---
title: Holographic Nodal Line Semimetals
url: https://www.emergentmind.com/topics/holographic-nodal-line-semimetals
type: topic
---

# Holographic Nodal Line Semimetals

Holographic nodal line semimetals (NLSMs) are topologically nontrivial metallic phases in strongly coupled systems, realized via gauge/gravity duality. In these phases, the gapless excitation manifold in boundary momentum space forms one or more closed loops (“nodal lines”) rather than isolated points, and these loops possess well-defined topological invariants. The holographic approach enables the exploration of such semimetals beyond weak-coupling band theory, revealing novel quantum phase transitions, scaling, and entanglement phenomena that are inaccessible by conventional methods [1801.09357, 1911.07978, 2109.07187, 2012.12602, 2509.15574, 2602.01545].

## 1. Holographic Construction of Nodal Line Semimetals

Holographic NLSMs are engineered in five-dimensional asymptotically AdS spacetimes with coupled matter fields. The minimal bulk action includes the Einstein–Hilbert term, two U(1) gauge fields (axial and vector), a complex scalar Φ dual to a fermion mass operator, and a complex or real two-form B_{ab} dual to an antisymmetric fermion bilinear. The essential couplings—particularly a quartic interaction λ|Φ|²B_{ab}^*B^{ab}—mediate competition between mass generation and Fermi surface topology, implementing the “mass ↔ nodal-loop” transition mechanism.

The typical bulk action is:
\[
S = \int d^5x\sqrt{-g}\left[
R+12-\frac{1}{4}F_{V}^{2}-\frac{1}{4}F_{A}^{2} 
+ \textrm{CS terms} 
- |D\Phi|^2 - m_1^2|\Phi|^2
- \frac{1}{3\eta}|\mathcal{D}_{[a}B_{bc]}|^2 - m_2^2|B_{ab}|^2
- \lambda|\Phi|^2|B_{ab}|^2
\right]
\]
with D_aΦ = (∇_a - iA_a)Φ, \(\mathcal{D}_{[a}B_{bc]} = \partial_{[a}B_{bc]} - iA_{[a}B_{bc]}\), and Chern-Simons terms encoding anomaly structure [1801.09357, 1911.07978]. Improved models employ a Chern-Simons plus mass term for the two-form to enforce duality relations for the corresponding boundary operators [2012.12602].

At zero temperature, the metric ansatz is a translationally invariant, anisotropic domain wall:
\[
ds^2 = u(r)(-dt^2 + dz^2) + dr^2/u(r) + f(r)(dx^2 + dy^2),
\]
with radial-dependent profiles
\[
\Phi = \phi(r),\quad B_{xy} = B(r),\quad B_{tz} = iB_{tz}(r),
\]
and other fields vanishing. The UV (boundary) expansions are
\[
\phi(r) \sim M/r,\qquad B_{xy}(r) \sim b r
\]
defining the tuning parameter \(M/b\) that controls the phase structure [1801.09357, 1911.07978, 2602.01545].

## 2. Fermionic Spectra and Nodal Loop Geometry

Fundamental fermionic excitations are introduced as probe Dirac spinors in the bulk, charged under the relevant gauge fields and coupled via Yukawa-type interactions to Φ and B_{ab}. The retarded boundary Green's function is extracted from the near-boundary asymptotics:
\[
\Psi(r,x) \sim \psi_0\, r^{-m_f} + \psi_1\, r^{m_f},\qquad 
G^R(\omega, \mathbf{k}) = -i\,\psi_1 / \psi_0
\]
subject to ingoing (IR) boundary conditions.

The defining property of the NLSM phase is the existence of multiple closed solutions to
\[
\det G^R{}^{-1}(0, \vec{k}) = 0
\]
in the k_z=0 plane. These equations yield a discrete set of Fermi momenta \(k_F^{(n)}\), forming one or more concentric circles in the (k_x, k_y) plane: \(k_x^2 + k_y^2 = (k_F^{(n)})^2\), corresponding to gapless band crossings (“nodal loops”) [1801.09357, 1911.07978, 2012.12602, 2109.07187]. At strong coupling, multiple rings (multi-Fermi-surfaces) can occur generically, and their radii exhibit an Efimov-like scaling [2109.07187].

## 3. Topological Invariants and Band Topology

The nodal rings in holographic NLSMs are stabilized by topological invariants. The central invariant is the Berry phase (or winding number) computed for the zero-frequency "topological Hamiltonian"
\[
H_{\rm topo}(\vec{k}) = -G^R{}^{-1}(0, \vec{k}).
\]
For a loop C encircling the nodal ring,
\[
\gamma = \oint_C i \langle n(\vec{k}) | d | n(\vec{k}) \rangle,
\]
and numerically one finds \(\gamma = \pi\) for each stable nodal loop [1801.09357, 1911.07978, 2012.12602].

Recent classifications [2509.15574] refine the topological structure:
- **Berry phase (\(\zeta_1\))**: quantized as \(1\) (mod 2) if topological.
- **Torus Wilson loop (\(\zeta_2\))**: a modulus-2 winding number from non-Abelian Berry holonomies.
- **Mirror invariants (\(\zeta_0, \widetilde\zeta_2\))**: stability indices tied to mirror symmetry, computed from mirror plane occupation and Wilson loop phases.

The full set \(\{\zeta_1, \zeta_2; \zeta_0, \widetilde\zeta_2\}\) characterizes whether the nodal line can be trivialized without breaking symmetry. In holography, all invariants are extracted from \(G^R(0, \vec{k})\) [2509.15574].

