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Electronic Raman scattering from orbital nematic fluctuations

We compute Raman scattering intensities via the lowest-order coupling to the bosonic propagator associated with orbital nematic fluctuations in a minimal model for iron pnictides. The model consists of two bands on a square lattice exhibiting four Fermi pockets and a transition from the normal to a nematic state. It is shown that the orbital fluctuations produce in the B1g channel strong quasi-elastic light scattering around the nematic critical temperature Tn, both above and below Tn. This holds for the A1g symmetry only below Tn whereas no low-energy scattering from orbital fluctuations is found in the B2g symmetry. Due to the nematic distortion the electron pocket at the X-point may disappear at low temperatures. Such a Lifshitz transition causes in the B2g spectrum a large upward shift of spectral weight in the high energy region whereas no effect is seen in the other symmetries.


I. INTRODUCTION
Electronic nematic states are electronic analogues of nematic liquid crystals, which break only the orientational symmetry, but retain the other symmetries of the system. Electronic nematicity is discussed in a number of correlated electron systems such as quantum spin systems, 1 two-dimensional electron gases, 2,3 cuprate superconductors, 4,5 bilayer ruthenates, 6 and iron pnictides. 7 Depending on electronic degrees of freedom responsible for nematic order, we may distinguish between three kinds of nematicity. i) The charge nematicity which is obtained either by partial melting of charge stripes 8 or by a Pomeranchuk instability, 9,10 ii) the spin nematicity which is driven by frustration between magnetic interactions, 11 and iii) the orbital nematicity due to orbital order caused, for example, by a spontaneous occupation difference between d yz and d zx orbitals in d-electron systems. 12,13 In iron pnictides, electronic nematicity is associated with the tetragonal-orthorhombic structural phase transition. 7 The structural transition is believed to be driven primarily by coupling to the electronic system. In fact, electronic resistivity exhibits a pronounced anisotropy by applying a uniaxial strain to the system. Moreover, angle-resolved photoemission spectroscopy 7,14 revealed directly a sizable energy difference of d yz and d zx orbitals, indicating the importance of orbital nematicity. [15][16][17][18][19] Since the structural transition occurs slightly above a spin-density-wave (SDW) phase, spin nematicity is also discussed as a plausible scenario [20][21][22] .
Quite recently a nematic instability was observed also in magnetic torque measurements, 23 which are very sensitive to the breaking of a fourfold symmetry. The observed critical temperature is much higher than the onset temperature of the SDW phase and extends to regions far away from the SDW phase. It seems therefore reasonable to associate such a nematic instability to orbital nematicity. 23 Measurements of the anisotropy of the resistivity and magnetic torque provide only indirect evidence of electronic nematicity. In the case of charge nematicity, it was shown theoretically 24 that the Raman spectroscopy in the B 1g channel in a tetragonal system measures directly the charge nematic correlation function. Since the nematic instability does not break translational symmetry and thus is characterized by momentum zero, it is natural to believe that the Raman spectroscopy can become a suitable method to detect also orbital nematicity.
In the present paper, we provide a microscopic understanding of Raman scattering by orbital nematic fluctuations. In Sec. 2 we introduce a minimal two-band model for iron pnictides, which exhibits an orbital nematic instability at low temperatures and four Fermi pockets. The Raman scattering intensity is then computed in the lowest-order of the bosonic propagator associated with nematic fluctuations. Numerical results both in the normal and nematic states and their interpretation are presented in Sec. 3. The effect of Coulomb screening is also studied. Our conclusions then follow in Sec. 4.

