PHYSICAL REVIEW B 101, 041201(R) (2020)Rapid Communications

Time-resolved ARPES spectra of nonequilibrium excitonic insulators: Revealing macroscopiccoherence with ultrashort pulses

E. Perfetto,1,2 S. Bianchi ,1 and G. Stefanucci 1,2

1Dipartimento di Fisica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133 Rome, Italy2INFN, Sezione di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133 Rome, Italy

(Received 3 December 2019; revised manuscript received 10 January 2020; published 24 January 2020)

The nonequilibrium (NEQ) excitonic-insulator (EI) phase is a dynamical state of matter for excited insulatorsor semiconductors, and it is characterized by self-sustained coherent oscillations of the excitonic condensate.In this Rapid Communication we highlight the main qualitative features of a NEQ-EI time- and angle-resolvedphotoemission spectroscopy (tr-ARPES) spectrum. We show that monochromatic probes yield a steady-statespectrum with excitonic structures originating from the dressing of conduction states with the coherent conden-sate. These structures are replicas of the NEQ valence bands but shifted upward in energy by the condensatefrequency. Reducing the probe duration below the condensate period, the band structure gradually fades awayand the tr-ARPES signal becomes proportional to the excitonic wave function. In addition, the signal amplitudebecomes periodic in the impinging time of the probe, with the same period of the oscillating condensate.

DOI: 10.1103/PhysRevB.101.041201

Normal band-gap insulators with bright bound excitonsmay turn into nonequilibrium (NEQ) excitonic insulators(EIs) when driven out of equilibrium by coherent pump pulseswith subgap frequencies larger than the exciton energy. TheNEQ-EI phase has been predicted long ago by Östreichandand Schönhammer [1]. More recently and independently theNEQ-EI phase has been shown to emerge using a time-dependent ansatz for the excitonic condensate [2]. The equiv-alence between the two approaches has been discussed inRef. [3] along with some remarkable properties of the NEQ-EI’s spectral function.

The realization and properties of the NEQ-EI phase haveattracted increasing attention over the last ten years [4–11]. Adistinctive feature of this phase is the self-sustained, coherentoscillation of the excitonic condensate (EC). Although deco-herence mechanisms due to electron-electron [9] or electron-phonon [6,12] scatterings eventually suppress this macro-scopic coherence, transient signatures can still be detected inphotoluminescence or photoabsorption experiments [8,13]. Inthis Rapid Communication we show that the NEQ-EI phaseleaves clear fingerprints also in time- and angle-resolved pho-toemission spectroscopy (tr-ARPES) spectra. The tr-ARPESsignal is highly sensitive to the probe-pulse duration as wellas the nature, BEC or BCS, of the EC. If we denote byεg the equilibrium gap and by εCMB the conduction-bandminimum with respect to the vacuum, then probe pulses ofenergy ωp � εg + |εCMB| and duration Tp � 2π/ωp yield asteady-state ARPES spectrum characterized by subgap ex-citonic structures. They originate from the dressing of theconduction states with the oscillating EC “field” [14], andappear as replicas of the NEQ valence bands shifted upward inenergy by the EC frequency. Furthermore, their shape changesconvexity while “moving” toward the BEC-BCS crossover.Using instead ultrafast probe pulses of duration Tp � 2π/εg,the tr-ARPES spectrum becomes periodic in the impinging

time of the probe, the period being the same as the EC period.In this case the band structure is no longer visible and the tr-ARPES signal is proportional to the excitonic wave function.

To demonstrate the aforementioned properties we con-sider a general many-body Hamiltonian for normal band-gapinsulators,