## 4. Phase Diagram and Quantum Phase Transitions

The NLSM–trivial phase structure is controlled by the UV ratio \(M/b\), corresponding to the mass deformation vs. topology-inducing source [1801.09357, 1911.07978, 2012.12602, 2109.07187, 2602.01545]. The generic behavior is:
- **For \(M/b < M_c\)**: topological nodal-line phase, with multiple Fermi surface rings and nontrivial Berry phases.
- **At \(M/b = M_c\)**: a critical Lifshitz solution with vanishing nodal ring radius; the phase transition is continuous (second order), with smooth free energy and sharp jumps in operator profiles or scaling exponents.
- **For \(M/b > M_c\)**: trivial gapped or partially-gapped semimetal, with no extended Fermi surfaces.

The critical point exhibits nontrivial scaling exponents (e.g., dynamical exponent \(z\)), and the phase transition is highly non-Landau, with similarities to BKT scaling near criticality [1801.09357, 2602.01545]. The location of \(M_c\) depends on model details, with reported values such as \(M_c/b \simeq 1.717\) [1801.09357, 1911.07978], or \(0.8597\) in the improved duality-constrained model [2012.12602, 2602.01545].

## 5. Finite-Temperature Effects and Transport

At finite temperature, multiple nodal rings broaden and melt sequentially as T increases. For elevated T, only the innermost nodal line remains visible, eventually disappearing into a gapped continuum, resulting in a finite window where the boundary spectral density exhibits a single apparent nodal ring [2109.07187].

Transport in the holographic NLSM is highly anisotropic:
- **Electrical conductivity**: At zero temperature, the longitudinal (x-y) DC conductivity vanishes, while the out-of-plane (z) conductivity is generically finite in the nodal-line phase. The AC conductivity exhibits model-dependent power-law scaling at low ω, with distinct exponents in the topological vs. trivial phases [2109.07187].
- **Thermal conductivity**: Ward identities are violated due to broken Lorentz invariance in the presence of B_{xy} background. The in-plane thermal conductivity exhibits Drude-like divergences at low temperature, while the out-of-plane channel is less anomalous.
- **Shear viscosity**: Multiple independent viscosity coefficients arise, with universal in-plane (xy,xy) viscosity s/4π, but strongly suppressed or enhanced out-of-plane components depending on phase and temperature.

A summary of the transport signatures is provided below.

| Channel         | Topological Phase | Trivial Phase   |
|-----------------|------------------|-----------------|
| σ_xx^{DC}       | 0                | 0               |
| σ_zz^{DC}       | nonzero          | 0               |
| η_{xy,xy}/s     | 1/4π             | 1/4π            |
| η_{xz,xz}/s     | 0 (T→0)          | 1/4π (T≫b)      |
| η_{zx,zx}/s     | ∞ (T→0)          | 1/4π (T≫b)      |

This anisotropy is a direct consequence of B_{xy} condensation and symmetry breaking in the IR [2109.07187].

## 6. Multipartite Entanglement and Nonlocal Order Parameters

Strongly coupled NLSMs exhibit short-range entanglement (SRE) in both topological and trivial phases, as all multipartite entanglement measures vanish in the long-strip (l→∞) limit. However, the exponents governing the decay of conditional mutual information, multi-entropy, entanglement wedge cross-section, and Markov gap display sharp and universal jumps at the quantum critical point.

For subsystem width l in the x or z directions, the scaling exponents α_x, β_z in
\[
\mathrm{Measure} \sim l^{-\alpha_x},\;l^{-\beta_z}
\]
depend on the IR dynamical exponent z, which is phase sensitive [2602.01545]. These exponents thus serve as robust nonlocal order parameters for the topological transition, providing a direct probe of emergent scaling and topology in the absence of weakly coupled quasiparticles.

## 7. Strong-Coupling Signatures, Band Topology, and Experimental Probes

Strongly coupled holographic NLSMs, compared to their weak-coupling band-theory analogs, exhibit several unique phenomena:
- **Multi-Fermi-surface (multiple nodal loops)**: Generically present at strong coupling due to the IR scaling of probe fermion equations. The number and spacing can be interpreted analogously to Efimov scaling [2109.07187].
- **Band-crossing ordering interchange**: The ordering of bands that cross along the nodal loop or Fermi surface may interchange as one moves around the ring, a phenomenon absent in free-fermion models [2509.15574].
- **Topological invariants without Bloch Hamiltonian**: All relevant invariants (\(\zeta_1,\zeta_2,\zeta_0,\tilde{\zeta}_2\)) are computed solely from the zero-frequency boundary Green's function, rather than a single-particle Hamiltonian [2509.15574].
- **Experimental implications**: ARPES could reveal multiple concentric nodal rings and drumhead surface states, while quantum oscillations (Shubnikov–de Haas) and Berry-phase interference give access to the topological indices. The presence of multiple nodal lines, strong anisotropy in transport, and discontinuous changes in scaling exponents are potential experimental signatures [2109.07187, 2509.15574].

Notably, explicit surface states (“drumhead modes”) and anomalous Hall transport have not yet been systematically studied in the holographic NLSM setup and remain important open problems [1911.07978]. 

---

**References:**  
- [1801.09357] Topological nodal line semimetals in holography  
- [1911.07978] Holographic Topological Semimetals  
- [2109.07187] Thermodynamics and transport of holographic nodal line semimetals  
- [2012.12602] An improved holographic nodal line semimetal  
- [2602.01545] Multipartite entanglement characterizing topological phase transitions in holographic nodal line semimetals  
- [2509.15574] Topological invariant for holographic Weyl-Nodal line coexisting semimetal

Source: https://www.emergentmind.com/topics/holographic-nodal-line-semimetals