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arXiv:1604.08730 · Physical Review A

Observability of Higgs mode in a system without Lorentz invariance

Xinloong Han, Boyang Liu, Jiangping Hu

Published Sep 8, 2016• 12 citations
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Abstract

We study the observability of the Higgs mode in BEC-BCS crossover. The observability of the Higgs mode is investigated by calculating the spectral weight functions of the amplitude fluctuation below the critical transition temperature. At zero temperature, we find that there are two sharp peaks on the spectral function of the amplitude fluctuation attributed to Goldstone and Higgs modes, respectively. As the system goes from the BCS to Bose-Einstein condensate (BEC) side, there is strong enhancement of spectral weight transfer from the Higgs to Goldstone mode. However, even at the unitary regime where the Lorentz invariance is lost, the sharp feature of the Higgs mode still exists. We specifically calculate the finite-temperature spectral function of the amplitude fluctuation at the unitary regime and show that the Higgs mode is observable at the temperature that present experiments can reach.

DOI
10.1103/physreva.94.033608

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We study the observability of the Higgs mode in BEC-BCS crossover. The observability of the Higgs mode is investigated by calculating the spectral weight functions of the amplitude fluctuation below the critical transition temperature. At zero temperature, we find that there are two sharp peaks on the spectral function of the amplitude fluctuation attributed to Goldstone and Higgs modes, respectively. As the system goes from the BCS to Bose-Einstein condensate (BEC) side, there is strong enhancement of spectral weight transfer from the Higgs to Goldstone mode. However, even at the unitary regime where the Lorentz invariance is lost, the sharp feature of the Higgs mode still exists. We specifically calculate the finite-temperature spectral function of the amplitude fluctuation at the unitary regime and show that the Higgs mode is observable at the temperature that present experiments can reach.
In this work we present a coherent picture for the evolution of Higgs mode in neutral sss-wave fermion superfluids. As the strength of attractive interaction between fermions increases from the BCS to the Bose-Einstein condensate (BEC) regime we find that the Higgs mode in the fermionic superfluids is pushed to higher energy while at the same time, gradually loses its spectral weight, which is because the system is tuned away from Lorentz invariance. Furthermore, when damping is taken into account, the spectral weight of the Higgs mode is significantly broadened due to coupling to phase mode in the whole BEC-BCS crossover.
Although substantial effort has been dedicated to analyzing the Higgs (amplitude) mode in superconducting systems, there are relatively few studies of the Higgs peak's dispersion and width, quantities which are observable in spectroscopic measurements. These properties can be obtained from the location of the pole of the amplitude (Higgs) susceptibility in the lower half plane of complex frequency. We analyze the behavior of the Higgs mode in a two-dimensional neutral fermionic superfluid at T=0T=0T=0 throughout the crossover from Bardeen-Cooper-Schrieffer (BCS) to Bose-Einstein condensation (BEC) behavior. This occurs when the dressed chemical potential $\ensuremath{\mu}$ changes sign from positive to negative. For $\ensuremath{\mu}>0$, we find a pole in the Higgs susceptibility in the lower half plane of frequency and demonstrate that it leads to a well-defined peak in spectroscopic probes. For $\ensuremath{\mu}<0$, the pole still exists but is ``hidden,'' not giving rise to a peak in spectroscopic probes. Extending this analysis to a charged superconductor, we find that the Higgs mode is unaffected by the long-range Coulomb interaction.
We discuss the derivation of a low-energy effective field theory of phase (Goldstone) and amplitude (Higgs) modes of the pairing field from a microscopic theory of attractive fermions. The coupled equations for Goldstone and Higgs fields are critically analyzed in the Bardeen–Cooper–Schrieffer (BCS)-to-Bose–Einstein condensate (BEC) crossover—both in three spatial dimensions and in two spatial dimensions. The crucial role of pair fluctuations is investigated, and the beyond-mean-field Gaussian theory of the BCS–BEC crossover is compared with available experimental data of the two-dimensional ultracold Fermi superfluid.
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.
The BCS-BEC crossover is studied in a systematic way in the broken-symmetry phase between zero temperature\nand the critical temperature. This study bridges two regimes where quantum and thermal fluctuations\nare, respectively, important. The theory is implemented on physical grounds, by adopting a fermionic selfenergy\nin the broken-symmetry phase that represents fermions coupled to superconducting fluctuations in weak\ncoupling and to bosons described by the Bogoliubov theory in strong coupling. This extension of the theory\nbeyond mean field proves important at finite temperature, to connect with the results in the normal phase. The\norder parameter, the chemical potential, and the single-particle spectral function are calculated numerically for\na wide range of coupling and temperature. This enables us to assess the quantitative importance of superconducting\nfluctuations in the broken-symmetry phase over the whole BCS-BEC crossover. Our results are relevant\nto the possible realizations of this crossover with high-temperature cuprate superconductors and with\nultracold fermionic atoms in a trap.
Collective modes in a superconductor correspond to fluctuations in the amplitude (Higgs mode) or the phase (Goldstone mode) of the order parameter. Starting from the BCS Hamiltonian, we derive a microscopic Hamiltonian for the coupled Higgs-Goldstone-electron system that contains all interactions among these degrees of freedom. The Higgs subsystem is a weakly interacting Bose gas: Higgs modes experience a mutual attraction mediated by the virtual exchange of Bogoliubov quasi-particle pairs. They also interact with external electromagnetic-fields with a coupling constant that we derive explicitly. We use this theory to examine whether an optically pumped gas of Higgs modes can undergo Bose-Einstein condensation. We identify a density window, bounded below by the equilibrium condensation criterion and above by collapse due to the attractive interaction, in which condensation can occur and we derive the expected density of Higgs-modes due to external pumping. This pumping depends crucially on non-parabolic corrections to the electron dispersion around the Fermi-energy. We show that the pumped mode density is compatible with condensation for realistic material parameters. We suggest possible experimental signatures of this non-equilibrium condensate.