Higgs Modes in Two-Dimensional Spatially-Indirect Exciton Condensates

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 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.

I. INTRODUCTION = V † The standard model of particle physics posits a bosonic Δ 𝑟𝑟⃗ 𝜓𝜓t 𝜓𝜓b Higgs field that provides elementary particles with mass ( ) = ( ) by breaking symmetries that would otherwise be present. 𝑖𝑖𝑖𝑖 𝑟𝑟⃗ The recent experimental detection [1, 2] of Higgs par- Δ 𝑟𝑟⃗ Δ 𝑟𝑟⃗ 𝑒𝑒 𝜑𝜑 𝑟𝑟⃗ 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 𝑟𝑟⃗ physics writ large, there has also been interest in excitations 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 common 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- † ∆(~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 condensates 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 † † 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 a† 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. Below we Y X ~ † |Φi = Z~k + z~k(Q)γ~ ~ γ~k,0 |XCi , (7) refer to n as the density of excitons; this terminology k+Q,1 ex ~k Q~ is motivated mainly by the low-density limit in which n a∗2 1. (Here a∗ = 2/me2 is the Bohr radius, ex B B ~ where γ† is a creation operator for a quasiparticle state which is the bound electron-hole pair size in the limit ~k,1 of small layer separations.) The exciton chemical poten- in the band that is empty in |XCi: tial parameter µ = Eg − Vb can be adjusted electrically by applying a gate voltage to alter the spatially indirect γ† = v a† − u a† (8) ~k,1 ~k c~k ~k v~k band gap Eg, provided that the barrier between conduction and valence band layers is sufficiently opaque, or by and applying a bias voltage Vb between layers [38]. The mean-field ground state is s X ~ 2 Z~k = 1 − |z~k(Q)| (9) Y † Y † † |XCi = γ |0i = (u a + v a ) |0i , (5) Q~ ~k,0 ~k c~k ~k v~k ~k ~k ~ is a normalization factor. The complex parameters z~k(Q) where are the amplitudes of all possible single-particle-hole excitations. r1 r1 u~k = (1 − ξ~k/E~k), v~k = (1 + ξ~k/E~k), (6) To characterize the quantum fluctuations of the mean- 2 2 field state in a physically transparent way, we define the observables and γ† is the creation operator for the dressed valence ~k,0 band quasiparticle states that are occupied in |XCi. 1 X † † ac~k τˆ (Q~ ) = (a , a ) σα . (10) Note that we have chosen u and v to be real and that α={x,y,z} c~k+Q~ v~k+Q~ ~k ~k 2 av~k ~k there is a family of degenerate states that differ only by ~ ~ ∗ a global shift in the phase difference between electrons Note that hΦ|τα(Q)|Φi = hΦ|τα(−Q)|Φi . For the inter- localized in different layers. layer phase choice we have made, the mean-field value of the order parameter ∆MF is real and spatially constant:

1 X B. Quadratic fluctuations ∆MF = u v , (11) A ~k ~k ~k We construct our theory of quantum fluctuations and collective excitations by starting from a many-body state where A is the sample area. When fluctuations are in- that incorporates arbitrary single-particle-hole excitation cluded, the order parameter becomes

1 X ∆(~r) = |hτ (Q~ )i| cos(Q~ · ~r − ϕ ) + i|hτ (Q~ )i| cos(Q~ · ~r − ϕ ) (12) A x Qx~ y Qy~ Q~

~ (2) where h...i = hΦ| ... |Φi and ϕQα~ is defined by hτα(Q)i = where δE is the harmonic fluctuation energy functional ~ [43] and B = hΦ|i ∂t|Φi is the Berry phase term which |hτα(Q)i| exp(iϕ ~ ). It follows that, to leading order, ~ Qα enforces bosonic quantization rules on the z (Q~ ) fluctu- fluctuations in the order parameter magnitude are pro- ~k ation parameters. The energy functional is obtained by portional to hτ (Q~ )i, while fluctuations in the order pa- x taking the expectation value of the many-body Hamilto- ~ ~ rameter phase are related to hτy(Q)i. hτz(Q)i measures nian and has the form fluctuations in the exciton density. We quantize fluctuations in the XC state by construct- ing the Lagrangian:

