Visibility of the amplitude (Higgs) mode in condensed matter

The amplitude mode is a ubiquitous collective excitation in condensed-matter systems with broken continuous symmetry. It is expected in antiferromagnets, short coherence length superconductors, charge density waves, and lattice Bose condensates. Its detection is a valuable test of the corresponding field theory, and its mass gap measures the proximity to a quantum critical point. However, since the amplitude mode can decay into low-energy Goldstone modes, its experimental visibility has been questioned. Here we show that the visibility depends on the symmetry of the measured susceptibility. The longitudinal susceptibility diverges at low frequency as $\mathsf{Im}{\chi }_{\sigma \sigma }\sim {\omega }^{-1}$ $(d=2)$ or $\log (1/|\omega \left|\right)$ $(d=3)$, which can completely obscure the amplitude peak. In contrast, the scalar susceptibility is suppressed by four extra powers of frequency, exposing the amplitude peak throughout the ordered phase. We discuss experimental setups for measuring the scalar susceptibility. The conductivity of the $O\left(2\right)$ theory (relativistic superfluid) is a scalar response and therefore exhibits suppressed absorption below the Higgs mass threshold, $\sigma \sim {\omega }^{2d+1}$. In layered, short coherence length superconductors, (relevant, e.g., to cuprates) this threshold is raised by the interlayer plasma frequency.

Figure 1

Visibility of the amplitude mode. The longitudinal and scalar susceptibilities of the two-dimensional $O\left(N\right)$ model at zero temperature are plotted in the large $N$ limit. $m$ is the renormalized amplitude mode mass. The coupling constant is $g=0.84g_{\mathrm{c}}^{\infty }$, which is within the broken symmetry phase. Note that dissipation into Goldstone modes has very different effects on the low-frequency behavior of the two susceptibilities and on the visibility of the amplitude mode around $\omega =m$.

Figure 2

Fluctuations in the Mexican hat potential. $\langle \mathit{\Phi }\rangle$ is the order parameter. The massive Higgs and massless Goldstone modes can be represented either in terms of longitudinal ($\sigma$) and transverse ($\mathit{\pi }$) degrees of freedom, respectively, or in terms of scalar ($\rho$) and direction (${\partial }_{\mu }\widehat{\mathit{n}}$) degrees of freedom.

Figure 3

Infrared divergences at weak coupling. Diagrams describing the infrared divergent weak coupling corrections to the longitudinal and scalar susceptibilities in Eqs. (7) and (11). Solid lines are $\sigma$ propagators; dashed lines represent $\mathit{\pi }$ propagators.

Figure 4

Weak coupling expansion for the dynamical conductivity. (a) Order $g^{-1}$ contribution to the weight of the superfluid $\delta$ function at $\omega =0$. (b) Order $g^0$ diagram has a threshold at the amplitude mode mass. (c),(d) Two-loop self-energy and vertex corrections which contribute to the subgap conductivity. As $N\to \infty$ diagram (c) dominates and yields $\sigma \sim {\omega }^{2d-3}$. For $N=2$ (the relativistic superfluid), cancellations between diagrams (c) and (d) suppress the subgap conductivity by four powers of $\omega$, and $\sigma \sim {\omega }^{2d+1}$ (see text).

Figure 5

Dynamical conductivity for $O\left(2\right)$ (relativistic bosons) in two dimensions. The arrow at zero frequency denotes is the superfluid $\delta$ function response. For neutral bosons, there is a weak $\mathcal{O}\left(g\right)$ subgap tail (see inset). The conductivity of a bosonic layered superconductor (red online) is plotted for using intralayer plasma frequency ${\omega }_{\mathrm{p}}=10m$ and a much smaller interlayer plasma frequency of ${\omega }_{\mathrm{p}}^c=0.1m$.

Figure 6

Dependence of the scalar peak on $g$. Scalar susceptibility ${\chi }_{\rho \rho }^{\prime \prime }\left(\omega \right)$ for different values of $g/g_c$ in $d=2$ dimensions. Results shown in the large $N$ limit for $\Lambda =2m_0$. (Inset) Renormalized mass $m$ (solid curve) and width $\gamma$ (dashed curve) of the peak in ${\chi }_{\rho \rho }^{\prime \prime }\left(\omega \right)$ as a function of the tuning parameter $g$, expressed in units of the bare mass $m_0$. For $g\to 0$, deep inside the ordered phase, the mass approaches its bare value $m_0$ and the peak is very sharp, $\gamma \to 0$. As $g$ approaches $g_c$, the mass is softened and the width grows, until for $g/g_c>0.96$ the peak energy is smaller than the width.

Figure 7

Diagrams in the large-$N$ limit. For $N=\infty$, only the RPA sums shown here contribute.

Figure 8

Order $g$ diagrams contributing to the optical conductivity for $\omega <2m$. Solid brown lines are $\sigma$ propagators while dashed blue lines are $\mathit{\pi }$ propagators. Note that diagram (c) does not contribute to the subgap conductivity and is therefore not shown in Fig. 4.

Figure 9

The range of $(k_1,k_2)$ integration is the gray triangle.