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arXiv:1205.2700 · Physical Review B

Spectral functions of the Higgs mode near two-dimensional quantum critical points

Daniel K. Podolsky, Subir Sachdev A

Published Aug 10, 2012• 111 citations
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Abstract

We study the Higgs excitation in the Goldstone phase of the relativistic O($N$) 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 $1/N$. 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 $1/N$, 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($N$)-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.

DOI
10.1103/physrevb.86.054508

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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 \ensuremathν\ensuremath{\nu}\ensuremathν 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 investigate the amplitude (Higgs) mode of the relativistic O(N)O\left(N\right)O(N) model in the vicinity of the Wilson-Fisher quantum critical point in $D=4\ensuremath{-}\ensuremath{\varepsilon}$ space-time dimensions. We compute the universal part of the scalar spectral function near the transition, to leading nontrivial order in the ordered phase, and to next to leading order in both the disordered phase and the quantum critical regime. We find that, in the disordered phase, the spectral function has a threshold behavior with no Higgs-like peak, whereas in the ordered phase, the Higgs mode appears as a well defined resonance. The pole associated with this resonance is purely real in the $D\ensuremath{\rightarrow}3+1$ limit, evolving smoothly with dimensionality to become purely imaginary at D=2+1D=2+1D=2+1 in the $N\ensuremath{\rightarrow}\ensuremath{\infty}$ limit. Our results complement previous studies of the scalar spectral function, and demonstrate that the resonance found in these studies can indeed be directly identified with the Higgs mode.
We study the ``Higgs'' amplitude mode in the relativistic quantum O(N)\mathrm{O}(N)O(N) model in two space dimensions. Using the nonperturbative renormalization group and the Blaizot--M\'endez-Galain--Wschebor approximation (which we generalize to compute four-point correlation functions), we compute the O(N)\mathrm{O}(N)O(N)-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=2N=2N=2 and 3 and determine its universal properties. No resonance is found for $N\ensuremath{\ge}4$. In the disordered phase, the spectral function exhibits a threshold behavior with no Higgs-like peak. We also show that for N=2N=2N=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 $\ensuremath{\epsilon}=4\ensuremath{-}(d+1)$ expansion near d=3d=3d=3.
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 study the ``Higgs'' amplitude mode in the relativistic quantum O(N)O(N)O(N) model in two space dimensions. Using the nonperturbative renormalization group we compute the O(N)O(N)O(N)-invariant scalar susceptibility in the vicinity of the zero-temperature quantum critical point. In the zero-temperature ordered phase, we find a well-defined Higgs resonance for N=2N=2N=2 with universal properties in agreement with quantum Monte Carlo simulations. The resonance persists at finite temperature below the Berezinskii-Kosterlitz-Thouless transition temperature. In the zero-temperature disordered phase, we find a maximum in the spectral function which is, however, not related to a putative Higgs resonance. Furthermore we show that the resonance is strongly suppressed for $N\ensuremath{\ge}3$.
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 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.