Holographic Weyl–Nodal Line Semimetal

• Holographic Weyl–nodal line semimetal is a strongly coupled topological phase that combines Weyl nodes and nodal rings through axial-vector and two-form deformations.
• The model employs a five-dimensional gauge/gravity duality framework to independently tune the Weyl and nodal-line sectors, enabling smooth topological transitions.
• Interacting topological invariants and fermionic Green’s functions expose a rich momentum-space structure with reliable signatures like nonzero Weyl charges and Berry phases.

A holographic Weyl–nodal line coexisting semimetal is a strongly coupled topological semimetal phase, realized through gauge/gravity duality, whose boundary fermionic response contains both Weyl-point topology and nodal-line topology at the same time. In the holographic construction, the Weyl sector is generated by an axial-vector deformation, while the nodal-line sector is generated by a complex two-form deformation; the resulting zero-temperature theory exhibits a finite coexistence region, neighboring critical and gapped sectors, and an interacting topological characterization based on the topological Hamiltonian $H(\mathbf{k})=-G^{-1}(0,\mathbf{k})$ rather than a microscopic Bloch Hamiltonian (Chu et al., 2024, Chen et al., 19 Sep 2025). The phase extends earlier holographic nodal-line models that established the IR competition mechanism between a mass operator and a topology-deforming operator, but those precursor models did not themselves realize coexistence with Weyl nodes (Liu et al., 2018, Liu et al., 2020).

1. Conceptual setting and defining features

The coexistence problem is nontrivial because the two topological structures are associated with different symmetry patterns. In the formulation used for the interacting holographic system, Weyl nodes require partial breaking of $TP$, whereas nodal lines require $TP$ protection and, in the 2025 invariant analysis, also exploit mirror symmetry in the nodal-line sector. In a strongly coupled setting there is no simple noninteracting band Hamiltonian from which one can directly read off point or line degeneracies, so the phase is defined instead through the zero-frequency interacting Green’s function and its derived topological Hamiltonian $H(\mathbf{k})=-G^{-1}(0,\mathbf{k})$ (Chen et al., 19 Sep 2025).

The weak-coupling template that motivates the holographic model uses an eight-component spinor with one block generating Weyl nodes and another generating a nodal ring. In the 2024 holographic construction this split structure is implemented by introducing two sets of gauge fields, two scalar mass deformations, and a complex antisymmetric two-form, so that the Weyl and nodal-line sectors can be tuned independently yet coexist in the same bulk solution (Chu et al., 2024).

Earlier holographic nodal-line work established the general mechanism that topology at strong coupling can be encoded by IR competition between a mass operator and an operator deforming the Fermi-surface topology. The 2018 model showed that a topological nodal line semimetal can undergo a continuous quantum phase transition to a topologically trivial semimetal, while the 2020 improved model enforced the duality relation between $\bar\psi\gamma^{\mu\nu}\psi$ and $\bar\psi\gamma^{\mu\nu}\gamma^5\psi$ through a first-order complex two-form sector (Liu et al., 2018, Liu et al., 2020). These ingredients later became the nodal-line building block inside the coexistence construction.

2. Bulk construction and holographic dictionary

The coexistence model combines the holographic ingredients of Weyl and nodal-line semimetals in a single five-dimensional bulk theory. The field content consists of the metric $g$, two vector gauge fields $V,\hat V$, two axial gauge fields $A,\hat A$, two scalars $\Phi_1,\Phi_2$, and a complex two-form $TP$0. In the 2024 presentation, the Weyl sector is encoded by $TP$1 together with $TP$2, while the nodal-line sector is encoded by $TP$3, $TP$4, and $TP$5; the 2025 invariant analysis uses the same combined structure and the same control by axial-vector and two-form deformations, although the scalar-sector labeling is presented differently (Chu et al., 2024, Chen et al., 19 Sep 2025).

The bulk action used for coexistence is

$TP$6

Here the two-form term is first order and is inherited from the improved nodal-line construction, where it enforces the self-duality structure appropriate to the tensor and pseudotensor operators of the boundary theory (Chu et al., 2024, Liu et al., 2020).

The zero-temperature ansatz turns on

$TP$7

together with

$TP$8

and the two-form components

$TP$9

The choice of $TP$0 plus imaginary $TP$1 is required by the self-duality relation and realizes a nodal ring in the $TP$2-$TP$3 plane, while $TP$4 realizes the Weyl sector (Chu et al., 2024).

