A Eigenstates of the non-interacting Weyl Hamiltonian

# Excitonic instability in optically-pumped three-dimensional Dirac materials

## Abstract

Recently it was suggested that transient excitonic instability can be realized in optically-pumped two-dimensional (2D) Dirac materials (DMs), such as graphene and topological insulator surface states. Here we discuss the possibility of achieving a transient excitonic condensate in optically-pumped three-dimensional (3D) DMs, such as Dirac and Weyl semimetals, described by non-equilibrium chemical potentials for photoexcited electrons and holes. Similar to the equilibrium case with long-range interactions, we find that for pumped 3D DMs with screened Coulomb potential two possible excitonic phases exist, an excitonic insulator phase and the charge density wave phase originating from intranodal and internodal interactions, respectively. In the pumped case, the critical coupling for excitonic instability vanishes; therefore, the two phases coexist for arbitrarily weak coupling strengths. The excitonic gap in the charge density wave phase is always the largest one. The competition between screening effects and the increase of the density of states with optical pumping results in a reach phase diagram for the transient excitonic condensate. Based on the static theory of screening, we find that under certain conditions for the value of the dimensionless coupling constant screening in 3D DMs can be weaker than in 2D DMs. Furthermore, we identify the signatures of the transient excitonic condensate that could be probed by scanning tunneling spectroscopy, photoemission and optical conductivity measurements. Finally, we provide estimates of critical temperatures and excitonic gaps for existing and hypothetical 3D DMs.

## I Introduction

In the past decade there has been a surge of interest in the so-called Dirac materials (DMs) which exhibit a linear, Dirac-like spectrum of quasiparticle excitations Wehling et al. (2014). This rapidly growing class encompasses a diverse range of quantum materials such as high-temperature -wave superconductors Balatsky et al. (2006), superfluid He Volovik (1992), graphene Castro Neto et al. (2009), topological insulators Hasan and Kane (2010); Qi and Zhang (2011), and Dirac Neupane et al. (2014) and Weyl semimetals Huang et al. (2015); Xu et al. (2015a). These materials are characterized by the presence of nodes in the quasiparticle spectrum and their properties can be understood within a unifying framework of DMs. The concept of DMs has recently been extended to bosonic DMs, e.g. bosonic systems with Dirac nodes in the excitation spectra which can be realized, for instance, in various artificial honeycomb lattices Fransson et al. (2016); Banerjee et al. (2016). The class of DMs also includes Dirac nodal line semimetals, in which two bands with linear dispersion are degenerate along a one-dimensional curve in momentum space Burkov et al. (2011); Sun et al. (2017).

An important topic that has emerged in the last few years is the study of non-equilibrium dynamics of DMs. One example is the interplay between light and the Dirac states in DMs Wang et al. (2016); Sánchez-Barriga et al. (2014); Kuroda et al. (2016). Understanding the non-equilibrium dynamics of Dirac carriers subject to perturbations by electromagnetic fields is crucial for applications in ultrafast photonics and high-mobility optoelectonics Otsuji et al. (2012); Wang et al. (2017). Experimental progress in this field is fueled by the availability of time-sensitive probes such as time-resolved pump-probe angular-resolved photoemission spectroscopy (ARPES) Gierz et al. (2013); Johannsen et al. (2013); Ulstrup et al. (2014); Johannsen et al. (2015); Gierz et al. (2015); Zhu et al. (2015); Neupane et al. (2015) and optical-pump terraherz(THz)-probe spectroscopy George et al. (2008); Gilbertson et al. (2011); Aguilar et al. (2015) that can study the electron dynamics on picosecond (ps) and even femtosecond (fs) time-scales.

Pump-probe experiments on Dirac states in graphene have demonstrated the existence of a broadband population inversion Li et al. (2012), a situation when highly excited electrons and holes form two independent Fermi-Dirac distribution with separate chemical potentials. This can generate optical gain and is promising for THz lasing applications Otsuji et al. (2012). The lifetime of population inversion in graphene is of the order of fs Li et al. (2012); Gierz et al. (2013); Johannsen et al. (2013); Ulstrup et al. (2014); Johannsen et al. (2015). Population inversion has also been demonstrated in three-dimensional topological insulators (3D TIs) with much longer lifetimes, ranging from few ps ( ps for SbTe Zhu et al. (2015)) to hundreds of ps ( ps for bulk-insulating (SbBi)Te Sumida et al. ()). Such long lifetimes are attributed to slow electron-hole recombination.

Motivated by these experimental results, we recently proposed to search for transient excitonic instability in optically-excited DMs with population inversion Triola et al. (2017). Given the Dirac nature of the spectrum, an inverted population allows the optical tunability of the density of states (DOS) of the electrons and holes, effectively offering control of the strength of the Coulomb interaction. The most promising candidate among two-dimensional (2D) materials is free-standing graphene pumped by circularly polarized light. 3D TIs with specially designed material parameters are also promising due to potentially long lifetimes of the optically-excited states.

