Higgs Mode in a Two-Dimensional Superfluid

We present solid evidence for the existence of a well-defined Higgs amplitude mode in two-dimensional relativistic field theories based on analytically continued results from quantum Monte Carlo simulations of the Bose-Hubbard model in the vicinity of the superfluid–Mott insulator quantum critical point, featuring emergent particle-hole symmetry and Lorentz invariance. The Higgs boson, seen as a well-defined low-frequency resonance in the spectral density, is quickly pushed to high energies in the superfluid phase and disappears by merging with the broad secondary peak at the characteristic interaction scale. Simulations of a trapped system of ultracold ${}^{87}\mathrm{Rb}$ atoms demonstrate that the low-frequency resonance is lost for typical experimental parameters, while the characteristic frequency for the onset of a strong response is preserved.

Figure 1

Universal scaling predictions for the scalar susceptibility (solid lines). The dash-dotted line depicts the prediction of Ref. 12 and misses most of the spectral density at the relevant energy scale $\Delta \propto (1-U/U_c{)}^{\nu }$. The two alternatives for connecting universal power laws are shown by dashed lines (one may also imagine multiple peaks in the crossover region).

Figure 2

Spectral density $S(\omega )$ of the kinetic energy correlation function for $U/J=16$ (thick solid line), 14 (dashed line), and 12 (thin solid line) at low temperature $T/J=0.1$. The Higgs amplitude mode (${\omega }_H$) emerges as a well-defined peak on the approach to the quantum critical point at $U_c=16.7424$.

Figure 3

Characteristic energies in the vicinity of the quantum critical point at $(U/J{)}_c=16.7424$. Circles for $U>U_c$ and $U<U_c$ stand for particle-hole gaps in the MI phase [17] and energies of the Higgs bosons, respectively. The squares above the circles denote the location of the broad secondary peak in $S(\omega )$ until it merges with the amplitude mode at interaction strength $U\le 12J$ to form a single peak (squares for $U/J\le 12$). Shaded regions indicate the characteristic broadening of peaks. The thick black line is the critical law $2.25J|(U-U_c)/J{|}^{\nu }$ obtained by fitting the smallest MI gaps; its mirror reflection is shown as a thin black line. The dashed red line indicates the typical interaction scale $U/J$.

Figure 4

Spectral densities in the SF and MI phases, with approximately the same amount of detuning from quantum criticality $|U_c-U|/U\approx 0.015$.

Figure 5

Evolution of spectral density with temperature at $U/J=16$. As temperature is increased from $T/J=0.1$ (thin black line) to $T/J=0.2$ (thin dashed red line) and $T/J=0.5$ (thick blue line), the peaks get broader but remain clearly identifiable. At $T/J=1$ (thick dashed green line), the two peaks merge.

Figure 6

Effects of the trapping potential and finite temperature for $U/J=14$ ($T_c/J\approx 1.04$). The spectral density of a homogeneous system at low temperature with Higgs resonance (dashed line) transforms into a broad (irregular) peak due to inhomogeneous broadening in a trapped system with $N=190$ particles at the unit filling factor in the middle (thin solid line). At a temperature $T/J=0.5$ (thick solid line), we observe a smooth single peak.