---
title: Dirac-line Fermi Surface
url: https://www.emergentmind.com/topics/dirac-line-fermi-surface
type: topic
---

# Dirac-line Fermi Surface

A Dirac-line Fermi surface is the momentum-space structure associated with band crossings that extend along a one-dimensional manifold rather than remaining isolated at points. In the reported material systems, the underlying Dirac nodal line may itself lie at the Fermi level, as in the vertical K–H crossings of SrGa$_2$ and BaGa$_2$, or it may generate observable Fermi-surface sheets such as the ring-torus of CaAgAs, the cage-like electron and hole pockets of ZrSiS, the elliptical nodal-line-derived contours of $\alpha$-RhSi, and the quasi-2D sheets of CaSb$_2$ [2510.05304] [1708.06874] [2002.04379] [2002.03020] [2206.15346]. The common organizing principle is symmetry protection—mirror, glide, screw, inversion, and $P\Theta$ symmetries constrain hybridization and determine whether the line node is exact, weakly gapped by spin–orbit coupling, or converted into nearby Fermi pockets.

## 1. Symmetry origin and protection

Dirac-line Fermi surfaces occur in several distinct symmetry settings. In ZrSiS, Schoop *et al.* identified the tetragonal space group $P4/nmm$ with a glide plane $\{M_x|\tfrac12,\tfrac12,0\}$ and a two-fold screw axis $\{C_{2\parallel z}|\tfrac12,\tfrac12,0\}$; the glide symmetry enforces four-fold degeneracies along Brillouin-zone boundary lines and produces a closed “diamond” of Dirac points in the $k_z=0$ plane [1509.00861]. In CaAgAs, the relevant symmetry is the horizontal mirror plane $M_z$, which protects a nodal ring on $k_z=0$ because the crossing bands carry opposite mirror eigenvalues [1708.06874]. In orthorhombic $\alpha$-RhSi, the nonsymmorphic screw axis on the $k_x=\pi$ plane, together with $P\Theta$, protects a four-band crossing along a closed line and prevents any SOC-induced gap [2002.03020]. In CaSb$_2$, inversion and a two-fold screw axis along $b$ enforce doubly degenerate quasi-2D sheets and Dirac lines in the $k_b=\pi$ plane [2206.15346]. In SrGa$_2$ and BaGa$_2$, the Dirac nodal line lies along the vertical K–H line, with six symmetry-equivalent copies around the hexagonal Brillouin zone [2510.05304].

Spin–orbit coupling modifies these line nodes in material-specific ways. In SrGa$_2$ and BaGa$_2$, Gao *et al.* reported that SOC opens only a small gap along K–H, about $5\,\mathrm{meV}$ in SrGa$_2$ and about $10\,\mathrm{meV}$ in BaGa$_2$ [2510.05304]. In CaAgAs, inclusion of SOC opens an essentially isotropic gap of about $75\,\mathrm{meV}$ and converts the negligible-SOC line-node semimetal into a narrow-gap strong topological insulator with $\nu_0=1$ [1708.06874]. In contrast, the $\alpha$-RhSi nodal line is described as symmetry protected against SOC-induced gapping on the screw-invariant plane [2002.03020].

These cases show that a Dirac-line Fermi surface is not tied to a single crystallographic archetype. The recurrent feature is that crystalline symmetry constrains the band representations strongly enough that a codimension-two band crossing persists along a line or loop.

## 2. Momentum-space geometries and low-energy structure

The geometry of a Dirac-line Fermi surface varies sharply among materials.

| System | Nodal-line locus | Reported Fermi-surface form |
|---|---|---|
| ZrSiS | closed “diamond” in $k_z=0$; low-energy pockets in the Z–R–A plane | diamond-shaped line nodes; cage-like pockets |
| CaAgAs | nodal ring on $k_z=0$ around $\Gamma$ | ring-torus |
| SrGa$_2$/BaGa$_2$ | vertical K–H line, six copies | neck/belly and spindle pockets near K–H |
| $\alpha$-RhSi | ellipse near $S$ on the $k_y$–$k_z$ plane at $k_x=\pi$ | elliptically shaped nodal line |
| CaSb$_2$ | closed Dirac loop in the $k_b=\pi$ plane | quasi-2D sheets |