In this paper, we focus on transient states in optically pumped 3D DMs such as the newly discovered Dirac and Weyl semimetals. These systems exhibit nodes formed by linearly dispersing bands in 3D momentum space. In a Dirac semimetal (DSM), the Dirac states are doubly degenerate. The degeneracy can be lifted by breaking either time-reversal or spatial-inversion symmetry, resulting in a Weyl semimetal (WSM) in which the nodes appear in pairs with opposite chirality. In addition, WSMs display the so-called topological Fermi arcs on their surfaces, which connect the bulk projections of the Weyl nodes. DSM have been observed in CdAs Liu et al. (2014a) and NaBi Liu et al. (2014b). WSM have been recently confirmed in TaAs Xu et al. (2015b) and signatures of the so-called type-II WSM with tilted cones have been seen in WTe Feng et al. (2016). These materials display a number of remarkable properties such as the solid state realization of the chiral anomaly and the resulting negative magnetoresistance effect Xiong et al. (2015); Ali et al. (2014).

So far experimental and theoretical efforts have been focused mostly on equilibrium and steady-state properties of 3D DMs. Characterization of the electronic structure and spin-texture e.g. by ARPES is used to verify the 3D Dirac nature of existing and predicted materials. Considerable amount of theoretical work has been done on magnetoelectrical transport Tabert et al. (2016) and optical properties Hosur et al. (2012); Ashby and Carbotte (2014); Tabert and Carbotte (2016) as well as on the role of disorder and interactions Wei et al. (2012, 2014a, 2014b) in DSMs and WSMs. In particular, excitonic instability in equilibrium WSM with chemical potential at the compensation point was studied for the case of short-range Wei et al. (2012) and long-range Wei et al. (2014a) interactions.

Non-equilibrium properties of 3D DMs is a much less explored topic. In contrast to 2D DMs, very little is known about non-equilibrium dynamics and relaxation of photoexcited carriers in 3D DMs. However, examples of pump-probe experiments similar to those done on graphene and 3D TIs have already appeared in the literature Manzoni et al. (2015); Ishida et al. (2016); Jadidi et al. (2017); Ma et al. (2017); Lu et al. (2017). Recent work revealed that ultrafast relaxation of Dirac fermions in CdAs DSM is qualitatively similar to that of graphene Lu et al. (2017). Given the growing interest in driven and non-equilibrium quantum states of matter and the potential of DMs for high-performance optoelectonic devices Otsuji et al. (2012); Wang et al. (2017), this trend will continue to gain momentum. In this context, we consider the possibility of realizing transient many-body states in optically-pumped 3D DMs. We find that external driving combined with the Dirac nature of quasiparticles create favorable conditions for transient exitonic instability.

Our theory is based on a low-energy effective model for a Dirac/Weyl system, which includes mean-field interactions and screening effects. Metallic screening which is a crucial factor in non-equilibrium, is treated within the static random phase approximation. We consider particle-hole instabilities in 3D DMs assuming the existence of non-equilibrium electron and hole populations which can be generated by optical pumping (see Fig. 1). Similar to the equilibrium case with unscreened interactions Wei et al. (2014a), we find that for screened Coulomb potential two possible excitonic phases exist, an excitonic insulator phase and the charge density wave phase originating from intranodal and internodal interactions, respectively. The main difference from the equilibrium case is that the critical coupling for excitonic instability vanishes for finite electron and hole chemical potentials. Hence, the two phases coexist for arbitrarily weak coupling strengths. However, the excitonic gap produced in the charge density wave phase is always larger than the one in the excitonic insulator phase.

Contrary to ordinary metals, we find that screening in 3D DMs can be weaker than in 2D DMs depending on the value of the the dimensionless coupling constant of the material. We present the phase diagrams for the transient excitonic condensate resulting from the competition between screening and the increase of the DOS with optical pumping. We propose several experimental measurements which could probe the existence of the transient excitonic condensate. Finally, we estimate critical temperatures and the size of excitonic gaps for a few realistic cases.

The paper is organized as follows. In Section II we present the details of our theoretical model. In particular, we discuss the form of the screened Coulomb potential in an optically-pumped 3D DM and derive the self-consistent equation for the excitonic gap. In Section III we present the spectroscopic features for the transient excitonic states in a pumped 3D DM, namely the spectral function, the density of states, and the optical conductivity. We also discuss the dependence of the size of the gap and the critical temperature on the material parameters such as the interaction strength, the chemical potential and the Dirac cone degeneracy. Finally, in Section IV we present concluding remarks.

## Ii Theoretical model

### ii.1 Hamiltonian of a 3D DM with interactions

A general interacting Hamiltonian for a DSM or WSM can be written as

 H=HD/W+V, (1)

where is the non-interacting Hamiltonian of a DSM/WSM and contains electron-electron interactions. Below we describe each of the terms in Eq. (1).

