Fermi-edge exciton-polaritons in doped semiconductor microcavities with finite hole mass
Dimitri Pimenov, Jan von Delft, Leonid Glazman, Moshe Goldstein
FFermi-edge exciton-polaritons in doped semiconductor microcavities with finite holemass
Dimitri Pimenov ∗ and Jan von Delft Arnold Sommerfeld Center for Theoretical Physics,Ludwig-Maximilians-University Munich, 80333 Munich, Germany
Leonid Glazman
Departments of Physics, Yale University, New Haven, Connecticut 06520, USA
Moshe Goldstein
Raymond and Beverly Sackler School of Physics and Astronomy, Tel Aviv University, Tel Aviv 6997801, Israel
The coupling between a 2D semiconductor quantum well and an optical cavity gives rise to com-bined light-matter excitations, the exciton-polaritons. These were usually measured when the con-duction band is empty, making the single polariton physics a simple single-body problem. Thesituation is dramatically different in the presence of a finite conduction band population, where thecreation or annihilation of a single exciton involves a many-body shakeup of the Fermi sea. Recentexperiments in this regime revealed a strong modification of the exciton-polariton spectrum. Pre-vious theoretical studies concerned with nonzero Fermi energy mostly relied on the approximationof an immobile valence band hole with infinite mass, which is appropriate for low-mobility samplesonly; for high-mobility samples, one needs to consider a mobile hole with large but finite mass.To bridge this gap we present an analytical diagrammatic approach and tackle a model with short-ranged (screened) electron-hole interaction, studying it in two complementary regimes. We find thatthe finite hole mass has opposite effects on the exciton-polariton spectra in the two regimes: In thefirst, where the Fermi energy is much smaller than the exciton binding energy, excitonic features areenhanced by the finite mass. In the second regime, where the Fermi energy is much larger than theexciton binding energy, finite mass effects cut off the excitonic features in the polariton spectra, inqualitative agreement with recent experiments.
I. INTRODUCTION
When a high-quality direct semiconductor 2D quantumwell (QW) is placed inside an optical microcavity, thestrong coupling of photons and QW excitations gives riseto a new quasiparticle: the polariton. The properties ofthis fascinating half-light, half-matter particle stronglydepend on the nature of the involved matter excitations.If the Fermi energy is in the semiconductor band gap,the matter excitations are excitons. This case is theoret-ically well understood [1, 2], and the first observation ofthe resulting microcavity exciton-polaritons was alreadyaccomplished in 1992 by Weisbuch et al. [3]. Severalstudies on exciton-polaritons revealed remarkable results.For example, exciton-polaritons can form a Bose-Einsteincondensate [4], and were proposed as a platform for high- T c superconductivity [5].The problem gets more involved if the Fermi energyis above the conduction band bottom, i.e., a conductionband Fermi sea is present. Then the matter excitationshave a complex many-body structure, arising from thecomplementary phenomena of Anderson orthogonality [6]and the Mahan exciton effect, entailing the Fermi-edgesingularity [7–11]. An experimental study of the result-ing “Fermi-edge polaritons” in a GaAs QW was first con-ducted in 2007 by Gabbay et al. [12], and subsequently ∗ [email protected] extended by Smolka et al. [13] (2014). A similar experi-ment on transition metal dichalcogenide monolayers wasrecently published by Sidler et al. [14] (2016).From the theory side, Fermi-edge polaritons have beeninvestigated in Ref. [15, 16]. However, in these worksonly the case of infinite valence band hole mass was con-sidered, which is the standard assumption in the Fermi-edge singularity or X-ray edge problem. Such a modelis valid for low-mobility samples only and thus fails toexplain the experimental findings in [13]: there, a high-mobility sample was studied, for which an almost com-plete vanishing of the polariton splitting was reported.Some consequences of a finite hole mass for polaritonswere considered in a recent treatment [17], but withoutfully accounting for the so-called crossed diagrams thatdescribe the Fermi sea shakeup, as we further elaboratebelow.The aim of the present paper is therefore to study theeffects of both finite mass and Fermi-edge singularity onpolariton spectra in a systematic fashion. This is doneanalytically for a simplified model involving a contact in-teraction, which nethertheless preserves the qualitativefeatures of spectra stemming from the finite hole massand the presence of a Fermi sea. In doing so, we distin-guish two regimes, with the Fermi energy µ being eithermuch smaller or much larger than the exciton binding en-ergy E B . For the regime where the Fermi energy is muchlarger than the exciton binding energy, µ (cid:29) E B , sev-eral treatments of finite-mass effects on the Fermi-edge a r X i v : . [ c ond - m a t . s t r- e l ] O c t singularity alone (i.e., without polaritons) are available,both analytical and numerical. Without claiming com-pleteness, we list [18–22]. In our work we have mainlyfollowed the approach of Ref. [18], extending it by go-ing from 3D to 2D and, more importantly, by addressingthe cavity coupling which gives rise to polaritons. Forinfinite hole mass the sharp electronic spectral featurecaused by the Fermi edge singularity can couple withthe cavity mode to create sharp polariton-type spectralpeaks [15, 16]. We find that the finite hole mass cuts offthe Fermi edge singularity and suppresses these polaritonfeatures.In the opposite regime of µ (cid:28) E B , where the Fermienergy is much smaller than the exciton binding energy,we are not aware of any previous work addressing themodification of the Fermi-edge singularity due to finitemass. Here, we propose a way to close this gap usinga diagrammatic approach. Interestingly, we find that inthis regime the excitonic singularities are not cut off, butare rather enhanced by finite hole mass, in analogy to theheavy valence band hole propagator treated in [23].This paper has the following structure: First, beforeembarking into technical details, we will give an intuitiveoverview of the main results in Sec. II. Detailed com-putations will be performed in subsequent sections: InSec. III, the full model describing the coupled cavity-QWsystem is presented. The key quantity that determinesits optical properties is the cavity-photon self-energy Π,which we will approximate by the electron-hole correlatorin the absence of a cavity. Sec. IV shortly recapitulateshow Π can be obtained in the regime of vanishing Fermienergy, for infinite and finite hole masses. Then we turnto the many-body problem in the presence of a Fermi seain the regimes of small (Sec. V) and large Fermi energy(Sec.VI). Using the results of the previous sections, po-lariton properties are addressed in Sec. VII. Finally, wesummarize our findings and list several possible venuesfor future study in Sec. VIII. II. SUMMARY OF RESULTS
In a simplified picture, polaritons arise from the hy-bridization of two quantum excitations with energiesclose to each other, the cavity photon and a QW res-onance [1, 2]. The resulting energy spectrum consists oftwo polariton branches with an avoided crossing, whoselight and matter content are determined by the energydetuning of the cavity mode from the QW mode.While the cavity photon can be approximated reas-onably by a bare mode with quadratic dispersion anda Lorentzian broadening due to cavity losses, the QWresonance has a complicated structure of many-body ori-gin. The QW optical response function is rather sensit-ive to nonzero density of conduction band (CB) electrons.Roughly, it tends to broaden QW spectral features, whichcontribute to the spectral width of polariton lines.A more detailed description of the polariton lines re- quires finding first the optical response function Π( Q , Ω)of the QW alone (without polaritons). Here, Q and Ωare, respectively, the momentum and the energy of anincident photon probing the optical response. The ima-ginary part of Π( Q , Ω), A ( Q , Ω) = − Im [Π( Q , Ω)] /π ,defines the spectral function of particle-hole excitationsin the QW. In the following, we discuss the evolutionof A ( Q , Ω) as the chemical potential µ is varied, con-centrating on the realistic case of a finite ratio of theelectron and hole masses. We assume that the temper-ature is low, and consider the zero-temperature limit inthe entire work. In addition, we will limit ourselves tothe case where the photon is incident perpendicular tothe QW, i.e. its in-plane momentum is zero, and study A (Ω) ≡ A ( Q = 0 , Ω).In the absence of free carriers ( µ is in the gap), a CBelectron and a hole in the valence band (VB) create ahydrogen-like spectrum of bound states. In the case of aQW it is given by the 2D Elliot formula (see, e.g., [24]).Being interested in the spectral function close to the mainexciton resonance, we replace the true Coulomb interac-tion by a model of short-ranged interaction potential ofstrength g [see Eqs. (10) and (12)]. As a result, there isa single bound state at an energy E G − E B ( g ), which weidentify with the the lowest-energy exciton state. Here, E G is the VB-CB gap, and energies are measured with re-spect to the minimum of the conduction band. A sketchof A (Ω) is shown in Fig. 1. A (Ω) Ω − E G − E B µ <
0M irrelevant
Figure 1. (Color online) Absorption spectrum for short-rangeelectron-hole interaction and µ <
0, given by the imaginarypart of Eq. (19).
