Quantitative first-principles theory of the Higgs amplitude-mode response across the strongly-to-weakly interacting crossover and the discrete trapped spectrum in the two-dimensional superfluid–Mott-insulator transition

The 2012 quantum-gas experiment of Endres et al. (Nature 487, 454) observed the Higgs (amplitude) mode in a two-dimensional Bose–Hubbard superfluid near the superfluid–Mott-insulator quantum critical point, saw the expected softening of the response at criticality, and found that the amplitude-mode response broadens and disappears as the system is tuned toward the weakly interacting regime. The authors' conclusion explicitly challenges theorists to develop a quantitative theory valid between the strongly interacting, approximately Lorentz-invariant critical regime and the weakly interacting, Galilean-invariant regime that predicts this disappearance, and to provide a first-principles treatment of the discrete, trap-induced spectrum of Higgs modes in the confined system. Published work treats each end of the crossover and parts of the trap problem separately, but no single quantitative, experimentally validated first-principles theory of the full crossover, and no first-principles calculation of the discrete trapped amplitude-like eigenmodes with their intrinsic widths and damping, has been published. The problem remains open.

Background

In the Bose–Hubbard model — lattice bosons with hopping amplitude $j$ and on-site repulsion $U$ — the ground state at commensurate filling is a superfluid at large $j/U$ and a Mott insulator (an incompressible state with a fixed integer number of bosons per site) at large $U/j$ , separated by a quantum phase transition at a critical hopping $j_{\mathrm{c}}$ ; the ratio $j/j_{\mathrm{c}}$ is the standard tuning parameter. Close to $j_{\mathrm{c}}$ on the superfluid side, the low-energy physics of this system is described by a relativistic $O(2)$ field theory in two spatial plus one imaginary-time dimension: spontaneous breaking of the continuous $U(1)$ symmetry produces a massless Nambu–Goldstone phase mode and a massive amplitude (‘Higgs’) mode, the analogue of the Higgs boson of particle physics. In this regime particle–hole symmetry — the approximate symmetry, near criticality, between adding and removing a boson, which makes the low-energy excitation spectrum approximately symmetric about zero energy — is emergent, and the dynamics are approximately Lorentz invariant. In the opposite, weakly interacting limit ( $j/j_{\mathrm{c}}\gg 1$ ), the same system is instead described by Gross–Pitaevskii theory (the mean-field theory of a weakly interacting bosonic condensate) for a dilute, Galilean-invariant continuum superfluid, in which Galilean invariance forbids a sharp gapped amplitude mode. The experiment of Endres, Fukuhara, Pekker, Cheneau, Schauß, Gross, Demler, Kuhr and Bloch (Nature 487, 454 (2012)) probed a two-dimensional gas of $^{87}$ Rb in an optical lattice (a periodic trapping potential for atoms created by interfering laser beams) by lattice-modulation spectroscopy: a periodic modulation of the lattice parameters drives transitions whose absorbed energy, read out through the resulting temperature increase, traces the spectral response as a function of modulation frequency $\nu_{\mathrm{mod}}$ and $j/j_{\mathrm{c}}$ . The measured response showed the gap-like onset of spectral weight softening on approach to $j_{\mathrm{c}}$ and the low-frequency behavior expected for a probe that couples in a rotationally invariant (scalar) way to the order-parameter amplitude, consistent with the $\omega^{3}$ scaling predicted by Podolsky, Auerbach and Arovas (2011) for two dimensions; a probe that instead couples longitudinally (along the order-parameter direction) would produce a qualitatively different, divergent low-frequency response in two dimensions. However, as $j/j_{\mathrm{c}}$ increased toward the weakly interacting limit, the observed response broadened and finally disappeared, and the experiment could only partially reproduce this behavior with a variational $2\times 2$ cluster wave-function calculation (a cluster generalization of the Gutzwiller mean-field ansatz that keeps correlations inside a four-site plaquette). The gas was moreover held in a harmonic trap, so that a discrete, finite-size spectrum of amplitude-like modes is expected, together with amplitude–phase coupling, inhomogeneous broadening (different local $j/j_{\mathrm{c}}$ across the cloud), and damping into low-energy phase modes; the experimental analysis relied on a heuristic trap model rather than a first-principles calculation. At the time, quantum Monte Carlo (QMC) simulations with analytic continuation — the numerical reconstruction of a spectral function from imaginary-time correlation data — had established a well-defined Higgs resonance at criticality in two dimensions (Pollet and Prokof'ev, 2012), while nonperturbative renormalization-group studies (fluctuation-resummation schemes that integrate out modes scale by scale, going beyond perturbation theory) (Rançon and Dupuis, 2014; Rose, Léonard and Dupuis, 2015) confirmed its universal properties near the $(2+1)$ -dimensional quantum critical point. Pekker and Varma's 2015 review summarized the cold-atom observations of the amplitude mode as a landmark but theory-limited result. The unresolved target is a quantitative, first-principles account of the measured amplitude-mode response that is valid across the whole crossover and in the actual trapped geometry.

Problem Statement

Construct a quantitative, first-principles theory of the amplitude (Higgs) mode response of a two-dimensional Bose–Hubbard superfluid in the regime of the Endres et al. (2012) experiment, and validate it against the measured lattice-modulation spectra. The theory must (i) be valid across the full crossover from the strongly interacting, approximately Lorentz-invariant quantum critical regime near $j/j_{\mathrm{c}}\approx 1$ to the weakly interacting, Galilean-invariant Gross–Pitaevskii regime at $j/j_{\mathrm{c}}\gg 1$ , and within a controlled microscopic (non-heuristic) framework predict the observed softening of the response near $j_{\mathrm{c}}$ and the broadening and eventual disappearance of the amplitude-mode response as $j/j_{\mathrm{c}}$ increases; (ii) treat the harmonic trap from first principles, yielding the discrete, finite-size spectrum of amplitude-like modes with their amplitude–phase coupling, intrinsic spectral widths and damping rates (including inhomogeneous broadening from the spatially varying local $j/j_{\mathrm{c}}$ and damping into low-energy phase modes and high-frequency continua), so that individual trapped amplitude-like eigenmodes and their contributions to the measured response are predicted; and (iii) account quantitatively for the dependence of the response on probe symmetry (rotationally invariant scalar versus longitudinal coupling to the order parameter), temperature (including the Berezinskii–Kosterlitz–Thouless physics — the vortex-unbinding transition that governs two-dimensional superfluids in place of true long-range order — of the two-dimensional superfluid), and the finite-amplitude experimental drive and readout. The theory must reproduce the published experimental spectra quantitatively across the full measured range of $j/j_{\mathrm{c}}$ and modulation frequency $\nu_{\mathrm{mod}}$ — matching the spectral onset, shape and width of the predicted response and their evolution with $j/j_{\mathrm{c}}$ — in particular the measured softening near $j_{\mathrm{c}}$ together with the broadening and eventual disappearance of the response at large $j/j_{\mathrm{c}}$ . Published spectra exist only for the rotationally invariant (scalar) probe. The three clauses (i)–(iii) are components of one objective: a single microscopic account of the amplitude mode's emergence, modification, and destruction across the superfluid crossover in the actual trapped geometry of the experiment.

References

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