#### Non-interacting Hamiltonian

We define a Weyl node as a linear crossing of two non-degenerate bands in 3D momentum space. A topological number, chirality , is assigned to each Weyl node. A WSM contains pairs of Weyl nodes with opposite chirality. In a DSM, the two Weyl nodes are degenerate in energy and momentum. At low energies, the Hamiltonian of a DSM can be mapped onto a Dirac Hamiltonian, which can be viewed as consisting of two copies of a Weyl Hamiltonian with opposite sign of

 HD Unknown environment 'tabular% (2) Hξ(k) =ξℏvσ⋅k. (3)

Here is the Hamiltonian of a Weyl node with chirality , is a set of Pauli matrices, is the 3D momentum, is the velocity of the Dirac states and is a four-component spinor. Since the Hamiltonian in Eq. (2) is block-diagonal, in order to describe a DSM, it is sufficient to find the eigenstates of . Then the eigenstates of are obtained by including a degeneracy factor .

In a WSM, the degeneracy between the nodes is lifted and the Hamiltonian can be written as

 HW =∑ξ∑kΦ†kHξ(k)Φk, (4) Hξ(k) =ξℏvFσ⋅(k−ξK)+ξIK0, (5)

where is a two-component spinor and is a identity matrix. Finite corresponds to broken time reversal symmetry while finite corresponds to broken inversion symmetry. For and , the Weyl nodes are located at the same energy but are shifted in momentum space by . For and , the Weyl nodes are located at the same momentum but are shifted in energy by . As a limiting case at and , the nodes become degenerate in energy and momentum as in a DSM [Eq. (2)].

Here we focus on the time reversal broken case. Inversion symmetry breaking can be included simply by introducing a rigid shift to the energy eigenvalues. To simplify notations, we assign labels R(right) and L(left) for the node located at with chirality and with chirality , respectively. The resulting Hamiltonian for R/L node reads

 HR/L(q)=±ℏvσ⋅q. (6)

The eigenvalues of this Hamiltonian are given by , where () stands for conduction (valence) band. The corresponding normalized eigenvectors for R/L node are given by

 Unknown environment '% (7) Unknown environment '% (8)

The Hamiltonian is diagonalized by a unitary transformation , where is composed of the eigenvectors in Eqs. (7-8) [see Appendix A for details]. The spinors can be expressed in the diagonal basis as , where and , are the fermionic creation(annihilation) operators corresponding to the bands .

#### Coulomb interactions in Dirac/Weyl Semimetal

We will now consider electron-electron interactions for a system of two, in general non-degenerate, Weyl nodes. The interacting Hamiltonian for a DSM can then be obtained in the limiting case when the nodes are degenerate in momentum space and energy. Starting from a general spin-independent particle-particle interaction potential

 V=∑σ,σ′∑k,k′,q∑αi=R,LΦα1†k′+q,σ′Φα2k′,σ′Φα3†k−q,σΦα4k,σ, (9)

we express the interaction in the diagonal basis of fermionic operators . Our derivation follows Refs. Wei et al. (2012, 2014a). For completeness, the main steps of the derivation are summarized in Appendix B. We change the notations as , where for R/L node and express the wavefunctions as . We also use the fact that . Considering pairing of the form , , the interaction potential is given by

 V=−∑q,q′n=±{V(q−q′)A(q,q′)∑α=R,Lcα†q,ncαq,−ncα†q′,−ncαq′,n+V(q−q′−2K)B(q,q′)cL†q,ncLq,−ncR†q′,−ncRq′,n Missing \left or extra \right (10)

where are momentum-dependent coefficients, defined as

 A(q,q′) = ^eq⋅^e∗q′+^e∗q⋅^eq′4 = sinθsinθ′2+1+cosθcosθ′2cos(ϕ−ϕ′),
 B(q,q′) = ^eq⋅^eq′+^e∗q⋅^e∗q′2 = sinθsinθ′2−1−cosθcosθ′2cos(ϕ−ϕ′),
 C(q,q′) = ^q⋅^q′+1. (13)

In the above expressions, and , where and are the azimuthal and polar angles of the spherical coordinate system respectively. In Eq. (10), is the distance in momentum space between the two Weyl nodes. DSM is obtained by setting . In Eq. (10), the first two terms correspond to intranodal interactions (pairing within a single node, L or R) while the last term corresponds to internodal interactions (pairing between L and R nodes).

Two forms of the interaction potential can be considered: (i) a simplified short-range, or contact interaction, i.e. , where is a constant in momentum space (delta function in real space), (ii) a more realistic Coulomb potential. In equilibrium, i.e. when the chemical potential is exactly at the Dirac or Weyl node, one can in principle consider the long-range unscreened Coulomb potential,  Wei et al. (2014a). However, for chemical potential away from the node, which is the case for doping and optical pumping, screening is important. In the following section we will derive the expressions for the screened Coulomb potential within the static random phase approximation for both 2D and 3D DMs.