H =∑

k,ν∈Vεkν v

†kν vkν +

∑k,α∈C

εkα c†kα ckα

+ 1

N∑

k1k2q

∑μν ∈ Vαβ ∈ C

U μαβνq v

†k1+qμ

c†k2−qα

ck2β vk1ν, (1)

where vkν (ckα) destroys a valence (conduction) electronfrom band ν ∈ V (α ∈ C) of quasimomentum k and N isthe number of k points. In Eq. (1) the intravalence andintraconduction interaction is assumed to only renormalizethe single-particle energies and it is therefore absorbed in thevalence and conduction bands. Coulomb integrals responsiblefor particle-hole recombinations are typically small and dis-carded altogether. Hence the numbers Nv and Nc of valenceand conduction electrons are separately conserved quantitiesand for large enough U ’s all valence (conduction) bandsare occupied (empty) in the ground state. The conservationof Nv and Nc is a direct consequence of the invariance ofH under the gauge transformation ckα → eiθ ckα . Relaxingthe approximations introduced in Eq. (1) does not changethe general picture [3]. A simplified version of the aboveHamiltonian has been used to study the equilibrium [15–21]and nonequilibrium [22–25] properties of ground-state EIs.

To calculate the tr-ARPES spectrum of the NEQ-EI de-scribed by the Hamiltonian in Eq. (1) we need the NEQGreen’s function G(t, t ′). The latter is obtained by averag-ing the time-order product of two fermionic operators inthe Heisenberg picture at times t and t ′ with the excited

2469-9950/2020/101(4)/041201(5) 041201-1 ©2020 American Physical Society

E. PERFETTO, S. BIANCHI, AND G. STEFANUCCI PHYSICAL REVIEW B 101, 041201(R) (2020)

density matrix ρ = exp [−β(H − μvNv − μcNc)]/Z , with β

the inverse temperature, Nv and Nc the number operators forvalence and conduction electrons, and μv, μc the valenceand conduction chemical potentials [26]. In Ref. [3] we haveshown that this NEQ Green’s function can also be obtained bydriving the system out of equilibrium with a properly chosencoherent pump pulse. The equilibrium Green’s function isrecovered by setting μv = μc = μ. Due to the invarianceunder discrete lattice translations the Green’s function Gk,i j

is diagonal in the quasimomentum k and, in general, offdiagonal in the band indices i, j ∈ V ∪ C. In the absence of EConly the V-V and C-C matrix elements are nonvanishing. TheNEQ-EI phase breaks the gauge symmetry associated with theconservation of Nv and Nc, giving rise to nonvanishing V-Cand C-V components.

In Ref. [3] we have shown that the removal part G<(t, t ′)of the excited Green’s function in the Hartree-Fock (HF)approximation reads

G<k (t, t ′) = i

∑λ

e−i(eλk−σz

δμ

2 )tρλkei(eλ

k−σzδμ

2 )t ′, (2)

where μ ≡ (μc + μv )/2 is the center-of-mass chemical po-tential, δμ ≡ μc − μv is the relative one, and σz is a diagonalmatrix with entries 1 (−1) in the V (C) subspace. The matrixρλ

k,i j ≡ f (eλk )ϕλ

kiϕλ∗k j , with f (ε) = 1/{exp[β(ε − μ)] + 1} the

Fermi function, is written in terms of the eigenvalues eλk and

eigenvectors ϕλk of the excited HF single-particle Hamilto-

nian Ek + Vk + σzδμ

2 where (Ek )i j = δi jεki, i ∈ V ∪ C, and Vkthe HF potential. Interestingly, the excited HF Hamiltoniancoincides with the Floquet HF Hamiltonian [14]. For theinteraction in Eq. (1) the Hartree potential contributes onlyto the V-V and C-C sectors and reads

Vk,i j = − i

N∑

mn∈C(V )

∑q

U imn j0 G<

q,nm(0, 0), i, j ∈ V (C),

(3)whereas the Fock potential contributes only to the V-C andC-V sectors and read

Vk,μβ = i

N∑α ∈ Cν ∈ V

∑q

U μαβν

k−q G<q,να (0, 0), μ ∈ V, β ∈ C,

(4)and Vk,βμ = V ∗

k,μβ . Equations (2)–(4) have to be solved self-consistently since the HF potential is a functional of G<.

