Higgs-like modes in two-dimensional spatially-indirect exciton condensates Fei Xue,1, 2 Fengcheng Wu,1, 3 and A.H. MacDonald1 1Department of Physics, University of Texas at Austin, Austin TX 78712, USA 2Institute for Research in Electronics and Applied Physics & Maryland Nanocenter, University of Maryland, College Park, MD 20742, USA 3Condensed Matter Theory Center and Joint Quantum Institute, Department of Physics, University of Maryland, College Park, Maryland 20742, USA (Dated: September 2, 2020) 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 conden- sate) 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. I. INTRODUCTION = V † The standard model of particle physics posits a bosonic Δ ⃗ t b Higgs field that provides elementary particles with mass ( ) = ( ) by breaking symmetries that would otherwise be present. ⃗ The recent experimental detection [1, 2] of Higgs par- Δ ⃗ Δ ⃗ ⃗ ticles, the elementary excitations of the Higgs field, is therefore an important advance in fundamental physics. Partly because of their importance to the foundations of Δ ⃗ physics writ large, there has also been interest in excita- tions that are analogous to Higgs particles in condensed matter, especially in superconducting metals [3]. Indeed, the absence of massless Goldstone boson excitations in superconductors[4, 5] in spite of their broken gauge sym- FIG. 1: Schematic illustration of bilayer exciton condensates metry, played an important role historically in the theo- and of Higgs-like amplitude mode excitations with a Mexican- retical work [6{8] that led to the Higgs field proposal. hat potential. Emergent symmetry-breaking bosonic fields are com- mon in condensed matter, where they typically arise from equilibrium or quasiequilibrium states of matter that interactions among underlying fermionic fields. Both have been extensively studied over the past couple of electron-electron pair fields, which condense in supercon- decades in semiconductor bilayer quantum wells [29{32], ductors, and electron-hole pair fields, which condense in including in the quantum Hall regime [33, 34]. SIXCs ferromagnets and in spin or charge density wave sys- have recently been observed in van der Waals hetero- tems, are common. An important difference between junction two-dimensional bilayer materials both in the the Higgs fields of particle physics and the symmetry- presence [35, 36] and in the absence [37, 38] of external breaking bosonic fields in condensed matter is the ab- magnetic fields. The bosonic order parameter field of a sence, in the former case, of an understanding of the spatially indirect exciton condensate field’s origin in terms of underlying degrees of freedom that might be hidden at present, akin to the understand- y ∆(~r) = Ψ (~r)Ψb(~r) (1) ing in condensed matter that the order parameter field t in a superconductor measures electron-electron pair am- has a nonzero expectation value in the broken-symmetry arXiv:2003.01185v3 [cond-mat.mes-hall] 1 Sep 2020 plitudes. Such an understanding might eventually be ground state, which is characterized by spontaneous in- achieved, and analogies to the observed properties of con- terlayer phase coherence and a suite of related anoma- densed matter might once again be valuable in suggest- lous transport properties [39]. The labels b; t on the ing theoretical possibilities. Motivated partly by that field operators in Eq. 1 refer to electrons in the bottom hope and partly by the goal of shedding new light on (b) and top (t) layers of a bilayer two-dimensional elec- the interesting literature [9{17] on Higgs-like excitation tron system, as illustrated schematically in Fig. 1. The in superconductors and in other condensed-matter sys- SIXC state can be described approximately using a mean- tems [18{28], we address the Higgs-like excitations of field theory [29{32, 40] analogous to the Bardeen-Cooper- two-dimensional spatially indirect exciton condensates. Schrieffer mean-field theory [41] of superconductors. Spatially indirect exciton condensates (SIXCs) are Superconductors break an exact gauge symmetry re- 2 lated to conservation of the electron number in the many- equal footing make qualitative errors in describing spa- body Hamiltonian. In electron-hole pair condensates the tially indirect exciton condensates. This comment ap- corresponding symmetry is only approximate but be- plies in particular to the short-range interaction mod- comes accurate when the electrons and holes are selected els that can be conveniently analyzed using Hubbard- from two different subsets of the single-particle Hilbert Stratonovich transformations (see, for example, Ref. [42, space whose electron numbers are approximately con- pp. 333{335]). In this section we briefly summarize that