Higgs-like modes in two-dimensional spatially indirect exciton condensates

Fei Xue, Fengcheng Wu, and A. H. MacDonald

Phys. Rev. B 102, 075136 – Published 25 August 2020

Abstract

Higgs-like modes in condensed-matter physics have drawn attention because of analogies to the Higgs bosons of particle physics. Here we use a microscopic time-dependent mean-field theory to study the collective mode spectra of two-dimensional spatially indirect exciton (electron-hole pair) condensates, focusing on the Higgs-like modes, i.e., those that have a large weight in electron-hole pair amplitude response functions. We find that in the low exciton density (Bose-Einstein condensate) limit, the dominant Higgs-like modes of spatially indirect exciton condensates correspond to adding electron-hole pairs that are orthogonal to the condensed pair state. We comment on the previously studied Higgs-like collective excitations of superconductors in light of this finding.

Received 16 March 2020
Revised 4 June 2020
Accepted 13 August 2020

DOI:https://doi.org/10.1103/PhysRevB.102.075136

Physics Subject Headings (PhySH)

Composite bosons Excitons Goldstone bosons Quasiparticles & collective excitations

Bose-Einstein condensates Two-dimensional electron system

Collective models Mean field theory

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Fei Xue ^1,2, Fengcheng Wu^1,3, and A. H. MacDonald¹

¹Department of Physics, University of Texas at Austin, Austin, Texas 78712, USA
²Institute for Research in Electronics and Applied Physics and Maryland Nanocenter,University of Maryland, College Park, Maryland 20742, USA
³Condensed Matter Theory Center and Joint Quantum Institute, Department of Physics, University of Maryland, College Park, Maryland 20742, USA

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Vol. 102, Iss. 7 — 15 August 2020

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Images

Figure 1
Schematic illustration of bilayer exciton condensates and of Higgs-like amplitude mode excitations with a Mexican-hat potential.
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Figure 2
Spectra of the magnitude of the imaginary part of $τ_{x} − τ_{x}$ (red lines), $τ_{y} − τ_{y}$ (blue lines), and $τ_{z} − τ_{z}$ (yellow lines) response functions. Black dots along $x$ axis represent the positive collective excitation energies which are square roots of eigenvalues of $Γ$ [Eq. (22)] and $K^{(+)} K^{(-)}$ [Eq. (25)]. The purple dashed lines denote the location of the electron-hole continuum, i.e., the minimum of $E_{\vec{k}} + E_{\vec{k} + \vec{Q}}$ . (a) and (b) show the results of $Q a_{B}^{*} = 0$ at low exciton density ( $n_{e x} a_{B}^{* 2} = 0.01$ ) and high exciton density ( $n_{e x} a_{B}^{* 2} = 0.1$ ), respectively. Insets in (a) and (b) show the mean-field energy bands (solid lines) and noninteracting bands (dashed lines) as a function of $k y$ ( $k x$ is at a fixed value) in these two cases. (c) and (d) show similar results at finite center-of-mass momentum $Q a_{B}^{*} = 1$ . Note that the yellow line for $Im χ_{z z}$ is absent at $Q = 0$ because its value is zero at all excitation energies.
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Figure 3
The squared modulus of wave functions of collective modes on the momentum grid at $\vec{Q} = 0$ in the BEC regime. (a) to (c) show $n s$ -like atomic orbital distribution corresponding to gapless Goldstone modes and the two Higgs-like modes denoted with arrows in Fig. 2. These three modes all have isotropic momentum distributions but with different numbers of nodes (the number of nodes is equal to $n - 1$ ). (d) to (f) show $2 p$ , $3 p$ , and $3 d$ -like atomic orbital distributions which do not contribute to the amplitude response functions. The energy sequence of all six collective modes is $1 s, 2 p, 2 s, 3 d, 3 p$ , and $3 s$ . Note that $2 p$ , $3 p$ , and $3 d$ are doubly degenerate, and one of each doublet is plotted here.
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Figure 4
Intensity $Im χ_{x x}$ as a function of center-of-mass momentum $Q$ and excitation energy $ω$ in both (a) BEC and (b) BCS regimes. The purple dashed line represents the electron-hole continuum. In the BEC regime in (a), three collective mode branches below the continuum dominating the responses are Goldstone mode ( $1 s$ ) and Higgs-like modes ( $2 s$ and $3 s$ ) plotted in Figs. 3 to 3. In the BCS regime in (b), only one gapped Higgs-like mode dominates the response at small $Q$ ( $Q a_{B}^{*} < 0.3$ ), and the second Higgs-like mode, which has lower energy than the first Higgs-like mode, appears as $Q$ increases.
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Figure 5
Spectra of the magnitude of the imaginary part of the $τ_{x} − τ_{x}$ response functions at $Q = 0$ with Dirac- $δ$ short-range interaction. Black dots along the $x$ axis represent the positive collective excitation energies which are square roots of eigenvalues of $Γ$ [Eq. (22)] and $K^{(+)} K^{(-)}$ [Eq. (25)]. The purple dashed line denotes the location of electron-hole continuum, i.e., the minimum of $E_{\vec{k}} + E_{\vec{k} + \vec{Q}}$ .
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