For µ >
0, electrons start to populate the CB. If thechemical potential lies within the interval 0 < µ (cid:28) E B ,then the excitonic Bohr radius r B remains small com-pared to the Fermi wavelength λ F of the electron gas,and the exciton is well defined. Its interaction with theparticle-hole excitations in the CB modifies the spectralfunction A (Ω) in the vicinity of the exciton resonance.The limit of an infinite hole mass was considered by Noz-i`eres et al. [8–10]: Due to particle-hole excitations of theCB Fermi sea, which can happen at infinitesimal energycost, the exciton resonance is replaced by a power lawspectrum, see inset of Fig. 2. In terms of the detuningfrom the exciton threshold, ω = Ω − Ω exc T , Ω exc T = E G + µ − E B , (1)the spectral function, A exc ( ω ) = − Im [Π exc ( ω )] /π , scalesas: A exc ( ω ) (cid:12)(cid:12)(cid:12)(cid:12) M = ∞ ∼ θ ( ω ) E B ω (cid:18) ωµ (cid:19) α , ω (cid:28) µ. (2)The effective exciton-electron interaction parameter α was found by Combescot et al. [11], making use of final-state Slater determinants. In their work, α is obtained interms of the scattering phase shift δ of Fermi level elec-trons off the hole potential, in the presence of a boundstate, as α = | δ/π − | . For the system discussed herethis gives [25]: α = 1 / (cid:12)(cid:12)(cid:12)(cid:12) ln (cid:18) µE B (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) . (3)We re-derive the result for α diagrammatically (seeSec. V), in order to extend the result of Combescot etal. to the case of a small but nonzero CB electron-VBhole mass ratio β , where β = m/M. (4)While the deviation of β from zero does not affect theeffective interaction constant α , it brings qualitativelynew features to A (Ω), illustrated in Fig. 2. The originof these changes is found in the kinematics of the inter-action of the exciton with the CB electrons. Momentumconservation for finite exciton mass results in phase-spaceconstraints for the CB particle-hole pairs which may beexcited in the process of exciton creation. As a result,the effective density of states ν ( ω ) of the pairs with pairenergy ω (also corresponding to the exciton decay rate)is reduced from ν ( ω ) ∼ ω at β = 0 [11] to ν ( ω ) ∼ ω / when ω is small compared to the recoil energy E R = βµ .A smaller density of states for pairs leads to a reducedtransfer of the spectral weight to the tail; therefore, thedelta function singularity at the exciton resonance sur-vives the interaction with CB electrons, i.e. β > A exc ( ω ) (cid:12)(cid:12)(cid:12)(cid:12) M< ∞ = A exc,incoh. ( ω ) θ ( ω ) + β α E B δ ( ω ) , (5a) A exc,incoh. ( ω ) ∼ E B α √ ωβµ β α ω (cid:28) βµ α ω (cid:16) ωµ (cid:17) α βµ (cid:28) ω (cid:28) µ. (5b)The main features of this spectral function are summar-ized in Fig. 2: As expected, the exciton recoil only playsa role for small frequencies ω (cid:28) βµ , while the infinitemass edge singularity is recovered for larger frequencies.The spectral weight of the delta peak is suppressed bythe interaction. For β → α (cid:54) = 0, we recover the infinite mass result, where no coherent part shows up. If,on the opposite, α → β (cid:54) = 0, the weight of thedelta peak goes to one: The exciton does not interactwith the Fermi sea, and its spectral function becomes apure delta peak, regardless of the exciton mass. A par-tial survival of the coherent peak at α, β (cid:54) = 0 could beanticipated from the results of Rosch and Kopp [23] whoconsidered the motion of a heavy particle in a Fermi gasof light particles. This problem was also analyzed byNozi`eres [22], and the coherent peak can be recovered byFourier transforming his time domain result for the heavyparticle Green’s function.At this point, let us note the following: for µ >
0, thehole can bind two electrons with opposite spin, givingrise to trion features in the spectrum. We will not focuson those, since, for weak doping, their spectral weight issmall in µ (more precisely, in µ/E T , where E T (cid:28) E B is the trion binding energy), and they are red detunedw.r.t. the spectral features highlighted in this work. Inthe regime of µ (cid:29) E B (cid:29) E T , trions should be neglibleas well. Some further discussion of trion properties canbe found in Appendix C. A ( ω ) βµ µ ∼ / √ ω ∼ ω α − ω = Ω − Ω exc T µ ωM = ∞ µ (cid:28) E B M < ∞ Figure 2. (Color online) Absorption for µ (cid:28) E B and finitehole mass, illustrating Eq. (5). The full green curve showsthe delta peak (broadened for clarity), while the dashed blueline is the incoherent part. Frequencies are measured fromthe exciton threshold frequency Ω exc T = E G + µ − E B . Theinset shows the infinite mass spectrum for comparison. Thedashed region in the inset indicates the continuous part ofthe spectrum, whose detailed form is beyond the scope of thispaper, as we only consider the leading singular parts of allspectra. Upon increase of chemical potential µ , the CB con-tinuum part (inset of Fig. 2) starts building up into thewell-known Fermi-edge singularity (FES) at the Burstein-Moss [26, 27] shifted threshold, Ω FES T = E G + µ . Forfinite mass ( β (cid:54) = 0), the FES will however be broadenedby recoil effects (see below). At the same time, the deltafunction singularity of Eq. (5a) at the absorption edgevanishes at some value of µ . So, at higher electron dens-ities, it is only the FES which yields a nonmonotonic be-havior of the absorption coefficient, while the absorptionedge is described by a converging power law with fixedexponent, see Eq. (8). This evolution may be contrastedto the one at β = 0. According to [11, 21], the counter-parts of the absorption edge and broadened FES are twopower law nonanalytical points of the spectrum which arepresent at any µ and characterized by exponents continu-ously evolving with µ . A more detailed discussion of theevolution of absorption spectra as µ increases from smallto intermediate to large values is presented in AppendixA.Let us now consider the limit µ (cid:29) E B , where the FESis the most prominent spectral feature, in closer detail.In the case of infinite hole mass ( β = 0), and in terms ofthe detuning from the FES threshold, ω = Ω − Ω FES T , Ω FES T = E G + µ, (6)the FES absorption scales as [8–10]: A FES ( ω ) (cid:12)(cid:12)(cid:12)(cid:12) M = ∞ ∼ θ ( ω ) (cid:18) ωµ (cid:19) − g , (7)as illustrated in the inset of Fig. 3. In the above formula,the interaction contribution to the treshold shift, whichis of order gµ , is implicitly contained in a renormalizedgap E G .What happens for finite mass? This question wasanswered in [18, 21, 22]: As before, the recoil comes intoplay, effectively cutting the logarithms contributing to(7). Notably, the relevant quantity is now the VB hole recoil, since the exciton is no longer a well defined entity.The FES is then replaced by a rounded feature, sketchedin Fig. 3, which sets in continuously: A FES ( ω ) (cid:12)(cid:12)(cid:12)(cid:12) M< ∞ ∼ (cid:16) ωβµ (cid:17) β − g · θ ( ω ) ω (cid:28) βµ (cid:18) √ ( ω − βµ ) +( βµ ) µ (cid:19) − g βµ (cid:28) ω (cid:28) µ. (8)Eq. (8) can be obtained by combining and extending to2D the results presented in Refs. [18, 21]. βµ µ ω = Ω − Ω T A ( ω ) ∼ ω ∼ (cid:16)p ( ω − βµ ) + ( βµ ) /µ (cid:17) − g µ (cid:29) E B µ ωM = ∞ Figure 3. (Color online) Finite mass absorption in the case E B (cid:28) µ . Frequencies are measured from Ω FES T = E G + µ .The inset shows the infinite mass case for comparison. The maximum of Eq. (8) is found at the so-called dir-ect threshold, ω D = βµ (see Fig. 4(a)). This shift isa simple effect of the Pauli principle: the photoexcitedelectron needs to be placed on top of the CB Fermi sea.The VB hole created this way, with momentum k F , cansubsequently decay into a zero momentum hole, scatter-ing with conduction band electrons [see Fig. 4(b)]. Theseprocesses render the lifetime of the hole finite, with a de-cay rate ∼ g βµ . Within the logarithmic accuracy of theFermi edge calculations, this is equal to βµ , the cutoffof the power law in Eq. (8) (See Sec. VI B for a moredetailed discussion). As a result, the true threshold ofabsorption is found at the indirect threshold, ω I = 0.Due to VB hole recoil, the CB hole-electron pair densityof states now scales as ν ( ω ) ∼ ω , leading to a similarbehavior of the spectrum, see Fig. 3. k Ω D βµ Ω I k F Ω k Ω k F (a) (b) Figure 4. (Color online) (a): The direct threshold Ω D =Ω FES T + βµ and the indirect threshold Ω I = Ω FES T [in the maintext, ω D/I = Ω
D/I − Ω FES T ] (b): The VB hole can undergoinelastic processes which reduces its energy, smearing the in-finite mass edge singularity. We note that at finite ratio β = m/M , raising thechemical potential µ from µ (cid:28) E B to µ (cid:29) E B resultsin a qualitative change of the threshold behavior from asingular one of Eq. (5b), to a converging power law, seethe first line of Eq. (8). Simultaneously, a broadened FESfeature appears in the continuum, at ω >
0. The differ-ence in the value of the exponent in the excitonic result[Eq. (5b)], as compared to the FES low-energy behavior[Eq. (8) for ω (cid:28) βµ ], can be understood from the differ-ence in the kinematic structure of the excitations: In theexciton case, the relevant scattering partners are an ex-citon and a CB electron-hole pair. In the FES case, onehas the photoexcited electron as an additional scatteringpartner, which leads to further kinematic constraints andeventually results in a different low-energy power law.In the frequency range βµ (cid:28) ω (cid:46) µ , the physics isbasically the same as in the infinite hole mass case ( β =0). There, the behavior near the lowest threshold (whichis exciton energy for µ (cid:28) E B and the CB continuum for µ (cid:29) E B ) is always ∼ ω (1 − δ/π ) − = ω ( δ/π ) − δ/π . But inthe first case ( µ (cid:28) E B ), δ ∼ π − α is close to π (due to thepresence of a bound state), so the threshold singularity isin some sense close to the delta peak , ∼ Im[1 / ( ω + i + )],that one would have for µ = 0, whereas in the secondcase ( µ (cid:29) E B ), δ ∼ g is close to zero, so the thresholdsingularity is similar to a discontinuity.Having discussed spectral properties of the QW alone,we can now return to polaritons. Their spectra A p ( ω ) canbe obtained by inserting the QW polarization as photonself-energy. While a full technical account will be given inSec. VII, the main results can be summarized as follows:In the first case of study, of µ (cid:28) E B and finite β ,the polaritons arise from a mixing of the cavity and thesharp exciton mode. The smaller the hole mass, the moresingular the exciton features, leading also to sharper po-lariton features. Furthermore, the enhanced exciton qua-siparticle weight pushes the two polariton branches fur-ther apart. Conversely, in the singular limit of infinitehole mass, the pole in the exciton spectrum turns intothe pure power law familiar from previous work, result-ing in broader polariton features. A comparison of theinfinite and finite hole mass versions of the polariton spec-tra A p ( ω ) when the cavity photon is tuned into resonancewith the exciton is presented in Fig. 5. Notably, the aboveeffects are rather weak, since the exciton is a relativelysharp resonance even for infinite hole mass. infinite massfinite mass, β = 0 . µ (cid:28) E B A p ( ω ) · ∆ π ω/ ∆ = (Ω − Ω exc T ) / ∆ Figure 5. (Color online) Comparison of the polariton spec-trum for µ (cid:28) E B , at zero cavity detuning. Frequencies aremeasured from the exciton threshold, Ω exc T = E G + µ − E B .The energy unit ∆ corresponds to the half mode splitting atzero detuning in the bare exciton case ( µ = 0). In the second case, µ (cid:29) E B , the matter component ofthe polaritons corresponds to the FES singularity, whichis much less singular than the exciton. Consequently,the polaritons (especially the upper one, which sees thehigh-frequency tail of the FES) are strongly washed outalready at β = 0. For finite hole mass, the hole recoil cutsoff the FES singularity, resulting in further broadening ofthe polaritons. In addition, there is an overall upward fre-quency shift by βµ , reflecting the direct threshold effect.Fig. 6 shows the two polariton spectra at zero detuning. A p ( ω ) · ˜∆ π ω/ ˜∆ = (Ω − Ω FES T ) / ˜∆infinite massfinite mass, β = 0 . µ (cid:29) E B Figure 6. (Color online) Comparison of the polariton spec-trum for µ (cid:29) E B , at zero cavity detuning . Frequenciesare measured from the indirect threshold, Ω FES T = E G + µ .The energy unit ˜∆, which determines the polariton splittingat zero detuning, is defined in Sec. VII, Eq. (76). The dot-ted vertical line indicates the position of the direct threshold, ω D = βµ . The cutoff of the lower polariton for finite masses iseven more drastic when the cavity is blue-detuned withrespect to the threshold: Indeed, at large positive cavitydetuning, the lower polariton is mostly matter-like, andthus more sensitive to the FES broadening. It thereforealmost disappears, as seen in Fig. 7. infinite massfinite mass, β = 0 . A p ( ω ) · ˜∆ π ω/ ˜∆ = (Ω − Ω FES T ) / ˜∆ µ (cid:29) E B Figure 7. (Color online) Comparison of the polariton spec-trum for µ (cid:29) E B , at large positive cavity detuning . Fre-quencies are measured from the indirect threshold, Ω FES T = E G + µ . III. MODEL
After the qualitative overview in the previous section,let us now go into more detail, starting with the precisemodel in question. To describe the coupled cavity-QWsystem, we study the following 2D Hamiltonian: H = H M + H L , (9) H M = (cid:88) k (cid:15) k a † k a k − (cid:88) k [ E k + E G ] b † k b k (10) − V S (cid:88) k , p , q a † k a p b k − q b † p − q ,H L = (cid:88) Q ω Q c † Q c Q − i d √S (cid:88) p , Q a † p + Q b p c Q + h.c. (11)Here, H M , adapted from the standard literature on theX-ray edge problem [18], represents the matter part ofthe system, given by a semiconductor in a two-bandapproximation: a k annihilates a conduction band (CB)electron with dispersion (cid:15) k = k m , while b k annihilatesa valence band (VB) electron with dispersion − ( E k + E G ) = − ( k M + E G ). E G is the gap energy, which isthe largest energy scale under consideration: In GaAs, E G (cid:39) S is the area of the QW, andwe work in units where (cid:126) = 1. Unless explicitly statedotherwise, we assume spinless electrons, and concentrateon the zero temperature limit.When a valence band hole is created via cavity photonabsorption, it interacts with the conduction band elec-trons with an attractive Coulomb interaction. Takinginto account screening, we model the interaction as point-like, with a constant positive matrix element V . Theeffective potential strength is then given by the dimen- sionless quantity g = ρV , ρ = m π , (12) ρ being the 2D DOS. The appropriate value of g will befurther discussed in the subsequent sections.Interactions of CB electrons with each other are com-pletely disregarded in Eq. (9), presuming a Fermi liquidpicture. This is certainly a crude approximation. It canbe justified if one is mostly interested in the form of singu-larities in the spectral function. These are dominated byvarious power laws, which arise from low-energy particlehole excitations of electrons close to the Fermi energy,where a Fermi-liquid description should be valid.The photons are described by H L : We study losslessmodes with QW in-plane momenta Q and energies ω Q = ω c + Q / m c , where m c is the cavity mode effective mass.Different in-plane momenta Q can be achieved by tiltingthe light source w.r.t. the QW. In the final evaluationswe will mostly set Q = 0, which is a valid approximationsince m c is tiny compared to electronic masses. The in-teraction term of H L describes the process of absorbing aphoton while creating an VB-CB electron hole pair, andvice versa. d is the interband electric dipole matrix ele-ment, whose weak momentum dependence is disregarded.This interaction term can be straightforwardly derivedfrom a minimal coupling Hamiltonian studying interb-and processes only, and employing the rotating wave andelectric dipole approximations (see, e.g., [28]).The optical properties of the full system are de-termined by the retarded dressed photon Green’s func-tion [16, 17]: D R ( Q , Ω) = 1Ω − ω Q + i + − Π( Q , Ω) , (13)where Π( Q , Ω) is the retarded photon self-energy. Thisdressed photon is nothing but the polariton. The spectralfunction corresponding to (13) is given by A ( Q , ω ) = − π Im (cid:2) D R ( Q , ω ) (cid:3) . (14) A ( Q , ω ) determines the absorption respectively reflec-tion of the coupled cavity-QW system, which are thequantities typically measured in polariton experimentslike [12, 13].Our goal is to determine Π( Q , Ω). To second order in d it takes the formΠ( Q , Ω) (cid:39) − i d S (cid:90) ∞−∞ dtθ ( t ) e i Ω t (15) × (cid:88) k , p (cid:104) | b † k ( t ) a k + Q ( t ) a † p + Q (0) b p (0) | (cid:105) , where | (cid:105) is the noninteracting electronic vacuum with afilled VB, and the time dependence of the operators isgenerated by H M . Within this approximation, Π( Q , ω )is given by the “dressed bubble” shown in Fig. 8. Theimaginary part of Π( Q , ω ) can also be seen as the lin-ear response absorption of the QW alone with the cavitymodes tuned away. Q , Ω Q , Ω Figure 8. The photon self-energy Π( Q , Ω) in linear response.Full lines denote CB electrons, dashed lines VB electrons, andwavy lines photons. The grey-shaded area represents the fullCB-VB vertex.
Starting from Eq. (15), in the following we will study indetail how Π( Q , ω ) behaves as the chemical potential µ isincreased, and distinguish finite and infinite VB masses M . We will also discuss the validity of the approximationof calculating Π to lowest order in d . IV. ELECTRON-HOLE CORRELATOR IN THEABSENCE OF A FERMI SEA
We start by shortly reviewing the diagrammatic ap-proach in the case when the chemical potential lies withinthe gap (i.e. − E G < µ < µ , Π is exactly given by the sum of the series of lad-der diagrams shown in Fig. 9, first computed by Mahan[29]. Indeed, all other diagrams are absent here sincethey either contain VB or CB loops, which are forbiddenfor µ in the gap. This is seen using the following expres-sions for the zero-temperature time-ordered free Green’sfunctions: G (0) c ( k , Ω) = 1Ω − (cid:15) k + i + sign( (cid:15) k − µ ) , (16) G (0) v ( k , Ω) = 1Ω + E G + E k + i + sign( − E G − E k − µ ) , (17)where the indices c and v stand for conduction andvalence band, respectively, and 0 + is an infinitesimal pos-itive constant. For − E G < µ <
0, CB electrons arepurely retarded, while VB electrons are purely advanced.Thus, no loops are possible. Higher order terms in d arenot allowed as well. + + Figure 9. The series of ladder diagrams. Dotted lines repres-ent the electron-hole interaction.
One can easily sum up the series of ladder diagramsassuming the simplified interaction V [18]. Let us startfrom the case of infinite VB mass ( β = 0), and concen-trate on energies | Ω − E G | (cid:28) ξ , where ξ is an appro-priate UV cutoff of order of CB bandwidth. Since theinteraction is momentum independent, all integrations inhigher-order diagrams factorize. Therefore, the n -th or-der diagram of Fig. 9 is readily computed:Π ( n )ladder (Ω) = d ρ ( − g ) n ln (cid:18) Ω − E G + i + − ξ (cid:19) n +1 . (18)Here and henceforth, the branch cut of the complex log-arithm and power laws is chosen to be on the negativereal axis. The geometric series of ladder diagrams can beeasily summed:Π ladder (Ω) = ∞ (cid:88) n =0 Π ( n )Ladder (Ω) = d ρ ln (cid:16) Ω − E G + i + − ξ (cid:17) g ln (cid:16) Ω − E G + i + − ξ (cid:17) . (19)A sketch of the corresponding QW absorption A ladder = − Im[Π ladder ] /π was already shown in Fig. 1.Π ladder (Ω) has a pole, the so-called Mahan exciton [18,29], at an energy ofΩ − E G = − E B = − ξe − /g . (20)In the following, we will treat E B as a phenomenologicalparameter. To match the results of the short-range in-teraction model with an experiment, one should equate E B with E , the energy of lowest VB hole-CB electronhydrogenic bound state (exciton). Expanding Eq. (19)near the pole, we obtain:Π ladder ( ω ) = d E B ρg G ( ω ) + O (cid:18) ωE B (cid:19) , (21) G ( ω ) = 1 ω + i + , where ω = Ω − E G + E B , and we have introduced thebare exciton Green’s function G , similar to Ref. [30].In this regime of µ , a finite hole mass only results in aweak renormalization of the energy by factors of 1 + β ,where β = m/M is the small CB/VB mass ratio. Fur-thermore, if finite photon momenta Q are considered, theexciton Green’s function is easily shown to be (near thepole): G ( Q , ω ) = 1 ω + Q /M exc + i + , (22)with M exc = M + m = M (1 + β ). V. ELECTRON-HOLE CORRELATOR FORSMALL FERMI ENERGYA. Infinite VB hole mass
Let us now slightly increase the chemical potential µ ,and study the resulting absorption. More precisely, weconsider the regime0 < µ (cid:28) E B (cid:28) ξ. (23)We first give an estimate of the coupling constant g = ρV Accounting for screening of the VB hole 2D Coulomb po-tential by the CB Fermi sea in the static RPA approxim-ation, and averaging over the Fermi surface [18, 29] onefinds: g ∼ (cid:40) − x/π x → , ln( x ) /x x → ∞ , (24)where x = (cid:112) µ/E with E being the true 2D bind-ing energy of the lowest exciton in the absence of a CBFermi sea. In the regime under study we may assume E B (cid:39) E (cid:29) µ , and therefore g (cid:46) g is meaningless. Instead, we will use µ/E B as our small parameter, and re-sum all diagramswhich contribute to the lowest nontrivial order in it.We will now restrict ourselves to the study of energiesclose to E B in order to understand how a small density ofCB electrons modifies the shape of the bound state reson-ance; we will not study in detail the VB continuum in thespectrum (cf. Fig. 2). We first compute the contributionof the ladder diagrams; as compared to Eqs. (21)–(22),the result solely differs by a shift of energies: ω = Ω − Ω exc T , Ω exc T = ( E G + µ ) − E B . (25)Also, the continuum now sets in when Ω equals Ω FES T = E G + µ , which is known as the Burstein-Moss shift[26, 27]. However, for finite µ one clearly needs to gobeyond the ladder approximation, and take into accountthe “Fermi sea shakeup”. To do so, we first consider thelimit of infinite M ( β = 0). In this regime, the QW ab-sorption in the presence of a bound state for the modelunder consideration was found by Combescot and Noz-i`eres [11], using a different approach [32].For finite µ , the physics of the Fermi-edge singularitycomes into play: Due to the presence