Higgs amplitude mode in the vicinity of a $(2+1)$-dimensional quantum critical point: A nonperturbative renormalization-group approach

We study the “Higgs” amplitude mode in the relativistic quantum $\mathrm{O}\left(N\right)$ model in two space dimensions. Using the nonperturbative renormalization group and the Blaizot–Méndez-Galain–Wschebor approximation (which we generalize to compute four-point correlation functions), we compute the $\mathrm{O}\left(N\right)$-invariant scalar susceptibility at zero temperature in the vicinity of the quantum critical point. In the ordered phase, we find a well-defined Higgs resonance for $N=2$ and 3 and determine its universal properties. No resonance is found for $N\ge 4$. In the disordered phase, the spectral function exhibits a threshold behavior with no Higgs-like peak. We also show that for $N=2$, the Higgs mode manifests itself as a very broad peak in the longitudinal susceptibility in spite of the infrared divergence of the latter. We compare our findings with results from quantum Monte Carlo simulations and $\epsilon =4-(d+1)$ expansion near $d=3$.

Figure 1

Diagrammatic representation of the scalar susceptibility (25) (a). A vertex ${\mathrm{\Gamma }}^{(n,m)}$ is represented by a black dot with $n$ solid lines and $m$ wavy lines (b). Solid lines connecting vertices stand for the propagator $G={\mathrm{\Gamma }}^{(2,0)-1}$.

Figure 2

Diagrammatic representation of the RG equations (30), (39), and (40). Signs and symmetry factors are not shown. The diagrammatic representation of the vertices ${\mathrm{\Gamma }}_k^{(n,m)}$ is the same as in Fig. 1 and the cross stands for ${\partial }_k{R}_k$.

Figure 3

Scalar susceptibility ${\chi }_s\left(\mathbf{p}\right)$ at the QCP for $N=1000$ (solid line) compared to the exact large-$N$ solution with $R_{\mathrm{\Lambda }}<\infty$ (circles) and $R_{\mathrm{\Lambda }}=\infty$ (squares). Here and in the following figures we use arbitrary units for the scalar and longitudinal susceptibilities.

Figure 4

Spectral functions ${\chi }_s^{\prime \prime }\left(\omega \right)$ and ${\chi }_{\mathrm{L}}^{\prime \prime }\left(\omega \right)$ in the ordered and disordered phases for $N=1000$ (solid line), compared to the exact large-$N$ solution (symbols). In the disordered phase, the exact solution for ${\chi }_{\mathrm{L}}^{\prime \prime }\left(\omega \right)={\chi }_{\mathrm{T}}^{\prime \prime }\left(\omega \right)\sim \delta (\omega -\mathrm{\Delta })$ is not shown.

Figure 5

$|{\chi }_s\left(\mathbf{p}\right)-{\chi }_s\left(0\right)|/p^{3-2/\nu }$ and ${\chi }_{\mathrm{L},\mathrm{T}}\left(\mathbf{p}\right)/p^{-2+\eta }$ at the QCP for $N=2$ and 3. The normalization is chosen to have a ratio equal to one for $p\to 0$.

Figure 6

${\chi }_{\mathrm{L},\mathrm{T}}\left(\mathbf{p}\right)$ and ${\chi }_{\mathrm{L},\mathrm{T}}^{\prime \prime }\left(\omega \right)$ in the disordered phase for $N=2$ and 3. The solid line and the symbols correspond to different values of $r_0-r_{0c}$.

Figure 7

Same as Fig. 6 but for the scalar susceptibility ${\chi }_s$.

Figure 8

Derivative $W_k\left(\rho \right)$ of the effective potential, shown in the range $[{\rho }_{\min ,k},{\rho }_{\max }]$, for various values of $k$ [from bottom to top: $ln(k/\mathrm{\Lambda })=-11.5$, $-12.25$, $-13$, and $-15.25]$ and $N=2$. The inset shows the corresponding flow of ${\rho }_{0,k}$ vs $ln(k/\mathrm{\Lambda })$.

Figure 9

${\chi }_s\left(\mathbf{p}\right)$ and ${\chi }_s^{\prime \prime }\left(\omega \right)$ in the ordered phase for $N=2$ and 3. The solid line and the symbols correspond to different values of $r_0-r_{0c}$.

Figure 10

Log-log scale plot of ${\chi }_s^{\prime \prime }\left(\omega \right)$ in the ordered phase for $N=2$ and 3, showing the asymptotic behavior ${\chi }_s^{\prime \prime }\left(\omega \right)\sim {\omega }^3$ at low energies.

Figure 11

Spectral function ${\chi }_s^{\prime \prime }\left(\omega \right)$ for various values of $N$ in the ordered phase.

Figure 12

${\chi }_{\mathrm{L}}\left(\mathbf{p}\right)$ and ${\chi }_{\mathrm{L}}^{\prime \prime }\left(\omega \right)$ in the ordered phase for $N=2$ and 3. The solid line and the symbols correspond to different values of $r_0-r_{0c}$.

Figure 13

Log-log scale plot of ${\chi }_{\mathrm{L}}^{\prime \prime }\left(\omega \right)$ in the ordered phase for $N=2$ and 3, showing the asymptotic behavior ${\chi }_{\mathrm{L}}^{\prime \prime }\left(\omega \right)\sim 1/\omega$ at low energies [42].