Orbital-dependent Fermi surface shrinking as a fingerprint of nematicity in FeSe

A large anisotropy in the electronic properties across a structural transition in several correlated systems has been identified as the key manifestation of electronic nematic order, breaking rotational symmetry. In this context, FeSe is attracting tremendous interest, since electronic nematicity develops over a wide range of temperatures, allowing accurate experimental investigation. Here we combine angle-resolved photoemission spectroscopy and theoretical calculations based on a realistic multiorbital model to unveil the microscopic mechanism responsible for the evolution of the electronic structure of FeSe across the nematic transition. We show that the self-energy corrections due to the exchange of spin fluctuations between hole and electron pockets are responsible for an orbital-dependent shrinking of the Fermi surface that affects mainly the $xz/yz$ parts of the Fermi surface. This result is consistent with our experimental observation of the Fermi surface in the high-temperature tetragonal phase, which includes the $xy$ electron sheet that was not clearly resolved before. In the low-temperature nematic phase, we experimentally confirm the appearance of a large ($\sim 50$ meV) $xz/yz$ splitting. It can be well reproduced in our model by assuming a moderate splitting between spin fluctuations along the $x$ and $y$ crystallographic directions. Our mechanism shows how the full entanglement between orbital and spin degrees of freedom can make a spin-driven nematic transition equivalent to an effective orbital order.

Figure 1

General sketch of the electronic structure. (a) Lines: Band structure of FeSe at $k_z=0$ along the $k_x$ direction of the 1Fe BZ obtained with a tight-binding model including SOC and mass renormalization (see text). Symbols show the low-energy model that we use in the calculation [Eq. (1)]. Dashed lines and open symbols denote the bands folded at $\mathrm{\Gamma }$ and $M_X$ in the 2Fe BZ. (b) FS cuts for the low-energy model in the 1Fe BZ. The exchange of spin fluctuations between hole and electron pockets at $M_X$ or $M_Y$ shrinks selectively the $yz$ or $xz$ orbitals, respectively. This effect, which is already present above $T_S$, modifies significantly the FS, as shown in panel (c) at 150 K. Here small symbols denote the bands folded in the 2Fe BZ.

Figure 2

Electronic structure at 150 K measured by ARPES. (a)–(c) Energy momentum plots along three $\mathrm{\Gamma }M$ directions indicated as thick dashed lines in panel (d). Lines are guides to the eyes indicating the dispersions of the different bands, with colors encoding the main orbital character (the even $xz/yz$ is $xz$ along $k_x$ and $yz$ along $k_y$). The data were measured at 40 eV photon energy ($k_z\sim 0$) with linear polarization along $k_x$. In (c), the area at each energy has been normalized to enhance the visibility of the electron pocket. (d) Fermi surface map obtained by integration of the ARPES spectral weight at +10 meV in a 4 meV window. (e) Image plot in gray scale of the spectral functions of the renormalized bands at 150 K obtained including self-energy corrections. Ticks along abscissa correspond to 0.1 $\mathrm{\Gamma }M$. Thin lines follow the experimental data shown in (a)–(c). As in Fig 1, symbols indicate the bare bands of Eq. (1).

Figure 3

Electronic structure at 20 K measured by ARPES. (a)–(c) Energy momentum plots in the same conditions as Fig. 2, but at 20 K. (d) Thin color line: experimental guides of the dispersions, also indicated in (a)–(c). Dashed lines: experimental guides at 150 K. (e), (f) Evolution with temperature of the bottom (e) and $k_F$ (f) of the three electron bands.

Figure 4

Temperature evolution at $M$ point. (a) Energy-momentum plots around $M$ in the same conditions as Fig. 2, but with all energy distribution curves normalized to constant area, which emphasizes the saddle bands. (b) Energy-momentum plots around $M$ in the same conditions as Fig. 2, showing the $xy$ electron band. The fit of this band (dotted blue line) is reported in (a). ${\mathrm{\Gamma }}_1$ and ${\mathrm{\Gamma }}_2$ are the $\mathrm{\Gamma }$ points at (0,0) and ($\pi ,\pi$), respectively.

Figure 5

Evolution of the spin fluctuation and nematic splitting with temperature. (a) Temperature evolution of the SF energies ${\omega }_{sf}^{X/Y}$ across the nematic transition. The SF propagator (5) becomes anisotropic below $T_S$, as shown in panel (b). (c)–(e) The anisotropy of the SF below $T_S$ induces a nematic splitting of the self-energy corrections for the $xz$ and $yz$ orbitals. Their real parts at zero frequency $\zeta$($\omega =0)$ are shown in panels (c) and (d) for the $\mathrm{\Gamma },M$ pockets. The resulting $xz/yz$ splitting below $T_S$ is shown in (d).

Figure 6

Electronic structure in the nematic state modeled in the spin-fluctuation scenario. Spectral functions of the renormalized bands at 20 K along $\mathrm{\Gamma }{M}_X$ (a) and $\mathrm{\Gamma }{M}_Y$ (b). Ticks along the abscissa correspond to 0.1 $\mathrm{\Gamma }M$. Thin lines reproduce the experimental data detailed in Fig. 3. The “break” at $-60$ meV in the dispersion of the outer hole band is due to the crossing of the $xy$ hole band. Along $k_x$ the main orbital character switches from $yz$ to $xz$ on the inner band and from $xz$ to $yz$ on the outer band, due to the crossing of the two bands hybridized through SOC. (c) Fermi surface calculated for low temperatures. (d) Map of the ARPES spectral weight integrated around $M$ at $-15$ meV over 4 meV and dispersions in the two perpendicular dispersions across the Dirac point $D$. (e), (f) Energy-momentum plots in two perpendicular cuts across the Dirac point.