In equilibrium (δμ = 0) and zero temperature no elec-trons occupy the conduction bands (Nc = 0). ConsequentlyG<(t, t ′) is block-diagonal in the V ∪ C space or, equivalently,invariant under the gauge transformation ckα → eiθ ckα . Thisis the normal band-insulator phase and, according to Eq. (2),the Green’s function depends only on the time difference. Ifthe system admits a bound exciton at energy εx, then a spon-taneous breaking of the gauge symmetry occurs in NEQ forδμ > εx [3]. This is the NEQ-EI phase and G<(t, t ′) acquiresoff-diagonal matrix elements in the V ∪ C space dependingon t and t ′ separately [see again Eq. (2)]. In particular, theequal-time off-diagonal G<

k,μα (t, t ) = eiδμt G<k,μα (0, 0). Below

we explore the dependence of the tr-ARPES spectra on theprobe-pulse duration in the NEQ-EI phase.

The tr-ARPES spectrum is given by the number of elec-trons Nk(ε) of energy ε and parallel momentum k ejectedby a probe pulse E(t ). Let Dki(ε) be the dipole matrix ele-ment between a Block state ki, i ∈ V ∪ C, and a continuumtime-reversed low-energy electron diffraction (LEED) state ofenergy ε and parallel momentum k, and let us define the scalarproduct T ki(ε, t ) = E(t ) · Dki(ε). Introducing the (retarded)ionization self-energy

�Rk,i j (ε; t, t ′) = −iθ (t − t ′)T ki(ε, t )e−iε(t−t ′ )T ∗

k j (ε, t ′), (5)

one can show that [27,28]

Nk(ε) = 2∫

dtdt ′ Re{Tr

[�R

k (ε; t, t ′)G<k (t ′, t )

]}. (6)

We observe that Nk(ε) is a functional of the probe pulse E(t )and, therefore, it depends on the time τ at which the pulseimpinges the NEQ system. Substituting Eqs. (2) and (5) intoEq. (6) we obtain an expression for Nk(ε) which is manifestlypositive,

Nk(ε) =∑

λ

f(eλ

k

)∣∣∣∣ϕλ†k E

(eλ

k1 − σzδμ

2− ε1

)· Dk(ε)

∣∣∣∣2

,

(7)where E(ω) = ∫

dt eiωt E(t ) is the Fourier transform of theprobe pulse (notice that E · Dk is a vector in the V ∪ C space).

For linearly polarized and monochromatic probes E(t ) =E sin[ωp(t − τ )], with ωp > 0, Eq. (7) yields a simple gener-alization of the standard photocurrent expression, i.e.,

Nk(ε) ∝∑

λ

f(eλ

k

){∣∣A(V )kλ

(ε)∣∣2

δ

(eλ

k − δμ

2− ε + ωp

)

+∣∣A(C)kλ

(ε)∣∣2

δ

(eλ

k + δμ

2− ε + ωp

)}, (8)

where A(V/C)kλ (ε) = ∑

i∈V/C ϕλ∗ki × [E · Dki(ε)]. As expected,

Eq. (8) is independent of τ . The equilibrium expression isrecovered by setting δμ = 0, which implies eλ

k = εkλ andϕλ

ki = δλi; at zero temperature∑

λ f (eλk − μ) → ∑

ν∈V andthe second row does not contribute. In the opposite limitof linearly polarized ultrafast probes E(t ) = E δ(t − τ ) theFourier transform E(ω) = E eiωτ and Eq. (7) yields

Nk(ε) =∑

λ

f(eλ

k

)∣∣A(V )kλ

(ε)ei δμ

2 τ + A(C)kλ

(ε)e−i δμ

2 τ∣∣2

. (9)

In the NEQ-EI phase δμ > εx and both A(V ) and A(C) arenonvanishing. Therefore the photocurrent oscillates with fre-quency δμ as we vary the impinging time τ . It is worthdiscussing those systems with the lowest-energy exciton (ofenergy εx) mainly localized in one valence band μ = v andone conduction band α = c. Then, it can be shown [3] that forδμ → εx only the λ for which limδμ→εx ϕλ

kv = 1 contributesto the sum in Eq. (9) and that in the same limit the conduc-tion components of the eigenfunction behave as ϕλ

kα ∝ δαcYk,where Yk is the excitonic wave function (the constant ofproportionality vanishes as δμ approaches εx). The latter isdefined as the bound-state solution of the excitonic eigen-value equation (εck − εvk − εx)Yk = 1