#### Screened Coulomb potential in 2D and 3D DM

The general expression for the frequency-dependent dielectric function in the random phase approximation, or the Lindhard dielectric function, reads Haug and Koch (2004)

 ε(q,ω)=1−V(q)∑kfk−q−fkω+iδ+ωk−q−ωk, (14)

where is the Coulomb potential, are the Fermi factors, and are the energies. In the limit , we obtain the static dielectric function , the statically screened Coulomb potential , and the expression for the screening wavevector . In the 2D case, we have

 ε2D(q,0) =1+κ2Dq, (15) V2Ds(q) =V2D(q)ε2D(q,0)=2πe2ε1q+κ2D, (16) κ2D =2πe2ε∂n∂μ, (17)

where is the electron density, is the chemical potential, is the electron charge and is the dielectric constant. Here is the unscreened Coulomb potential in 2D momentum space, which is obtained by Fourier transform from the bare real space potential .

Analogously, in the 3D case, we have

 ε3D(q,0) =1+κ3D2q2, (18) V3Ds(q) =V3D(q)ε3D(q,0)=4πe2ε1q2+κ3D2 (19) κ3D =√4πe2ε∂n∂μ, (20)

where is the unscreened Coulomb potential in 3D momentum space.

So far we assumed the presence of one type of carriers, say electrons, defined by density and chemical potential . In the case of population inversion generated by optical pumping, we have electron and hole plasmas which exist at different densities and chemical potentials. Therefore, one should define the global screening wavevector Klingshirn (2005)

 κ2D =2πe2ε∑i=e,h∂ni∂μi, (21) κ3D = ⎷4πe2ε∑i=e,h∂ni∂μi. (22)

where and () are the electron/hole density and electron/hole chemical potential, respectively. It is instructive to re-write the global screening wavevector in terms of the screening vectors of electron and hole plasmas

 κ2D =κe2D+κh2D, (23) (κ3D)2 =(κe3D)2+(κh3D)2, (24)

where and are given in Eq. (17) and (20), respectively. Assuming equal densities for electrons and holes, one can see that in 2D the screening wavevector increases by a factor of while in 3D it increases by a factor of , compared to electron/hole screening wavevector.

Using the general expressions obtained above, we calculated the screening wavevector in 2D and 3D for a system with Dirac dispersion. The results are summarized in Table 1 (the details of the calculation are presented in Appendix C, where we also show the results for 2D and 3D electron gas). Only the results for a single type of carriers (electrons or holes) are presented. The total screening can be then obtained from Eqs. (23) and (24). We take the zero temperature limit, , which is referred to as the Thomas-Fermi approximation Haug and Koch (2004); Das Sarma et al. (2011, 2015).

In Table 1, we defined the dimensionless coupling constant and the Fermi wavevector . One can see that in both 2D and 3D DM, the screening wavevector scales linearly with . However, the prefactors in the linear dependence are different. The dimensionless coupling constant and the degeneracy factor can be different for 2D and 3D DM. One needs to know the values for these parameters in order to make quantitative predictions about the strength of the screening effects. Assuming for simplicity equal chemical potentials and velocities of the Dirac states in 2D and 3D DM, we find that, in general, the screening in 2D DM is stronger than in 3D DM if the following conditions are satisfied

 α3D <πg22Dα22D/2g3D,forg2Dα2D>1, (25) α3D >πg22Dα22D/2g3D,forg2Dα2D<1, (26)

where the subscript refers to 2D/3D DM.

### ii.2 Excitonic instability in a pumped 3D DM: quasi-equilibrium model

Before discussing excitonic instability in pumped 3D DM, we should note that conditions for excitonic condensation can be realized without pumping in WSM with broken spatial inversion symmetry, when the nodes are shifted symmetrically in energy with respect to the original Dirac node. In this case, there exist perfectly nested electron and hole Fermi surfaces, similarly to the case of graphene in parallel magnetic field Aleiner et al. (2007). Hence the excitonic order can be established at arbitrary weak coupling strength. Such situation has been considered for instance in Ref. Zyuzin and Burkov (2012). The case of optical pumping considered here is unique in a sense that the excitonic states are of transient nature. In contract to the equilibrium case, the electron and hole DOS, and hence the strength of the effective interaction and the value of critical temperature, is controlled by optical pumping Triola et al. (2017).

We will now consider a simple model of an optically pumped DM, in which electrons in conduction and valence bands are described by two separate Fermi-Dirac distributions with different chemical potentials and , respectively, as illustrated in Fig. 1. We assume that these non-equilibrium populations have been established after some time and we will solve for the excitonic order parameter self-consistently assuming quasi-equilibrium, i.e. that the lifetime of the inverted population is infinitely long. The relaxation of the populations and the order parameter towards equilibrium using a dynamical method based on rate equations Triola et al. (2017) will be considered in future work.