of the CB Fermi sea,CB electron-hole excitations are possible at infinitesimalenergy cost. As a result, the exciton Green’s function, which weanalogously to (21) define as proportional to the dressedbubble in the exciton regime,Π exc ( ω ) = d E B ρg G exc ( ω ) + O (cid:18) ωE B (cid:19) , (26) G exc ( ω ) = 1 ω − Σ exc ( ω ) , (27)gets renormalized by a self-energy Σ exc ( ω ). This self-energy turns the exciton pole turns into a divergent powerlaw [11]: G exc ( ω ) ∼ ω + i + · (cid:18) ω + i + − µ (cid:19) ( δ/π − , (28)where δ is the scattering phase shift of electrons at theFermi-level off the point-like hole potential. One shouldnote that no delta-peak will appear for δ/π (cid:54) = 1. A sketchof the resulting absorption A is shown in Fig. 10. A/M ρπ ω E B + µ µ ∼ ω ( δ/π − − µ (cid:28) E B Figure 10. (Color online) QW Absorption for µ (cid:28) E B and M = ∞ . The power law (28) is valid asymptotically closeto the left peak. The dashed region indicates the continuouspart of the spectrum, compare caption of Fig. 2. Let us further discuss the result (28). It was obtainedin [11] using an elaborate analytical evaluation of finalstate Slater determinants, and actually holds for anyvalue of µ . A numerical version of this approach for theinfinite VB mass case was recently applied by Baeten andWouters [16] in their treatment of polaritons. In addition,the method was numerically adapted to finite masses byHawrylak [19], who, however, mostly considered the masseffects for µ (cid:29) E B .However, due to the more complicated momentumstructure, it seems difficult to carry over the methodof [11] to finite masses analytically. Instead, we will nowshow how to proceed diagrammatically. Our analysis willgive (28) to leading order in the small parameter µ/E B ,or, equivalently, α = δ/π − δ = π for µ = 0 due to the presence of abound state — the exciton): G exc ( ω ) (cid:39) ω + i + (cid:18) α ln (cid:18) | ω | µ (cid:19) − iα πθ ( ω ) (cid:19) . (29)The merit of the diagrammatical computation is twofold:First, it gives an explicit relation between α and theexperimentally-measurable parameters µ , E B . Second,the approach can be straightforwardly generalized to fi-nite masses, as we show in the next subsection.Let us note that a similar diagrammatic method wasalso examined by Combescot, Betbeder-Matibet et al. in a series of recent papers [30, 33–36]. Their modelHamiltonians are built from realistic Coulomb electron-hole and electron-electron interactions. As a result, theyassess the standard methods of electron-hole diagramsas too complicated [30], and subsequently resort to ex-citon diagrams and the so-called commutation technique,where the composite nature of the excitons is treatedwith care. However, the interaction of excitons with aFermi sea is only treated at a perturbative level, assum-ing that the interaction is small due to, e.g., spatial sep-aration [33]. This is not admissible in our model, wherethe interaction of the VB hole with all relevant electrons(photoexcited and Fermi sea) has to be treated on thesame footing. Rather, we stick to the simplified form ofcontact interaction, and show how one can use the frame-work of standard electron-hole diagrams to calculate allquantities of interest for infinite as well as for finite VBmass. The results presented below then suggest that for µ (cid:28) E B the finite mass does not weaken, but ratherstrengthens the singularities, which is in line with resultson the heavy hole found in [23].Here we only present the most important physical in-gredients for our approach, and defer the more technicaldetails to Appendix B. In the regime of interest, we canperform a low-density computation, employing the smallparameter µ/E B . Since all energies are close to E B , theleading-order exciton self-energy diagrams is then thesum of all diagrams with one CB electron loop. Onecan distinguish two channels: direct and exchange, to bedenoted by D and X , as depicted in Fig. 11. All suchdiagrams with an arbitrary number of interactions con-necting the VB line with the CB lines in arbitrary orderhave to be summed. Factoring out E B ρ/g · G ( ω ) ,the remaining factor can be identified as the exciton self-energy diagram. (a) (b) Figure 11. Leading-order direct self-energy diagrams: (a) dir-ect contribution D and (b) exchange contribution X . An evaluation of these diagrams is possible either inthe time or in the frequency domain. Of course, both ap-proaches must give same result. In practice, however, the time domain evaluation is more instructive and requiresless approximations, which is why we will discuss it first.The frequency domain evaluation, however, is far moreconvenient for obtaining finite mass results, and will bediscussed thereafter.The time domain approach is similar in spirit to theclassical one-body solution of the Fermi-edge problem byNozi`eres and de Dominicis [10]. Since the infinite-masshole propagator is trivial, G v ( t ) = iθ ( − t ) e iE G t , the directdiagrams just describe the independent propagation oftwo electrons in the time-dependent hole potential. Thus,in the time domain the sum of all direct diagrams D ( t )factorizes into two parts representing the propagation ofthese two electrons: D ( t ) = (cid:90) k
0. It is instructive to study the pole-pole combin-ation, which corresponds to a would be “trion” (boundstate of the exciton and an additional electron) and isfurther discussed in Appendix C. Adding to it the pole-continuum contributions we find, for small ω : D ( ω ) = ρE B g ω + i + ) Σ Dexc ( ω ) . (32)This corresponds to a contribution to the exciton self-energy which reads:Σ Dexc ( ω ) = − ρ (cid:90) k To perform the evaluation, we make use of the fol-lowing simplification: To begin with, we often encountercomplicated logarithmic integrals; however, the imagin-ary part of the integrand is just a delta function, so, uponintegration, one finds step functions. Since the integrandis retarded, it is then possible to recover the full expres-sion from the imaginary part using the Kramers-Kronigrelation; the step functions then become logarithms.With that, the sum over diagrams appearing in Fig. 12assumes the form D ( ω ) = E B g ω + i + ) (cid:90) k We are now in a position to tackle finite VB mass M . Let us also consider a finite incoming momentum Q . Clearly, the one-loop criterion for choosing diagramsstill holds, since we are still considering the low-densitylimit, µ (cid:28) E B . We also disregard any exchange contri-butions for the same reasons as for the infinite mass case.As a result, we only have to recompute the series of directdiagrams of Fig 12. We start with the first one. It gives: I = − E B V g (cid:90) k >k F d k (2 π ) − ω + E B + E ( k − Q ) + (cid:15) k − µ − i + ) (cid:16) − E B + ω − ( Q − q ) / M exc − (cid:15) k + (cid:15) k + i + − E B (cid:17) , (46)where q = k − k . The imaginary part of (46) reads:Im[ I ] = − V g (cid:90) k >k F d k (2 π ) πδ (cid:18) ω − ( Q − q ) M exc − (cid:15) k + (cid:15) k (cid:19) + O (cid:18) µE B (cid:19) . (47)By Eq. (47), I can be rewritten in a simpler form (ensur-ing retardation), valid for small ω : I (cid:39) V g (cid:90) k >k F d k (2 π ) ω − ( Q − q ) M exc − (cid:15) k + (cid:15) k + i + . (48)This form can be integrated with logarithmic accuracy,which, however, only gives Re[ I ]. Specializing to Q (cid:28) k F for simplicity, one obtains:Re[ I ] (cid:39) ln (cid:18) max( | ω + (cid:15) k − µ | , βµ ) E B (cid:19) . (49)As for the infinite mass case, the higher order diagramsof Fig. 12 give higher powers of I . Similarly to Eq. (44),one then obtains for the self-energy part, to leading log-arithmic accuracy:Σ exc ( Q , ω ) = − (cid:90) k This phase space reduction also affects the excitonspectral function, and hence the absorption: We first re-strict ourselves to the leading behavior, i.e., we disreg-ard any small renormalizations that arise from includ-ing Re[Σ exc ] or from higher-loop corrections. InsertingEq. (53) into Eq. (27) we then obtain, for small energies ω : A ( Q = 0 , ω ) (cid:39) − ∆ Im[Σ( ω )] ω ∼ ∆ α θ ( ω ) √ βµ · ω , (54)with ∆ = d ρE B g . (55)The factor ∆ (with units of energy) determines the po-lariton splitting at zero detuning, and will be discussedin Sec. VII. The 1 / √ ω divergence seen in (54) was alsofound by Rosch and Kopp using a path-integral approach[23] for a related problem, that of a heavy hole propagat-ing in a Fermi sea. In addition, Rosch and Kopp find aquasi particle delta peak with a finite weight. This peakcan also be recovered within our approach upon inclusionof the correct form of Re[Σ exc ]. From Eqs. (49) and (50)we may infer it to beRe[Σ exc ( Q = 0 , ω )] = α ω ln (cid:32) (cid:112) ω + ( βµ ) µ (cid:33) , (56)where we have rewritten the maximum-function with log-arithmic accuracy using a square root. This cut-off oflogarithmic singularities (which are responsible for edgepower laws) by recoil effects is a generic feature of ourmodel, and will reoccur in the regime of µ (cid:29) E B presen-ted in Sec. VI. In qualitative terms, this is also discussedin Ref. [22] (for arbitrary dimensions). Our results are infull agreement with this work.We may now deduce the full photon self-energy Π exc as follows: In the full finite-mass version of the power law(28), the real part of the logarithm in the exponent willbe replaced by the cut-off logarithm from Eq. (56). The3imaginary part of this logarithm will be some function f ( ω ) which continuously interpolates between the finite-mass regime for ω (cid:28) βµ [given by Eq. (53) times ω − ],and the infinite mass regime for ω (cid:29) βµ . Therefore, wearrive atΠ exc ( Q = 0 , ω ) = (57)∆ ω + i + exp (cid:34) α (cid:32) ln (cid:32) (cid:112) ω + ( βµ ) µ (cid:33) − if ( ω ) (cid:33)(cid:35) , where f ( ω ) = (cid:40) π (cid:113) ωβµ θ ( ω ) ω (cid:28) βµπ ω (cid:29) βµ. (58)It is seen by direct inspection that (57) has a delta peakat ω = 0 with weight ∆ β α .One can also asses the weight of the delta peak bycomparing the spectral weights of the exciton spectralfunction in the infinite and finite mass cases: The weightof the delta peak must correspond to the difference inspectral weight as the absorption frequency power law ischanged once β becomes finite. In the infinite mass case,the absorption scales as A ∞ ( ω ) ∼ ∆ α ω (cid:18) ωµ (cid:19) α θ ( ω ) , (59)as follows from Eq. (28) above. Thus, the spectral weightin the relevant energy region is given by (cid:90) βµ dωA ∞ ( ω ) = ∆ β α . (60)In contrast, using Eq. (53), the spectral weight of thefinite mass case is (cid:90) βµ dωA ( Q = 0 , ω ) = ∆ α . (61)For scattering phase shifts δ close to π (i.e., α → β > 0, a pole with weight proportional to β α [Eq. (60)] at ω = 0 should be present in the spectrum,if β is not exponentially small in α . This weight is exactlythe same as for the heavy hole when computed in a secondorder cumulant expansion [23].The full imaginary part of Π exc ( Q = 0 , ω ) was alreadygiven explicitly in Eqs. (5a) and (5b), and plotted inFig. 