ZrSiS: Layered Topological Nodal-Line Semimetal

• ZrSiS is a layered tetragonal nodal-line semimetal characterized by Dirac-like band crossings and nonsymmorphic symmetry protection from its square-net Si sublattice.
• Its bulk electronic structure exhibits an extended 2 eV linear dispersion window and a three-dimensional nodal network revealed by ARPES and quantum oscillation studies.
• Magnetotransport and optical conductivity measurements highlight high carrier mobility, giant magnetoresistance, and disorder-broadened Dirac electrodynamics.

ZrSiS is a layered tetragonal topological nodal-line semimetal in the PbFCl structure type whose low-energy electronic structure is dominated by Dirac-like band crossings organized into extended nodal lines rather than isolated point nodes. Across angle-resolved photoemission, quantum oscillation, optical, scanning-tunneling, and transport studies, it is characterized by a square-net Si sublattice, weak spin–orbit coupling (SOC), a broad linear-dispersion window extending up to about $2\ \mathrm{eV}$ in parts of the Brillouin zone, and a combination of bulk nodal-line states with surface-derived bands that are unusually prominent for a nonsymmorphic semimetal (Schoop et al., 2015, Neupane et al., 2016, Fu et al., 2017).

1. Crystal structure, nonsymmorphic symmetry, and topological classification

ZrSiS crystallizes in the tetragonal nonsymmorphic space group $P4/nmm$ (No. 129) and is usually described as a layered PbFCl-type compound built from Zr–S layers and a square net of Si atoms. The crystals cleave naturally along the $(001)$ plane, yielding a surface without reconstruction in ARPES and STM studies. The Si square net is the central structural element in the topological description because it sits on a glide plane, while the full space group also contains screw symmetries that enforce band degeneracies on specific Brillouin-zone boundaries and high-symmetry lines (Schoop et al., 2015, Neupane et al., 2016).

The symmetry mechanism is explicitly nonsymmorphic. In the standard notation, the glide can be written as $G=\{M|\mathbf{t}\}$, combining a mirror with a fractional translation. In ZrSiS this structure produces “band sticking” at $X$ and $M$, protects Dirac-like crossings, and stabilizes nodal-line features in the spinless limit. Low-energy theoretical descriptions in the cited literature therefore treat ZrSiS as a nodal-line semimetal with a Berry phase of $\pi$ on loops encircling isolated nodal lines, while also emphasizing that the relevant protection is tied primarily to glide-mirror and screw symmetries rather than to a bulk $\mathbb{Z}_2$ insulating invariant (Neupane et al., 2016, Topp et al., 2017).

SOC is weak but not negligible. Several studies describe small SOC-induced gaps, with values depending on the momentum-space location and the probe: calculations and ARPES work cited a gap below experimental resolution and estimated it to be less than $10\ \mathrm{meV}$ in the spinless-nodal-fermion regime, while other band-structure and STM-based studies discuss gaps of order $20$–$P4/nmm$0 near low-energy Dirac crossings, and magneto-optical analysis reported $P4/nmm$1 for Dirac-like bands near the low-energy cage (Neupane et al., 2016, Schoop et al., 2015, Su et al., 2018, Gudac et al., 2022). This suggests that “the SOC gap of ZrSiS” is not a single probe-independent number, but a momentum- and surface-sensitive quantity.

2. Bulk nodal-line network and electronic structure

Early ARPES and first-principles studies established the basic Fermiology: a diamond-shaped Fermi-surface pocket centered at $P4/nmm$2, an ellipsoidal pocket at $P4/nmm$3, and a small electron-like pocket centered at $P4/nmm$4. Photon-energy-dependent ARPES showed that the $P4/nmm$5-centered feature does not disperse with $P4/nmm$6, identifying it as surface-derived, while the $P4/nmm$7- and $P4/nmm$8-centered structures were assigned to bulk nodal-line states. The same work emphasized that linear dispersion survives over an unusually large energy window, up to approximately $P4/nmm$9 above and below $(001)$0 in parts of the Brillouin zone (Neupane et al., 2016, Schoop et al., 2015).

Bulk-sensitive soft-x-ray ARPES later separated the true bulk electronic structure from the surface states that dominate vacuum-ultraviolet measurements. Those data resolved two groups of bulk nodal lines: closed nodal rings on the $(001)$1 and $(001)$2 planes, and extended nodal lines along the high-symmetry lines $(001)$3 and $(001)$4. The horizontal rings are connected by vertical nodal lines on the $(001)$5 and $(001)$6 planes, forming a three-dimensional “cage”-like Fermi-surface network. In that formulation, the entire Fermi surface is constituted by nodal-line fermions, with carrier compensation pinning the nodal-line crossings near the Fermi level (Fu et al., 2017).

