Multiple optical gaps and laser with magnonic pumping in 2D Ising superconductors

Mikhail SilaevComputational Physics Laboratory, Tampere University, Tampere 33720, FinlandMoscow Institute of Physics and Technology, Dolgoprudny, 141700 Russia and

Institute for Physics of Microstructures, Russian Academy of Sciences, 603950 Nizhny Novgorod, GSP-105, Russia(Dated: November 8, 2021)

Ising superconductivity has been recently discovered in 2D transition metal dichalcogenides. Wereport that such superconductors have unusual optical properties controlled by the in-plane Zeemanfield. First, we find several optical gaps visible as peaks of the conductivity and the Raman suscep-tibility. Moreover, we find that the Ising spin splitting in the spectrum of Bogolubov quasiparticlesenables strong population inversion generated by the time-dependent Zeeman field. Ultimately thisleads to the possibility of the superconducting laser with magnonic pumping which can be realized inthe van der Waals structures consisting of the Ising superconductor and the ferromagnetic insulatorlayers.

PACS numbers:

It is difficult to overestimate the importance of theoptical probes for studies of superconducting materials.The existence of superconducting gap in the quasiparti-cle spectrum has been discovered with the help of thefar-infrared optical conductivity1–3. Later on the su-perconducting gap has been observed by the inelasticRaman scattering4–6, which has become one of the ba-sic tools in studying the anisotropic7,8 and multibandsuperconductivity9–11 as well as the superconducting col-lective modes4,12–19.

Previous studies of the optical properties have beenfocused on the 3D materials. Recently the fam-ily of 2D superconducting materials has appeared20,in twisted bilayer graphene21–23, rhombohedral trilayergraphene24 and few-layer transition metal dichalco-genides (TMD)25–31. The in-plane symmetry break-ing combined with the heavy transition-metal atomsleads to the extremely large spin-orbital coupling inTMD32–36 (SOC). It has the Ising character having theform of the effective out-of-plane Zeeman field chang-ing the sign between electron pockets at K and K ′

points of the Brillouin zone25,26,34–37. Recently dis-covered so-called Ising superconductivity (IS)25,26,37–39

features strong enhancement of the in-plane criti-cal field26,30,37,40–42, giant thickness-dependent tran-sition state spin Hall signal43, electric field effecton superconductivity26,44, unconventional supercurrentphase45,46, multiple gaps47, collective modes48, and non-trivial interplay with magnetism46,49–55. Theoreticallysuch superconductors have been studied in the contextof the unconventional pairing mechanisms56–63, magneticproperties57,58,64, parity58,59,62,65,66 and singlet-tripletmixing superconductivity58,59,61,62,65,66, topological37,transport properties67,68 and magnetic properties66,69,70.

One can expect that 2D IS should have unusual opticalproperties but there have been no studies in this direc-tion. In the present Letter we predict multiple peaks inthe spectrum of the optical conductivity σ(Ωph) and the

Raman scattering χRam(ΩRam) where ΩRam = Ω(f)ph −

Ω(i)ph is the Raman shift, see Fig.1a,b. These features are

FIG. 1: Optical properties of the IS. (a) Photon reflectionwith the frequency Ωph. (b) Raman scattering from the inci-

dent to the reflected photon of the frequencies Ω(i)ph and Ω

(f)ph ,

respectively. (c) Superconducting laser with magnon pump-ing generated by the ferromagnetic insulator (FI) with themagnetization M precessing at frequency Ωm.

controlled by the on-plane Zeeman field which can be in-duced either by the external magnetic field or by the mag-netic proximity effect in the Van der Waals (VdW) struc-ture consisting of the superconductor and ferromagneticinsulator (FI) layers. Recently the VdW IS/FI struc-tures like NbSe2/ CrB3 or other combinations has beenstudied experimentally46,49–55 and theoretically70. Theoptical response of IS pumped by magnons Fig.1c, thatis by the time-dependent Zeeman field features the lasingeffect. Ising spin splitting of Bogolubov spectrum allowsto generate very strong quasiparticle population inversionresulting in the negative conductivity Re(σ)(Ωph) < 0 ofthe IS.