2. That plot illustrates the main conclusion of thissection: For finite mass, Fermi sea excitations with largemomentum transfer are energetically unfavorable, andare therefore absent from the absorption power law. Asa result, the pole-like features of the absorption are re-covered. C. Validity of the electron-hole correlator as aphoton self-energy Let us now assess the validity of the expressions forthe CB electron-VB hole correlator [Eqs. (28) and (57)] as a photon self-energy. Using them, one assumes thatonly electron-hole interactions within one bubble are ofrelevance, and electron-hole interactions connecting twobubbles (an example is shown in Fig. 14) can be disreg-arded. V Figure 14. Two dressed bubbles, connected by one electron-hole interaction (dotted line). This is an example of a photonself-energy diagram that is not contained in our approxima-tion for Π( Q , ω ). The regime where such an approximation is valid maybe inferred from the following physical argument: Elec-tronic processes (i.e. electron-hole interactions) happenon the time scale of Fermi time 1 /µ . On the other hand,the time scale for the emission and reabsorption of aphoton (which is the process separating two bubbles) isgiven by 1 /ρd (where d is the dipole matrix element).If the second scale is much larger than the first one, elec-trons and holes in distinct bubbles do not interact. Thus,the our approach is valid as long as ρd (cid:28) µ. (62)Under this condition, the following physical picture is ap-plicable: an exciton interacts with the Fermi sea, givingrise to a broadened exciton, which in turn couples to thecavity photons. When Eq. (62) is violated, one shouldthink in different terms: excitons couple to photons, lead-ing to exciton-polaritons. These then interact with theFermi sea. The second scenario is, however, beyond thescope of this paper.The above discussion is likewise valid for the regime oflarge Fermi energy, which is studied below. VI. ELECTRON-HOLE CORRELATOR FORLARGE FERMI ENERGY We now switch to the opposite regime, where µ (cid:29) E B ,and excitons are not well-defined. For simplicity, wealso assume that µ is of the order of the CB bandwidth.Hence, E B (cid:28) µ (cid:39) ξ . Within our simplified model, thefinite mass problem in 3D was solved in [18]. This treat-ment can be straightforwardly carried over to 2D [42].To avoid technicalities, we will, however, just show howto obtain the 2D results in a “Mahan guess” approach [7],matching known results from [21]. To this end, we willfirst recapitulate the main ingredients of the infinite masssolution.4 A. Infinite hole mass The FES builds up at the Burstein-Moss shiftedthreshold Ω FES T = E G + µ . Its diagrammatic derivationrelies on a weak-coupling ansatz: The parameter g = ρV is assumed to be small. As seen from Eq. (24), this is in-deed true for µ (cid:29) E . In principle, below the FES therewill still be the exciton peak; however, this peak will bebroadened into a weak power law, and thus merge withthe FES. For finite mass (see below), the position of thewould-be exciton may even be inside FES continuum,which makes the exciton disappear completely. What ismore, the exciton weight, being proportional to E B , isexponentially small in g (since µ (cid:39) ξ ). We may thereforesafely disregard the exciton altogether (see also discus-sion in Appendix A).To leading order in g ln( ω/µ ), the dominant contribu-tion comes from the so called “parquet” diagrams, con-taining all possible combinations of ladder and crosseddiagrams [8, 9]. The value of the pure ladder diagrams isgiven by Eq. (18), with Ω − E G replaced by ω = Ω − Ω FES T .The lowest-order crossed diagram is shown in Fig. 15.With logarithmic accuracy the contribution of this dia-gram is easily computed:Π crossed = − d ρg [ln( ω/µ )] . (63)This is − / Figure 15. Lowest order crossed diagram contributing to theFES. In his original paper Mahan computed all leading dia-grams to third order and guessed the full series from anexponential ansatz [7]. The corresponding result for thephoton self-energy Π FES ( ω ) readsΠ FES ( ω ) = d ρ g (cid:18) − exp (cid:20) − g ln (cid:18) ω + i + − µ (cid:19)(cid:21)(cid:19) . (64)Relying on coupled Bethe-Salpeter equations in thetwo channels (ladder and crossed), Nozi`eres et al. thensummed all parquet diagrams, where a bare vertexis replaced by (anti-)parallel bubbles any number of times [8, 9]. The result corresponds exactly to Mahan’sconjecture, Eq. (64).By the standard FES identification δ/π = g + O ( g ),the power law in Eq. (64) coincides with the one given inEq. (28); the phase shift is now small. One should alsopoint out that the peaks in the spectra in the regimesof small µ (Fig. 2) and large µ (Fig. 3) are not continu-ously connected, since the FES arises from the continuousthreshold, whereas the exciton does not.Let us finally note that since µ is a large scale, Eq. (64)should be a good approximation for the photon self-energy, since the condition (62) is easily satisfied. B. Finite hole mass As in the regime of the exciton, in the finite masscase the result (64) will be modified due to the recoilenergy βµ . However, it will now be the VB hole recoil(or the hole lifetime, see below) instead of the exciton re-coil — the latter is meaningless since the exciton is not awell defined entity anymore. This is most crucial: SinceCB states with momenta smaller than k F are occupied,VB holes created by the absorption of zero-momentumphotons must have momenta larger than k F . Therefore,the hole energy can actually be lowered by scatteringswith the Fermi sea that change the hole momenta to somesmaller value, and these scattering processes will cut offthe sharp features of Π FES ( ω ). The actual computationof the photon self-energy with zero photon momentum,Π FES ( Q = 0 , ω ), proceeds in complete analogy to the3D treatment of [18]. Limiting ourselves to the “Mahanguess” for simplicity, the main steps are as follows.The first major modification is the appearance of twothresholds: As easily seen by the calculation of the ladderdiagrams, the finite mass entails a shift of the pole of thelogarithm from ω = 0 to ω = βµ , which is the minimalenergy for direct transitions obeying the Pauli principle.Correspondingly, ω D = βµ is called the direct threshold.Near this threshold, logarithmic terms can be large, anda non-perturbative resummation of diagrams is required.However, the true onset of 2DEG absorption will actuallybe the indirect threshold ω I = 0. There, the valence bandhole will have zero momentum, which is compensatedby a low-energy conduction electron-hole pair, whose netmomentum is − k F . The two thresholds were shown inFig. 4. It should be noted that for E B < βµ the excitonenergy ≈ ω D − E B , is between ω I and ω D . Hence, inthis case the exciton overlaps with the continuum and iscompletely lost.Near ω I , the problem is completely perturbative. Inleading (quadratic) order in g , the absorption is determ-ined by two diagrams only. The first one is the crosseddiagram of Fig. 15. The second one is shown in Fig. 16.When summing these two diagrams, one should take intoaccount spin, which will simply multiply the diagram ofFig. 16 by a factor of two (if the spin is disregarded, thediagrams will cancel in leading order). Up to prefactors5of order one, the phase-space restrictions then result in a2DEG absorption (see [21] and Appendix E): A ( Q = 0 , ω ) = d g (cid:18) ωβµ (cid:19) θ ( ω ) . (65)The phase space power law ω is specific to 2D . Its 3Dcounterpart has a larger exponent, ω / [21], due to anadditional restriction of an angular integration. Figure 16. (Color online) Second diagram (in additionto Fig. 15) contributing to the absorption at the indirectthreshold ω I . The blue ellipse marks the VB self-energy in-sertion used below. Let us now turn to the vicinity of ω D , where one hasto take into account the logarithmic singularities and thefinite hole life-time in a consistent fashion. Regardingthe latter, one can dress all VB lines with self-energydiagrams as shown in Fig. 16. The self-energy insertionat the dominant momentum k = k F readsIm[Σ VB ( k F , ω )] = 1 √ θ ( ω ) g βµ ω ( βµ ) , ω (cid:28) βµ. (66)As can be shown by numerical integration, this expressionreproduces the correct order of magnitude for ω = βµ ,such that it can be safely used in the entire interestingregime ω ∈ [0 , βµ ]. The power law in Eq. (66) is againspecific to 2D. In contrast, the order of magnitude of theinverse lifetime is universal,Im[Σ VB ( k F , βµ )] ∼ g βµ. (67)Disregarding the pole shift arising from Re[Σ], the self-energy (67) can be used to compute the “dressed bubble”shown in Fig. 17. With logarithmic accuracy, the dressedbubble can be evaluated analytically. In particular, itsreal part reads:Re [Π db ] ( ω ) (cid:39) ρd ln (cid:113) ( ω − βµ ) + ( g βµ ) µ . (68)This is just a logarithm whose low-energy divergence iscut by the VB hole life time, in full analogy to Eq. (56),and in agreement with Ref. [22]. Figure 17. The CB electron-VB hole bubble, with the holepropagator dressed by the self-energy, Eq. (67). For the computation of polariton spectra later on, itturns out to be more practical to obtain both the real andthe imaginary parts of Π db ( ω ) by numerically integratingthe approximate form [42]:Π db ( ω ) (cid:39) (69) d (2 π ) (cid:90) k>k F d k ω − ( (cid:15) k − µ ) − k M + i Im[ ˜Σ VB ( ω − (cid:15) k + µ )] , Im[ ˜Σ VB ( x )] = (cid:40) g √ θ ( x ) x ( βµ ) x < βµ g √ βµ x > βµ, to avoid unphysical spikes arising from the leadinglogarithmic approximation. A corresponding plot of − Im [Π db ] is shown in Fig. 18. The numerical expres-sion − Im [Π db ] simplifies to the correct power law (65) inthe limit ω → 0, and approaches the infinite mass value d ρπ for large frequencies.Higher-order diagrams will contain higher powers ofthe rounded logarithm (68). The parameter controllingthe leading log scheme now reads l ≡ g ln( βg ) . (70)One can distinguish different regimes of l . The simplestis l (cid:28) 1, which holds in the limit g → β is not exponentially small in g ). In this limit,no singularity is left. The large value of the Fermi energy(small g ) and the large value of the hole decay βµ havecompletely overcome all interaction-induced excitonic ef-fects. A decent approximation to the 2-DEG absorptionis then already given by the imaginary part of the dressedbubble. Fig. 18 shows the corresponding absorption.6 − Im [Π db ( ω )] /d ρπ ωβµ g = 0 . g = 0 . ∼ ω Figure 18. (Color online) Imaginary part of the dressedbubble for two values of g , obtained from numerical integ-ration of Π db , using the hole self-energy insertion of (66). The more interesting regime corresponds to g ln( βg ) (cid:38) 1, where arbitrary numbers of conduc-tion band excitations contribute to the absorption alike[43]. A non-perturbative summation is needed, whichis, however, obstructed by the following fact: As foundby straightforward computation, the crossed diagramsare not only cut by g βµ due to the hole decay, butalso acquire an inherent cutoff of order βµ due to thehole recoil. A standard parquet summation is onlypossible in a regime where these two cutoffs cannotbe distinguished with logarithmic accuracy, i.e. where β (cid:28) g . For small enough g this will, however, alwaysbe the case in the truly non-perturbative regime where β must be exponentially small in g .As a result of these considerations, the logarithms ofthe parquet summation have to be replaced by the cut-offlogarithms (68), with g βµ replaced by