The line-node geometry is not uniform. The $(001)$7 nodal ring is nearly ideal and lies close to $(001)$8, whereas the $(001)$9 ring oscillates around $G=\{M|\mathbf{t}\}$0 and forms alternating electron and hole pockets. The $G=\{M|\mathbf{t}\}$1 nodal line is described as “saddle”-type, with crossing energy oscillating between $G=\{M|\mathbf{t}\}$2 at $G=\{M|\mathbf{t}\}$3 and $G=\{M|\mathbf{t}\}$4 at $G=\{M|\mathbf{t}\}$5, while the low-energy nodal lines close to $G=\{M|\mathbf{t}\}$6 form the cage that dominates transport and optics (Fu et al., 2017, Schilling et al., 2017).

A recurring source of confusion in the literature is the relation between surface and bulk bands. In ZrSiS, surface-derived states are not merely weak perturbations: VUV ARPES, slab DFT, and STM-based studies all show strong surface features near $G=\{M|\mathbf{t}\}$7 and around $G=\{M|\mathbf{t}\}$8. Their visibility does not negate the bulk nodal-line interpretation; rather, it reflects the fact that cleavage breaks the nonsymmorphic constraints locally and creates additional surface bands on top of an already unusual bulk semimetal (Topp et al., 2017, Su et al., 2018).

3. Magnetotransport, quantum oscillations, and butterfly magnetoresistance

Transport measurements on high-quality single crystals established ZrSiS as a high-mobility semimetal with extremely large, nonsaturating magnetoresistance (MR). Representative values include $G=\{M|\mathbf{t}\}$9 at $X$0 and $X$1 for $X$2, $X$3, nearly $X$4 at $X$5 and $X$6 around $X$7, and $X$8 at $X$9 and $M$0 in pulsed fields. The MR is strongly anisotropic and gives rise to the characteristic “butterfly” angular pattern, with the detailed angular maximum depending on sample geometry and device conditions (Singha et al., 2016, Ali et al., 2016, Wang et al., 2016).

Hall and quantum-oscillation analyses show that this behavior is multiband. A two-band Hall fit reported $M$1, $M$2, $M$3, and $M$4 at $M$5, while flake-device studies argued that near-perfect electron–hole compensation, tuned by the Zeeman effect, is the primary origin of the butterfly MR. In that account, the direct causal link between butterfly MR and Berry phase remains uncertain; the most robust explanation is compensation physics in an anisotropic multiband semimetal (Singha et al., 2016, Voerman et al., 2019).

Shubnikov–de Haas, de Haas–van Alphen, and thermoelectric oscillation studies consistently resolve several extremal orbits. Frequently reported frequencies include a low-frequency orbit near $M$6–$M$7 and a larger orbit near $M$8–$M$9, together with additional harmonics and higher-frequency pockets. Reported cyclotron masses for the principal low- and high-frequency pockets are typically light, e.g. $\pi$0 and $\pi$1 in one SdH study, while thermoelectric oscillations yielded $\pi$2 for a $\pi$3 3D hole-like Dirac cone and $\pi$4 for an $\pi$5 2D electron-like Dirac cone (Singha et al., 2016, Matusiak et al., 2017).

Berry-phase assignments are not completely uniform across measurements. One magnetotransport analysis reported non-trivial $\pi$6 Berry phase for both principal pockets, whereas another found the low-frequency $\pi$7 mode nontrivial but the $\pi$8 $\pi$9 mode trivial or near-trivial. A separate angular SdH study identified an abrupt change in the Berry phase of the quasi-2D $\mathbb{Z}_2$0 orbit between $\mathbb{Z}_2$1 and $\mathbb{Z}_2$2, interpreting it as an angle-induced topological phase transition that coincides with the dip region of the butterfly MR (Singha et al., 2016, Wang et al., 2016, Ali et al., 2016). This suggests that Berry-phase extraction in ZrSiS is orbit- and geometry-dependent, rather than reducible to a single universal phase label.

Longitudinal transport adds another layer. Negative longitudinal MR for $\mathbb{Z}_2$3 was fitted to a semiclassical form consistent with a chiral-anomaly contribution, and thermoelectric quantum oscillations remain visible up to $\mathbb{Z}_2$4, reflecting the exceptionally light masses and broad linear-dispersion window of the Dirac-like carriers (Singha et al., 2016, Matusiak et al., 2017).