Our model consists of the multiband 2D superconduc-tor with the pairing between the electronic states situ-ated at K and K ′ points of the Brillouin zone as shownin Fig.2a, with the Zeeman field is h. It is described bythe Hamiltonian

H = τ3(ξk − iΛ) (1)

Λ = i(βτ0σz + τ3hσ) + τ1∆. (2)

where τ1,2,3 and σx,y,z are the Pauli matrices in Nambuand spin spaces, respectively. The kinetic energy relativeto the Fermi level is ξk = (k2 − k2

F0)/2m = vFk, wherethe Fermi velocity vF in general depends on the momen-

arX

iv:2

111.

0362

3v1

[co

nd-m

at.s

upr-

con]

5 N

ov 2

021

2

FIG. 2: (a) Schematic Fermi surface of the two-dimensionalmetal with Ising SOC field oriented along z causing the spin-splitting between spin-up (blue) and spin-down (red) states.The parts of Fermi surface in the second Brillouin zone areshown by dashed lines. (b) The model piecewise-constantdependence of the SOC strength β(θΓ). (c) Bogolubov quasi-particle spectrum in the Fermi surface pocket near the Kpoint. The red and blue lines correspond to the spin-up andspin-down Fermi surfaces as in (a).

tum direction. The SOC field which in general has to bethe odd function of momentum direction relative to the Γpoint β(kΓ) = −β(−kΓ) is taken in the model piece-wiseconstant form shown in Fig.2b. The order parameter ∆ isassumed to the isotropic. The in-plane Zeeman field hxis induced either by the external magnetic field h = µBBor by the magnetic proximity effect with the adjacent FI.

The Hamiltonian (1 ) yields Bogolubov spectrum con-sisting of two spin-split branches

E↑(↓)(k) =

√E2bdg + β2 + h2 ± 2

√h2E2

bdg + ξ2kβ

2 (3)

Ebdg =√ξ2k + ∆2 is the Bogolubov spectrum in the usual

superconductor. For h = 0 the Ising SOC splits theusual Bogolubov spectrum by shifting the spin-up/downbranches left/right as shown in Fig.2c. The spin-up/down branch intersection at k = kF0 shown in Fig.2c,that is at ξk = 0 is removed by the Zeeman field h ⊥ z.For h 6= 0 the spectrum (3) has two types of gaps. The or-

dinary gap has the magnitude of ∆ord = ∆√

1− h2/β2

and is located at ξk =√β2 − h2∆2/β. Note that the

spectrum becomes gapless ∆ord = 0 at h > β while theorder parameter remains non-zero ∆ 6= 0. The addi-tional so-called ”mirage” gap64 is located at ξp = 0 and

separates the gap edges at ∆± =√

(h±∆)2 + β2. Thespectrum in the illustrative case β = 3∆, h = 2∆ isshown in Fig.3a.

The average 〈Sz〉 spin projection is shown by the colorcode of the spectrum branches in Fig.3a. Near the usualgap edge the spin state is almost 〈Sz〉 = ±1/2. On theother hand, near the mirage gap edges ξk = 0 the spinstate 〈Sz〉 = 0, which indicates the dominating role ofthe spin-triplet pairing at this point.

Several van Hove singularities in the spectrum (3) re-sults in the multiple optical gaps visible as peaks in theoptical responses. In Fig.3a three most important of

them are indicated by the dashed arrows I, II and III.The excitation process I corresponds to the breaking ofthe usual spin-singlet Cooper pairs, while II and IIIcorresponds to the breaking of the spin-triplet Cooperpairs. The corresponding optical gaps are ΩI = 2∆ord,ΩII = ∆ord + ∆− and ΩII = ∆ord + ∆+ respectively.At such frequencies the external perturbation creates thepair of quasiparticles at the van Hove singularities whichwill be shown to result in peaks in the optical responses.