(2) L = hΦ|i~∂t − H|Φi ≈ B − δE , (13) 4

X 1 δE(2) = hΦ|H|Φi = {E (Q~ )z∗(Q~ )z (Q~ ) + Γ (Q~ )[z (Q~ )z (−Q~ ) + z∗(Q~ )z∗(−Q~ )]}. (14) ~k,~p ~k ~p 2 ~k,~p ~k ~p ~k ~p Q,~ ~k,~p

(2) ~ ~ δE accounts for variations in band kinetic energy and Note that x~k(Q) and y~k(Q) are also complex, but sat- Hartree and exchange interaction energy as the many- isfy x (Q~ ) = x∗ (−Q~ ) and y (Q~ ) = y∗ (−Q~ ) so that ~k −~k ~k −~k electron state fluctuates. Explicit forms for the matrices ~ ~ E and Γ are given in Appendix B. Q and −Q fluctuations are not independent. Order pa- We separate the fluctuation Hamiltonian into ampli- rameter amplitude and exciton density fluctuations are tude and phase fluctuation contributions by making the both related to fluctuations in the x fields, while phase change in variables fluctuations are related to fluctuations in the y fields:

~ 1 ~ ~ z~ (Q) = √ [x~ (Q) + iy~ (Q)], (15) k 2 k k ∗ ~ 1 ~ ~ z (−Q) = √ [x~ (Q) − iy~ (Q)]. (16) −~k 2 k k

~ ~ ~ ~ ~ ~ Note that although τx(Q) and exciton density τz(Q) ables, X(Q) ≡ (x~k (Q), . . . , x~k (Q),...) and Y(Q) ≡ ~ 1 i ﬂuctuations are both related to x~k(Q), they have dif- (y (Q~ ), . . . , y (Q~ ),...), whose elements are labeled by ~k1 ~ki ferent ~k-dependent weighting factors. For each wave the particle-hole pair’s hole momentum. In terms of these vector transfer Q~ we deﬁne vectors of x and y vari- vector variables the action

Z 1 X Z S = dt(B − δE(2)) = dt Y†(Q~ )∂ X(Q~ ) − X†(Q~ )∂ Y(Q~ ) − X†(Q~ )K(+)X(Q~ ) − Y†(Q~ )K(−)Y(Q~ ) , (18) 2 ~ t ~ t Q~

where K(±)(Q~ ) = E ± Γ (Q~ ) is real and symmet- dependent Hartree-Fock theory for the exciton conden- ~k,~p ~k,~p ~k,−~p ric for both sign choices and we have used the fact that sate state response functions. It is in the same spirit R † R † as auxiliary field functional integral theories of harmonic dtY ∂tX = − dtX∂tY . Minimizing the action yields the following equations of quantum fluctuations but unlike those approaches treats motion: Hartree and exchange energy contributions on an equal footing [42], an attribute that is necessary if the bilayer (−) ~ ~∂tX = K Y(Q), exciton condensate is to be described directly. (+) ~ ~∂tY = −K X(Q). (19) This theory of fluctuations is equivalent to time- 5