The UV data define independent deformations of the two topological sectors: $TP$5

$TP$6

Thus $TP$7 control the Weyl sector and $TP$8 control the nodal-line sector in the 2024 phase-diagram analysis. For numerics, the coexistence papers fix

$TP$9

$H(\mathbf{k})=-G^{-1}(0,\mathbf{k})$0

$H(\mathbf{k})=-G^{-1}(0,\mathbf{k})$1

These choices define the specific strongly coupled model in which coexistence is studied (Chu et al., 2024, Chen et al., 19 Sep 2025).

3. Zero-temperature IR solutions and phase diagram

A principal result of the 2024 construction is the classification of nine different IR geometries, corresponding to nine boundary states: four stable phases, four phase boundaries, and one double critical point. In the two-dimensional slice with

$H(\mathbf{k})=-G^{-1}(0,\mathbf{k})$2

the phase diagram is qualitatively similar to the weak-coupling one, but the shapes and locations of the boundaries are quantitatively different, and strong coupling reshapes the phase regions near criticality (Chu et al., 2024).

IR solution	Boundary state	Physical meaning
Weyl-nodal	Weyl-nodal phase	Weyl nodes and nodal ring coexist
Weyl-critical	Weyl-Dirac boundary	Nodal ring critical, Weyl survives
Weyl-gap	Weyl-gap phase	Weyl survives, nodal line gapped
Critical-nodal	Dirac-nodal boundary	Weyl critical, nodal ring survives
Double critical	Double critical point	Both sectors critical
Critical-gap	Dirac-gap boundary	Weyl critical, nodal gapped
Gap-nodal	Gap-nodal phase	Weyl gapped, nodal ring survives
Gap-critical	Gap-Dirac boundary	Nodal critical, Weyl gapped
Gap-gap	Gap-gap phase	Both sectors gapped

The Weyl–nodal phase is the genuine coexistence phase. Its IR solution combines a finite axial gauge profile in the Weyl sector with a Lifshitz-like anisotropic nodal-line geometry. For the parameters quoted in the paper,

$H(\mathbf{k})=-G^{-1}(0,\mathbf{k})$3

This phase occupies the lower-left region of the phase diagram at $H(\mathbf{k})=-G^{-1}(0,\mathbf{k})$4 (Chu et al., 2024).

The double critical point occurs at

$H(\mathbf{k})=-G^{-1}(0,\mathbf{k})$5

for $H(\mathbf{k})=-G^{-1}(0,\mathbf{k})$6. It is a Lifshitz-type geometry in which both the Weyl and nodal-line sectors are simultaneously critical, and it is the point where all four phase boundaries meet. The Weyl–Dirac boundary runs between $H(\mathbf{k})=-G^{-1}(0,\mathbf{k})$7 and $H(\mathbf{k})=-G^{-1}(0,\mathbf{k})$8, while the Dirac–nodal boundary runs between $H(\mathbf{k})=-G^{-1}(0,\mathbf{k})$9 and $\bar\psi\gamma^{\mu\nu}\psi$0 (Chu et al., 2024).

The transition structure is continuous. The free energy obtained from the renormalized Euclidean action is smooth across the double critical point. Along the cut

$\bar\psi\gamma^{\mu\nu}\psi$1

the free energy as a function of $\bar\psi\gamma^{\mu\nu}\psi$2 is smooth, supporting continuous topological phase transitions (Chu et al., 2024).

Transport distinguishes the Weyl content of the phases through the anomalous Hall conductivity,

$\bar\psi\gamma^{\mu\nu}\psi$3

Along the cut

$\bar\psi\gamma^{\mu\nu}\psi$4

the system crosses the double critical point at $\bar\psi\gamma^{\mu\nu}\psi$5; for $\bar\psi\gamma^{\mu\nu}\psi$6 the phase is Weyl-gap and $\bar\psi\gamma^{\mu\nu}\psi$7, whereas for $\bar\psi\gamma^{\mu\nu}\psi$8 the phase is gap-nodal and $\bar\psi\gamma^{\mu\nu}\psi$9. Near criticality,

$\bar\psi\gamma^{\mu\nu}\gamma^5\psi$0

This provides an order parameter for the Weyl sector even when a nodal-line sector coexists or competes with it (Chu et al., 2024).

4. Fermionic Green’s function and the spectral definition of coexistence

The coexistence phase is defined operationally through the fermionic Green’s function rather than directly through a microscopic free Hamiltonian. The 2025 analysis introduces two eight-component bulk fermions, reflecting the eight-component weak-coupling template and allowing simultaneous treatment of Weyl and nodal-line sectors in the interacting response (Chen et al., 19 Sep 2025).