N∑

q U vccvk−q Yq. Thus,

if we define the complex numbers �veiθv ≡ E · Dkv (ε) and

041201-2

TIME-RESOLVED ARPES SPECTRA OF NONEQUILIBRIUM … PHYSICAL REVIEW B 101, 041201(R) (2020)

FIG. 1. Color plot of the tr-ARPES spectra Nk (ε) vs ω = ε − ωp and k for probe pulses of different duration Tp impinging the system atdifferent times τ . The system is in the BEC NEQ-EI phase characterized by δμ = 0.82. Energies are measured in units of the bare gap εg.Decreasing the duration of the probe pulse (slides from right to left) the photoemission signal loses information on the NEQ band structureand it acquires a dependence on the impinging time τ of the pulse (slides from top to bottom). Notice that for long pulses (right most slides)the valence band is only slightly distorted.

�ceiθc ≡ E · Dkc(ε), then for δμ → εx, Eq. (9) implies

N (ac)k (ε) ∝ �v�cYk cos(δμτ + θv − θc), (10)

where N (ac)k (ε) is the τ -dependent part of Nk(ε). Assuming

that the dipoles have a weak dependence on k, the amplitudeof the cosine is, for any given ε, proportional to the excitonicwave function.

We illustrate the characteristic features of a NEQ-EI tr-ARPES spectrum for a one-dimensional two-band model atzero temperature and with a valence band εkv = −W

2 [1 −cos(k)] − εg/2 separated by a direct gap of magnitude εg fromthe conduction band εkc = −εkv , and a short-range interactionU vccv

q = U . For simplicity we also take the dipole matrixelements Dki(ε) = D independent of momentum, energy, andband index for ε larger than the vacuum energy. This alsoimplies that the amplitudes A(V/C)

k are independent of ε.Henceforth all energies are measured in units of the bare gapεg and times in units of 1/εg. For a bandwidth W = 2 theground state of this model is a band insulator for U < Uc �2.3 and an EI for larger U ’s [3]. We have therefore chosen therepresentative value U = 1 to describe the NEQ-EI features.The system admits an exciton at energy εx � 0.76; hence thetransition to the NEQ-EI phase occurs at δμ � 0.76. To studythe evolution from monochromatic to ultrafast probe pulses

we use the electric field

E(t ) = E sin

(π (t − τ )

Tp

)2

sin[ωp(t − τ )] (11)

active from t = τ until t = τ + Tp and zero otherwise. Thelimiting cases in Eqs. (8) and (9) are recovered for Tp → ∞and Tp → 0+, respectively.

In Fig. 1 we consider a NEQ-EI phase with low exci-ton density (BEC regime). The relative chemical potentialis δμ = 0.82 > εx corresponding to a density in the con-duction band nc = Nc/N = 0.07. We show the color plotof Nk (ε) vs ω = ε − ωp for ωp = 40 � W , different valuesof the probe duration Tp, and for three representative timesτ = 0, π

2δμ, π

δμ. For deltalike probes, Tp = 0+, the spectrum

is independent of the photoelectron energy ε [see Eq. (9)],and strongly dependent on τ . As the probe pulse acquires afinite duration, a dependence on ε appears but the spectrumcontinues to change quite dramatically for different τ . Onlyfor durations Tp � 10 does the τ dependence become weakand the spectrum approaches the removal part of the spectralfunction [see Eq. (8)]. In particular, for Tp = 40 the spectrumexhibits an excitonic structure just below the conduction-bandminimum—no sign of the conduction band in the normal-insulator phase is visible. The exciton contribution to Nk (ε) is

041201-3

E. PERFETTO, S. BIANCHI, AND G. STEFANUCCI PHYSICAL REVIEW B 101, 041201(R) (2020)

FIG. 2. Color plot of the tr-ARPES spectra Nk (ε) vs ω = ε − ωp and k for probe pulses of different duration Tp impinging the system atdifferent times τ . The system is in the BCS NEQ-EI phase characterized by δμ = 1.05. Energies are measured in units of the bare gap εg.Notice that for long pulses (right most slides) the valence band is strongly distorted, its shape turning convex.

the replica of the NEQ valence band but it is shifted upward byδμ [see again Eq. (8)]. In the language of Floquet theory theexciton structure emerges from the dressing of the conductionstates with the oscillating EC field. For any fixed k, the excitonand the valence-band state can then be seen as the analog ofthe Autler-Townes doublet in atomic physics [14,29]. We alsonotice that at low exciton densities (BEC regime) the valenceband is only slightly distorted and therefore the excitonicstructure is, in this case, concave.