C. Particle-hole correlation functions Noting that the eigenvalues of Γ are identical to the eigenvalues of K(+)K(−) and that the equation of motion Collective modes give rise to poles in particle-hole for phase fluctuations can be written in the form channel Greens functions. As in the case of BCS super- − 2∂2Y = K(+)K(−)Y, (25) conductors [3, 10], those collective modes that have large ~ t residues in (ˆτ , τˆ ) particle-hole Greens functions can be 2 x x we have identified them as the squares ωi of the elemen- identified as Higgs-like modes. Because the fluctuation tary excitation frequencies. Hamiltonian δE(2) is the sum of quadratic contributions When expressed in terms of the normal mode fields, in the X and Y fields, which are canonically conjugate, the action in Eq. 18 has the form we can apply the generalized Bogoliubov transformation Z [44] described in detail below to write the fluctuation 1 X † † † † S = dt ~Ψ ∂tΠ − ~Π ∂tΨ − Π Π − Ψ ΛΨ . Hamiltonian in a free-boson form: 2 Q~ Q~ X X ~ † ~ ~ (26) H = E0 + ~ωi(Q)Bi (Q)Bi(Q). (20) The time-ordered Green’s function at each Q~ can be cal- Q~ i culated directly from the action of fields φ, where ω (Q~ ) is an excitation energy and B†(Q~ ) and −1 ~ i i −Λ i~ω B (Q~ ) are linear combinations of the x and y fields. To G(ω) = i ~k ~k −i~ω −I (27) evaluate correlation functions involving the τα(Q~ ) fields 2 2 −1 −I −i~ω we reexpress x (Q~ ) and y (Q~ ) in Eqs. 17 in terms of these = (det|Λ − ~ ω |) . ~k ~k i~ω −Λ free boson fields. The character of collective excitations is revealed by the residues of response functions at poles Note that the Green’s function is a 2 × 2 matrix in the that lie below particle-hole continua. basis of fields {Ψ, Π}. The τα fields can be expressed in To carry out this procedure explicitly, we start from terms of the normal-mode fields for each Q~ using the assumptions that the amplitude/density kernel K(+) X T −1 is positive definite at any Q~ and that the phase kernel τx = [Tx(L ) S]i Πi ≡ τx,iΠi, K(−) is positive definite for Q~ 6= 0 and positive semidef- i ~ X T −1 inite for Q = 0. These assumptions are satisfied when- τz = [Tz(L ) S]i Πi ≡ τz,iΠi, (28) ever the mean-field condensate is metastable. The zero i eigenvalue of K(−) at Q~ = 0 arises from the broken U(1) X τ = (T LS) Ψ ≡ τ Ψ , symmetry associated with spontaneous interlayer phase y y i i y,i i coherence. Because K(+) is real symmetric and positive i definite, it is possible [45] to perform a Cholesky decom- where the Tα on the right-hand sides of these equations position for each Q~ by writing are the matrix forms of Eq. 17. The linear response functions are related to the time-ordered Green’s functions, K(+) = LLT (21) i 0 χAB = − hT [Aˆ(t), Bˆ(t )]i . (29) and then to diagonalize ~ The response functions of operators expressed in terms of Γ = LT K(−)L = ΓT . (22) fields Πi can be evaluated by performing the average in T Eq. 29 using the quadratic action weighting factor with Writing Γ = SΛS , where S is an orthogonal matrix and the result that Λ is a diagonal matrix, we define new fields X ωi A0iBi0 Ai0B0i T −1 χAB = ( − ), (30) Ψ = S L Y, 2 ω − ωi + iη ω + ωi + iη (23) i Π = ST LT X, where Amn is the matrix element in a complete set of where X, Y are the density and phase fluctuation vectors fields Πi. We identify Higgs-like modes by finding isolated eigenvalues |ω |2 with large values of |τ |2 in the labeled by wavevector ~k introduced above. The trans- i x,i imaginary part of the response functions for positive fre- formed Hamiltonian is quencies:

X ωi 2 1 1 Imχxx(ω) = −π |τx,i| δ(ω − ωi). (31) H = (X†K(+)X + Y†K(−)Y) = (Π†Π + Ψ†ΛΨ) 2 2 2 i 1 X 2 2 2 2 Note that Imχ is similar to Eq. 31 by replacing |τ |2 = (|Πi| + ~ ωi |Ψi| ). zz x,i 2 2 i with |τz,i| . For Imχyy, the factor ωi/2 also needs to be (24) replaced by 1/(2ωi). 6