After Fourier transformation,

$\bar\psi\gamma^{\mu\nu}\gamma^5\psi$1

the coupled radial Dirac equations are solved with infalling IR boundary conditions. UV source and response coefficients are assembled into matrices, producing the retarded Green’s function

$\bar\psi\gamma^{\mu\nu}\gamma^5\psi$2

and the topological Hamiltonian

$\bar\psi\gamma^{\mu\nu}\gamma^5\psi$3

This is the interacting-system substitute for a band Hamiltonian and the basis for all later invariant calculations (Chen et al., 19 Sep 2025).

In the Weyl–nodal phase, the effective band structure extracted from $\bar\psi\gamma^{\mu\nu}\gamma^5\psi$4 shows Weyl nodes on the $\bar\psi\gamma^{\mu\nu}\gamma^5\psi$5-axis and nodal rings in the $\bar\psi\gamma^{\mu\nu}\gamma^5\psi$6 plane. Along $\bar\psi\gamma^{\mu\nu}\gamma^5\psi$7, the nodal-line sector is fully gapped while the Weyl-sector bands exhibit multiple crossings and multiple Fermi surfaces. Along $\bar\psi\gamma^{\mu\nu}\gamma^5\psi$8 at $\bar\psi\gamma^{\mu\nu}\gamma^5\psi$9, the Weyl sector is gapped while the nodal-line sector has twofold degenerate zero crossings corresponding to nodal rings. The paper emphasizes that the momentum-space structure is richer than in weakly coupled minimal models, featuring multiple Fermi surfaces, multiple sets of bands, and band crossing ordering interchange (Chen et al., 19 Sep 2025).

These spectral features were anticipated by the earlier holographic nodal-line literature. The 2018 and 2020 nodal-line models already showed that strong coupling can produce multiple nodal loops or nodal rings in the fermionic spectrum, rather than the single loop of the simplest weak-coupling model. In those precursor studies the topological phase was diagnosed by probe-fermion spectral functions with poles at finite-radius momenta in the $g$0-$g$1 plane, and the improved 2020 model further showed that at least one family of nodal rings carries Berry phase $g$2 (Liu et al., 2018, Liu et al., 2020). The coexistence phase inherits that strong-coupling spectral richness and adds Weyl-sector crossings in the same interacting Green’s function.

5. Interacting topological invariants

The 2025 work computes five topological invariants for the holographic coexistence phase: the Weyl charge, $g$3, $g$4, $g$5, and $g$6. All are defined from the occupied eigenstates of the topological Hamiltonian $g$7, not from a bare noninteracting Hamiltonian (Chen et al., 19 Sep 2025).

The Weyl charge is the Berry-curvature monopole charge integrated over a closed sphere $g$8 around a Weyl node,

$g$9

For the nodal ring, the $V,\hat V$0 Berry-phase invariant is

$V,\hat V$1

with $V,\hat V$2 a loop linked with the ring. The higher invariant $V,\hat V$3 is defined by Wilson-loop winding on a torus $V,\hat V$4 surrounding the ring, while the mirror-protected invariants $V,\hat V$5 and $V,\hat V$6 compare, respectively, occupied mirror eigenstructure at points inside and outside the ring and Wilson-loop average phases on mirror-plane loops inside and outside the ring (Chen et al., 19 Sep 2025).

For the Weyl–nodal coexistence phase the numerical results are

$V,\hat V$7

These values show simultaneous nonzero Weyl monopole charge and nontrivial nodal-line topology. They also show that the nodal ring is protected against small perturbations through $V,\hat V$8 and is mirror protected through $V,\hat V$9, but there is no higher obstruction forcing a ring that has shrunk to a critical configuration to re-expand, because $A,\hat A$0 (Chen et al., 19 Sep 2025).

The paper also studies critical phases. In the critical phase of the holographic Weyl semimetal, the effective band structure shows a fourfold Dirac point where two Weyl nodes merge, and the Weyl charge on an enclosing sphere is zero. In the critical phase of the holographic nodal-line semimetal, strongly coupled behavior differs from simple weak-coupling intuition: two nodal rings coincide into a single critical nodal ring structure and there remain infinitely many discrete critical nodal rings in the effective band description, yet $A,\hat A$1 vanishes, so these structures are topologically trivial. In the Weyl–Critical regime of the coexistence model, the Weyl charges remain $A,\hat A$2 while $A,\hat A$3, establishing that the Weyl topology survives whereas the nodal-line sector has become critical or trivial (Chen et al., 19 Sep 2025).