For h = 0 in Eq.(3) the Ising spin-splitting makes theminimal energy difference between empty spin-up andoccupied spin-down branches to be min(E↑ + E↓) = 2β.Therefore, magnon pumping with frequency Ωm ≈ 2βcreates the hot spot in the quasiparticle population in-version near ξk = 0. This leads to the possibility ofthe quasiparticle transitions between ξk = 0 and ξk 6= 0states assisted by the impurity scattering and accom-panied either by the photon radiation or absorption.Since the density of states increases at lower energiesN(ε < β) > N(ε > β) there are more emission processesresulting in the lasing which shows up as the negativeconductivity Reσ < 0.

FIG. 3: (a) Bogolubov spectrum of the IS with in-planeZeeman field hx 6= 0. The color code of the branches is deter-mined by the average spin projection 〈Sz〉 projection rangingfrom +1/2 to −1/2 shown by the blue and red colors respec-tively. The dashed arrows I, II, III show excitation processescorresponding to the optical gaps ΩI , ΩII , ΩIII . (b) Lasingeffect in the IS with magnon pumping. The vertical blackarrow shows the excitation of quasiparticles by the oscillatingZeeman field hm cos(Ωmt)x generating hot spot in the quasi-particle population (pink shading). The violet arrows showphoton emissions or absorptions with quasiparticle down/uptransitions assisted by the impurity scattering.

In each of the six Fermi pockets in Fig.2a we introducethe quasiclassical propagator71 g = g(t, t′, r,vF ) whichsatisfies by the Eilenberger equation72

ivF [τ3A, g]t + [AγA, g]t = (4)

− τ3∂t, gt + [Λ, g]t +[〈g〉, g]t

2τimp.

In general it contains the time-dependent vector poten-tial A(t) and the Zeeman field h = h(t). The last termin Eq.4 is the collision integral describing the intraband

3

impurity scattering with the rate τ−1imp. We denote the

commutators [X, g]t = X(t)g(t, t′)− g(t, t′)X(t′) and theconvolution is given by 〈g〉 g =

∫dt1〈g〉(t, t1)g(t1, t

′).The angle-averaging is done over the Fermi pocket cen-tered at the K point at θΓ = 0 shown in Fig.2a 〈g〉 =∫ 2π

0g(θK)dθK , where θK is the angle which characterises

the direction of the momentum k relative to the K point.Here we take into account the contribution of all three Kpoints in Fig.2a by extending the integral over the angleθK from 0 to 2π.

The first term in the l.h.s. of Eq.4 is the usual para-magnetic vartex which determines the conductivity andthe Meissner effect. The second term is the diamag-netic vertex describing the electron density modulationby the time-dependent A(t). This term comes with thecoefficient73 γij = ∂E/∂ki∂ki − 〈∂E/∂ki∂ki〉 where thesubtraction of Fermi pocket average term takes into ac-count Coulomb interaction. The charge neutrality inEq.4 is maintained since 〈γ〉 = 0. However the direction-dependent density variation responsible for the Ramanscattering74,75is non-zero.

(b)(a)

FIG. 4: (a) The order parameter ∆(T, h)/∆0. (b) The den-sity of states N(ε, h) taken at T = 0.01. In both panelsβ = 4∆0.

The equilibrium GF g0(ω)eiω(t−t′) is determined bythe stationary Eilenberger Eq.4 in the imaginary timedomain t → it with A = 0 and h = hx = const. Ithas an analytic solution71. With the equilibrium GF wecan study the order parameter determined by the self-consistency equation

∆ = λπT

4

∑

ω

Tr[τ1g0(ω)] (5)

where ω runs over fermionic Matsubara frequencies. Thedependence ∆(h, T ) Fig.4a for β = 4∆0 strongly exceeds

the Chandrasekhar-Clogston limit76,77 ∆/√

2. Here weuse the usual units of ∆0 = 1.76Tc0, where Tc0 is the crit-ical temperature at h = 0. The density of states is givenby the analytic continuation N(ε) = ReTr[τ3g0(−iε)]/4.The function N(ε, h) at T = 0.01Tc0 is shown in Fig.4b.It demonstrates the peaks both at the usual spectral gapand at the ”mirage” gap edges in accordance with thevan Hove singularities of the spectrum (3).

