Antisymmetric Raman response

The gist The theory of antisymmetric Raman response establishes it as a unique tool to probe microscopic features such as interband energy scales and to detect reflection symmetry breaking in quantum materials.

Theory and Definition

Antisymmetric Raman response is defined as the difference between the Raman signals of two scattering geometries related by an exchange of mutually perpendicular incoming and the outgoing photon polarizations It is characterized by the absence of intraband terms, unlike standard Raman responses which measure auto-susceptibilities where both intra- and interband processes contribute The antisymmetric response is realized by comparing the signal Ixθyθ(omega) with Iyθxθ(omega) having the polarizations exchanged, defined as IA(omega, θ) ≡ Ixθyθ(omega)−Iyθxθ(omega) This response is non-zero in orthorhombic or lower symmetry systems because it probes the cross-susceptibility of two Raman operators.

Key Properties of the Response

The theory connects IA(omega, θ) to finite frequency cross-susceptibilities of two Raman operators via a fluctuation-dissipation relation. A key finding is that these cross-susceptibilities have spectroscopic “gaps” due to the complete absence of intraband terms even in metals. Furthermore, the frequency integrated cross-susceptibilities are thermodynamic order parameters for reflection symmetry breaking. The response is an even function of an external magnetic field, satisfying a reciprocity relation IA(omega, θ) is an even function of an external magnetic field.

Probing Electronic Systems

The antisymmetric Raman response contains information that cannot be accessed using standard Raman response. The theory shows that the real and imaginary parts of the cross-susceptibility ϕ iA2g(omega+iη) are odd and even functions of omega, respectively, which is opposite to the behavior seen in standard Raman response. The frequency integral Z ∞−∞ dω 2π ϕ iA2g(ω + iη) = i⟨[Γˆi, ΓˆA2g]⟩, which acts as an order parameter for reflection symmetry breaking that can be extracted from spectroscopy.

Examples and Applications

The theory is illustrated with examples from the charge density wave rare-earth tritellurides and the excitonic insulator Ta2NiSe5. In a normal metal, the absence of intraband Drude contribution in Imϕ B1gA2g(omega) is imposed by ΓˆA2g, and this response is zero below a threshold frequency omegath which measures the minimum interband excitation energy in metals. For the amplitude mode of a CDW, IA(E0, 0) > 0 implies Imϕ B2gA2g(E0) > Imϕ B1gA2g(E0) indicates that the CDW order parameter is predominantly of the inter-orbital type.

Conclusion

Antisymmetric Raman response measures cross-susceptibilities of two Raman operators, of which one is ΓˆA2g that filters out intraband excitations. The spectra have “gaps” below a threshold which measures minimum interband excitation energy in metals and the charge gap in insulators. It can detect subtle mirror symmetry breaking such as CDW transitions in tritellurides and excitonic insulator transition in Ta2NiSe5. In symmetry broken phases a collective mode can appear inside the gap which has microscopic information such as the orbital character of the order parameter.

How it works

The derivation starts from the standard description of nonresonant Raman response, assuming that in most cases of interest D/ω¯ ∼ 0.1 or less. The angle dependent antisymmetric Raman intensity is expressed as IA(omega, θ) = 2 cos(2θ)I B2gA2g(omega) − 2 sin(2θ)I B1gA2g(omega). The fluctuation-dissipation relation is derived from the imaginary time ordered function χ iA2g(τ) and its Fourier transform. The reciprocity relation IA(omega, θ) is an even function of an external magnetic field because C iA2g(t, h) = -C A2g i(t, −h).

Reciprocity Relation

The reciprocity relation is established by showing that the Fourier transforms satisfy the relation C iA2g(omega, h) = −C A2g i(omega, −h), from which Eq. (7) of the main text follows. This relationship holds because Raman operators are time reversal invariant and ΓˆA2g is antihermitian.

Example (1), antisymmetric response from electronic continuum

In the electronic continuum example, the bare scale of ΓˆA2g is set by (V0, ϵm), while that of ΓˆB1g by (t, t′). The threshold for interband transitions is given by omegaph = 2V0 in the γ ≃ 0 limit. This implies that interband particle-hole excitations need a minimum energy omegath = 2V0 in this model.