A plausible implication is that strong coupling can substantially alter the geometric appearance of criticality while leaving the topological classification intact: the effective bands may display multiple or coincident critical nodal structures even when the relevant invariants vanish.

6. Precursor nodal-line constructions and their role as building blocks

The coexistence phase did not emerge in isolation. The 2018 holographic nodal-line semimetal provided the first strong-coupling realization of a topological nodal line semimetal and established the bulk mechanism that later informed coexistence models. Its five-dimensional bulk theory contains a scalar $A,\hat A$4 dual to the mass operator and a massive antisymmetric two-form $A,\hat A$5 dual to an operator deforming the topology of the Fermi surface. The central interaction,

$A,\hat A$6

forces competition between the two channels in the IR. In that model the topological NLSM phase occurs for $A,\hat A$7, the critical solution at $A,\hat A$8, and the trivial phase for $A,\hat A$9; the fermion spectral function shows multiple closed nodal loops in the topological phase, and the free energy is smooth across the transition (Liu et al., 2018).

That 2018 model did not realize coexistence with Weyl nodes. It explicitly compared its mechanism with earlier holographic Weyl semimetals and proposed a common IR framework: include one bulk field dual to a mass operator and another dual to an operator deforming the Fermi-surface topology, arrange their interaction so that they cannot both dominate at leading IR order, and obtain topological, trivial, and critical classes of IR solutions. This general design principle directly foreshadowed later coexistence constructions (Liu et al., 2018).

The 2020 improved holographic nodal-line model refined the tensor sector by enforcing the operator identity

$\Phi_1,\Phi_2$0

through a first-order complex two-form action. In that model the UV source relation

$\Phi_1,\Phi_2$1

follows automatically, and the phase transition occurs at

$\Phi_1,\Phi_2$2

The topological phase again exhibits multiple nodal rings confined to $\Phi_1,\Phi_2$3, with linear dispersion near the rings, and at least one family of rings carries Berry phase $\Phi_1,\Phi_2$4 (Liu et al., 2020).

For coexistence physics, the importance of the improved model is structural. It supplies a more faithful holographic tensor sector, already embedded in an action that also contains the axial gauge and anomaly ingredients familiar from holographic Weyl semimetals. The later coexistence model can therefore be viewed as the simultaneous activation, in one backreacted background, of the axial-vector channel responsible for Weyl nodes and the complex two-form channel responsible for nodal rings (Liu et al., 2020, Chu et al., 2024).

7. Interpretation, scope, and unresolved issues

The combined picture from the 2024 and 2025 studies is that topological semimetal coexistence survives at strong coupling and is not restricted to a multicritical fine-tuned point. For fixed $\Phi_1,\Phi_2$5, the Weyl–nodal phase occupies a finite region of the $\Phi_1,\Phi_2$6 plane, with neighboring mixed phases in which one topological sector survives while the other is gapped. The 2024 paper established this phase structure through bulk IR classification, anomalous Hall conductivity, and free energy; the 2025 paper then supplied the direct interacting topological diagnosis through Green’s functions and invariants (Chu et al., 2024, Chen et al., 19 Sep 2025).

Strong coupling also generates spectral phenomena not typical of minimal weak-coupling band models. The 2025 paper emphasizes multiple Fermi surfaces, multiple sets of bands, and band crossing ordering interchange in both Weyl and nodal sectors, together with unusual critical nodal-ring structures. The authors treat these as notable and unique features inherent to strongly coupled topological semimetals within this holographic framework, but do not claim full universality beyond the class of models studied (Chen et al., 19 Sep 2025).

Several limitations are also explicit. The 2024 coexistence construction did not yet compute direct fermionic spectral functions or topological invariants for the coexistence phase; instead it inferred the nodal-line content from IR structure and analogy with the earlier nodal-line models. Those computations were performed later in the 2025 work. The 2024 paper also identifies further directions, including topological entanglement entropy, triple-degenerate nodal points, multiple Weyl pairs, and disorder (Chu et al., 2024).

Within the development of holographic topological matter, the holographic Weyl–nodal line coexisting semimetal therefore represents an overview of two previously separate strong-coupling mechanisms. The axial-vector sector provides a holographic route to Weyl-node separation, the complex two-form sector provides a holographic route to nodal rings consistent with tensor-operator duality, and the topological Hamiltonian $\Phi_1,\Phi_2$7 furnishes a unified interacting diagnostic of both structures in a single phase (Liu et al., 2020, Chen et al., 19 Sep 2025).