An alternative approach to obtain the same results is than 1 × 10−8. Because our momentum space grids are to map the diagonalized action in Eq. 26 to that of a set of necessarily discrete, corresponding to applying periodic independent harmonic oscillators, defining the oscillator boundary conditions to a finite area system, the num- ladder operators Bi by ber of particle-hole pairs at a given excitation momentum residing on our k-space grid is finite. The distinc- r ω tions between particle-hole continua and isolated collec- Π = i ~ i (B† − B ) i 2 i i tive modes made below are qualitative but, for the most r (32) part, unambiguous. 1 † Ψi = (Bi + Bi), 2~ωi

0† (a) 2 (b) 2 where Bi satisfy [Bi,Bi ] = δi,i0 . The ﬂuctuation Hamil- 푄 = 0 1 푄 = 0 1 0 0 ~ BEC BCS tonian for each Q is then 1 1

2 2 2 1 0 1 2 2 1 0 1 2 X † H = E0 + ~ωi Bi Bi. (33) 2s 3s i 1s From the general linear response theory, the Lehmann (c) (d) representation of the response function is [43] 푄 = 1 푄 = 1 BEC BCS 1 X Pm − Pn χ (ω) = A B , (34) AB ω − ω + iη mn nm ~ mn nm

e−βEn where Pn = P −βEn (β = 1/kBT ) is the occupation ∗ ∗ n e ℏ휔/푅푦 ℏ휔/푅푦 probability, ωnm = (En − Em)/~ is the excitation fre- ˆ quency, and Amn ≡ hψm| A |ψni is the matrix elements FIG. 2: (Color online) Spectra of the magnitude of the imag- ˆ in a complete set of exact eigenstates |ψni of H. At zero inary part of τx − τx (red lines), τy − τy (blue lines), and temperature, the imaginary part of the response function τz − τz (yellow lines) response functions. Black dots along x- is axis represent the positive collective excitation energies which are square roots of eigenvalues of Γ [Eq. 22] and K(+)K(−) π X ImχAB = − Pm[AmnBnmδ(ω − ωnm) [Eq. 25]. The purple dashed lines denote the location of the ~ nm electron-hole continuum, i.e., the minimum of E~k +E~k+Q~ . (a) ∗ − A B δ(ω + ω )] and (b) show the results of QaB = 0 at low exciton density nm mn nm (n a∗2 = 0.01) and high exciton density (n a∗2 = 0.1), re- π X ex B ex B = − [A0nBn0δ(ω − ωn0) − An0B0nδ(ω + ωn0)]. spectively. Insets in (a) and (b) show the mean-field energy ~ n bands (solid lines) and noninteracting bands (dashed lines) as (35) a function of ky (kx is at a fixed value) in these two cases. (c) and (d) show similar results at finite center-of-mass momen- ∗ tum QaB = 1. Note that the yellow line for Imχzz is absent III. RESULTS at Q = 0 because its value is zero at all excitation energies.