4. Optical conductivity and thermodynamic signatures

Optical spectroscopy provided a direct bulk probe of the low-energy Dirac bands. ZrSiS exhibits a frequency-independent interband conductivity over $\mathbb{Z}_2$5–$\mathbb{Z}_2$6 ($\mathbb{Z}_2$7–$\mathbb{Z}_2$8), with $\mathbb{Z}_2$9 at all measured temperatures between $10\ \mathrm{meV}$0 and $10\ \mathrm{meV}$1. This behavior was assigned to interband transitions between quasi-two-dimensional Dirac bands derived from the low-energy nodal-line cage. Unlike graphene, the conductivity is not universal; in the simple circular nodal-line model,

$10\ \mathrm{meV}$2

so its magnitude scales with total nodal-line length $10\ \mathrm{meV}$3. Using the measured plateau gives $10\ \mathrm{meV}$4, consistent with band-structure estimates of the nodal-line extent (Schilling et al., 2017).

The same optical study resolved a low-frequency Drude contribution, a Pauli-blocking edge, and a very small low-energy gap. The observed dip in $10\ \mathrm{meV}$5 near $10\ \mathrm{meV}$6 implies $10\ \mathrm{meV}$7, giving an optical upper bound $10\ \mathrm{meV}$8. Upon cooling, the momentum-relaxation rate collapses: Drude fits below $10\ \mathrm{meV}$9 yielded $20$0–$20$1, corresponding to $20$2–$20$3, and with $20$4 this implies a momentum-relaxation length $20$5 (Schilling et al., 2017).

A first-principles multi-orbital theory sharpened the interpretation of the flat optical response. In that calculation, the clean-limit dynamical conductivity is not frequency independent; it rises up to about $20$6 and then drops sharply because the realistic nodal-line network is anisotropic and the vertical nodal lines disperse in energy. A nearly flat $20$7 up to about $20$8 emerges only after including disorder broadening through $20$9, with $P4/nmm$00 producing the closest match to experiment (Habe et al., 2018). A plausible implication is that the observed optical plateau is a disorder-broadened manifestation of nodal-line electrodynamics, not the exact universal response of an ideal Dirac model.

Thermodynamic measurements revealed a different signature of nodal-line physics. Magnetic susceptibility data on $P4/nmm$01 showed an out-of-plane dia- to paramagnetic crossover in $P4/nmm$02, while the in-plane susceptibility remained weakly temperature dependent. The anomaly was modeled as an orbital contribution associated with a degeneracy point $P4/nmm$03 where two near-$P4/nmm$04 nodal lines cross along $P4/nmm$05: $P4/nmm$06 For low Hf content, the extracted $P4/nmm$07 was approximately constant at $P4/nmm$08, and the same study identified Lifshitz transitions near $P4/nmm$09 and $P4/nmm$10 through changes in the dHvA frequencies (Gudac et al., 2022).

5. Surface bands, quasiparticle interference, and local electric-field control

The $P4/nmm$11 surface of ZrSiS hosts distinctive surface bands that are neither conventional Shockley/Tamm states nor topological surface states in the strong-$P4/nmm$12 sense. A symmetry-based slab description showed that cleavage lowers the bulk $P4/nmm$13 symmetry to the symmorphic surface group $P4/nmm$14, thereby lifting nonsymmorphic bulk degeneracies and creating detached “floating” two-dimensional bands. These floating bands are intense in ARPES, appear prominently around $P4/nmm$15, remain visible up to $P4/nmm$16 photon energy, and are sensitive to potassium adsorption, which changes their connectivity and opens an avoided crossing near $P4/nmm$17 (Topp et al., 2017). A common misconception is therefore that all prominent surface bands in ZrSiS are topologically protected; the cited surface-state literature instead identifies a substantial subset as symmetry-lifted, topologically trivial surface descendants of bulk bands.

STM and QPI measurements added orbital and termination sensitivity to that picture. Both S- and Si-terminated surfaces show spectroscopic signatures of the diamond-shaped Dirac bulk band, but the surface bands depend strongly on termination because their spectral weight is tied to different orbital components. On the S-terminated surface, QPI resolves a spin-polarized $P4/nmm$18-centered surface band near $P4/nmm$19, and the disappearance of the $P4/nmm$20 channel between about $P4/nmm$21 and $P4/nmm$22 was interpreted as suppressed backscattering in a helical, time-reversal-symmetric surface state. The same study reported $P4/nmm$23 for the bulk diamond band and $P4/nmm$24 for the $P4/nmm$25-centered surface band (Su et al., 2018).