βµ . The imagin-ary part of the logarithm is then given by the functionplotted in Fig. 18. The resulting full photon self-energyin the non-perturbative FES regime reads:Π FES ( Q = 0 , ω ) (cid:39) − d ρ g (cid:18) exp (cid:20) − g (cid:18) Π db ( ω ) ρd (cid:19)(cid:21) − (cid:19) . (71)A sketch of Im [Π FES ] is shown in Fig. 3. VII. POLARITON PROPERTIES When the cavity energy ω c is tuned into resonancewith the excitonic 2DEG transitions, the matter and lightmodes hybridize, resulting in two polariton branches. Wewill now explore their properties in the different regimes. A. Empty conduction band To gain some intuition, it is first useful to recapitulatethe properties of the exciton-polariton in the absence ofa Fermi sea. Its (exact) Green’s function is given byEq. (13), with ω Q=0 = ω c and Π( ω ) = ∆ / ( ω + i + ),where ∆ is a constant (with units of energy) which de-termines the polariton splitting at zero detuning. Interms of our exciton model, one has ∆ = (cid:112) d ρE B /g . ω is measured from the exciton pole. A typical densityplot of the polariton spectrum A p = − Im (cid:2) D R ( ω, ω c ) (cid:3) /π ,corresponding to optical (absorption) measurements ase.g. found in [13], is shown in Fig. 19. A finite cavityphoton linewidth Γ c = ∆ is used. The physical picture istransparent: the bare excitonic mode (corresponding tothe vertical line) and the bare photonic mode repell eachother, resulting in a symmetric avoided crossing of twopolariton modes.For analytical evaluations, it is more transparent toconsider an infinitesimal cavity linewidth Γ c . The lowerand upper polaritons will then appear as delta peaks inthe polariton spectral function, at positions ω ± = 12 (cid:16) ω c ± (cid:112) ω c + 4∆ (cid:17) , (72)and with weights W ± = 11 + ( ω c ± √ + ω c ) . (73)We note that the maximum of the polariton spectra scalesas 1 / Γ c for finite Γ c . Our spectral functions are normal-ized such that the total weight is unity. From Eq. (73) itis seen that the weight of the “excitonic” polaritons (cor-responding to the narrow branches of Fig. 19) decays as∆ /ω c for large absolute values of ω c . ω/ ∆ ω c / ∆ A p · ∆ π Figure 19. (Color online) µ = 0: Exciton-polariton spectrumas function of cavity detuning ω c and energy ω , measured inunits of the half polariton splitting ∆, with Γ c = ∆. B. Large Fermi energy Let us study polariton properties in the presence of aFermi sea. Reverting the order of presentation previouslytaken in the paper, we first turn to the regime of largeFermi energy, E B (cid:28) µ . This is because for E B (cid:28) µ theinequality ρd (cid:28) µ (62) is more easily satisfied than inthe opposite limit of E B (cid:29) µ , facilitating experimentalrealization. We compute the polariton properties usingthe electron-hole correlators as cavity photon self-energy.A similar approach was applied recently by Averkiev andGlazov [15], who computed cavity transmission coeffi-cients semiclassically, phenomenologically absorbing theeffect of the Fermi-edge singularity into the dipole matrixelement. Two further recent treatments of polaritons fornonvanishing Fermi energies are found in [16] and [17].In the first numerical paper [16], the Fermi-edge singu-larity as well as the excitonic bound state are accountedfor, computing the electron-hole correlator as in [11], butan infinite mass is assumed. The second paper [17] isconcerned with finite mass. However, the authors onlyuse the ladder approximation and neglect the crosseddiagrams, partially disregarding the physical ingredientsresponsible for the appearance of the Fermi-edge powerlaws. We aim here to bridge these gaps and describethe complete picture in the regime of large Fermi energy(before turning to the opposite regime of µ (cid:28) E B ).In the infinite mass limit we will use Eq. (64) as thephoton self-energy. It is helpful to explicitly write downthe real and imaginary parts of the self-energy in leadingorder in g :Re [Π FES ] ( ω ) = ˜∆ (cid:32) − (cid:18) | ω | µ (cid:19) − g (cid:33) , (74)Im [Π FES ] ( ω ) = − ˜∆ · πg (cid:18) ωµ (cid:19) − g θ ( ω ) (75)˜∆ ≡ d ρ g , (76)where we have introduced the parameter ˜∆, which de-termines the splitting of the polaritons, playing a similarrole to ∆ in the previous case of empty CB. In the fol-lowing, ˜∆ will serve as the unit of energy.For a cavity linewidth Γ c = 1 ˜∆, a typical spectral plotof the corresponding ”Fermi-edge polaritons” is shown inFig. 20. It is qualitatively similar to the results of [15].A quantitative comparison to the empty CB case is ob-viously not meaningful due to the appearance of the ad-ditional parameters µ (units of energy) and g (dimen-sionless). Qualitatively, one may say the following: Thelower polariton is still a well-defined spectral feature. Forzero cavity linewidth (see below), its lifetime is infinite.The upper polariton, however, is sensitive to the high-energy tail of the 2DEG absorption power law (75), andcan decay into the continuum of CB particle-hole excita-tions. Its linewidth is therefore strongly broadened. Onlywhen the 2DEG absorption is cut off by finite bandwidth effects (i.e., away from the Fermi-edge), a photonic-likemode reappears in the spectrum (seen in the upper rightcorner of Fig. 20). A p · ˜∆ πω/ ˜∆ ω c / ˜∆ Figure 20. (Color online) µ (cid:29) E B : Infinite hole mass Fermi-edge-polariton spectrum A p ( ω, ω c ) as function of cavity de-tuning ω c and energy ω , measured in units of the effectivesplitting ˜∆. It was obtained by inserting Eqs. (74) and (75)into Eq. (14). Parameter values: µ = 30 ˜∆, Γ c = 1 ˜∆, and g = 0 . For more detailed statements, one can again considerthe case of vanishing cavity linewidth Γ c . A spectral plotwith the same parameters as in Fig. 20, but with smallcavity linewidth, Γ c = 0 . 01 ˜∆, is shown in Fig. 21(a). ω/ ˜∆(a) (b) A p ˜∆ πω c / ˜∆ (a) (b) Figure 21. (Color online) µ (cid:29) E B : (a) Fermi-edge-polaritonspectrum with the same parameters as in Fig. 20, but Γ c =0 . 01 ˜∆. The white dashed lines denote the location of the spec-tral cuts presented in Fig. 22. (b) Spectrum with a nonzeromass-ratio β = 0 . 2, and otherwise the same parameters asin (a). This plot was obtained by inserting the finite massphoton self-energy of Eq. (71) into Eq. (14), with ω c replacedby ω c + βµ to make sure that the cavity detuning is meas-ured from the pole of the photon self-energy. Note that thefrequency range of panel (b) is shifted as compared to (a). We first examine the lower polariton (assuming zerolinewidth), which is a pure delta peak. Its position is8 A p ( ω ) · ˜∆ π ω/ ˜∆ infinite massfinite mass, β = 0 . ω c = − ω c = 0 ω c = 8 ˜∆(a) (b) (c) A p ( ω ) · ˜∆ π A p ( ω ) · ˜∆ π ω/ ˜∆ ω/ ˜∆ ω Figure 22. (Color online) µ (cid:29) E B : Spectral cuts at fixed cavity detuning through the polariton spectra of Fig. 21, for bothinfinite (continuous blue lines) and finite (dashed orange lines) hole mass. (a) Large negative cavity detuning. The dottedvertical line line always indicates the position of the direct threshold at ω = βµ . The inset is a zoom-in on the absorption onsetat the indirect threshold. (b) Zero cavity detuning. (c) Large positive cavity detuning. determined by the requirement ω − ω c − Re [Π FES ( ω )] = 0 . (77)One may study the solution of this equation in three dis-tinct regimes, corresponding to ω c → −∞ , ω c = 0, and ω c → + ∞ .For ω c → −∞ , the solution of Eq. (77) approaches ω = ω c , and the lower polariton acquires the full spectralweight (unity): For strong negative cavity detunings, thebare cavity mode is probed. The corresponding spectralcut is shown in Fig. 22(a) (continuous line). We will re-frain from making detailed statements about the way thebare cavity mode is approached, since this would requirethe knowledge of the photon self-energy at frequenciesfar away from the threshold. As the cavity detuning isdecreased, the lower polariton gets more matter-like. Atzero detuning [see Fig. 22(b)], and for g not too small(w.r.t. g ˜∆ /µ ), the weight of the lower polariton is ap-proximately given by 1 / (1+2 g ). For large positive cavitydetunings [see Fig. 22(c)], the position of the matter-likelower polariton approaches ω = 0, ω ∼ − ω − / (2 g ) c as ω c → ∞ . (78)The lower polariton weight also scales in a power lawfashion, ∼ ω − − / (2 g ) c , distinct from the excitonic regime,where the weight falls off quadratically [Eq. (73)].Due to the finite imaginary part of the self-energyΠ FES ( ω ), the upper polariton is much broader than thelower one: the photonic mode can decay into the con-tinuum of matter excitations. At large negative detun-ings [see the inset to Fig. 22(a)], the upper polariton hasa power law like shape (with the same exponent as theFermi-edge singularity), and for ω c → −∞ its maximumapproaches ω = 0 from the high-energy side. As the de-tuning is increased (made less negative), the maximumshifts away from ω = 0, approaching the free cavity modefrequency ω = ω c for ω c → ∞ . Since the weight and height are determined by the value of Im[Π FES ] at themaximum, they increase correspondingly.Let us now consider the case of finite mass. Using thefinite mass photon self-energy (64) instead of (71), theFermi-edge-polariton spectrum with a nonzero mass-ratioof β = 0 . ω = 0 in the infinite mass case to ω = βµ in thefinite mass case, reflecting the Burstein-Moss shift in the2DEG absorption. (ii) As opposed to the infinite masscase, the lower polariton is strongly broadened at largepositive detunings.These points are borne out more clearly in Fig. 22(a)–(c) (dashed lines), which presents cuts through Fig. 21(b)at fixed detuning. The situation at large negative de-tuning is shown in Fig. 22(a): Compared to the infinitemass case, shown as full line, the polaritons are shiftedtowards higher energies. In addition, the shape of the up-per polariton is slightly modified — its onset reflects theconvergent phase-space power law ω of Eq. (65) foundfor the 2DEG absorption. This is emphasized in the in-set. At zero cavity detuning [Fig. 22(b)], the situationof the finite and infinite mass cases is qualitatively sim-ilar. When the cavity detuning is further increased, theposition of the pole-like lower polariton approaches thedirect threshold at ω = βµ (indicated by the vertical dot-ted line). When the pole is in the energy interval [0 , βµ ],the lower polariton overlaps with the 2DEG continuumabsorption, and is therefore broadened. This is clearlyseen in Fig. 22(c): Instead of a sharp feature, there isjust a small remainder of the lower polariton at ω = βµ .As a result, one may say that in the regime of the Fermi-edge singularity, i.e., large µ , the finite mass will cut offthe excitonic features from the polariton spectrum – in-stead of the avoided crossing of Fig. 19, Fig. 21(b) exhib-its an almost photonic-like spectrum, with a small (cav-ity) linewidth below the threshold at ω = βµ , and a lar-9ger linewidth above the threshold, reflecting the step-like2DEG absorption spectrum of Fig. 3. The finite massthus leads to a general decrease of the mode splittingbetween the two polariton branches. This trend contin-ues when the Fermi energy is increased further.It is instructive to compare this behavior with the ex-perimental results reported in [13]. There, two