Fate of the Higgs Mode Near Quantum Criticality

We study a relativistic $O(N)$ model near the quantum critical point in $2+1$ dimensions for $N=2$ and $N=3$. The scalar susceptibility is evaluated by Monte Carlo simulation. We show that the spectrum contains a well-defined peak associated with the Higgs mode arbitrarily close to the critical point. The peak fidelity and the amplitude ratio between the critical energy scales on both sides of the transition are determined.

Figure 1

Critical energy scales near the quantum phase transition in relativistic $O(N)$ field theory for $N=2$, 3. $\delta g\equiv (g-g_c)/g_c$ is the dimensionless tuning parameter. $m_H$ is the Higgs peak energy (mass) in the ordered phase $\delta g<0$, and $\Delta$ is the gap in the disordered phase $\delta g>0$. Solid lines describe the critical behavior $m_H={\mathcal{B}}_{-}|\delta g{|}^{{\nu }_N}$ and $\Delta ={\mathcal{B}}_{+}|\delta g{|}^{{\nu }_N}$. Inset: Phase diagram in the microscopic $g$, $\mu$ parameter space [defined in Eq. (1) for $N=2$]. Two choices of $\mu$, denoted by (a) and (b), were studied in this Letter.

Figure 2

(a) The scalar susceptibility as a function of Matsubara frequency for $N=2$ and $\mu =0.5$. Lower curves, different $\delta g>0$ (disordered phase). Upper curves, $\delta g<0$ (ordered phase) (b) Collapse of the same susceptibility onto scaling functions ${\tilde{\Phi }}_{\pm }$. Results include two values of $\mu$ (see inset of Fig. 1).

Figure 3

Scalar susceptibility ${\chi }_s^{\prime \prime }(\omega )$ (blue online), in arbitrary units, for different values of $\delta g<0$ at $\mu =0.5$. Curves shifted vertically by $\delta g$. The Higgs energy $m_H$ as a function of $\delta g$ is extracted from the peak positions (dots) as the solid curve (red online).

Figure 4

Rescaled spectral function vs $\omega /\Delta$ for $N=2$, 3, at $\mu =0.5$. At low values of $\omega /\Delta$, these curves collapse to the universal scaling function ${\Phi }_{-}^{\prime \prime }(\omega /\Delta )$, in accordance with Eq. (5).