More recent QPI work resolved a symmetry-selection rule directly. Native point defects preserve the glide symmetry and suppress small-$P4/nmm$26 scattering between states of different symmetry eigenvalues, so the inter-square vector $P4/nmm$27 is absent. Atomic step edges break glide symmetry locally, lift the selection rule, and generate a strong $P4/nmm$28 channel connecting the inner and outer floating-band squares. The collapse of $P4/nmm$29 to zero at $P4/nmm$30 gave a direct experimental determination of the surface nodal-line energy along $P4/nmm$31 (Lodge et al., 2024).

The same STM study demonstrated local electric-field tunability of the ZrSiS surface electronic structure. Tip-induced vertical fields shifted the floating-band surface-state peaks at $P4/nmm$32 and $P4/nmm$33 downward by up to $P4/nmm$34, and increased a SOC-induced avoided crossing near $P4/nmm$35 from $P4/nmm$36 to $P4/nmm$37, i.e. a nearly $P4/nmm$38 enhancement (Lodge et al., 2024). This establishes ZrSiS as a system in which both symmetry-selective scattering and surface SOC splittings can be manipulated locally.

6. Correlation effects, pressure tuning, lattice response, and interfacial superconductivity

Beyond single-particle topology, ZrSiS has been used as a platform for correlation-driven scenarios centered on excitonic physics. One model study reduced the low-energy problem to two nested square lattices with strong electron–hole symmetry and interaction parameters $P4/nmm$39, $P4/nmm$40, and $P4/nmm$41, and predicted an excitonic instability with pseudogap formation at $P4/nmm$42 in the zero-doping limit (Rudenko et al., 2017). A separate renormalization-group treatment proposed that ZrSiS may lie in the quantum critical region between semimetal and excitonic insulator, where excitonic fluctuations reduce $P4/nmm$43 and $P4/nmm$44 without opening a gap, producing $P4/nmm$45 for representative parameters and a finite quasiparticle residue characteristic of a strongly correlated Fermi liquid rather than a non-Fermi liquid (Wang et al., 2019). These theories were motivated in part by the unconventional mass enhancement reported by Pezzini et al. and by the absence of a spectroscopic gap.

Pressure studies indicate that the band topology is tunable but do not support intrinsic bulk superconductivity under compression. In SdH measurements up to about $P4/nmm$46, the small orbit near $P4/nmm$47 showed an abrupt area drop and a Landau-fan intercept shift of $P4/nmm$48 between $P4/nmm$49 and $P4/nmm$50, interpreted as a possible pressure-induced topological quantum phase transition, while the larger orbit near $P4/nmm$51 retained a Berry phase of $P4/nmm$52 and changed only weakly with pressure. Separate resistivity measurements found no evidence for pressure-induced superconductivity to at least $P4/nmm$53 and $P4/nmm$54 (VanGennep et al., 2019).

High-pressure Raman and synchrotron x-ray diffraction extended the structural picture. Those measurements identified a tetragonal-to-orthorhombic transition near $P4/nmm$55, the emergence of a monoclinic phase near $P4/nmm$56, and prolonged phase coexistence under compression. The $P4/nmm$57 mode softens subtly near $P4/nmm$58, its linewidth peaks near $P4/nmm$59, and the tetragonal $P4/nmm$60-axis shows a discontinuity near $P4/nmm$61, all of which were discussed as signatures of coupled structural and possible electronic-topological changes (Singha et al., 2017).

A distinct and more local phenomenon is tip-induced superconductivity. Point-contact spectroscopy with a non-superconducting Ag tip produced a mesoscopic superconducting phase with $P4/nmm$62, $P4/nmm$63–$P4/nmm$64, $P4/nmm$65, and $P4/nmm$66. Interface DFT attributed the effect to an approximately $P4/nmm$67–$P4/nmm$68 enhancement of the density of states at $P4/nmm$69, while preserving Dirac features along $P4/nmm$70 and $P4/nmm$71 for moderate interface perturbation (Aggarwal et al., 2018). The contrast with the absence of bulk pressure-induced superconductivity indicates that superconductivity in ZrSiS is, at least in current experiments, an interfacial or mesoscopic instability rather than an established bulk phase.

ZrSiS therefore occupies a distinctive position among topological semimetals: a chemically stable, nonsymmorphic nodal-line system in which bulk and surface Dirac-like states coexist, optical conductivity is flat over an unusually wide low-energy window, magnetotransport is dominated by giant and angle-sensitive MR, and both correlation effects and local symmetry breaking can be probed with unusual clarity. The resulting literature spans bulk topology, surface-state engineering, compensation-driven transport, and interaction-driven instabilities without reducing the material to any single one of those phenomena.