Increasing δμ and hence the exciton density, the averageelectron distance becomes smaller than the width of theexcitonic wave function. In this BCS-like regime the excitonsoverlap. In Fig. 2 we show that tr-ARPES spectra for thesame parameters except that δμ = 1.05 > εg, correspondingto a density in the conduction band nc = 0.2. In this casetoo the spectrum shows a strong τ dependence for probedurations Tp < 10 and it becomes independent of τ for largerdurations. However, at a difference with the BEC regime, forTp > 10 the top of the valence band is strongly distorted,its shape turning convex. Correspondingly [see Eq. (8)], alsothe shape of the excitonic structure experiences a concave-to-convex transition. We also notice that for Tp < 2 and at anyfixed photoelectron energy the signal does not have a singlemaximum or minimum at k = 0. Rather, the spectrum exhibitsa double maximum or minimum symmetrically positionedwith respect to k = 0. This feature is particularly evident for

Tp = 0+. In Fig. 3 we show the tr-ARPES spectrum integratedover energy for deltalike probes impinging at different τ ∈(0, 2π/δμ) in the BEC (left) and BCS (right) regimes. On theleft face of the box we project (blue line) the τ -independenttr-ARPES spectrum generated by long probes [see Eq. (8)].Evidently, the information brought by monochromatic probesdoes not get lost when using ultrafast pulses. In particular,the concave-to-convex transition at the BEC-BCS crossoveras well as the shape of the excitonic structure can be

FIG. 3. tr-ARPES spectrum integrated over energy for δ-likeprobes vs the impinging time τ . Left: BEC regime with δμ = 0.82and hence nc = 0.07. Right: BCS regime with δμ = 1.05 and hencenc = 0.2. In both cases the shape around k = 0 at τ = π/(2δμ)resembles the shape of the excitonic structure appearing in thephotoemission spectrum for long probe pulses (Tp → ∞) (see theblue curves).

041201-4

TIME-RESOLVED ARPES SPECTRA OF NONEQUILIBRIUM … PHYSICAL REVIEW B 101, 041201(R) (2020)

predicted by properly choosing the impinging times τ . Re-markably, in the BEC regime (δμ → εx) this information iscontained in the excitonic wave function Yk since, as discussedbelow Eq. (9), the integrated spectrum is proportional toYk cos(δμτ ).

In conclusion, we have highlighted the main qualitativefeatures of the tr-ARPES spectrum of a system in the NEQ-EIphase. This phase is characterized by coherent oscillations ofthe EC at frequency δμ, and it can be generated by pumpinga normal insulator or semiconductor with a coherent light offrequency below the gap and slightly larger than the excitonenergy [3]. Using ultrafast probes of duration Tp < 2π/δμ

we have shown that the photocurrent signal is strongly de-pendent on the impinging time τ . The relation between theperiod 2π/δμ of the spectrum and the energy shift δμ of

the excitonic structure can be used as a fingerprint of theNEQ-EI phase. We also observe that the time periodicityof the tr-ARPES spectrum is, in principle, experimentallyaccessible. For materials with a quasiparticle gap in the eVrange it is sufficient to use an attosecond probe pulse and varythe impinging times τ in a window of a few femtoseconds—decoherence mechanisms due to phonons or electronic corre-lations typically take place on longer timescales. Besides theτ dependence, the tr-ARPES spectrum is also dependent onthe EC density and for small densities and ultrafast probes itis proportional to the excitonic wave function.