We now apply the theory outlined above to bilayer Bilayer exciton condensates have a BEC-BCS crossover exciton condensates. The length and energy units we use that can be tuned by varying not the strength of inter- in our calculations are those appropriate for Coulomb actions, [47–52] as in cold-atom systems, but the Fermi ∗ 2 2 energy of the underlying electrons and holes [36, 53–55]. interactions, the Bohr radius aB = ~ /(me ), and the effective Rydberg Ry∗ = e2/(2a∗ ). Typical values of In two dimensions the binding energy of a single electron- B ∗ these parameters in transition metal dichalcogenides[40] hole pair is 4Ry∗ (d=0), and the Fermi energy in Ry ∗ ˚ ∗ units is 2πna∗2. The bilayer exciton condensate there- bilayers are aB ≈ 10A, Ry ≈ 100meV, while typical B ∗ fore approaches a BEC limit for small values of na2 when values for GaAs bilayer quantum wells [46] are aB ≈ B 100A˚, Ry∗ ≈ 5meV. For all the numerical calculations, the chemical potential is positive but below the exciton ∗ binding energy shown in the inset of Fig. 2(a). As the we assume me = mh, and use d/aB = 0.5. The time-dependent mean-field theory is expected to chemical potential becomes zero or even negative, the be most reliable in the dilute density limit that resem- condensate approaches a BCS limit shown in the inset 2 bles a two-dimensional hydrogenlike problem. We con- of Fig. 2(b) for naB & 0.1 at µ = 0. In Figs 2(a) and sider two cases where the chemical potential parameter (c), we plot the magnitude of the imaginary part of re- µ is just below the binding energy and µ = 0. In both sponse functions Imχxx (red lines), Imχyy (blue lines), ∗ and Imχ (yellow lines) as a function of positive excita- two cases, we use the momentum cutoff kcaB = 6 and zz ~ ~ ∗ a 200 × 200k mesh. The momentum cutoff is chosen to tion energies for Q = 0 and QaB = 1 for a low exciton ∗2 make sure the exciton density at the cutoff is smaller density in the BEC regime, nexaB = 0.01. Black dots 7 along the x axis represent discrete collective mode spec- In the dilute limit the matrices K(+) = K(−) reduce tra ωi, and we denote the particle-hole continuum (the to the two-particle electron-hole relative motion Hamil- ~ minimum of E~k + E~k+Q~ ) with a vertical purple dashed tonian matrices at center-of-mass wavevector Q. This line. To get smooth lines, we use Lorentzian functions to very dilute exciton condensate limit becomes a standard plot the Dirac-δ functions (Eq. 31) with a width equal to two-dimensional hydrogen-like problem where each exci- 2 the smallest energy scale in our calculation, i.e., kmesh/2. tation can be characterized by atomic-like orbitals, such These calculations identify certain collective modes at as 1s, 2s, 2p, etc. Note that in this limit, the χzz response energies below the particle-hole continuum that have a weighting factor projects out relative-motion states that large weight in the (τx, τx) pair amplitude response func- are orthogonal to the pair state that is macroscopically tions. This result is reminiscent of the finding in earlier occupied in the ground state - 1s hydrogenic pair states work [3, 10] that for superconductors there is a collec- in the Coulomb interaction case. We have computed tive mode at the edge of the excitation continuum with the momentum space wavefunctions, i.e., eigenvectors a large residue in the pair amplitude response function. of K(+)K(−), for the six lowest-energy collective modes At finite excitation wavevector Q~ additional modes have in the BEC regime at Q~ = 0 (Fig. 2(a)) and find that significant pair amplitude character. they resemble hydrogenic atomic orbitals very well, as

The corresponding results for χyy (exciton phase) and shown in Fig. 3. The energy sequence of these modes is χzz (exciton density) are presented as blue and yellow 1s, 2p, 2s, 3d, 3p, and 3s where 1s is the gapless Goldstone lines in Fig 2. All three lines share similar peak positions mode and 2s and 3s are Higgs-like modes with peaks in due to the coupling between different channels except χxx responses (denoted with arrows in Fig. 2(a)). Note that χzz is absent in Q = 0. The differences in coupling that 2p, 3p, and 3d states are doubly degenerate and only between exciton density fluctuations and amplitude fluc- one state of each doublet is shown in Figs. 3(d)–(f). In tuations is due to the different ~k-dependent weighting this way we have found that the large-weight amplitude factors in Eq. 17 although both responses are related to response corresponds to the addition of an electron-hole ~ pair to the system, not in the 1s pair state which is con- the changes in the x~k(Q) fields. The strong mixing of phase and amplitude fluctuations is also observed in su- densed, but in higher-energy orbitals. The lowest-energy perconductors with particle-hole symmetry breaking [24]. high-weight state in the BEC limit at Q = 0 corresponds to adding an electron-hole pair in a 2s state, which in two The Goldstone mode energy vanishes as Q~ → 0 in the dimensions has a binding energy relative to the particle- electron-hole pair case because these modes are neutral, hole continuum that is smaller by a factor of 9. The whereas it has a finite energy in the three-dimensional second-highest weight state corresponds to the 3s state electron-electron pair case of superconductors because of ~ and higher n excitations are not fully identifiable only be- the divergence in the Coulomb interactions as Q → 0. cause of the finite density of the momentum space grids The collective modes that have large weight in the used in our calculations. In a SIXC, therefore, the gapped χxx response function are entirely different in character. Higgs-like modes are excitations in which one electron- At Q~ = 0, the Goldstone mode contribution to any re- hole pair is added in a state that is orthogonal to the sponse functions Imχαα vanishes because of the mode fre- pair state present in the condensate in the BEC regime. quency factor in Eq. 31 even though the matrix elements As we see in Fig. 2 (c), at finite Q~ , the Goldstone mode