differ-ential reflectivity measurements were conducted, whichcan be qualitatively identified with the polariton spec-tra. The first measurement was carried out using a low-mobility GaAs sample (which should behave similarly tothe limit of large VB hole mass), and moderate Fermienergies. A clear avoided crossing was seen, with the up-per polariton having a much larger linewidth than thelower one (see Fig. 2(A) of [13]). In the second meas-urement, the Fermi energy was increased further, and ahigh-mobility sample was studied, corresponding to fi-nite mass. A substantial reduction of the mode splittingbetween the polaritons was observed (Fig. 2(C) of [13]).While a detailed comparison to the experiment of [13]is challenging, due to the approximations we made andthe incongruence of the parameter regimes (in the exper-iment one has µ (cid:39) E B ), the general trend of reducedmode splitting is correctly accounted for by our theory. C. Small Fermi energy We now switch to the regime of of small Fermi en-ergy discussed in Sec. V, a regime in which the polaritonspectra have not been studied analytically before. Weagain assume that the condition (62), required for theapproximating the photon self-energy by Eq. (15), is ful-filled. This may be appropriate for systems with a largeexciton-binding energies, e.g., transition metal dichalco-genide monolayers as recently studied in [14].For infinite mass, we may use Eq. (28) as photon self-energy, multiplied by a prefactor ∆ = d ρE B /g [cf.Eq. (55)], and expand the real and imaginary parts toleading order in α = ( δ/π − . The energy ω isnow measured from the exciton pole: ω = Ω − Ω exc T ,Ω exc T = E G + µ − E B . The corresponding polariton spec-trum for a small cavity linewidth is shown in Fig. 23(a).Qualitatively, it strongly resembles the bare exciton caseas in Fig. 19 (note that in Fig. 23 the cavity linewidthwas chosen to be 100 times smaller than in Fig. 19), butwith a larger linewidth of the upper polariton. This isdue to the possible polariton decay into the particle holecontinuum contained in the excitonic power law, Eq. (28). ω/ ∆(a) (b) A p · ∆ πω c / ∆ (a) (b) Figure 23. (Color online) µ (cid:28) E B : Exciton-polariton spec-trum for small Fermi energy. The white dashed lines denotethe location spectral cuts presented in Fig. 24. (a) Infin-ite mass. This plot was obtained by inserting the ExcitonGreen’s function for µ (cid:38) 0, given by Eq. (28) multiplied by∆ = d ρE B /g , into the photon Green’s function, Eq. (14).Parameters: µ = 10∆, Γ c = 0 . α = ( δ/π − = 0 . β = 0 . 4. In this plot, thefinite mass Exciton Green’s function, Eq. (57), was used, withthe same parameters as in (a). The detailed discussion of polariton properties in theregime of µ (cid:28) E B parallels the previous discussion inthe regime E B (cid:28) µ . For small negative detuning ω c [Fig. 24 (a)], the lower polariton is found at approxim-ately ω = ω c . The upper polariton has a significantlysmaller weight, its shape reflects the excitonic power lawof Eq. (28). However, compared to the previous spectralcuts (Fig. 22) the upper polariton peak is much morepronounced. This results from the exciton being nowpole-like, as compared to the power law Fermi-edge sin-gularity. Increasing the detuning, weight is shifted tothe upper polariton. At zero detuning [Fig. 24(b)], theweight of the lower polariton is only order O (cid:0) α (cid:1) largerthan the weight of the upper polariton. At large positivedetuning, the position of the lower polariton is found atapproximately ω ∼ − ω − / (1 − α ) c as ω c → ∞ . (79)The lower polariton thus approaches the exciton linefaster than in the pure exciton case, but slower than inthe Fermi-edge regime [Eq. (78)]. A similar statementholds for the weight of the lower polariton, which scalesas ω − − α c .The spectrum in the finite mass case is qualitativelysimilar, see Fig. 23(b). Quantitatively, a stronger peakrepulsion can be seen, which may be attributed to the en-hanced excitonic quasiparticle weight in the finite masscase. A comparison of spectral cuts in the finite masscase [Fig. 24(a)–(c)] further corroborates this statement[especially in Fig. 24(c)]. Indeed, one finds that the po-sition of the lower polariton at large cavity detuning isapproximately given by ω ∼ − β α · ω − c as ω c → ∞ , (80)i.e., the excitonic line at ω = 0 is approached more slowlythan in the infinite mass case, Eq. (79). The correspond-ing weight falls off as ω − c . Thus, the lower polariton0 A p ( ω ) · ∆ π ω/ ∆ infinite massfinite mass, β = 0 . ω c = − ω c = 0 ω c = 4∆(a) (b) (c) A p ( ω ) · ∆ π A p ( ω ) · ∆ π ω/ ∆ ω/ ∆ Figure 24. (Color online) µ (cid:28) E B : Spectral cuts at fixed cavity detuning through the polariton spectra of Fig. 23, for bothinfinite (continuous blue lines) and finite hole mass (dashed orange lines). (a) Large negative cavity detuning. The inset showsa zoom onto the upper polaritons. (b) Zero cavity detuning. (c) Large positive cavity detuning. has a slightly enhanced weight compared to the infinitemass case. In addition, in the spectral cut at large neg-ative detuning, [inset to Fig. 24(a)], the upper polaritonappears as a sharper peak compared to the infinite masscase, which again results from the enhanced quasi particleweight of the finite mass case. VIII. CONCLUSION In this paper we have studied the exciton-polaritonspectra of a 2DEG in an optical cavity in the presenceof finite CB electron density. In particular, we have elu-cidated the effects of finite VB hole mass, distinguishingbetween two regimes. In the first regime (small Fermienergy as compared to the exciton binding energy), wehave found that excitonic features in the 2DEG absorp-tion are enhanced by the exciton recoil and the resultingsuppression of the Fermi edge singularity physics. In con-trast, in the second regime of Fermi energy larger thanthe exciton binding energy, it is the VB hole which re-coils at finite mass. This cuts off the excitonic features.These modifications also translate to polariton spectra,especially to the lower polariton at large cavity detuning,which is exciton-like. Our findings reproduce a trend seenin a recent experiment [13].We would like to mention several possible extensionsof this work. To begin with, it would be promising tostudy the effect of long-range interactions on the powerlaws, and hence on polariton spectra, from an analyt-ical perspective. Long-range interactions are expected tobe most important in the regime of small Fermi energy,leading to additional bound states and to the Sommer-feld enhancement effects [24]. Moreover, one should tryto explore trionic features, for which it is necessary to in-corporate the spin degree of freedom (to allow an electronto bind to an exciton despite the Pauli principle). An-other interesting direction would be to tackle the limit of equal electron and hole masses, which is relevant totransition metal dichalcogenides, whose polariton spec-tra in the presence of a Fermi sea where measured in arecent experiment [14]. Lastly, one should address thebehavior of the polariton in the regime of small Fermienergy and strong light-matter interactions. Then, notthe exciton, but rather the polariton interacts with theFermi sea, and different classes of diagrams have to beresummed to account for this change in physics. ACKNOWLEDGMENTS This work was initiated by discussions with A. Im-amo˘glu. The authors also acknowledge many helpfulcomments from F. Kugler, A. Rosch, D. Schimmel, andO. Yevtushenko. This work was supported by fundingfrom the German Israeli Foundation (GIF) through I-1259-303.10. D.P. was also supported by the German Ex-cellence Initiative via the Nanosystems Initiative Munich(NIM). M.G. received funding from the Israel ScienceFoundation (Grant 227/15), the US-Israel Binational Sci-ence Foundation (Grant 2014262), and the Israel Min-istry of Science and Technology (Contract 3-12419), whileL.G. was supported by NSF Grant DMR-1603243. Appendix A: Evolution of absorption spectra withincreasing chemical potential In this Appendix, we present an extended overviewof how the absorption spectra evolve inbetween the con-trolled extremal limits of µ (cid:28) E B and µ (cid:29) E B .For µ (cid:28) E B , the dominant spectral feature is the ex-citon. For finite mass ( β (cid:54) = 0), it has a coherent delta-likepart and an incoherent tail, see Eq. (5), while the infinitemass exciton ( β = 0) is a purely incoherent power law,see Eq. (2). These pronounced excitonic features are well1separated from the CB continuum part at Ω FES T = E G + µ (see inset to Fig. 2).As µ is increased, the incoherent exciton part [Eqs.(5b) and (2)] starts to overlap with the CB continuumpart. Moreover, the overall relative weight of both thecoherent and incoherent portions of the exciton part ofthe spectrum (which are both proportional to E B ) willdiminish. Still, within our simplified model which neg-lects CB electron-CB electron interactions, and for β = 0,this exciton feature will never disappear completely, sincein this model an infinite mass VB hole is simply a localattractive potential for the CB electrons, and such a po-tential will always have a bound state in 2D. However,for finite VB hole mass, the exciton energy (location ofthe coherent delta peak) will penetrate into the CB con-tinuum when µ becomes larger than E B /β (cid:29) E B (i.e.,when E B crosses the indirect threshold, see Fig. 4(a)).More importantly, CB electron-CB electron interactionswould screen the hole potential, and will thus reduce theexciton binding energy and presumably eliminate the ex-citon part of the spectrum completely as soon as µ (cid:29) E B .To describe this situation, it has been customary inthe literature [11, 21] to still employ the same simpli-fied model neglecting CB electron-CB electron interac-tions, but assume that the hole potential does not createa bound state for large enough µ , a practice we followin this work as well. Then, for µ (cid:29) E B , one shouldconcentrate on the remaining, CB continuum part of thespectrum, which will evolve into the Fermi-edge singu-larity (FES), cut off by the VB hole recoil energy for β (cid:54) = 0. A putative evolution of absorption spectra withincreasing µ is sketched in Fig. 25. A (Ω) ΩΩ FES T µ % µ (cid:28) E B µ (cid:29) E B Figure 25. (Color online): Putative evolution of absorptionspectra as µ is increased. The colored arrows represent delta-function peaks, their height corresponds to the relative weightof those peaks. The (hand-sketched) plots of this figure com-prise the effects of a (large) finite VB hole mass ( β (cid:54) = 0) andelectron-electron interactions, beyond what’s actually com-puted in this paper. For clarity, the shift of the spectra withincreasing µ is disregarded. For µ even larger than shown inthe sketch, the FES will reduce to a step-like feature again. Appendix B: Evaluation of the exciton self-energy inthe time-domain In this Appendix, we present the time-domain eval-uation of the exciton self-energy diagrams of Fig. 11.These diagrams contain one CB electron loop only, andtherefore yield the leading contribution when µ/E B issmall. We will start with the direct diagrams [Fig. 11(a)],and then turn to the exchange series [Fig. 11(b)]. 