We consider the Hamiltonian for a system of two Weyl nodes, with interactions given by Eq. (10). For the case of optical pumping, the band dispersions of conduction electrons and valence electrons (holes) need to be modified as follows

 ε+q = ℏv|q|−μe (27) ε−q = −ℏv|q|−μh. (28)

The self-consistent equation for the order parameter, or gap , of the excitonic condensate in this system reads (see Appendix D for the derivation)

 Δ(q)=∑q′~V(q,q′)Δ(q′)ω+(q′)−ω−(q′)[nF(ω+)−nF(ω−)], (29)

where

 ω±(q)=ε+q+ε−q2±12√(ε+q−ε−q)2+4|Δ(q)|2 (30)

are the excitonic bands, is the Fermi-Dirac distribution, is the temperature (assumed to be the same for photoexcited electrons and holes), and is the Boltzmann constant. The form of the interaction potential depends on the particular case we are considering (intranodal or internodal scattering) and the approximations made to the Coulomb potential. The order parameter is of the form

 Δ(q)=∑q′~V(q,q′)⟨cα†q′,ncβq′,−n⟩, (31)

where . The case corresponds to intranodal interactions and leads to an excitonic insulator (EI) phase. The case corresponds to internodal interactions and leads to a charge density wave (CDW) phase with the modulation momentum equal to the distance between the Weyl nodes. Excitonic phases in a WSM for contact and unscreened Coulomb potential in equilibrium () have been studied in detail in Refs. Wei et al. (2012, 2014a). It was shown that for contact (short-range) interaction, the EI phase is more energetically favorable and is accompanied by gap opening at the nodes. In contract, for long-range unscreened interactions the CDW phase is more energetically favorable. Below we specify the form of the order parameter and the gap equation for intranodal and internodal interactions in the case of a pumped (non-equilibrium) 3D DM. (The equilibrium case is considered in Appendix D as a limit of Eq. (29) for .)

#### Excitonic Insulator (intranodal interactions)

The intranodal part of the interaction potential is given by the first two terms in Eq. (10). Since , we can neglect the second term. The interaction potential becomes

 V=−∑q,q′,n=±~Vintra(q,q′)∑α=L,Rcα†q,ncαq,−ncα†q′,−ncαq′,n, (32)

where

 ~Vintra(q,q′) = V(q−q′)[sinθsinθ′2+1+cosθcosθ′2 (33) V(q−q′)[sinθsinθ′2×cos(ϕ−ϕ′)].

Furthermore, we only keep the slowest varying term in the angular-dependent part of the potential given by . The mean-field order parameter can be defined as

 Δα(q)=∑q′~Vintra(q,q′)⟨cα†q′,+cαq,−⟩,α=L,R. (34)

The mean-field Hamiltonian of the system reads

 H = ∑q,α=L/Rn=±εnqcα†q,ncαq,n−∑q,α=L,R~Δα(q)cα†q,−cαq,+ (35) − ∑q,α=L,R~Δα∗(q)cα†q,+cαq,−,

where and for are given in Eqs. (27)-(28). The first term in Eq. (35) is the non-interacting Hamiltonian of the two nodes while the last two terms describes interactions within each node.

The self-consistent gap equation becomes

 Δα(q)=∑q′2~VintraΔ(q′)ω+(q′)−ω−(q′)[nF(ω+)−nF(ω−)], (36)

where are defined in Eq. (30).

#### Charge density wave (internodal interactions)

The internodal part of the interaction potential is given by the last term in Eq. (10). Since , the leading term is is given by and the interaction potential can be written as

 V=−∑q,q′,n=±~Vinter(q,q′)cL†q,ncRq,−ncR†q′,−ncLq′,n, (37)

where

 ~Vinter(q,q′)=−[2V(2K)−V(q−q′)(1+^q⋅^q′)]. (38)

Furthermore, we neglect the angular-dependent part proportional to . In this case, the mean-field order parameter is defined as

 Δn(q)=∑q′~Vinter(q,q′)⟨cR†q′,−ncLq,n⟩, (39)

and the mean-field Hamiltonian is given by

 H = ∑q,α=L/Rn=±εnqcα†q,ncαq,n−∑q,n=±Δn(q)cL†q,ncRq,−n (40) − ∑q,n=±Δ∗n(q)cR†q,−ncLq,n.

The corresponding self-consistent gap equation reads

 Δn(q)=∑q′~Vinter(q,q′)Δ(q′)ω+(q′)−ω−(q′). (41)

#### Numerical solution of the gap equation

Since the order parameter is momentum-dependent, we will solve the self-consistent gap equation numerically. In order to simplify the numerical analysis, we will use dimensionless units , , , where is the momentum cutoff of the Dirac model in Eqs. (27)-(28). The corresponding cutoff energy scale is .