τx(y,z),ωGS are nonzero. However, a few peaks appear also makes a nonzero contribution to the pair amplitude in the τx − τx response below the particle-hole contin- response which is even larger in magnitude due to the uum. These are identified as Higgs-like modes because mixing of phase and amplitude fluctuations. The 2s-like they produce poles in the amplitude-amplitude response Higgs-like mode still has a very large weight in the spectra functions and correspond to the Higgs-like modes iden- and is located below the electron-hole continuum even in tified in studies of superconductors. The property that ~ ∗ the large wave vector QaB = 1 case. In the low exciton they appear below the particle-hole continuum is the key density limit, the wavevector dependence of the Gold- difference from superconductors with short-range inter- stone collective mode is consistent with the Bogoliubov action. theory of weakly interacting bosons. Figure 4(a) shows To understand the character of the Higgs-like modes the intensity of Imχxx as a function of Q and excitation more fully, we examine the low carrier density limit in energy ω in the BEC regime. We can identify three dom- ~ inant branches of collective modes having large weight which u~k has small values at all k. From Eq. 17 it follows in amplitude-pair fluctuations. Note that 3s becomes that, to lowest order in u~k, fainted at large Q due to its closeness to the continuum (purple dashed line). The 2s-like Higgs-like mode is one (±) 1 ~ K = δ~ (E~ + E~ ~ ) − U(k − ~p), of our main findings in the SIXC. k,~p k k+Q A ~ X ~ As the BCS regime at large exciton densities is ap- τx,y(Q) = x~ (Q), k (36) proached, the spectra of the Higgs-like modes change. ~k In Figs 2(b) and (d), we find that different collective ~ X ~ τz(Q) = (u~k + u~k+Q~ )x~k(Q). modes have a larger weight in the response function as ~k wave vector increases. At zero wave vector, we iden- 8

(a) “1s” (b) “2s” (c) “3s”

∗

퐵

푎

푦 푘

(d) “2p” (e) “3p” (f) “3d”

∗

퐵

푎

푦 푘

∗ ∗ ∗ 푘푥 푎퐵 푘푥 푎퐵 푘푥 푎퐵

FIG. 3: (Color online) The squared modulus of wavefunctions of collective modes on the momentum grid at Q~ = 0 in the BEC regime. (a) to (c) show ns-like atomic orbital distribution corresponding to gapless Goldstone modes and the two Higgs-like modes denoted with arrows in Fig. 2(a). 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 2p, 3p, and 3d-like atomic orbitals distributions which do not contribute to the amplitude response functions. The energy sequence of all six collective modes is 1s, 2p, 2s, 3d, 3p, and 3s. Note that 2p, 3p, and 3d are doubly degenerate, and one of each doublet is plotted here.

(a) (b)

