Probing emergent QED in quantum spin ice via Raman scattering of phonons: Shallow inelastic scattering and pair production

We present an unconventional mechanism for Raman scattering of phonons, which is based on the linear magnetoelastic coupling present in non-Kramers magnetic ions. This provides a direct coupling of Raman-active phonons to the magnet's quasiparticles. We propose to use this mechanism to probe the emergent magnetic monopoles, electric charges, and photons of the emergent quantum electrodynamics (eQED) of the U(1) quantum spin liquid known as quantum spin ice. Detecting this eQED in candidate rare-earth pyrochlore materials, or indeed signatures of topological magnetic phases more generally, is a challenging task. We show that the Raman scattering cross section of the phonons directly yields relevant information, with the broadening of the phonon linewidth, which we compute, exhibiting a characteristic frequency dependence reflecting the two-particle density of states of the emergent excitations. Remarkably, we find that the Raman linewidth is sensitive to the details of the symmetry fractionalization and hence can reveal information about the projective implementation of symmetry in the quantum spin liquid, thereby providing a diagnostic for a $\pi$-flux phase. The Raman scattering of the phonons thus provides a useful experimental tool to probe the fractionalization in quantum spin liquids that turns out to closely mirror pair production in quantum electrodynamics and the deep inelastic scattering of quantum chromodynamics. Indeed, the difference to the latter is conceptual more than technical: the partons (quarks) emerge from the hadrons at high energies due to asymptotic freedom, while those in eQED arise from fractionalization of the spins at low energies.

Figure 1

Correspondence of scattering diagrams and schematic for the Raman response: (a) The high energy photon can lead to creation of a positron and an electron via pair production, (b) in non-Kramers quantum spin ice, the phonon flips a spin and creates two magnetic monopoles of opposite charges, (c) in deep inelastic scattering, a photon emitted from a lepton scatters off a parton, a quark $q$, contained in the hadron, a $q\overline{q}$-pion as a free particle at high energies, (d) in shallow inelastic scattering, an optical phonon emitted from a photon scatters off a parton, a magnetic monopole or electric charge, emerging from the spin degrees of freedom from fractionalisation at low energies, (e) schematic of the energy scales of different density of states (DOS) of the emergent excitations contributing to the phonon linewidth.

Figure 2

Feynman diagrams for the interactions between the excitations of the quantum spin ice and the Raman active phonons in a non-Kramers system due to the linear spin-phonon coupling [Eqs. (6) and (7)]. (a) The vertex corresponds to the phonon-magnetic monopole interaction described by Eqs. (18) and (19). Dotted, solid, and curly lines denote phonon, monopole and emergent photons, respectively. Thin and thick solid lines represent two flavors of monopoles, $A$ and $B$, respectively. (b) Vertex for the phonon-(emergent) photon interaction described by Eq. (24). The circle represents the dipolar form factors [see Eq. (25)] that make the vertex gauge invariant. (c) Vertex for the phonon and electric charge interaction. The dashed line denotes the electric charge.

Figure 3

Sublattices of an up tetrahedron. 0,1,2,3 denote the four sublattices and ${\widehat{z}}_0$, ${\widehat{z}}_1$, ${\widehat{z}}_2$, ${\widehat{z}}_3$ represent the four respective local quantization axes [see Eqs. (A2) in Appendix pp1-s1].

Figure 4

GMFT Feynman diagram for phonon and magnetic monopole interaction [described by Eqs. (20) and (21)] in the zero flux phase [see Fig. 2 for further details].

Figure 5

GMFT Feynman diagram for phonon and electric charge interaction [described by Eq. (26)]; (see Fig. 2 for further details.

Figure 6

Self-energy of the phonon due to the phonon-magnetic monopole interaction (see Fig. 4).

