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arXiv:1402.3098 · Physical Review B

Higgs amplitude mode in the vicinity of a (2+1) -dimensional quantum critical point

Adam Rançon, N. Dupuis

Published May 6, 2014• 57 citations
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Abstract

We study the ``Higgs'' amplitude mode in the relativistic quantum $O(N)$ model in two space dimensions. Using the nonperturbative renormalization group we compute the $O(N)$-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 $N=2$ 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 $N\ensuremath{\ge}3$.

DOI
10.1103/physrevb.89.180501

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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≥3N\ensuremath{\ge}3N\ensuremath≥3.
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.
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 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.
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We study the thermodynamics of the relativistic quantum O(N)(N)(N) model in two space dimensions. In the vicinity of the zero-temperature quantum critical point (QCP), the pressure can be written in the scaling form $P(T)=P(0)+N({T}^{3}/{c}^{2}){\mathcal{F}}_{N}(\ensuremath{\Delta}/T)$, where ccc is the velocity of the excitations at the QCP and $|\ensuremath{\Delta}|$ a characteristic zero-temperature energy scale. Using both a large-NNN approach to leading order and the nonperturbative renormalization group, we compute the universal scaling function FN{\mathcal{F}}_{N}FN​. For small values of NNN ($N\ensuremath{\lesssim}10$) we find that FN(x){\mathcal{F}}_{N}(x)FN​(x) is nonmonotonic in the quantum critical regime ($|x|\ensuremath{\lesssim}1$) with a maximum near x=0x=0x=0. The large-NNN approach---if properly interpreted---is a good approximation both in the renormalized classical ($x\ensuremath{\lesssim}\ensuremath{-}1$) and quantum disordered ($x\ensuremath{\gtrsim}1$) regimes, but fails to describe the nonmonotonic behavior of FN{\mathcal{F}}_{N}FN​ in the quantum critical regime. We discuss the renormalization-group flows in the various regimes near the QCP and make the connection with the quantum nonlinear sigma model in the renormalized classical regime. We compute the Berezinskii-Kosterlitz-Thouless transition temperature in the quantum O(2) model and find that in the vicinity of the QCP the universal ratio ${T}_{\mathrm{BKT}}/{\ensuremath{\rho}}_{s}(0)$ is very close to $\ensuremath{\pi}/2$, implying that the stiffness ${\ensuremath{\rho}}_{s}({T}_{\mathrm{BKT}}^{\ensuremath{-}})$ at the transition is only slightly reduced with respect to the zero-temperature stiffness ${\ensuremath{\rho}}_{s}(0)$. Finally, we briefly discuss the experimental determination of the universal function F2{\mathcal{F}}_{2}F2​ from the pressure of a Bose gas in an optical lattice near the superfluid--Mott-insulator transition.