A. Nematic transition in a minimal two-band model
Our model Hamiltonian has the form H = H 0 + H 1 where the interaction part H 1 is given by The difference density operator n i− is defined by n i− = n i1 − n i2 with the density operator n iα = σ c † iασ c iασ . i and σ are site and spin indices, respectively, and α = 1, 2 is a band index. g is a coupling constant which is considered as a parameter in our model. An expression for H 0 suitable for pnictides is 25,26 with Reasonable values for the hopping amplitudes are 26 t = −t 1 , t 2 /t = 1.5, t 3 /t = −1.2, t 4 /t = −0.95, which we will also use in our calculations. The band indices α = 1 and 2 originate mainly from the d zx and d yz orbitals, respectively. In the following energies are always given in units of t.
Taking H 1 into account the Green's function matrix in band space is given in mean-field approximation byĜ and n − = n i− . ... denotes the expectation value, µ is the chemical potential, and δ an infinitesimally small positive quantity. Carrying out the inversion in Eq. (6), rearranging terms and using Pauli matrixesτ 1 ,τ 2 ,τ 3 as well as the 2 × 2 unit matrixτ 0 we find Here we used the abbreviations n − satisfies the following nonlinear equation, The dispersion of the two bands in the normal state is shown in the upper panel in Fig. 1. The special k-points are Γ = (0, 0), X = (π, 0), M = (π, π), and Y = (0, π), and the dispersion is drawn along the lines passing through these points. The band λ + k (λ − k ) crosses the Fermi energy around X and Y (Γ and M) points. Consequently, the Fermi lines form two electron pockets at X and Y and two hole pockets at Γ and M, as shown in the right from C 4v to C 2v . As a result the pockets at the X and Y points are no longer related by symmetry. The electron pocket at the X point shrinks while that at the Y point expands.
When n − becomes sufficiently large a Lifshitz transition can occur at low temperatures where the pocket at the X point vanishes. This is illustrated in Fig. 1(b) and in the left inset of Fig. 1(c).

B. Raman scattering intensity
The Raman scattering intensity S γ (ω) is given by where b(ω) is the Bose function (e βω − 1) −1 and β −1 = T the temperature. χ γ (ω) is the retarded Green's function where N is the number of lattice sites, [·, ·] the commutator, and ρ γ (t) the operator in the Heisenberg picture. γ αβ (k) is the Raman vertex in the effective mass approximation given by e i and e f are the polarization vectors of the incident and scattered light, respectively.
The underlying point group C 4v gives rise to three independent cross sections corresponding to the representations B 1g , B 2g , and A 1g . The B 1g contribution is obtained by taking the polarization vectors e i = 1 √ 2 (1, 1) and e f = 1 It is convenient to consider γ B 1g αβ as a matrix in band spaceγ B 1g and to express the dependence on α and β in terms of Pauli matricesτ 1 ,τ 2 ,τ 3 . Inserting the explicit expressions for ǫ αβ k we obtainγ with In a similar way the B 2g contribution to the Raman vertex is obtained by taking e i = (1, 0) and e f = (0, 1):γ For the polarization vectors e i = e f = 1 √ 2 (1, 1), both A 1g and B 2g channels contribute to the Raman intensity. Subtracting the latter the A 1g contribution becomeŝ Using the random phase approximation χ γ (ω) is given by the bubble diagrams depicted in Fig. 2. The labelγ stands for one of the three Raman verticesγ B 1g ,γ B 2g orγ A 1g . The Here Π γ3 (Π 33 ) denotes a bare bubble with the verticesγ andτ 3 (τ 3 andτ 3 ).
Let us calculate Π αγ (ω) for general verticeŝ Performing the internal frequency sum by means of the spectral function γ denotes Raman vertices andτ 3 a vertex coming from the interaction H 1 . and carrying out an analytic continuation we obtain for the imaginary part of Π αγ (ω), where f is the Fermi function. Writinĝ we obtain from Eqs. (9)-(12) the following expressions for the coefficients, Carrying out the Tr in Eq. (37) the imaginary part of the considered bubble can be written as where the scalar product of two four-dimensional vectors G α and G γ′ appears. The components of these vectors are given by and All the susceptibilities Π occurring in Eqs. (32) and (33) can be obtained from the general expression Eq. (42) as special cases. Explicit expressions are given in the appendix. The calculation of the various susceptibilities in the Raman scattering intensity S γ (ω) [Eq. (17)] has thus been reduced to one frequency and one two-dimensional momentum integration which have to be carried out numerically. The corresponding real parts can be obtained by a Kramers-Kronig transformation.