Example (2), amplitude mode of the CDW phase

The amplitude mode of the CDW appears in an antisymmetric channel, and its contribution to the Raman response is obtained as the vertex correction in the amplitude channel of the bare electronic response. The computational details for this example are given in SM, Sec. H. The amplitude mode propagator is well described by a Lorentzian function D(omega) = 2E0/[(omega + iγ0)2 − E 0 2], with E 0 = 0.005 eV and γ 0 = 0.2E 0 The resulting Fermi surface is shown in Fig. 2(a) of the main text. The amplitude mode propagator is well described by a Lorentzian function D(omega) = 2E0/[(omega + iγ0)2 − E 0 2], with E 0 = 0.005 eV and γ 0 = 0.2E 0. This implies that the CDW order parameter is predominantly of the inter-orbital type, with an xy or B2g character in the orbital space (px, py).

Exciton insulator

In the excitonic insulator phase, Imϕ B1gA2g(omega) measures electronic monoclinicity. The imaginary part of Imϕ B1gA2g(omega) in Fig. 3 shows that omegath ∼ 2Vmin is now a measure of the charge gap. This allows for the extraction of electronic monoclinicity from the integrated spectrum using Eq. (6). The imaginary part Imϕ B2gA2g(E0) < 0 is consistent with r < 0 in the model. This suggests that the CDW phase can be described by the mean field Hamiltonian H = H0 + ∆P k[c†k+Q,xck−Q,y + c†k+Q,yck−Q,x + h.c.]. The amplitude mode describes fluctuations of ∆, hybridized with the phonon that triggers the CDW instability.

Conclusion

Antisymmetric Raman response measures cross-susceptibilities of two Raman operators, of which one is ΓˆA2g that filters out intraband excitations.

1. textbfFundamental Probe of Symmetry Breaking in Materials Systems: Antisymmetric Raman Response (IA(omega, θ)). This tool can detect reflection symmetry breaking by establishing that IA(omega, θ) is non-zero in orthorhombic or lower symmetry systems, allowing for the direct spectroscopic identification of structural transitions like Charge Density Wave (CDW) phases in rare-earth tritellurides or excitonic insulator transitions in Ta2NiSe5.

2. textbfSpectroscopic Mapping of Electronic Gaps and Collective Modes. The theory shows that The spectra have 'gaps' below a threshold which measures minimum interband excitation energy, and under suitable conditions, a collective mode [two loop graph in Fig. 1(b)] can appear inside this spectroscopic 'gap', providing microscopic information such as the orbital character of the order parameter.

3. textbfQuantifying Orbital Character via Cross-Susceptibilities. By measuring cross-susceptibilities of two Raman operators, specifically comparing signals like "IA(E0, 0) > 0 and Imϕ B2g(E0) ≫ Imϕ B1g(E0), the system can determine if the CDW order parameter is predominantly of the inter-orbital type, with an xy or B2g character in the orbital space (px, py)."

4. textbfDetection of Subtleties in Metallic Systems. The theory distinguishes between standard Raman response and antisymmetric response by noting that for a normal metal, the absence of intraband Drude contribution in Imϕ B1g(omega), which is imposed by ΓˆA2g, allowing the detection of a threshold frequency omegath = 2V0 even in metallic systems, which is not accessible via standard Raman channels.

5. textbfCharacterization of Order Parameter Tilt and Monoclinicity. The analysis of the CDW amplitude mode shows that "a finite r is necessary to explain the monoclinicity of the T < Tc phase, where 'r' is a tilt parameter, enabling AI systems to map how the ordering wavevector 2Q is tilted by an angle r compared to the direction x = y."

We develop the theory of antisymmetric Raman response, defined as the difference between the Raman signals of two scattering geometries related by an exchange of mutually perpendicular incoming and the outgoing photon polarizations. Such responses, finite in orthorhombic or lower symmetry systems, are related to cross-susceptibilities of two Raman operators and are characterized by the absence of intraband terms. This is in contrast to standard Raman responses which measure auto-susceptibilities where both intra- and interband processes contribute. We illustrate the theory with examples from the charge density wave rare-earth tritellurides and the excitonic insulator Ta 2 NiSe 5. Our theory establishes antisymmetric Raman response as a unique tool to probe microscopic features such as interband energy scales and to detect reflection symmetry breaking.