Linear response conductivity. The conductivity ten-sor is given by the Fermi pocket average of the com-bination σij = νe2〈vFivFj〉Kσ, where ν is the normal

density of states, e is the electron charge. The direction-independent conductivity amplitude σ is determined bythe correction to the GF linear in the applied ac fieldAeiΩpht which can be written in the form (vFA)gA.Then the conductivity is given by the expression in termsof the Keldysh component gKA

σ

σn=

1

16Ωph

∫ ∞

−∞dεTr[τ3g

KA ] (6)

where σn = τimp/4 is the normal state conductivity. Thefunction gA can be found analytically from the Eilen-berger Eq.4 linearized with respect to A. The particu-larly simple expression is obtained in the limit of large im-purity scattering τimp∆ 1. We use this approximationto find gA for Matsubara frequencies and and implement-ing the analytical continuation to the real frequencies71.The resulting conductivity spectrum σ(Ωph) is shown inthe Fig.5a.

(a) (b)

(c) (d)

FIG. 5: (a) Conductivity σ(Ωph) of the IS. (b) Frequencyderivative of conductivity dRe(σ)/dΩph . (c) Raman sus-ceptibility χRam(ΩRam) of the IS. (d) Frequency derivativeof Raman susceptibility Im(dχRam/dΩRam). Arrows I, II,III point to the features corresponding to the resonant pair-breaking processes shown in Fig.3a. Black solid lines in (b,d)show the optical gaps ΩI(h), ΩII(h), ΩIII(h). In all pan-els T = 0.01Tc0, β/∆0 = 4. In (a,c) h/∆0 = ±2.8 whichcorresponds to the crossection along dashed lines in (b,d).

Raman susceptibility Now let us consider the Ra-man susceptibility determined by the generalized density-density correlation function73–75,78,79 ν〈(eiγef )2〉χRam.Here the prefactor is the Raman vertex determined bythe initial Ai = eiΩitAiei and final Af = eiΩf tAfefphoton polarizations. The isotropic amplitude which wedenote for brevity χRam can be found by solving the

4

Eilenberger Eq.4 linearized with respect to the diamag-netic driving term (AiγA

∗f )eiΩRamt. The corresponding

correction to the GF has the form (AiγA∗f )gAA. It can

be found analytically in the limit of large impurity scat-tering τimp∆ 1, see details in71. Then the Ramansusceptibility is given by the expression in terms of theKeldysh component gKAA

χRam

χ(n)Ram

=1

16ΩRam

∫ ∞

−∞dεTr[gKAA] (7)

where χ(n)Ram = ΩRamτimp/4 is the normal metal Raman

susceptibility amplitude in the dirty limit. Implement-ing the analytical continuation to the real frequencies71

ΩRam → iΩRam yields the Raman susceptibility shownin the Fig. 5c,d.

Both σ(Ωph) and χRam(ΩRam) have three distinct fea-tures marked by the arrows I, II and III in Fig.5 corre-sponding to excitation processes across the optical gapswith the same numbering in Fig.3a. To make these fea-tures more visible we plot the derivatives dReσ/dΩph inFig.5b and dχRam/dΩRam in Fig.5d. One can see thatthe these derivatives have three peaks which perfectlycoincide with the optical gaps shown by the black solidlines ΩI(h), ΩII(h), ΩIII(h). Note that these dependen-cies are symmetric by h→ −h.