served separately. In the case of spatially indirect exciton theory and generalize it in a way that makes evaluation condensates, the electrons and holes are selected from of the two-particle Green's functions that characterize separate two-dimensional layers. Exceptionally among the system's particle-hole excitations particularly con- electron-hole pair condensates, the Hamiltonian terms venient. The SIXC's elementary excitation energies are that break separate particle-number conservation can be identified with the poles of those Green's functions and made arbitrarily weak simply by placing an insulating are the eigenvalues of a matrix constructed from the ker- barrier between the two-dimensional subsystems. Phe- nel of the quadratic-fluctuation energy functional. The nomena associated with broken gauge symmetries can be character of given elementary excitations is classified by realized as fully as desired by suppressing single-particle determining which particle-hole pair response functions processes that allow electrons to move between b and t have large residues at its poles. layers. The properties of spatially indirect exciton con- densates are therefore very closely analogous to those of two-dimensional superconductors, as we shall emphasize A. Mean-field theory again below. The main difference between the two cases is that the condensed pairs are charged in the super- For simplicity we neglect the spin and valley degrees of conducting case, altering how the ordered states interact freedom that often play a role in realistic SIXC systems. with electromagnetic fields. The self-consistent Hartree-Fock mean-field Hamiltonian In this paper we employ a time-dependent mean-field of the broken-symmetry bilayer exciton condensate state weak-coupling theory description of the bilayer exciton is then [40] condensate's elementary excitations to identify Higgs-like modes and to demonstrate that in the low-density Bose- X y y ac~k HMF = (a ; a )(ζ + ξ σz − ∆ σx) : (2) c~k v~k ~k ~k ~k Einstein condensate (BEC) limit they have a simple in- av~k ~k terpretation as excitations in which electron-hole pairs are added to the system in electron-hole pair states that Here a and ay are fermionic annihilation and creation n~k n~k are orthogonal to the 1s pair state that is condensed in operators for the conduction (n = c) band electrons lo- the many-body ground state. In Sec. II we first briefly de- calized in the top layer and valence (n = v) electrons lo- scribe some details of our theory of the SIXC's collective calized in the bottom layer, σz;x are Pauli matrices that excitations. In Sec. III we summarize and discuss numer- 2 2 act in the band space, and ζ~k = ~ k [1=(4me)−1=(4mh)] ical results we have obtained by applying this theory to accounts for the difference between conduction and va- bilayer two-dimensional electron-hole systems. Finally, lence band effective masses, which plays no role in the in Sec. IV we conclude by commenting on similarities temperature T = 0, charge-neutral limit that we con- and differences between bilayer exciton condensates and sider. For convenience, we take me = mh in the calcula- other systems in which Higgs-like modes have been pro- tions described below. The dressed band parameters ξ~ posed and observed. k and ∆~k are obtained by solving the self-consistent-field equations: 2k2 µ~ 1 X II. COLLECTIVE EXCITATION THEORY ξ = ~ + − V (1 − ξ =E ); ~k 4m 2 2A ~k−~k0 ~k0 ~k0 ~k0 The mean-field theory of the bilayer exciton conden- ∆ 1 X ~k0 (3) sate is a generalized Hartree-Fock theory in which trans- ∆~k = U~k−~k0 ; 2A E~k0 lational symmetry is retained but spontaneous interlayer ~k0 phase coherence, which breaks separate conservation of q E = ξ2 + ∆2 ; the particle number in the two layers, is allowed. In ~k ~k ~k Ref. [40] we presented a theory of the bilayer exciton condensate's elementary collective excitations and quan- where m = memh=(me + mh) is the reduced mass and A is the area of the two-dimensional system. In Eq. 3, tum fluctuations that accounts for quadratic variations 2 of the Hartree-Fock energy functional. Importantly for V~q = 2πe =(q) and U~q = V~q exp(−qd) are the intralayer the findings that are the focus of the present paper, the and interlayer Coulomb interactions, theory fully accounts for the long-range Coulomb inter- 2 µ~ = µ + 4πe nexd/, actions among electrons and holes. Theories that do not 1 X recognize the Coulomb's interactions' long range or which n = (1 − ξ =E ); (4) ex 2A ~k ~k do not treat electrostatic and exchange interactions on an ~k 3 µ is the chemical potential parameter for excitons, and corrections to the mean-field state: n is equal to both the density of conduction band elec- ex h i trons and the density of valence band holes.