3s ∗

푅푦 2s

휔 ℏ

∗ ∗ Q 푎퐵 Q 푎퐵

FIG. 4: (Color online) Intensity Imχxx 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 (1s) and Higgs-like modes (2s and 3s) plotted in Figs. 3(a) to (c). In the BCS regime in (b), only one gapped Higgs-like mode dominates the response at ∗ small Q (QaB < 0.3) and the second Higgs-like mode which has lower energy than the first Higgs-like mode, appears as Q increases. 9 tify the first Higgs-like mode shown in Fig. 2(b) and find quantum fluctuations in the electron-hole pair amplitude. that its qualitative interpretation as the excitation of a We find that in the low exciton density BEC regime the noncondensed pair is unchanged. As Q~ varies, a sec- strongest response to Higgs-like perturbations is one in ond peak belonging to a different excitation energy below which an electron-hole pair is added in a state that is the first Higgs-like mode appears, and the new Higgs-like orthogonal to the pair state present in the ground-state mode shows higher peak at Q~ increases, as illustrated condensate. This interpretation retains qualitative valid- in Fig 2(d). Figure 4(b) shows that only one prominent ity when the exciton density is increased and the BCS gapped Higgs-like mode below the continuum has large limit is approached. These findings shed new light on intensity in Imχxx response functions at small wavevec- previous work that has studied Higgs-like modes in su- tor besides the Goldstone mode. As Q increases, the perconductors, in which the Higgs-like response appears, second gapped Higgs-like mode below the first Higgs- mysteriously perhaps, at the edge of the particle-hole like mode appears which is qualitatively different from continuum. In light of the present calculations it is clear the BEC case. We suspect that the large weight in the that this property just reflects the absence in the BCS first Higgs-like mode spreads to other modes as the first models used for these studies of a higher-energy electron- Higgs-like mode disperses closer to the flat particle-hole electron pair bound state and begs the question as to continuum, while the hydrogenic 2s-like Higgs-like mode whether or not higher-energy bound states do exist in disperses similar to the continuum in the BEC regime. some superconductors. Since Higgs-like excitations in su- We, nevertheless, find that even in the BCS regime the perconductors change the total electron number, they can SIXC supports Higgs-like modes below the particle-hole be observed only indirectly [56–63]. One possible strat- continuum. This behavior is in contrast to the case of egy to detect these higher-energy bound states where the BCS models commonly used for superconductors in they are suspected is to look for resonant features in which Higgs-like modes are located exactly at the edge of the bias voltage dependent subgap currents of Joseph- the particle-hole continuum 2∆ [3]. The Higgs-like modes son junctions. here are distinct modes, higher in energy than the Gold- Two different cases need to be distinguished when dis- stone modes but still in the excitation gap. The source cussing the detection of Higgs-like modes. When a spa- of the difference is the nature of the attractive interac- tially indirect exciton condensate is formed from equi- tion between electrons and holes, which supports several librium populations of electrons and holes in two sepa- bound states. The spectrum of amplitude fluctuation rate layers, the operator τ corresponds to tunneling be- can reflect the spectrum of collective particle-hole exci- x tween layers. The presence of a spatially indirect exciton tations, including bound states, if any. If we replace the condensate or incipient condensate then appears as an interlayer Coulomb potential by a δ-function attractive anomaly in the interlayer tunneling current-voltage rela- interaction with a cutoff, as commonly employed in the tionship near zero bias [33–38]. We anticipate that Higgs- theory of superconductivity, the Higgs-like modes evolve like modes will appear as finite-bias voltage anomalies at into resonances at the bottom of the particle-hole con- energies below the particle-hole continuum. tinuum (shown in Appendix. A) – the resonances that have [3, 9–17] been identified as the Higgs-like modes of The case in which an exciton condensate is formed in superconductors. quasiequilibrium systems of electrons and holes, either in the same layer or in adjacent layers, generated by opti- cal pumping is perhaps simpler experimentally. In this IV. DISCUSSION case the coherent excitons are routinely [64] examined by measuring the photoluminescence (PL) signal. Emitted In this paper we have applied time-dependent mean- photons with energy ~ω can be generated by transitions field theory to spatially indirect exciton condensates with between initial N-exciton states and final N − 1 exciton the goal of identifying collective modes associated with states which satisfy

~ω = Ei(N) − Ej(N − 1) = µex + [Ei(N) − E0(N)] − [Ej(N − 1) − E0(N − 1)]. (37)