In what follows we will assume that is given by the 3D screened Coulomb potential

 V(q−q′) = 4πε1(q−q′)2+κ2 (42) ≡ 4παℏv(q−q′)2+κ2,

where is a 3D Thomas-Fermi screening wavevector (see Section II.1.3). Hence, taking into account only the leading terms in the interaction potential, the gap equation in dimensionless units for intranodal an internodal interactions is given by respectively

 Δα(q) =4πα(2π)3∫1(q−q′)2+κ2Δα(q′)cos(ϕ−ϕ′)ω+(q′)−ω−(q′) (43) =4πα(2π)3∫1(q−q′)2+κ2×[nF(ω+)−nF(ω−)]dV,
 Δn(q) =4πα(2π)3∫1(q−q′)2+κ2Δn(q′)ω+(q′)−ω−(q′) (44) =4πα(2π)3∫1(q−q′)2+κ2×[nF(ω+)−nF(ω−)]dV,

where is the volume element.

First, we consider the gap equation for the CDW phase, Eq. (44). In this case, the order parameter is isotropic and the Coulomb potential can be replaced by its angle-average, which only depends on the lengths of vectors and and can be calculated analytically (see Appendix D for details).

The EI gap equation, Eq. (43), is explicitly angle-dependent. However, it can be re-written in the form similar to the CDW case. For this we express as and the EI gap as . The resulting self-consistent equation for the magnitude of the gap is identical to the one in the CDW case with additional factor of . Therefore the EI gap is always smaller than the CDW gap for the same model parameters.

Using the self-consistently calculated order parameter, we can obtain the observable quantities, such as the DOS and the spectral function as well as the phase diagrams for the excitonic condensate in pumped 3D DM. The results of the calculations are presented in the next section.

## Iii Results

### iii.1 Density of states and spectral function

The spectral function and the DOS for a single Weyl node in the normal state (no interactions included) and in equilibrium CDW excitonic phase are shown in Fig. 2 and 3, respectively. The excitonic phase is characterized by a gap which opens up at the equilibrium chemical potential, . In this case, we calculated the gap self-consistently using Eq. (44) for  num (a) (see also Appendix D). Both EI and CDW phase is accompanied by a gap opening at the Weyl/Dirac node. The excitonic gap is larger in the CDW phase compared to the EI phase.

Figure 4 shows the spectral function and the DOS in a pumped 3D DM with population inversion. In this case, the gap is calculated self-consistently using Eqs. (43)-(44) for screened Coulomb potential and . At this point, it is important to describe the role of the Dirac cone degeneracy . The degeneracy factor is the number of Dirac cones in the system. In the effective model of a 3D DM considered here, for a DSM and for a WSM with two nodes, and for a single Weyl node. In real materials the degeneracy can be large, for example in TaAs WSM. In this work we take into account metallic screening for situations when the chemical potential is away from the compensation point. This is precisely the case for population inversion generated by optical pumping (see Fig. 1). The degeneracy factor enters the definition of the Thomas-Fermi screening wavevector [see Table 1]. In fact, the screening wavevector increases with increasing the chemical potential, the dimensionless coupling constant or the degeneracy factor. Therefore screening becomes stronger for larger .

For optical pumping in a 3D DM, we consider the following cases, (i) all nodes are equally affected by pumping, (ii) pumping is realized selectively for a certain number of nodes. In a WSM with two non-degenerate nodes, the first case corresponds to in the calculation of the screening wavevector while the second case to . This can be generalized to WSM with the total number of nodes , where is an even integer, and pumping on nodes, where . At the same time, one needs to keep track of the type of interactions (internodal or intranodal) that are possible in the two cases. For , both types of interactions are present and therefore both the EI and CDW phases can be realized. For , one of the cones has a population inversion, with finite Fermi surfaces for electrons and holes, while the other cone is in equilibrium and its corresponding Fermi surface shrinks to a point. The strongest pairing is realized for intranodal interactions. Internodal interactions are in principle possible but the resulting excitonic gap vanishes rapidly as a function of the mismatch between the equilibrium and non-equilibrium chemical potentials. (The same holds for intranodal interactions with mismatched electron and hole chemical potentials,  Triola et al. (2017)). In Fig. 4 we plot the results for the transient EI phase since the gap in the CDW phase is vanishingly small for the present choice of parameters ( and in dimensionless units). In a DSM with the minimal degeneracy only intranodal interactions are included, however for , both intranodal and internodal interactions are possible.

In the pumped 3D DM, the excitonic phase is characterized by gaps that open up at the non-equilibrium electron and hole chemical potentials as illustrated in Fig. 4. The gaps can be seen in the spectral function, which is indirectly probed in ARPES experiments. In the DOS, the gaps separate occupied and non-occupied states in the valence and conduction bands. Such spectroscopic features could be probed by scanning tunneling microscopy.

### iii.2 Phase diagrams of the excitonic condensate

The dependence of the excitonic gap on the dimensionless coupling constant in equilibrium and in pumped 3D DM is shown in Figs. 5 and 6 respectively. In equilibrium, there is a critical coupling for CDW(EI) phase such that for , the gap becomes different from zero. The values of the critical coupling that we obtained numerically using Eq. (43) and (44) with  num (b), are in agreement with analytical results of Ref. Wei et al. (2014a).