III. RESULTS
We first present a general symmetry argument which shows that orbital nematic fluctuations couple in the normal state only to the B 1g channel and in the nematic state to both B 1g and A 1g channels. We then present numerical results for the Raman spectra by computing the diagrams shown in Fig. 2. The most interesting effect due to orbital nematic fluctuations is the appearance of a central mode in some of the Raman spectra. We will show that its main properties can be understood from analytic considerations. We will also study the effect of Coulomb screening on our obtained results.

A. Selection rules
As seen in Fig. 2(a), orbital nematic fluctuations couple to the Raman susceptibility via a bubble diagram with verticesγ andτ 3 . This diagram determines our selection rules for nematic fluctuations.
In the normal state the symmetry group of our system is C 4v . To see how the band basis transforms under its group elements, we consider each term on the right-hand side of Eq. (28). In the first term it is clear that both γ has B 2g symmetry whereas the product of γ A 1g 1 andτ 1 should transform as A 1g , which means thatτ 1 has B 2g symmetry. In the third term, γ A 1g 3 has B 1g symmetry and thuŝ τ 3 should also have B 1g symmetry so that their product transforms as A 1g . Sinceτ 3 has B 1g symmetry, the bubble with verticesγ andτ 3 is finite forγ B 1g and vanishes otherwise. This leads to our selection rule that orbital nematic fluctuations can be observed in the normal state only in the B 1g channel.
In the nematic phase the point group is reduced to C 2v . A 1g and B 1g denote then the same representation. Hence not only the bubble withγ B 1g andτ 3 but also that withγ A 1g andτ 3 can be nonzero. That is, in the nematic state orbital nematic fluctuations can be observed in both B 1g and A 1g channels but not in the B 2g channel.
The obtained selection rules can also be verified by computing Π γ3 (ω) directly. In the Appendix we indeed find in this way that Π A 1g 3 is zero in the normal state and that Π B 2g 3 is zero both in the normal and nematic states. The left panels in Fig. 3 show Imχ B 1g (ω) for various temperatures below and above the transition temperature T n , respectively, over a large energy interval of the order of the band width. In such a representation no distinct temperature dependence is visible even if the sys-     the total intensity is given approximately by the first term in Eq. (32) and is small and practically constant at very low energies.

Figures 4(a) and 4(c)
show Imχ A1g (ω) on a large energy scale for temperatures below and above T n , respectively. On this energy scale the temperature dependence of the spectra is quite weak. Since Π A 1g 3 = 0 for T > T n (see Sec. III A) the curves in Fig. 4(c) represent only the imaginary part of Π A 1g A 1g (ω). They exhibit two well pronounced peaks. The main peak at ω ≈ 7.5t arises from interband transitions near the k-points (π/4, 3π/4) and (3π/4, π/4) and the other peak at low energy is due to intraband transition. The curves in Fig. 4(a) include both terms in Eq. (32). However, the contribution from the second term is minor in the energy interval considered in the figure and the curves describe essentially Π A 1g A 1g (ω).

Figures 4(b) and 4(d) show the A 1g Raman intensity in a very small frequency interval
near ω = 0. In these plots the low-frequency behavior of the A 1g Raman scattering becomes visible which cannot be seen in the left panels because of their large energy scales. For   linearly in the frequency, passes through a maximum near ω ≈ 2δ and then decays rapidly with increasing frequency. On the other hand, the interband contribution is very small at small frequencies, rises with increasing frequency and shows a sharp maximum near ω ≈ 7t due to strong transitions between the two bands near the point k = (π/2, π/2), see the band structure shown in Fig. 1(a). The interband contribution extends over a large frequency region comparable to the total band width. The lower panel in Fig. 6 shows ImΠ 33 (ω) for several choices of temperatures. It depends in general only weakly on temperature and this holds both for the intra-and the interband contributions.
The second term in Eq. (32) exhibits a pole which describes in the static limit a nematic phase transition with a transition temperature T n determined by It is easy to show that T n coincides with the largest temperature where Eq. (16) has a nonvanishing solution for n − , which is the usual definition of the transition temperature. Since the central peak emerges near T = T n and ω = 0, we take T − T n and ω as small quantities.
Since Π 33 (ω) depends only weakly on T one may put in the expression everywhere T = T n except in the first term in the denominator. Writing for small frequencies we obtain with where we denote the temperature dependence explicitly as a second argument in Π 33 .
Putting in numbers one realizes that m(0) is very small compared to one for the parameter range considered by us. One reason for this is that ReΠ 33 depends only weakly on temperature which makes the numerator of m 2 (ω) small. As a result the second term in Eq. (32) represents a low-energy contribution to the Raman susceptibility. Going over to the Raman scattering intensity Eq. (17) and taking the classical limit for the Bose function, (1 + b(ω))/π → T /(πω), we obtain approximately for the low-energy Raman response, Neglecting the ω-dependence of m 2 (ω) the Raman intensity consists at low frequencies of a central peak of a Lorentzian shape with width m(0); its peak height S γ (0) is proportional to