For ZrSiS, the diamond-shaped line in the $k_z=0$ plane can be written implicitly as
$$
\cos(k_x a)+\cos(k_y a)=0,\qquad k_z=0,
$$
and the local low-energy Hamiltonian near a point on the loop is
$$
H(q)\approx v_1 q_2 \sigma_1 + v_3 q_3 \sigma_2,
$$
with linear dispersion transverse to the loop and no leading dispersion along the tangential direction [1509.00861]. In CaAgAs, the minimal two-band model
$$
H(\mathbf k)=\epsilon_0(\mathbf k)\sigma_0+M(\mathbf k)\sigma_z+A k_z \sigma_x
$$
produces a nodal ring through the condition
$$
M_0-B_1(k_x^2+k_y^2)=0,\qquad k_z=0,
$$
and shifting the chemical potential by $\mu$ generates the toroidal Fermi surface
$$
\frac{(\sqrt{k_x^2+k_y^2}-k_0)^2}{(\mu/2B_1k_0)^2}+\frac{k_z^2}{(\mu/A)^2}=1
$$
[1708.06874].

In $\alpha$-RhSi, Mozaffari *et al.* describe an elliptically shaped nodal line near the $S$ point as the intersection of two upside-down Dirac cones [2002.03020]. Their symmetry-based four-band model is
$$
H(k)=\epsilon_0 \mathbb{1}_4+\Delta_0 \tau_z\otimes \mathbb{1}_2+v_y k_y \tau_x\otimes \sigma_x+v_z k_z \tau_x\otimes \sigma_z,
$$
and the nodal-line locus can be written as
$$
v_y^2 k_y^2+v_z^2 k_z^2=\Delta_0^2.
$$
Using $\hbar v_y\approx 1.13\,\mathrm{eV\cdot \AA}$, $\hbar v_z\approx 0.23\,\mathrm{eV\cdot \AA}$, and $\Delta_0\approx 2.5\,\mathrm{meV}$, they estimate a semi-major axis $a\approx 0.01\,\mathrm{\AA^{-1}}$ and a semi-minor axis $b\approx 0.002\,\mathrm{\AA^{-1}}$ [2002.03020].

In SrGa$_2$ and BaGa$_2$, the nodal line is not a ring around $\Gamma$ but a vertical line of the form
$$
k=(1/3,\,2/3,\,\xi),\qquad \xi\in[0,\,1/2],
$$
with six equivalent copies. The observed Fermi surfaces are then formed by pockets attached to this line, including neck and belly sections of a quasi-cylindrical pocket and, in BaGa$_2$, a spindle-shaped $\beta$ pocket around K–H [2510.05304].

These examples establish that “Dirac-line Fermi surface” is geometrically heterogeneous: it may denote a torus enclosing a nodal ring, a closed diamond of four-fold crossings, quasi-2D sheets tied to a screw-protected loop, or electron and hole pockets on opposite sides of a small SOC gap.

## 3. Experimental determination

The determination of a Dirac-line Fermi surface has relied on a recurrent combination of ARPES, quantum oscillations, and DFT.

In ZrSiS, ARPES supported by *ab initio* calculations showed several Dirac cones forming a Fermi surface with a diamond-shaped line of Dirac nodes, and the linearly dispersed bands extend over roughly $-2.0\,\mathrm{eV}$ to $+2.0\,\mathrm{eV}$ around $E_F$ [1509.00861]. A later high-field study resolved the low-energy cage-like Fermi surface with six fundamental extremal orbits by combining Shubnikov–de Haas and de Haas–van Alphen measurements up to $35\,\mathrm{T}$ with DFT; the experimental $F(\theta)$ curves agree with DFT to within about $5\%$ over the full angular range [2002.04379].

