Landau and fractionalized theories of periodically driven intertwined orders

Abstract

We obtain the phase diagrams of field theories of intertwined orders in the presence of periodic driving by an external field which preserves all symmetries. We consider both a conventional Landau theory of competing orders, and a fractionalized theory in which the order parameters are distinct composites of an underlying multi-component Higgs field. We work in the large $N$ limit and couple to a Markovian bath. The long time limits are characterized by non-zero average values, oscillations with the drive period and/or half the drive period, quasi-periodic oscillations, or chaotic behavior.

Figures (10)

• Figure 1: Equilibrium phase diagram of the fractionalized theory \ref{['eq:Lagrangian']} in the absence of gauge fields in the limit $N\to\infty$Christos2024. The red dashed lines indicate parameter cuts along which we study the effect of a periodic drive. The horizontal cut can be mapped to a theory with a single order parameter, corresponding to the driven $O(N)$ model analyzed in Ref. Diessel25. In this work we focus on the vertical cut, which probes the regime of two competing orders.
• Figure 2: Equilibrium phase diagrams of the $O(N)\times O(M)$ model \ref{['Eq:Landau_model']} as a function of the static mass parameters $R_0$ and $\Delta r_0$ for different inter-component couplings $v=-0.5$, $v=0.5$, and $v=1.5$. The red dashed line at $R_0/\Omega^2=-1.6$ corresponds to the parameters along which we study the effect of periodic driving.
• Figure 3: Phase diagram of the fractionalized theory in the $v_0/\Omega^2-v_1/\Omega^2$ plane, obtained from solutions of Eqs. \ref{['eq:EoM']}. Different colors denote the metallic phase (gray), superconducting (SC) phase (blue), and charge-density-wave (CDW) phase (yellow). Regions with simultaneous SC and CDW order are shown in pink, with the labels indicating the dominant frequency components of the order parameters: $\Omega$ and/or $\Omega/2$. Hatched areas mark parameter regimes where neither simple $\Omega$-locked nor $\Omega/2$-locked responses occur, but instead signatures of quasiperiodic or chaotic dynamics are found (cf. Fig. \ref{['Fig:Chaos']}). The dashed–dotted and dotted lines mark cuts along which the parameter $r_1/\Omega^2$ is additionally driven (see Fig. \ref{['Fig:PD_r1v1']}).
• Figure 4: Time evolution of the order parameters during the final 10 drive periods of a simulation run over 9000 driving periods in the metallic (gray), SC (blue) and CDW (yellow) phases. From top to bottom, the panels correspond to $(v_0, v_1)=(0.0, 0.4)$, $(0.5,0.2)$, $(-0.3,0.08)$, and $(-0.3,0.12)$.
• Figure 5: Representative time evolution of the superconducting order parameter $\phi_{\text{SC}}$ (blue) and the charge-density-wave order parameter $\phi_{\text{CDW}}$ (yellow) in the four different coexisting phases. From top to bottom the panels show oscillations at $(\Omega,\Omega),(\Omega,\Omega/2),(\Omega/2,\Omega),$ and $(\Omega/2,\Omega/2)$, respectively. The corresponding parameter values are $(v_0,v_1) = (0.5,0.38), (-0.21,0.24), (1.2,0.8), (0,0.18)$. (b–e) Corresponding stroboscopic plots, where the order parameter is sampled at integer multiples of the drive period, $t=nT$, over the last $t/T=2000$ cycles. Each point corresponds to the pair $(\phi_{\text{CDW}}(nT),\,\phi_{\text{SC}} (nT))$. Panel (b) shows the phase where both order parameters oscillate at $\Omega$. Panel (c) illustrates the coexistence of both components oscillating at different frequencies ($\Omega,\Omega/2$), while panel (d) shows the complementary mixed case with $(\Omega/2,\Omega)$ contributions. Panel (e) corresponds to the phase where both order parameters are period-doubled ($\Omega/2,\Omega/2$).