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Acta Physica Sinica

Higgs mode near superfluid-to-Mott-insulatortransition studied by the quantum Monte Carlo method

Kun Chen, Youjin Deng

Published Jan 1, 2015• 1 citation
Publisher

Abstract

In additional to the phonon (massless Goldstone mode) in Galilean invariant superfluid, there is another type of mode known as the Higgs amplitude mode in superfluid with emergent Lorentz invariance. In two dimensions, due to the strong decay into phonons, whether this Higgs mode is a detectable excitation with sharp linear response has been controversial for decades. Recent progress gives a positive answer to this question. Here, we review a series of numerical studies of the linear response of a two-dimensional Lorentz invariant superfluid near the superfluid-Mott insulator quantum critical point (SF-MI QCP). Particularly, we introduce a numerical procedure to unbiasedly calculate the linear response properties of strongly correlated systems. The numerical procedure contains two crucial steps, i.e., one is to use a highly efficient quantum Monte Carlo method, the worm algorithm in the imaginary-time path-integral representation, to calculate the imaginary time correlation functions for the system in equilibrium; and then, the other is, based on the imaginary time correlation functions, to use the numerical analytical continuation method for obtaining the real-time (real-frequency) linear response function. Applying this numerical procedure to the two-dimensional Bose Hubbard model near SF-MI QCP, it is found that despite strong damping, the Higgs boson survives as a prominent resonance peak in the kinetic energy response function. Further investigations also suggest a similar but less prominent resonance peak near SF-MI QCP on the MI side, and even on the normal liquid side. Since SF-MI quantum criticality can be realized by ultracold aotms in optical lattice, the Higgs resonance peak can be directly observed in experiment. In addition, we point out that the same Higgs resonance peak exists in all quantum critical systems with the same universality, namely (2 + 1)-dimensional relativistic U(1) criticality, as SF-MI QCP.

DOI
10.7498/aps.64.180201

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In additional to the phonon (massless Goldstone mode) in Galilean invariant superfluid, there is another type of mode known as the Higgs amplitude mode in superfluid with emergent Lorentz invariance. In two dimensions, due to the strong decay into phonons, whether this Higgs mode is a detectable excitation with sharp linear response has been controversial for decades. Recent progress gives a positive answer to this question. Here, we review a series of numerical studies of the linear response of a two-dimensional Lorentz invariant superfluid near the superfluid-Mott insulator quantum critical point (SF-MI QCP). Particularly, we introduce a numerical procedure to unbiasedly calculate the linear response properties of strongly correlated systems. The numerical procedure contains two crucial steps, i.e., one is to use a highly efficient quantum Monte Carlo method, the worm algorithm in the imaginary-time path-integral representation, to calculate the imaginary time correlation functions for the system in equilibrium; and then, the other is, based on the imaginary time correlation functions, to use the numerical analytical continuation method for obtaining the real-time (real-frequency) linear response function. Applying this numerical procedure to the two-dimensional Bose Hubbard model near SF-MI QCP, it is found that despite strong damping, the Higgs boson survives as a prominent resonance peak in the kinetic energy response function. Further investigations also suggest a similar but less prominent resonance peak near SF-MI QCP on the MI side, and even on the normal liquid side. Since SF-MI quantum criticality can be realized by ultracold aotms in optical lattice, the Higgs resonance peak can be directly observed in experiment. In addition, we point out that the same Higgs resonance peak exists in all quantum critical systems with the same universality, namely (2 + 1)-dimensional relativistic U(1) criticality, as SF-MI QCP.
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)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.
We present solid evidence for the existence of a well-defined Higgs amplitude mode in twodimensional 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 ultra-cold Rb atoms demonstrate that the low-frequency resonance is lost for typical experimental parameters, while the characteristic frequency for the onset of strong response is preserved.
We present spectral functions for the magnitude squared of the order parameter in the scaling limit of the two-dimensional superfluid to Mott insulator quantum phase transition at constant density, which has emergent particle-hole symmetry and Lorentz invariance. The universal functions for the superfluid, Mott insulator, and normal liquid phases reveal a low-frequency resonance which is relatively sharp and is followed by a damped oscillation (in the first two phases only) before saturating to the quantum critical plateau. The counterintuitive resonance feature in the insulating and normal phases calls for deeper understanding of collective modes in the strongly coupled (2+1)-dimensional relativistic field theory. Our results are derived from analytically continued correlation functions obtained from path-integral Monte Carlo simulations of the Bose-Hubbard model.
We study the Higgs excitation in the Goldstone phase of the relativistic O(NNN) model in two spatial dimensions at zero temperature. The response functions of the order parameter, and its magnitude squared, become universal functions of frequency in the vicinity of the quantum critical point described by the Wilson-Fisher fixed point, and we compute them to next-to-leading order in 1/N1/N1/N. The Higgs particle has an infrared singular decay to gapless Goldstone excitations, and its response functions are characterized by a pole in the lower half of the complex frequency plane. The pole acquires a nonzero real part only at next-to-leading order in 1/N1/N1/N, demonstrating that the Higgs excitation has an oscillatory component even in the scaling limit. Both the real and imaginary parts of the pole position vanish with the correlation length exponent $\ensuremath{\nu}$ upon approaching the critical point. We present evidence that the spectral density of the O(NNN)-invariant amplitude-squared of the order parameter has a peak at a nonzero frequency in the scaling limit. We connect our results to recent experimental studies of the superfluid-insulator quantum phase transition of ultracold bosonic atoms in optical lattices.
We study the Higgs mode of superfluid Bose gases in a three-dimensional optical lattice, which emerges near the quantum phase transition to the Mott insulator at commensurate fillings. Specifically, we consider responses of the Higgs mode to temporal modulations of the onsite interaction and the hopping energy. In order to calculate the response functions including the effects of quantum and thermal fluctuations, we map the Bose-Hubbard model onto an effective pseudospin-1 model and use a perturbative expansion based on the imaginary-time Green's function theory. We also include the effects of an inhomogeneous trapping potential by means of a local density approximation. We find that the response function for the hopping modulation is equal to that for the interaction modulation within our approximation. At the unit filling rate and in the absence of a trapping potential, we show that the Higgs mode can exist as a sharp resonance peak in the dynamical susceptibilities at typical temperatures. However, the resonance peak is significantly broadened due to the trapping potential when the modulations are applied globally to the entire system. We suggest that the Higgs mode can be detected as a sharp resonance peak by partial modulations around the trap center.
Relativistic O(NNN) field theories are studied near the quantum-critical point in two space dimensions. We compute dynamical correlations by large-scale Monte Carlo simulations and numerical analytic continuation. In the ordered side, the scalar spectral function exhibits a universal peak at the Higgs mass. For N=3N=3N=3 and 4, we confirm its ${\ensuremath{\omega}}^{3}$ rise at low frequency. On the disordered side, the spectral function exhibits a sharp gap. For N=2N=2N=2, the dynamical conductivity rises above a threshold at the Higgs mass (density gap), in the superfluid (Mott insulator) phase. For charged bosons (Josephson arrays), the power-law rise above the Higgs mass increases from two to four. Approximate charge-vortex duality is reflected in the ratio of imaginary conductivities on either side of the transition. We determine the critical conductivity to be ${\ensuremath{\sigma}}_{\mathrm{c}}^{*}=0.3(\ifmmode\pm\else\textpm\fi{}0.1)\ifmmode\times\else\texttimes\fi{}4{e}^{2}/h$ and describe a generalization of the worm algorithm to N>2N>2N>2. We use a singular value decomposition error analysis for the numerical analytic continuation.