D. Coulomb screening
It is known that the long-range Coulomb interaction may strongly screen the first diagram in Fig. 2 Fig. 2(a) by replacingτ 3 byτ 0 and the coupling constant g by the bare Coulomb potential V (q) where q is the momentum of the incident photon. Taking the limit q → 0 the additional contribution due to Coulomb screening is given by The explicit calculation of Eq. (57) shows that the bubbles in this expression contain only intraband scattering processes and thus may become relevant only at low energy, similar as in the case of Fig. 6(a). Figure 7 shows for a representative case A 1g spectra with (solid line) and without (dashed line) Coulomb screening. As expected the two curves are very similar or practical identical at energies larger than about t whereas the height of the lowenergy maximum near 0.2t is noticeably reduced by screening. The inset in Fig. 7 exhibits the scattering intensity S A 1g (ω) on a very low energy scale. In this range S A 1g is mainly determined by the second term in Eq. (32) which is unaffected by Coulomb screening. This and B 2g so that these channels are unaffected by the Coulomb interaction in this case.
On the other hand, Π γ0 (ω) is nonzero for γ = B 1g in the nematic state (see Sec. III A).
However, we find that the resulting changes in Fig. 3 would be invisibly small. We also have considered corrections of the nematic fluctuations due to the Coulomb interaction, i.e., where the propagator D 33 (ω) in Eq. (33) contains contributions from Π 00 (ω) due to the non-vanishing Π 30 (ω) in the nematic state. We found that these effects are very small and do not change substantially the presented results in the nematic state.

IV. CONCLUSIONS
Our analysis shows that low-energy orbital fluctuations near the nematic transition temperature T n can produce a central peak in the Raman intensity S γ (ω). Depending on the symmetry this central peak may appear both above and below T n as in the case of the B 1g spectrum, only below T n as for the A 1g or not at all as for the B 2g spectrum. The area under the central peak diverges as |T − T n | −1 and the peak width becomes narrower as |T − T n |. While our theory contains the damping δ of the electrons, we have found that the integrated spectral weight of the central peak does not depend essentially on δ. The predicted selection rules and properties of the central peak may be helpful to detect orbital fluctuations in Raman spectra.
After the present work was completed we became aware of recent Raman scattering experiments in Ba(Fe 1−x Co x ) 2 As 2 near the SDW phase. 33 The authors found a strong enhancement of the scattering intensity at low energies around the tetragonal-orthorhombic structural phase transition in the B 1g channel. Their data are consistent with our results Quite recently magnetic torque measurements revealed the breaking of the fourfold symmetry far away from the SDW instability. As an explanation the occurrence of an orbital nematic instability was discussed. 23 It is highly desirable to perform Raman scattering measurements around the nematic critical temperatures measured by the magnetic torque experiments and to confirm the presence of nematic fluctuations.
Raman scattering in the high energy region involves mainly individual particle-hole excitations. At low temperatures the nematic distortion may become large enough to induce a Lifshitz transition where the Fermi pocket at the X-or the Y -point disappears. As a result particle-hole excitations at the X-point are allowed to occur. The resulting upward shift of spectral weight in the Raman intensity should be observable at high frequencies in the B 2g , but not in the A 1g or B 1g Raman spectra.