In CaAgAs, soft-x-ray ARPES with photon energies between $500\,\mathrm{eV}$ and $975\,\mathrm{eV}$ was used to map the three-dimensional bulk valence bands. At $h\nu=550\,\mathrm{eV}$, the measurements cut through $\Gamma$ in the 13th Brillouin zone and showed a bright continuous loop at $E_F$ in the $k_x$–$k_y$ plane, with no other pockets appearing elsewhere in the three-dimensional Brillouin zone [1708.06874].

In SrGa$_2$ and BaGa$_2$, Gao *et al.* combined angle-dependent dHvA, ARPES, and DFT. ARPES cuts through $\Gamma$–K–M and A–H–L reveal the Dirac point at K very close to $E_F$, while photon-energy scans locate K–H in $k_z$ and directly observe the linear crossing and its SOC splitting along the nodal line [2510.05304]. For the oscillatory sector, torque magnetometry from $\theta=0^\circ$ to $90^\circ$ identifies in SrGa$_2$ the frequencies $\alpha=75.5\,\mathrm{T}$, $\gamma=510.4\,\mathrm{T}$, and $\eta\sim 335\,\mathrm{T}$, and in BaGa$_2$ the frequencies $\alpha=36.8\,\mathrm{T}$, $\beta=54.8\,\mathrm{T}$, $\gamma=356.5\,\mathrm{T}$, and $\eta\approx 500\,\mathrm{T}$ [2510.05304].

In $\alpha$-RhSi, dHvA oscillations measured by magnetic torque at $T=350\,\mathrm{mK}$ and fields up to $30\,\mathrm{T}$ yield FFT peaks between roughly $0.5\,\mathrm{kT}$ and $2.0\,\mathrm{kT}$. Their angular dependence matches unit-cell-rotated elliptical cross-sections of the DFT Fermi-surface sheets to within typical DFT uncertainty below $10\%$, with best agreement obtained after shifting $E_F$ downward by about $2\,\mathrm{meV}$ [2002.03020].

The significance of this experimental program is methodological as much as topological. In all of these systems, no single probe is sufficient: ARPES determines the band crossings directly, quantum oscillations isolate the extremal orbits, and DFT organizes both into a three-dimensional Fermiological picture.

## 4. Quantum oscillations, magnetic breakdown, and transport lifetimes

The standard semiclassical entry point is Onsager quantization. For an extremal orbit,
$$
F=\frac{\hbar}{2\pi e}A_F,
$$
or equivalently $A_{\rm ext}=(2\pi e/\hbar)F$, and the oscillatory response is analyzed with Lifshitz–Kosevich damping factors [2002.04379] [2510.05304]. In the notation used by Gao *et al.*,
$$
R_T=\frac{r a T\mu}{\sinh(r a T\mu)},\qquad
R_D=\exp[-r a T_D\mu/B],\qquad
R_S=\cos(r\pi g\mu/2),
$$
with $a\simeq 14.69\,\mathrm{T/K}$ and $\mu\equiv m^*/m_0$ [2510.05304].

ZrSiS is the canonical case for magnetic breakdown on a Dirac-line-derived Fermi surface. For $B\parallel c$, the principal hole and electron pockets in the Z–R–A plane are the $\alpha$ and $\beta$ orbits. In the high-field study, their experimental frequencies are $240\pm 10\,\mathrm{T}$ and $420\pm 10\,\mathrm{T}$, corresponding to cross-sectional areas $0.023\,\mathrm{\AA^{-2}}$ and $0.040\,\mathrm{\AA^{-2}}$ [2002.04379]. In the strain study, the same pockets are reported as $F_\alpha=245\,\mathrm{T}$ with $A_\alpha\approx 2.34\times10^{-2}\,\mathrm{\AA^{-2}}$ and $F_\beta=418\,\mathrm{T}$ with $A_\beta\approx 4.17\times10^{-2}\,\mathrm{\AA^{-2}}$ [2405.13601]. The two pockets sit on opposite sides of a tiny SOC-induced momentum-space gap, reported as $\Delta k\approx 0.005\,\mathrm{\AA^{-1}}$ in the strain study and $\Delta k\approx 4.9\times 10^{-3}\,\mathrm{\AA^{-1}}$ in the high-field study, enabling breakdown orbits such as $2\beta-\alpha$ and $\alpha+\beta$ [2405.13601] [2002.04379]. Above about $5\,\mathrm{T}$, additional frequencies between $7.7\,\mathrm{kT}$ and $8.9\,\mathrm{kT}$ appear and are assigned to orbits that encircle the entire nodal loop [2002.04379].