In the pumped regime with finite chemical potentials for electrons and holes, the critical coupling for excitonic instability vanishes (see Fig. 6). This numerical result was proven analytically using a model with contact interaction in the case of 2D DM Triola et al. (2017). Therefore, for any value of the coupling , both the CDW and EI phase can be realized in the pumped case. Both the internodal and intranodal interactions contribute to the gap opening at the non-equilibrium chemical potentials. However, due to the structure of the self-consistent gap equation the value of the excitonic gap in the EI phase is always smaller. In our model calculations we are able to consider the two phases separately and to calculate the corresponding contributions to the excitonic gap.

In the equilibrium case with chemical potential exactly at the node, the size of the gap is controlled by the strength of the coupling . In the pumped regime, there are two additional factors, the degeneracy which affects the screening and the chemical potentials, and , which control both the screening and the DOS at the electron and hole Fermi surfaces. Figure 7 shows the dependence of the gap on for fixed and equal in magnitude chemical potentials and few different values of . One can see the reduction in the size of the gap with increasing due to screening.

An important question to answer is whether optical pumping creates more favorable conditions for excitonic instability compared to equilibrium. In principle, in the pumped regime excitonic condensation occurs for arbitrary weak coupling strengths due to finite DOS at the electron and hole chemical potentials (see Fig. 6). However, the size of the gap decreases with increasing the chemical potentials due to screening. As a result of this interplay, pumping is efficient only in a certain segment of the parameter space defined by material parameters as illustrated in Figs. 8 and 9. Here we introduce for convenience a single quasi-equilibrium chemical potential assuming . Equal chemical potential for electrons and holes can be realized if the equilibrium chemical potential is at the Dirac point before pumping (see Fig. 1).

The phase diagram in Fig. 8(a) shows the maximum of the gap as a function of the chemical potential and the dimensionless coupling in the range for a WSM with . Clearly, there is a reduction of the critical coupling for . By looking at the scans of the phase diagram, vs curves for few different ’s in Fig. 8(b), one can see that for the gap is zero in equilibrium () and its size increases monotonically with increasing . In this segment of the parameter space, optical pumping promotes excitonic instability while the equilibrium system remains gapless.

For larger and , screening becomes stronger. As a result the size of the gap in the pumped regime decreases and becomes smaller than the equilibrium gap [Fig. 8(b) for ], provided that exceeds the critical coupling for excitonic instability in equilibrium. Similar behavior is observed for larger degeneracy factor as shown in Fig. 9(a) and (b). The gap decreases further with increasing .

The effect of pumping can also be seen in the behavior of the critical temperature as a function of the chemical potential . We define as a value of the temperature such that for the excitonic gap is different from zero. Figure 10 shows the maximum of the gap as a function of and for a WSM with [Fig. 10(a)] and [Fig. 10(b)]. The values of are chosen to be just below the equilibrium critical coupling for the EI and CDW phases. This is the coupling regime considered in Fig. 8 and Fig. 9 in which pumping is most efficient (for the present choice of parameters). The line separating the dark () and bright () regions of the phase diagram defines the dependence of on the chemical potential. As shown in Fig. 10(a), increases with the chemical potential until it reaches a maximum at . Further increasing leads to reduction of due to screening. Similar behaviour is observed for [see Fig. 10(b)]; however, reaches a maximum at a slightly larger due to different values of and .

### iii.3 Estimates of critical temperature and excitonic gap

In order to provide an estimate of the size of the effects proposed in this work, we consider some examples of real material realizations of DSM and WSM (see Table 2). Anticipating future material discoveries, we also consider several examples of 3D DMs with parameters tuned in such a way as to reduce the screening effects and maximize the size of the gap and . Although many 3D DMs have been proposed in the last few years, here we focus on two examples, CdAs and TaAs, for which extensive ARPES data and detailed density functional theory (DFT) calculations are available. This allowed us to extract material parameters for numerical calculations.

There are several important parameters that control the size of the gap and , (i) the Dirac cone velocity and the dielectric constant of the material which determine the value of the dimensionless coupling constant , (ii) the energy scale over which the 3D Dirac states exist (the cutoff energy scale in our calculations); this energy scale limits the range of chemical potentials of the inverted electron/hole populations that can be achieved by pumping, (iii) the Dirac cone degeneracy . In the majority of 3D DMs the Dirac dispersion is anisotropic, with the velocity in the -direction () typically different from in-plane velocities (). Also the velocities in the upper and lower Dirac cones might differ slightly. For our numerical estimates, we use the average velocity. The Dirac cone degeneracy in real materials can be quite large, which is detrimental for excitonic effects in pumped 3D DM due to metallic screening. Here we present optimistic estimates assuming selective pumping with small Dirac cone degeneracy that gives the maximum gap and ( for a WSM and for a DSM in the CDW phase). For hypothetical 3D DM, we consider a larger range of ’s to show the reduction of the gap with increasing the degeneracy.