Superconducting laser with magnon pumping. Thehigh-frequency electromagnetic pumping generates thepopulation inversion in the spectrum of Bogolubov quasi-particles which leads to the non-trivial effects such as thestimulated superconductivity80–84. However this type ofthe population inversion is not sufficient for the super-conducting laser, that is for the stimulated light missiondue to the dipolar transitions between the different quasi-particle states.

FIG. 6: Re(σ)(Ωph,Ωm) of the IS with magnon pumpingh = xhm cos(Ωmt), hm =

√ΓTc0, Γ = 0.01Tc0, T = 0.01Tc0,

β = 4∆0. The lasing regime corresponds to Re(σ) < 0 in(a,b) according to the mechanism in Fig.3b.

The situation changes qualitatively in the presenceof the Ising spin splitting in the Bogolubov spectrumshown in Fig.3b. In this configuration one can generatea very strong quasiparticle imbalance by inducing transi-tions between spin-down and spin-up branches shown bythe vertical dashed line corresponding to the magnonicpumping.

We assume that the superconductor is pumped by Zee-man field h = hm cos(Ωmt)x and calculate the linear re-sponse conductivity σ(Ωm,Ωph). The total conductivitycan be written as the two parts σ = σeq + σhh. The firstterm here σeq is the conductivity of the equilibrium su-perconductor given by the Eq.6. The second term σhhis determined by the correction to the GF linear in theapplied ac field AeiΩpht and of the second order in theoscillating Zeeman field amplitude (vFA)h2

mgAhh. Thiscorrection can be found analytically in the limit of largeimpurity scattering rate τimp∆ 1. For this we use iter-ative solution71 of the Eilenberger Eq.4 to get the station-ary second-order correction driven by the Zeeman h2

mghh.We use at as a source to find the correction driven by theelectromagnetic field gAhh. Then, similarly to the usualconductivity (6), the magnon-driven correction σhh(Ωph)is given by

σhhσn

=h2m

16Ωph

∫ ∞

−∞dεTr[τ3g

KAhh] (8)

Note that the magnitude of the correction σhh is deter-mined by the ratio of pumping strength and the rate ofthe inelastic relaxation Γ, so that σhh/σn ∼ h2

mΓ. Sincethe inelastic relaxation is slow Γ/∆0 1 we can achievelarge σhh ∼ σeq with weak pumping hm ∼

√Γ∆0 .

The total conductivity Reσ(Ωph,Ωm) is shown in theFig.6 for the narrow interval of magnon frequencies Ωm ∼2β. There is a sharp negative peak at Ωm ≈ 2β andΩph ∆ corresponding to the superconducting laser ac-cording to the mechanism shown in Fig.3b. At largerpumping frequency Ωm ≈ 2.4β the hot spot in quasi-particle distribution becomes larger which leads to thestronger photon absorption than emission. In result thereappears a sharp positive peak Reσ(Ωph,Ωm) > 0. Thepeaks shown in Fig.6a at Ωm ≈ 2β are very sharp be-cause of the peculiar resonant nesting of the branches3. Indeed the interlevel distance is almost constantmin(E↑ + E↓) ≈ 2β at |ξk| < β which leads to the exci-tation of many momentum states by the same pumpingfrequency Ωm ≈ 2β.

To conclude, we predict unusual optical properties ofIS resulting from the interplay of the superconductiv-ity, SOC and ferromagnetism. Multiple optical gaps inconductivity and Raman susceptibility offer the directway to detect the finite-energy spin-triplet pairing cor-relations and the ”mirage” gap64,68 controlled by the in-plane magnetic field. Moreover, we have shown that theIsing spin splitting of Bogolubov quasiparticle spectrumallows for the generation of quasiparticle population in-version driven by the oscillating Zeeman field. This ef-fect leads to the predicted superconducting laser whichpotentially can be realized in VdW IS/FI structures.

Author thanks Alexander Mel’nikov for useful discus-sions. This work was supported in part by the Russianscience foundation.