Figure 7

Frequency dependence of linewidth of (a) ${\mathbf{e}}_{\mathbf{g}}$ and (b) ${\mathbf{t}}_{\mathbf{2}\mathbf{g}}$ phonons in the zero-flux phase due to the phonon-magnetic monopole coupling. For both plots, we have chosen $\frac{\lambda }{2J_{zz}}=0.7$ and $\frac{J_{\pm }}{2J_{zz}}=0.1$ for illustrative purpose. The inset of (b) shows the two-particle density of states (DOS) of magnetic monopoles for the same values of $\lambda /2J_{zz}$ and $J_{\pm }/2J_{zz}$.

Figure 8

Density of states of different bands contributing to the phonon linewidth in the $\pi$-flux phase. We have chosen $\lambda /2J_{zz}=0.7,J_{\pm }/2J_{zz}=0.3$ for illustrative purpose. Red and blue curves denote the two-particle density of states for upper $({\epsilon }_{+}^{\pi }$) and lower $({\epsilon }_{-}^{\pi }$) bands, respectively. Black and magenta curves denote density of states of ${\epsilon }_{+}^{\pi }+{\epsilon }_{-}^{\pi }$ and ${\epsilon }_{+}^{\pi }-{\epsilon }_{-}^{\pi }$, respectively.

Figure 9

Self-energy of the magnetic monopoles due to the gauge fluctuation.

Figure 10

Self-energy of the phonon due to the phonon-(emergent) photon interaction [see Fig. 2].

Figure 11

Energy dependence of the linewidth of the ${\mathbf{e}}_{\mathbf{g}}$ phonons due to phonon-(emergent) photon coupling. The energy dependence is shown at different temperatures. $c_e=\sqrt{UK}$ is the velocity of the emergent photons and its density of states is plotted in the inset.

Figure 12

Self-energy of the phonon due to the phonon-electric charge interaction (see Fig. 5).

Figure 13

Two-particle density of states of the charges. For illustrative purposes, we have chosen $t/m=0.2$ where ${\mathrm{\Delta }}_c/m=0.43$.

Figure 14

Feynman diagram for phonon and magnetic monopole interaction due to the spin-phonon coupling quadratic in spin operators. (a) and (b) are the contributions from $H_1$ and (c) and (d) are the contributions from $H_2$ [see Eqs. (F3) and (F4)].

Figure 15

Feynman diagram for phonon and photon interaction due to the spin-phonon coupling quadratic in spin operators. The circles represent the form factor that makes the vertex gauge invariant. (a) and (b) are contributions due to $H_1$ and $H_2$, respectively [see Eqs. (F3) and (F4)].

Figure 16

Feynman diagram for the phonon mediated Loudon-Fleury vertex for magnetic monopoles. The curly lines denote the external Raman photons. The first and second diagrams show the coupling of phonon to external photon [see Eq. (C1)] and magnetic monopole, respectively. Integrating out the phonons, the external photon-monopole [see Eq. (H3)] vertex is obtained which is shown in the rightmost panel.

Figure 17

Feynman diagram contributing to the Raman intensity due to the phonon mediated Loudon-Fleury vertex for magnetic monopoles.

Figure 18

Unit cell of the diamond lattice with $\pi$ flux. The gauge mean field is chosen such that $A_{\mathbf{r},\mathbf{r}+{\mathbf{e}}_{\mu }}=\pi$ on the yellow bonds and $A_{\mathbf{r},\mathbf{r}+{\mathbf{e}}_{\mu }}=0$ on the black bonds. $A1,A2,B1,B2$ denote the four sublattices in the enlarged unit cell of the $\pi$-flux phase.

Figure 19

Self-energy bubble diagrams for phonon in $\pi$-flux phase. The label $A\mu \to A\nu$ ($B\alpha \to B\beta$) implies the $A\mu$ ($B\alpha$) monopole is created in the left vertex and $A\nu$ ($B\beta$) monopole is annihilated at the right vertex. $\mu ,\nu ,\alpha ,\beta =1,2$, hence, there are 16 possible distinct diagrams.

Figure 20

Feynman diagram for phonon-mediated Loudon-Fleury vertex for emergent photons.