The results of numerical calculations for and are summarized in Table 2. The first two rows correspond to CdAs DSM and TaAs family of WSMs which includes TaAs,TaP,NbAs, and NbP. Values of the dimensionless coupling constant for these materials are based on the values of Dirac velocities and dielectric constants found in the literature. The energy scale of the 3D Dirac states is extracted from ARPES and DFT (see Table 2). The rest of the results refer to 3D DMs with theoretical parameters. As one can see from Table 2, in TaAs WSM the excitonic gap can reach  meV with critical temperature of few K ( K). In CdAs, we find a gap to be only a small fraction of meV which might be difficult to observe experimentally. This is mainly due to relatively large dielectric constant, close to that of a typical 3D TI, which leads to small coupling constant .

Our theory allows us to derive a set of general criteria for material candidates in which transient excitonic instability can be realized and possibly observed experimentally. The criteria are (i) large coupling (i.e. small Dirac cone velocity and small dielectric constant), (ii) large (the energy extent of the Dirac states), and (iii) small Dirac cone degeneracy. This is illustrated by our results for hypothetical 3D DM (see the last three rows in Table 2) where we assumed similar to that of graphene and . For degeneracy , a gap of tens of meV’s and of the order of  K can be achieved.

### iii.4 Optical conductivity in pumped 3D DM

The real part of the frequency-dependent optical conductivity can be calculated using the spectral form of the Kubo formula

 Re[σαβ(Ω)] = e2πΩ∫+∞−∞dω[nF(ω)−nF(ω+Ω)] (45) ×∑kTr[^vαA(k,ω)vβA(k,ω+Ω)],

where is the velocity operator and is the spectral function. In this section we will consider a DM in the normal state, both in equilibrium and with population inversion, and in the exciton phase. In the normal state, represents the spectral function of the quasiparticles. In the exciton phase, is the full spectral function, which is a matrix whose diagonal (off-diagonal) elements give the quasiparticle (anomalous) spectral function Schachinger and Carbotte (2017). In what follows, we only present the final results of the calculations for different cases considered (for details of the calculations see Appendix E.)

#### Equilibrium

We start with a brief reminder on the optical properties of non-interacting Dirac fermions in equilibrium (no population inversion). Optical conductivity of a WSM have been studied in recent work Hosur et al. (2012); Timusk et al. (2013); Ashby and Carbotte (2014); Tabert and Carbotte (2016). Here we consider a non-interacting Weyl Hamiltonian for one node with a fixed chirality (). For concreteness we assume that the chemical potential . After calculating the spectral function of a WSM (see Appendix E), we find that the optical conductivity is given by

 Re[σxx(Ω)] = e26πvFΩ∫+∞−∞dω[nF(ω)−nF(ω+Ω)] (46) ×∫∞0dεε2[A+A′++A−A′−+2(A+A′− ×∫∞0ε2[A+A′++A−A′−+A−A′+)],

where and . The first two terms in the kernel of the energy integral in Eq. (46) give the intraband contribution to the optical conductivity, while the last two terms give the interband contribution.

At , we obtain the expressions for the intraband and interband conductivity, respectively

 Re[σintraxx(Ω)] =e2μ26πvFδ(Ω) (47) Re[σinterxx(Ω)] =e2Ω24πvFΘ(Ω−2μ), (48)

where is the chemical potential of the system and is a step function ( for and for ).

As one can see from Eq. (48), in 3D DM the interband conductivity vanishes linearly with , unlike in graphene where the interband piece is constant and equal to , the universal optical conductivity of clean graphene. The linear frequency dependence of the optical conductivity of 3D Dirac fermions has been observed experimentally Timusk et al. (2013).

#### Pumping: normal state

We will now consider the case of optical pumping in which population inversion is realized. Optical conductivity of pumped grapheme with population inversion has been studied in Refs. Svintsov et al. (2014a, b). In the case of population inversion, the Fermi-Dirac distributions of conduction band (electrons) and valence band (holes) are given by

 ne/hF(ω)=1/(e(ω−μe/h)/T+1), (49)

where is the corresponding chemical potential. Note that and are defined on a segment and , respectively. Assuming equal in magnitude chemical potentials for electrons and holes, the quasi-equilibrium chemical potential .

Interband optical conductivity for a pumped WSM with population inversion is given by

 Re[σinterxx(Ω),T] =e224πv[nhF(−Ω/2)−neF(Ω/2)]. (50)

In the equilibrium case [Eq. (48)], direct interband transitions are possible only if , while in the pumped case interband conductivity is positive for and negative for . This is also illustrated in Fig. 11. The analytical expressions for optical conductivity for 2D (grapheme) and 3D (single Weyl node) in equilibrium and under population inversion are summarized in Table 3. Numerical calculations of the optical conductivity with momentum-dependent gap including quasiparticle and anomalous contributions will be considered elsewhere.