Transport lifetimes on Dirac-line Fermi surfaces are highly tunable even when the extremal areas are nearly unchanged. In ZrSiS under uniaxial strain along the $a$ axis, all fundamental frequencies remain unchanged within $\pm 1\,\mathrm{T}$ for $|\varepsilon|\le 0.3\%$, and DFT confirms negligible change of the nodal-line radius or magnetic-breakdown gap [2405.13601]. What changes strongly is the Dingle temperature and hence the quantum mobility of the $\alpha$ orbit: $T_D=6.2\,\mathrm{K}$ at $-0.28\%$, $9.1\,\mathrm{K}$ at zero strain, and $10.9\,\mathrm{K}$ at $+0.34\%$, corresponding to $\mu_q=2.1$, $1.5$, and $1.2\times 10^3\,\mathrm{cm^2/Vs}$ [2405.13601]. Compression sharpens SdH and magnetic-breakdown peaks, whereas under tension both $\alpha$ and $\beta$ peaks weaken and the magnetic-breakdown peaks vanish [2405.13601].

SrGa$_2$ and BaGa$_2$ illustrate a different transport scale. For $B\parallel c$, the full multi-harmonic LK fits yield in SrGa$_2$ $m^*/m_0=0.117$ and $T_D=4.1\,\mathrm{K}$ for the $\alpha$ pocket, giving $\mu_q\approx 4.4\times10^3\,\mathrm{cm^2/Vs}$, and in BaGa$_2$ $m^*/m_0=0.174$ and $T_D=0.62\,\mathrm{K}$ for the $\beta$ pocket, giving $\mu_q\approx 2.0\times10^4\,\mathrm{cm^2/Vs}$ [2510.05304].

These data show that Dirac-line Fermi surfaces are not defined only by topology. They are equally distinguished by their susceptibility to coupled-orbit dynamics, unusually small breakdown gaps, and strong variation of the quantum scattering time with strain or orbit character.

## 5. Topological invariants and Berry-phase interpretation

The topological content of a Dirac-line Fermi surface is often summarized by the Berry phase accumulated on a loop linking the nodal line. In ZrSiS, any small closed path that links the line node once acquires a quantized Berry phase
$$
\Gamma_C=\oint_C A(k)\cdot dk=\pi \pmod{2\pi},
$$
and the same $\pi$ Berry phase appears in the $\alpha$-RhSi description for any loop on the screw-invariant plane that encircles the nodal line once [1509.00861] [2002.03020]. In CaAgAs, the topological description is cast in mirror sectors: a mirror-Chern number can be defined in the absence of SOC, while with SOC the gapped system carries a strong $\mathbb Z_2$ index $\nu_0=1$ [1708.06874].

A more general integer protection appears in the theoretical Dirac-line criticality of a Weyl Lifshitz transition. For the tilted Weyl Hamiltonian
$$
H(p)=c\,\boldsymbol{\sigma}\cdot p-f c p_z \mathbb 1,
$$
the critical tilt $f=1$ produces a zero-energy manifold along the $p_z$ axis, and the line is protected by the winding invariant
$$
N_2=\oint_C \frac{dl}{2\pi i}\,D(B)^{-1}\partial_l D(B),
$$
which equals $1$ for $D(B)\propto p_x+i p_y$ [2605.27453].