5

Appendix A: Calculation of the equilibrium Greenfunction (GF)

The full GF is determined by the Gor’kov equation forimaginary frequeincies

G−10 G = 1 (A1)

G−10 = H + iω (A2)

We will need the quasiclassical GF which can be ob-tained from the full GF by integrating over the kineticenergy in the vicinity of the Fermi surface

g =i

π

∫dξk τ3G (A3)

The full GF determined by (A1) can be written as

G =C0 + ξkC1 + ξ2

kC2 + ξ3kC3

(ξ2k − E2

1)(ξ2k − E2

2)(A4)

where

E1,2 =

√h2 + β2 −∆2 − ω2 ± 2i

√β2(∆2 + ω2) + h2ω2

(A5)

Then the integration over ξp yields

∫dξk

(ξ2k − E2

1)(ξ2k − E2

2)=

iπ

2E1E2(E1 + E2)(A6)

∫dξk

ξ2k

(ξ2k − E2

1)(ξ2k − E2

2)= − iπ

2(E1 + E2)(A7)

Hence

g0 =1

2(E1 + E2)

(C2 −

C0

E1E2

)(A8)

It is instructive to check the structure of quasiclassicalGF in spin-Nambu space. In general it is given by

g0 = gz0σz τ0 + gy2σy τ2+ (A9)

(g01σ0 + gx1σx)τ1 + (g03σ0 + gx3σx)τ3

The last four terms in the r.h.s. are the usual ones forthe superconductor with the Zeeman field. The first twoterms appear due to the interplay of the Ising SOC andthe Zeeman spin splitting fields.

Appendix B: Calculation of the conductivity andRaman susceptibility

To find the conductivity we need to determine the lin-ear correction to the GF (vFA)gA(ω+, ω) generated bythe applied ac field AeiΩpht where ω+ = ω + Ωph. For

that we need to solve the linearized Eilenberger equationfor imaginary frequencies

i[τ3g0(ω)− g0(ω+)τ3] = (B1)

[Λ, gA] + [ω+τ3gA − ωgAτ3]+

τ−1imp[g0(ω+)gA − gAg0(ω)]

The solution can be found analytically in the limit ofτimp∆0 1 in the form

gA =iτimp

2[g(ω+)τ3g(ω)− σ0τ3] (B2)

Implementing the analytical continuation to the real fre-quencies we obtain the Keldysh component gKA and hencethe conductivity.

The Raman susceptibility is given by the linear re-sponse relation written in terms of the quasiclassicalGF correction linear in the diamagnetic term in theEilenberger equation (AiγA

∗f )gAA(ω+, ω) where ω+ =

ω + ΩRam. The Eilenberger equation for the correctiongAA reads as

g0(ω)− g0(ω+) = (B3)

[Λ, gAA] + [ω+τ3gAA − ωgAAτ3]+

τ−1imp[g0(ω+)gAA − gAAg0(ω)]

where we take into account that 〈(eiγef )〉 = 0. The solu-tion can be found analytically in the limit of τimp∆0 1in the form

gAA =iτimp

2[g(ω+)g(ω)− σ0τ0] (B4)

Implementing the analytical continuation to the real fre-quencies we obtain the Keldysh component gKAA andhence the Raman susceptibility.

Appendix C: Conductivity of the Isingsuperconductor with magnon pumping

We assume that the Ising superconductor is subjectedto the ac electromagnetic field AeiΩpht and the Zeemanfield with two tones h(t) = hm(eiΩm1t+ eiΩm2t). In orderto construct correctly the analytical to real frequencieswe need to consider from the beginning the general casewith Ωm1 6= −Ωm2. We introduce the following notationfor frequencies

ω1 = ω, (C1)

ω2 = ω + Ωm1,

ω3 = ω + Ωm2

ω4 = ω + Ωm1 + Ωm2

ω5 = ω + Ωph

ω6 = ω + Ωm1 + Ωm2 + Ωph.