1. Direct diagrams First, we note that the bare Green’s functions in thetime domain read G (0) c ( k , t ) = − i ( θ ( t ) − n k ) e − i(cid:15) k t , (B1) G (0) v ( t ) = iθ ( − t ) e iE G t , (B2)with the zero temperature Fermi function n k = θ ( k F − k ).Using these, we will evaluate the series of direct diagramsof Fig. 11(a). The temporal structure of a generic direct2diagram is illustrated via the example of Fig. 26. T T T m m, n = 3 t t t n k q n +1 k m k q q q t Figure 26. A direct self-energy diagram in the time-domain.The Green’s function with an arrow indicates the CB electronpropagating backwards in time. To compute such a diagram, we make the following ob-servation: Since the VB propagator has no momentumdependence, all VB phase factors simply add up to give a total factor of e − iE G t . Then, the step functions inthe VB propagators enforce time ordering for the in-termediate time integrals. In the specific case shownin Fig. 26, 0 < T < t < T < t < T m < t n < t with m = n = 3 ( m and n count the number of in-teraction lines above and below the dashed VB line,respectively). However, there are also diagrams with m = n = 3, but with a different relative ordering of theinteraction lines. Summing over all those diagrams for m and n fixed, one needs to integrate over the time ranges0 < t < ... < t n < t ∩ < T < ... < T m < t . Thismeans that the time integration for the direct diagramssplits into a product of two functions, representing thepropagation of a Fermi sea electron (above the VB linein Fig. 26) and a photoexcited electron (below the VBline) in the time-dependent potential.We are now in the position to write down the full ex-pression for the sum of direct diagrams D to all ordersin the interaction, fixing the signs with Wick’s theorem: D ( t ) = − (cid:90) k 2. Exchange diagrams The computation of the exchange diagrams, thoughtechnically sligthly more involved, essentially proceedsalong the same lines. The general time-structure of anexchange diagram is illustrated in Fig. 27. k T T T m t t t n q q q n k k m k m +1 m = n = 30 t Figure 27. An exchange self-energy diagram in the time-domain. The Green’s function shown with an arrow indicatesthe CB electron propagating backwards in time. As for the direct diagrams, the VB propagators just en-force a time ordering. In addition, there is the condition t n > T . When this condition is violated, the diagramreduces to a ladder diagram, which must be excluded toavoid double counting. Taking this into account, the fullexpression for the sum of exchange diagrams reads:4 X ( t ) = ∞ (cid:88) m,n =1 ( − V ) m + n e − iE G t (cid:90) 11 + g ln (cid:16) ω − µ + i + − ξ (cid:17) − E G − ω + i + (cid:90) ∞−∞ dω π ( − g ) ln (cid:18) ω − µ + i + − ξ (cid:19) 11 + g ln (cid:16) ω − µ + i + − ξ (cid:17) − ω + Ω − E G + i + ω + ω − Ω + E G − (cid:15) k − i + . (B20)This expression can be evaluated as before, splitting itinto pole-pole, pole-branch and branch-branch contribu-tions using Eq. (B15). In complete analogy to the dir-ect diagrams, the imaginary part of the branch-branchcontribution can be shown not to contribute in the re-gime of interest to us, and we therefore disregard itcompletely. Straight-forwardly evaluating the pole-poleand pole-branch contributions, one ultimately arrives atEq. (34) in the main text. Appendix C: Trion contribution to the excitonself-energy diagrams The pole-pole contribution to the direct self-energy D ( ω ) [Eq. (30)] physically represents two electronstightly bound to the hole potential. Indeed, it assumesthe form: D pole-pole ( ω ) = (cid:90) k The exchange contribution to the exciton self-energy,Eq. (35), can be understood by the following considera-tions. The ground state energy of an N -particle systemin the presence of an attractive delta function potentialstrong enough to form a bound state is lower than the N -particle ground state energy of the system without thepotential by an amount∆ E = − E B − (1 − α ) µ, (D1)which is the sum of the bound state energy E B , and asecond term which arises from the rearrangement of theFermi sea, described by Fumi’s theorem [41] [recallingthat 1 − α = δ/π , cf. Eq. (40)]. We find that the ex-change diagrams give the contribution µ , while the term αµ stems from the direct diagrams [Eq. (38)]. To createsuch an attractive potential, one has to lift one electronfrom the VB to the CB, which costs E G + µ . In our treat-ment, the extra cost µ appearing here is contained in theshift of the pole of the ladder diagrams, Eq. (25). Thus,the minimal absorption energy predicted by our model is E G − E B + αµ ≈ E G − E B .At first sight this seems to contradict the experimentalresults (e.g., [14]), according to which the minimal ab-sorption energy is E G − E B + µ (or 2 µ for equal electron-hole masses). This is attributed to “phase-space fillingeffects”, or, in other words, the Burstein-Moss shift [26],which precisely correspond to the shift of the ladder pole,without the Fumi contribution. The reason for this dis-crepancy is that our model ignores the CB electron-CB electron interaction, which would render the exciton elec-trically neutral and suppress the Fumi shift. Thus, as alsopointed out in the literature on the X-ray edge problem,neglecting electron-electron interactions gives the rightpower law scalings of the spectra only, but not the cor-rect threshold energies.Another aspect of Eq. (35) is its lack of dependence onthe frequency ω . In other words, the Anderson orthogon-ality power law of the exciton Green’s function does notdepend on X ( ω ). This could have been anticipated byan argument based on Hopfield’s rule of thumb [44] andthe results of [11]. Consider the spinful case, and studythe absorption spectral function for, e.g., right-hand cir-cularly polarized light at the exciton threshold, creatinga spin down electron and a spin up hole. The spectrumshould have the form1 ω · ω (1 − δ ↓ /π ) +(1 − δ ↑ /π ) . (D2)For the spin down electrons, the exponent is (1 − δ ↓ /π ) rather than ( δ ↓ /π ) because of the Hopfield rule: oneelectron is lifted from the valence band to the conductionband. For the spin up electron, no electron is lifted. How-ever, the exciton is the secondary threshold in the spinfulcase (the primary one is the trion). As seen from [11], thespin up exponent should therefore also be as in Eq. (D2).Now, in the spinful case all direct diagrams will comewith a spin factor of 2, while the exchange diagrams willnot. However, we see that the exponent in (D2) is ex-actly 2 times the exponent the spinless case, Eq. (28),when recalling that δ ↑ = δ ↓ = δ for our spin-independentpotential. This shows that the exchange diagrams shouldindeed not contribute to Anderson orthogonality, at leastto leading order. Appendix E: Computation of phase-space integralsfor the particle-hole pair density of states To clarify the different role of the recoil in the exciton(section V B) and FES cases (section VI B), let us presentthe computation of two important phase space integrals. 1. Exciton recoil We start with the evaluation of the imaginary part ofthe exciton self-energy Im[Σ]( ω ) given in Eq. (52), focus-ing on zero exciton momentum. Im[Σ] reads:Im[Σ exc ] (cid:39) − πV ρg α (cid:90) k 0, and thesephase space restrictions pile up to give Im[Σ exc ] ∼ ω / .To perform the calculation in detail, we substitute x = k √ m , y = k √ m . Switching the integrals for convenience,we can rewrite (E1), to leading order in the mass ratio β , asIm[Σ exc ] = − α π (cid:90) x> √ µ d x (cid:90) y< √ µ d y (E2) δ (cid:0) ω − ( x − µ ) + ( y − µ ) − β ( x − y ) (cid:1) . First, it is obvious that (E2) is proportional to θ ( ω ), sinceall terms subtracted from ω in the delta function are pos- itive, hence there cannot be any cancellations. Second, itis clearly seen that x (cid:39) √ µ , y (cid:39) √ µ to yield a nonzerocontribution for small ω . Thus, we may linearize the dis-persion relation, starting with y : y = ( √ µ + γ y ) e y , (E3) y = µ + 2 √ µγ y + O ( γ y ) . (E4)In doing so, we effectively disregard subleading terms oforder O ( ω /µ ) in the argument of the delta function.Introducing the notation φ = (cid:93) ( x , y ) , c = cos( φ ) , (E5)we arrive at:Im[Σ exc ] = (E6) − α θ ( ω ) π (cid:90) x> √ µ d x (cid:90) − √ − c (cid:90) −√ µ dγ y ( √ µ + γ y ) δ (cid:18) ω − ( x − µ ) − βx + 2 βx √ µc − βµ (cid:124) (cid:123)(cid:122) (cid:125) = A + γ y (2 βxc − β √ µ + 2 √ µ ) (cid:124) (cid:123)(cid:122) (cid:125) = B (cid:19) . Since the only contribution comes from γ y close to theupper boundary, we can write √ µ + γ y (cid:39) √ µ . Using B (cid:39) √ µ , the trivial integral over γ y then results inIm[Σ exc ] = − α π (cid:90) x> √ µ d x (cid:90) − dc √ − c θ ( A ) . (E7)To find the leading power law in ω of this expression, weassume that ω (cid:28) βµ . Then, we rewrite θ ( A ) as θ ( = C (cid:122) (cid:125)(cid:124) (cid:123) ω − ( x − µ ) − βx − βµ +2 βx √ µc ) = θ ( c − ( − C/ βx √ µ )) . (E8)We now use x (cid:39) √ µ . Thus, we can write − C/ βx √ µ (cid:39) − (cid:18) ω βµ − x − µ βµ (cid:19) + O ( ω/µ ) . (E9)Going back to (E7) givesIm[Σ exc ] = (E10) − α θ ( ω ) π (cid:90) x> √ µ d x θ ( ω − ( x − µ )) (cid:90) − ( ω − ( x − µ )) / βµ dc √ − c . Using that for 0 < t < (cid:90) − t (cid:112) − y dy = arccos(1 − t ) = √ t + O ( t / ) , (E11) we obtainIm[Σ exc ] = − α θ ( ω ) (cid:90) √ µ + ω √ µ xdx (cid:115) ω − ( x − µ ) βµ . (E12)This can be integrated exactly to give:Im[Σ exc ]( ω ) = − α √ βµ · θ ( ω ) ω / . (E13)The numerical prefactor should be correct, but is of noparametric relevance and is set to unity for convenience,thereby giving formula (53) of the main text. 2. FES regime: VB hole recoil In the regime of the FES, not the exciton, but thevalence band hole recoils. Near the direct thresholdat ω = βµ , the quantity describing the hole decay isIm[Σ VB ( k F , ω )] as given in (66), which scales differentlycompared to the exciton decay because the VB hole has Q = k F unlike the Q = 0 exciton (we do not presentthis computation here since the power law is of not muchrelevance for the 2DEG absorption we are interested in;see [42] for details).Near the indirect threshold, the VB hole again has mo-mentum Q = 0, and the resulting 2DEG absorption A ( ω )as given in (65) scales as ∼ ω . This result was alreadypresented in [21], though without derivation. Since thecomputation is very similar to the previous one for the7exciton decay, let us just sketch it: By performing fre-quency integrals in Figs. 15 and 16, and momentum sub-stitutions as for the exciton, one arrives at: A ( ω ) ∼ (cid:90) x >µ d x (cid:90) z >µ d z (cid:90) y <µ d y (E14) δ (cid:16) ω − (cid:0) x − µ (cid:1) + (cid:0) y − µ (cid:1) − (cid:0) z − µ (cid:1) − β ( x + z − y ) (cid:17) , which is similar to the previous expression (E2) except foran additional scattering partner, the photoexcited elec-tron (corresponding to the z -integral). Again, there canbe no cancellations in the deltafunction, and the compu-tation proceeds analogously to sec. E 1. Effectively, thesummands ( x − µ ) , ( y − µ ) and ( z − µ ) contribute afactor of ω to A ( ω ). One factor is fixed by the delta func-tion, such that in total one has ω . In addition, there is the hole recoil term β ( x + z − y ) . 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