady state is non-periodic but the correlation function $C_\mathbf{q}(t)$ diverges at finite $\mathbf{q}$. We labeled this region as quasi-periodic PDW (qp-PDW). Panel (b): Representative dynamics $\bar{b}(t)$ over the last ten periods, plotted versus $t/T_0$, for the parameter points indicated by the symbols in panel (a). Top: metal ($\bar{b}=0$), middle: SC (small oscillations with the drive period about a nonzero mean), bottom: period-doubling SC (oscillations with period $2T_0$ around zero).
• Figure 2: Lines in $\lambda_0$-$\lambda_1$ plane where the damped Mathieu equation supports oscillating steady states at $\omega=0$ (blue lines) and at $\omega=\Omega/2$ (red lines) for $\gamma=0.3\Omega$. In the gray shaded regions, the solutions are decaying to zero, whereas in the unshaded regions solutions grow indefinitely with time.
• Figure 3: Panel (a): Location of the different finite mean-field solutions of the simplified two-modes model \ref{['eq:K kernel analyitic']} within the $r_0$-$r_1$ plane. The solid black curve denotes the curve \ref{['eq:stability_condition']}. Panel (b): Phase diagram for the simplified two-mode model \ref{['eq:K kernel analyitic']}. The dashed line indicates a second-order phase transition, while the dashed one indicates a first order phase transitions. The large black dot indicates the exceptional point. The symbols (star, triangle, diamond) denotes points where we analyze the fluctuation dispersions (see Fig. \ref{['Fig:Dispersions']}).
• Figure 4: Real and imaginary parts of the quasiparticle dispersion relations for the two-mode model, Eq. \ref{['eq:K kernel analyitic']}. Panels (a)–(c) correspond to the parameter points marked by a star, triangle, and diamond, respectively, in the phase diagram of Fig. \ref{['Fig:OPMeanField']} (b).
• Figure 5: Behavior of the magnetic field inside the material for a superconducting state with an order parameter that is either uniform or carries a finite momentum in the form of Eq. \ref{['eq: FF state']}. The $x$-axis coordinate indicates the distance from an interface to vacuum. In panels (a) and (b) we show a state with period doubling, whereas in panels (c) and (d) we have a superconducting state without period doubling, with an order parameter that oscillates around a finite offset. Panels (a) and (c) display the magnetic field profiles near the vacuum interface, in contrast to panels (b) and (d), which show the field behavior within the bulk of the material. The curves have been calculated using Eq. \ref{['eq: London time dep']} for $4c^2\kappa_1/\Omega^2=1000$ for $\kappa_0=2|\kappa_1|$ (period-doubling state) and $\kappa_0=2.5|\kappa_1|$ (non period-doubling state). Different colors refer to different times in units of the half period $T_0/2=\pi/\Omega$, as indicated in the colorbars. The black curves in both panels indicate the period-averaged magnetic field.