The ‘Higgs’ amplitude mode at the two-dimensional superfluid/Mott insulator transition

Abstract

Spontaneous symmetry breaking plays a key role in our understanding of nature. In a relativistic field theory, a broken continuous symmetry leads to the emergence of two types of fundamental excitations: massless Nambu-Goldstone modes and a massive 'Higgs' amplitude mode. An excitation of Higgs type is of crucial importance in the standard model of elementary particles [1] and also appears as a fundamental collective mode in quantum many-body systems [2]. Whether such a mode exists in low-dimensional systems as a resonance-like feature or becomes over-damped through coupling to Nambu-Goldstone modes has been a subject of theoretical debate [2][3][4][5][6][7]. Here we experimentally reveal and study a Higgs mode in a two-dimensional neutral superfluid close to the transition to a Mott insulating phase. We unambiguously identify the mode by observing the expected softening of the onset of spectral response when approaching the quantum critical point. In this regime, our system is described by an effective relativistic field theory with a two-component quantum-field [2,8,9], constituting a minimal model for spontaneous breaking of a continuous symmetry. Additionally, all microscopic parameters of our system are known from first principles and the resolution of our measurement allows us to detect excited states of the many-body system at the level of individual quasiparticles. This allows for an in-depth study of Higgs excitations, which also addresses the consequences of reduced dimensionality and confinement of the system. Our work constitutes a first step in exploring emergent relativistic models with ultracold atomic gases.Higgs modes are amplitude oscillations of a quantum field and appear as collective excitations in quantum many-body systems as a consequence of spontaneous breaking of a continuous symmetry. Close to a quantum critical point, the low-energy physics of such systems is in many cases captured by an effective Lorentz invariant critical theory [2]. The minimal version of such a theory describes the dynamics of a complex order parameter Ψ = |Ψ|e iφ near a quantum phase transition between an ordered (|Ψ| > 0) and a disordered phase (|Ψ| = 0). Within the ordered phase, the classical energy density has the shape of a Mexican hat (Fig. 1a) and the order parameter takes on a non-zero value in the minimum of this potential. Hereby, its phase φ acquires a definite value through spontaneous breaking of the rotation symmetry (i.e., U (1) symmetry). Expanding the field around the symmetry broken ground state leads to two types of modes: a Nambu-Goldstone mode and a Higgs mode related to phase and amplitude variations of Ψ, respectively (Fig. 1a). In contrast to the phase mode, the amplitude mode has a finite excitation gap (i.e., a finite mass), which is expected to show a characteristic softening when approaching the disordered phase (Fig. 1a). The sketched minimal model of an order parameter with N = 2 components belongs to a class of O(N ) relativistic * Electronic address: manuel.endres@mpq.mpg.de field t...

Low-frequency conductivity in many-body localized systems

“…In principle one can also measure a.c. conductivity in ultracold atomic systems such as optical lattices by applying a periodically modulated tilt to the entire lattice [45]. However, in the current optical-lattice MBL experiments [27] it is more convenient to study relaxation in the time domain; we now show how our results generalize to such experiments.…”

“…Moreover, our results here should also describe, e.g., thermal transport, in systems where the only conserved quantity is the energy. However, our analysis of the conductivity relies on the fact that the conductivity is related to the spectral function of a current (i.e., a quantity associated with a globally conserved charge) and does not extend to generic spectral functions, such as those probed using optical lattice modulation spectroscopy [45]. In fact for more general local spectral functions, the operator's matrix element between states within a MB Mott or Griffiths resonance is not restricted to vanish in proportion to ω so may remain of order one, resulting in a divergent low-frequency spectral function S(ω) ∼ ω (α−2) in the MBL phase.…”

Higgs mode and its decay in a two-dimensional antiferromagnet

“…Therefore, its rapid decay in the longitudinal susceptibility measured by INS does not necessarily imply its instability in two dimensions. In fact, it has been shown in other two-dimensional systems, such as disordered superconductors 3 and superfluids of cold atoms 2 , that the Higgs mode is clearly visible in the scalar susceptibility with its characteristic ∼ω 3 onset in the energy spectrum. Indeed, theory predicts a contrasting behaviour of the Higgs mode in the scalar and longitudinal susceptibilities; in the latter, the Higgs mode quickly loses its coherence by decaying into a pair of Goldstone modes 25,26 .…”

Collective Modes in the Cooperative Jahn–Teller Model: Path Integral Approach

“…of the bosonic fields entering the path integral (7). Here the conjugate variables ζ γ, j and δ ρ γ, j describe, respectively, the local phase and the density fluctuation of the bosonic condensates around the mean-field solutionρ = (g/2∆ 0 ) 2 sin 2 (θ ).…”

“…Such amplitude modulation of the order parameter is referred to as Higgs mode generated by a physical mechanism analogous as the Higgs boson in high energy physics. Recently, the amplitude mode was experimentally observed in a system of strongly interacting condensate of ultracold atoms near the superfluid-insulator phase transition [6,7] opened fascinating prospect for exploring the condensed matter excitations under controlled conditions.…”