First, we find the second-order correction to the GFghh(41) ∼ h2

m, where we introduce notation ghh(41) ≡

6

ghh(ω4, ω1), so that the in the time domain it is

ghh(t, t′) = ghh(ω4, ω1)ei(ω4t−ω1t′). After the analytical

continuation we will put Ωm1 = −Ωm2 = Ωm to getthe stationary correction ghh(t, t′) = ghh(t − t′). Wesearch this correction by solving the Usadel equation byiterations with respect to the Zeeman field in the imagi-nary frequencies. We start with the first-order correctiongh = hm(gxσx+ gyσx). For that we need to solve the lin-earized Eilenberger equation for imaginary frequencies

iσx[τ3g0(ω)− g0(ω+)τ3] = (C2)

[Λ, gh] + [ω+τ3gh − ωghτ3]

where ω+ = ω2 for gh(21) or ω+ = ω3 for gh(31). Thesolution is given by

gx(21) =i(s1 + s2)

(s1 + s2)2 + 4β2[g0(2)τ3g0(1)− τ3] (C3)

gy(21) =2iβ

(s1 + s2)2 + 4β2[τ3g0(1)− g0(2)τ3] (C4)

For gh(31) we need to replace 2 → 3. The second ordercorrection ghh(41) can be found by solving the equation

s4g0(4)ghh − s1ghhg0(1) = (C5)

i[τ3gx(21)− gx(43)τ3] + i[τ3gx(31)− gx(42)τ3]

It is supplemented by the normalization condition

g0(4)ghh + ghhg0(1)+ (C6)

gx(43)gx(31) + gx(42)gx(21)+

gy(43)gy(31) + gy(42)gy(21) = 0

The solution is given by

(s1 + s4)ghh(41) = (C7)

ig0(4)[τ3(gx(21) + gx(31))− (gx(43) + gx(42))τ3]

− s1g0(4)[gx(42)gx(21) + gx(43)gx(31)]

− s1g0(4)[gy(42)gy(21) + gy(43)gy(31)]

To find the contribution to electric current in the pres-ence of magnon pumping we need to find the anisotropiccorrection to the GF which is linear in A and of the

second order in Zeeman field (vFA)h2mgAhh. We con-

sider the limit of strong impurity scattering when theanisotropic correction driven by the field AeiΩpht to theGF is in general given by (vFA)gA

gA(t, t′) = (C8)

iτimp2

[g(t, t1)eiΩpht1 τ3 g(t1, t′)− τ3σ0δ(t− t′)eiΩpht]

where g(t1, t′) is in general the non-equilibrium the GF in

the absence of electromagnetic field. Taking into accountthat in the presence of the magnon pumping g = g0 + ghhand going to the Fourier representation we get from (C8)

gAhh(t, t′) = gAhh(61)ei(ω6t−ω1t′) where

gAhh(61) =iτimp

2[g0(6)τ3ghh(41) + ghh(65)τ3g0(1)]

(C9)

Using this expression we can use the analytical continua-tion to real energies to find the Keldysh component gKAhhwhich determines the correction to the conductivity σhh.

Appendix D: Analytical continuation

In order to find the real-frequency response we needto implement the analytical continuation. The analyti-cal continuation of the sum by Matsubara frequencies isdetermined according to the general rule85

T∑

ω

g1(ω1)g2(ω2)→ (D1)

∫dε

4πin0(ε1)

[g1(−iεR1 )− g1(−iεA1 )

]g2(−iεA2 )+

∫dε

4πin0(ε2)g1(−iεR1 )

[g2(−iεR2 )− g2(−iεA2 )

]

where n0(ε) = tanh(ε/2T ) is the equilibrium distributionfunction. In the r.h.s. of (D1) we substitute ε1 = ε,ε2 = ε + Ω and εR/A = ε ± iΓ. Here the term withΓ > 0 is added to shift of the integration contour intothe corresponding half-plane.

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Optical signatures of Ising superconductivity(Dated: November 8, 2021)

We report that Ising superconductivity in 2D superconductors shows up as peaks in both thelinear conductivity and Raman scattering amplitude induced by the external in-plane magneticfield.