At the same time, Berry-phase extraction from quantum oscillations is not straightforward. Gao *et al.* emphasize that in centrosymmetric SrGa$_2$ and BaGa$_2$ the orbital magnetic moment contribution is negligible, but the Zeeman term remains, so the total phase shift is $\lambda=\phi_B+\phi_Z$ [2510.05304]. Because multiple $(g,\phi_B)$ pairs can reproduce the same oscillatory signal, $\Delta M(B)$ alone cannot uniquely fix the Berry phase and $g$ factor. They therefore conclude that even with higher harmonics included in the LK fit, the Berry phases cannot be unambiguously determined when the Zeeman effect is included [2510.05304]. The ZrSiS high-field study reached a parallel conclusion from a different direction: the coexistence of multiple orbits, magnetic breakdown, and tunneling phase shifts rendered a reliable extraction of a nontrivial $\pi$ Berry phase inconclusive [2002.04379].

A common misconception is therefore that an odd multiple of $\pi$ inferred from a Landau-fan analysis is, by itself, decisive evidence for a nontrivial nodal-line topology. The reported work shows that this inference can fail when Zeeman splitting, higher harmonics, or magnetic breakdown are appreciable.

## 6. Tunability, superconductivity, and critical extensions

Dirac-line Fermi surfaces are experimentally tunable without necessarily shifting the Fermi-surface area. In ZrSiS, uniaxial strain of order $|\varepsilon|\lesssim 0.3\%$ modifies the prominence of SdH and magnetic-breakdown peaks while leaving the fundamental frequencies essentially unchanged, leading to the conclusion that the scattering time along a Dirac-nodal loop is highly sensitive to layer spacing through the ratio $c/\sqrt{ab}$ [2405.13601]. The same study states that strain offers a clean handle, free from doping or hydrostatic pressure-induced phase transitions, for modulating topological-semimetallic properties [2405.13601].

In CaSb$_2$, the Dirac-line-derived quasi-2D Fermi surface participates directly in superconductivity. The measured dHvA and SdH frequencies for $H\parallel c^*$ are $F_1=73.8\,\mathrm{T}$, $F_2=78.0\,\mathrm{T}$, and $F_3=112.6\,\mathrm{T}$, with the $\sim 75\,\mathrm{T}$ branch following $F(\theta)\simeq F(0)/\cos\theta$ and effective masses near $m^*/m_e\approx 0.16$ for band C [2206.15346]. The superconducting upper critical fields are $\mu_0 H_{c2\parallel c^*}(0)=12$–$15\,\mathrm{mT}$ and $\mu_0 H_{c2\perp c^*}(0)=35$–$42\,\mathrm{mT}$; the resulting Ginzburg–Landau parameters are $\kappa_{c^*}\approx 1.1$–$1.4$ and $\kappa_{ab}\approx 3.2$–$3.9$, placing the material near type-I behavior for $H\parallel c^*$ [2206.15346]. The two-band analysis and the agreement between superconducting anisotropy and quasi-2D mass enhancement lead to the conclusion that the Dirac-line band contributes to pairing [2206.15346].

Field tuning is also relevant in $\alpha$-RhSi. Because the two Dirac points of the participating Kramers-degenerate bands are only about $5\,\mathrm{meV}$ apart, the Zeeman energy can drive a crossing between spin-split partners at fields of order
$$
B_c\simeq \frac{5\,\mathrm{meV}}{2\mu_B}\simeq 40\,\mathrm{T},
$$
which was proposed as a possible explanation for anomalies in magnetic torque [2002.03020].

At the theoretical limit of a Weyl Lifshitz transition, the Dirac-line Fermi surface becomes the critical state itself. At $f=1$, the density of states scales as $N(E)\sim |E|$, and the specific heat scales as $C_v\sim T^2$, in contrast to the $E^2$ and $T^3$ behavior of a point node [2605.27453]. Chowdhury *et al.* further map this critical state onto the Painlevé–Gullstrand horizon, identifying $f(r)=v(r)/c$ so that $f=1$ at the emergent event horizon, with analogue Hawking temperature
$$
T_{\rm H}=\frac{\hbar}{2\pi}\left|\frac{dv}{dr}\right|_{r=r_h}.
$$
This suggests that the Dirac-line Fermi surface is not only a materials-specific fermiological object but also a universal critical configuration in topological Lifshitz transitions [2605.27453].

Source: https://www.emergentmind.com/topics/dirac-line-fermi-surface