The matrix elements for these processes are proportional Ei(N) − E0(N) is the excitation relative to the ground to the operator τx, which changes the number of electron- state in the N-exciton initial state, and Ej0(N − 1) = hole pairs present in the system by one, and can there- Ej(N −1)−E0(N −1) is the excitation energy relative to fore generate Higgs-like excitations. In Eq. 37, µex is the ground state in the N − 1 exciton final state. A simi- the chemical potential of excitons which is non-zero in lar analysis applies in the case of polariton condensates in non-equilibrium condensed exciton systems, Ei0(N) = which the exciton system is coupled to two-dimensional 10 cavity photons [65]. The PL spectrum consists of a seg- Measurement Laboratory, Award No. 70NANB14H209, ment for which ~ω > µex due to thermal excitations in through the University of Maryland. F.W. is supported the initial state and a so-called ghost segment in which by the Laboratory for Physical Sciences. ~ω < µex due to excitations being generated in the final state when the exciton number changes. Because they have a high energy, Higgs-like modes are not likely to Appendix A: Case of short-range interlayer be thermally populated, but they can be visible in the interaction ghost mode spectrum when exciton-exciton interactions are strong. Indeed very recent work [66] which appeared By replacing the long-range Coulomb interaction be- as this paper was under preparation has claimed that a −qd Higgs-like excitation is present in the PL spectrum of a tween electrons and holes U(~q) ∝ e /q with a Dirac-δ polariton condensate at energy ω = µ − E , and short-range interaction U(~q) ∝ δ(~q), we find that only ~ ex Higgs the gapless Goldstone mode exists below the continuum has made the numerical observation that EHiggs is close to the energy difference between the cavity-dressed 2s and Higgs-like modes are right at the edge of the electron- and 1s excitonic bound states. The present paper ap- hole continuum shown in Fig. 5. This is very similar to pears to explain this observation. the case of a BCS superconductor where the Higgs-like mode is located exactly at the particle-hole continuum.

Note that the mean-ﬁeld gap ∆~k is a constant 2∆0 and V. ACKNOWLEDGMENTS the continuum is 2∆0 at Q = 0.

F.X. thanks M. Lu and X.-X. Zhang for helpful discus- ~ sions. This work was supported by the Army Research Appendix B: explicit expressions for E~k,~p(Q) and Oﬃce under Award No. W911NF-17-1-0312 (MURI) and ~ Γ~k,~p(Q) by the Welch Foundation under Grant No. F1473. F.X. acknowledges support under the Cooperative Research ~ ~ Agreement between the University of Maryland and the Below are explicit expressions for E~k,~p(Q) and Γ~k,~p(Q), National Institute of Standards and Technology Physical which appear in the energy variation δE(2) in Eq. (14).

~ E~k,~p(Q) = δ~k,~p(ζ~k+Q~ − ζ~k + E~k + E~k+Q~ ) 1 + V (Q~ ) − V (~k − ~p)(u u v v + v v u u ) A ~k ~p ~k+Q~ ~p+Q~ ~k ~p ~k+Q~ ~p+Q~ 1 − U(Q~ )(v u u v + u v v u ) A ~k ~p ~k+Q~ ~p+Q~ ~k ~p ~k+Q~ ~p+Q~ 1 − U(~k − ~p)(u u u u + v v v v ), A ~k ~p ~k+Q~ ~p+Q~ ~k ~p ~k+Q~ ~p+Q~ (B1) 1 Γ (Q~ ) = V (Q~ ) − V (~k + Q~ − ~p)(u u v v + v v u u ) ~k,~p A ~k ~p ~k+Q~ ~p−Q~ ~k ~p ~k+Q~ ~p−Q~ 1 − U(Q~ )(v u u v + u v v u ) A ~k ~p ~k+Q~ ~p−Q~ ~k ~p ~k+Q~ ~p−Q~ 1 + U(~k + Q~ − ~p)(v u v u + u v u v ), A ~k ~p ~k+Q~ ~p−Q~ ~k ~p ~k+Q~ ~p−Q~

~ where u~k and v~k are deﬁned in Eq. (6), and V (Q) and interactions. U(Q~ ) are respectively intralayer and interlayer Coulomb

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