PACS numbers:

I. INTRODUCTION

II. RESULTS

A. Spectrum and order parameter

The full GF is determined by the Gor’kov equation

G−10 G = 1 (1)

G−10 = τ3(ξp + βσ3) + τ1∆ + τ0(iω + hσx) (2)

The Zeeman field is induced by the external magneticfield h = µBB.

We will need the quasiclassical GF which can be ob-tained from the full GF by integrating over the kineticenergy in the vicinity of the Fermi surface

g =i

π

∫dξpτ3G (3)

The equilibrium quasiclassical GF is determined by theEilenberger equation

[Λ, g] = 1 (4)

Λ = βσ3 + τ1∆ + τ3(iω + hσx) (5)

The order parameter is determined by the self-consistency equation

∆ = πT∑

ω

Tr[τ1g(ω)] (6)

The density of states is N(ε) = Tr[τ3g(−iε)]. It is shownin Fig.1.

FIG. 1: (Color online) (Left) The order parameter∆(T, h)/∆0 and (Right) The density of states N(ε, h), T =0.01. β = 7.

B. Conductivity: Usadel theory

The conductivity is given by

σ

σn=πT

Ω

∑

ω

Tr[τ3g(ω + Ω)τ3g(ω)] (7)

where σn is the normal state conductivity.

FIG. 2: (Color online) Conductivity and Raman response ofthe Ising superconductor, T = 0.01Tc0.

FIG. 3: (Color online) Frequency derivative of conductivitydRe(σ)/dω and Raman response dIR/dω of the Ising super-conductor, T = 0.01, β = 7.

C. Raman scattering: dirty limit

Raman scattering vertex does not break the localcharge neutrality, i.e. it does not change the total elec-tron density. Therefore it can be calculated using qua-siclassical theory with the time-dependent diamagneticterm as the source γ(p)A2(t), where the coefficient γ(p)takes into account corrections to the electron mass due

arX

iv:2

111.

0362

3v1

[co

nd-m

at.s

upr-

con]

5 N

ov 2

021

2

to the interband transitions? . The charge neutrality ismaintained by adding the isotropic electric potential termequal to 〈γ〉FSA

2(t) where 〈γ〉FS is the average over theFermi surface. In the non-stationary situation the dia-magnetic term does not disappear from the Eilenbergerequation which then in the dirty limit reads as

e2

c2[γA2, g]t =

[〈g〉, g]

2τimp(8)

where γ = γ− γ. The solution can be written in the form

g = AΩ2A−Ω1

gAA (9)

gAA =e2

c2τimpγg(ω + Ω)g(ω) (10)

The Raman scattering amplitude is given by

IR = νπT∑

ω

Tr[〈γgAA〉FS ] (11)

D. Analytical continuation

In order to find the real-frequency response we needto implement the analytical continuation of Eq.(??,??).

These third-order responses are obtained by the sum-mation of expressions which depend on the four shiftedfermionic frequencies such as g = g(ω1, ω2, ω3, ω4). Theanalytical continuation of the sum by Matsubara frequen-cies is determined according to the general rule?

T∑

ω

g(ω1, ω2)→ (12)

∫dε

4πin0(ε1)

[g(−iεR1 ,−iεR2 )− g(−iεA1 ,−iεR2 )

]+

∫dε

4πin0(ε2)

[g(−iεA1 ,−iεR2 )− g(−iεA1 ,−iεA2 )

]

where n0(ε) = tanh(ε/2T ) is the equilibrium distributionfunction. In the r.h.s. of (12) we substitute in each termωk<l = −iεRk and ωk>l = −iεAk for k = 1, 2, 3, 4, denoteεk = ε + (3 − k)Ω and εR = ε + iΓ, εA = ε − iΓ. Herethe term with Γ > 0 is added to shift of the integrationcontour into the corresponding half-plane.