We acknowledge useful discussions with A. Marini andD. Sangalli. We also acknowledge funding from MIUR PRINGrant No. 20173B72NB and from INFN17−nemesys project.

[1] T. Östreich and K. Schönhammer, Z. Phys. B: Condens. Matter91, 189 (1993).

[2] M. H. Szymanska, J. Keeling, and P. B. Littlewood, Phys. Rev.Lett. 96, 230602 (2006).

[3] E. Perfetto, D. Sangalli, A. Marini, and G. Stefanucci, Phys.Rev. Materials 3, 124601 (2019).

[4] C. Triola, A. Pertsova, R. S. Markiewicz, and A. V. Balatsky,Phys. Rev. B 95, 205410 (2017).

[5] A. Pertsova and A. V. Balatsky, Phys. Rev. B 97, 075109 (2018).[6] R. Hanai, P. B. Littlewood, and Y. Ohashi, J. Low Temp. Phys.

183, 127 (2016).[7] R. Hanai, P. B. Littlewood, and Y. Ohashi, Phys. Rev. B 96,

125206 (2017).[8] R. Hanai, P. B. Littlewood, and Y. Ohashi, Phys. Rev. B 97,

245302 (2018).[9] K. W. Becker, H. Fehske, and V.-N. Phan, Phys. Rev. B 99,

035304 (2019).[10] M. Yamaguchi, K. Kamide, T. Ogawa, and Y. Yamamoto, New

J. Phys. 14, 065001 (2012).[11] M. Yamaguchi, K. Kamide, R. Nii, T. Ogawa, and Y.

Yamamoto, Phys. Rev. Lett. 111, 026404 (2013).[12] K. Hannewald, S. Glutsch, and F. Bechstedt, J. Phys.: Condens.

Matter 13, 275 (2000).[13] Y. Murotani, C. Kim, H. Akiyama, L. N. Pfeiffer, K. W. West,

and R. Shimano, Phys. Rev. Lett. 123, 197401 (2019).[14] E. Perfetto and G. Stefanucci, Phys. Rev. A 91, 033416

(2015).

[15] J. M. Blatt, K. W. Böer, and W. Brandt, Phys. Rev. 126, 1691(1962).

[16] L. V. Keldysh and Y. U. Kopaev, Fiz. Tverd. Tela. 6, 2791(1964) [Sov. Phys. Solid State 6, 2219 (1965)].

[17] A. N. Kozlov and L. A. Maksimov, JETP 21, 790 (1965).[18] D. Jérome, T. M. Rice, and W. Kohn, Phys. Rev. 158, 462

(1967).[19] B. Halperin and T. Rice, Solid State Phys. 21, 115 (1968).[20] L. V. Keldysh and A. N. Kozlov, JETP 27, 521 (1968).[21] C. Comte and P. Nozières, J. Phys. (France) 43, 1069 (1982).[22] D. Golež, P. Werner, and M. Eckstein, Phys. Rev. B 94, 035121

(2016).[23] S. Mor, M. Herzog, D. Golež, P. Werner, M. Eckstein, N.

Katayama, M. Nohara, H. Takagi, T. Mizokawa, C. Monneyet al., Phys. Rev. Lett. 119, 086401 (2017).

[24] Y. Murakami, D. Golež, M. Eckstein, and P. Werner, Phys. Rev.Lett. 119, 247601 (2017).

[25] R. Tuovinen, D. Golež, M. Schüler, P. Werner, M. Eckstein, andM. A. Sentef, Phys. Status Solidi B 256, 1800469 (2019).

[26] G. Stefanucci and R. van Leeuwen, Nonequilibrium Many-Body Theory of Quantum Systems: A Modern Introduction(Cambridge University Press, Cambridge, UK, 2013).

[27] E. Perfetto, D. Sangalli, A. Marini, and G. Stefanucci, Phys.Rev. B 94, 245303 (2016).

[28] J. K. Freericks, H. R. Krishnamurthy, and T. Pruschke, Phys.Rev. Lett. 102, 136401 (2009).

[29] S. H. Autler and C. H. Townes, Phys. Rev. 100, 703 (1955).

041201-5