Holographic Weyl Semimetal

• Holographic Weyl semimetal is a strong-coupling realization where axial and mass deformations drive a transition between topological and trivial phases without relying on weakly coupled band structures.
• The minimal bottom-up model uses a five-dimensional AdS framework with gauge fields and a charged scalar to reproduce key features like nonzero anomalous Hall conductivity and quantum criticality.
• Infrared observables, particularly the behavior of the axial gauge field at the horizon, serve as proxies for topological order, linking anomaly-induced transport to robust holographic RG flow.

A holographic Weyl semimetal is a strongly coupled AdS/CFT realization of Weyl-semimetal physics in which a four-dimensional boundary theory is deformed by a time-reversal-breaking axial source $b$ and a mass deformation $M$, and the resulting phase is diagnosed not by weakly coupled band topology but by anomaly-induced transport and infrared bulk data. In the foundational constructions, the defining signatures are a nonzero anomalous Hall conductivity and a quantum phase transition to a topologically trivial or insulating phase as $M/b$ is increased; the crucial conceptual point is that this persists even when the strongly coupled theory has no quasiparticle excitations, no sharp Weyl cones, and no Berry-monopole description in momentum space (Landsteiner et al., 2015, Landsteiner et al., 2015).

1. Origins and conceptual framework

An early precursor to the subject studied single-particle fermionic response in a Lifshitz background and argued that, especially for $z=2$, the boundary system behaves like an undoped interacting Weyl semimetal. That construction emphasized chiral, gapless, particle-hole-symmetric spectral structure and a quantum phase transition between a non-Fermi-liquid phase and a Fermi-liquid phase with two spontaneously formed Fermi surfaces at zero chemical potential, but it did not explicitly realize anomaly-induced Hall transport or a momentum-space node-separation order parameter in the later sense of holographic Weyl semimetals (Gursoy et al., 2012).

The modern anomaly-based formulation takes as weak-coupling benchmark the Lorentz-violating Dirac model

$\mathcal{L}=\bar\Psi\left(i\slashed\partial-e\slashed A-\gamma_z\gamma_5 b+M\right)\Psi,$

for which the system is gapless when $|b|>|M|$, with effective node separation

$2b_{\mathrm{eff}}=2\sqrt{b^2-M^2},$

and gapped when $|b|<|M|$. In that regime the anomalous Hall current is

$$\mathbf J=\frac{e^2}{2\pi^2}\,\mathbf b_{\mathrm{eff}}\times \mathbf E.$$

Holographic models were built to reproduce precisely the two low-energy features encoded in these formulas: a nonzero anomalous Hall effect and a transition driven by the competition between $b$ and $M$0, while abandoning the weak-coupling reliance on Weyl cones, Berry curvature monopoles, or quasiparticle band structure (Landsteiner et al., 2015, Landsteiner et al., 2015).

A central conceptual departure follows from strong coupling. In the holographic theory, the topological distinction is not defined through singularities of single-particle wavefunctions in momentum space. Instead, the distinction between phases is encoded in the infrared fate of bulk fields and in transport coefficients fixed by anomaly data. This is why the phrase “holographic Weyl semimetal” refers less to a literal holographic band structure than to an interacting topological semimetal phase whose low-energy response mirrors that of a Weyl semimetal (Landsteiner et al., 2015, Landsteiner et al., 2015).

2. Minimal bottom-up construction

The minimal bottom-up model is a five-dimensional asymptotically AdS theory containing a vector gauge field $M$1, an axial gauge field $M$2 or $M$3, and a complex scalar $M$4 charged under the axial field. In its minimal form the action is

$M$5

with

$M$6

and $M$7, so that the dual scalar operator has dimension three. The Chern-Simons term is essential because it reproduces the axial anomaly while preserving the non-anomalous vector current (Landsteiner et al., 2015).

The ultraviolet dictionary is fixed by the boundary conditions

$M$8

Thus $M$9 is the source for the time-reversal-breaking axial deformation and $M/b$0 is the source for the mass operator. In the probe-limit version the metric is fixed to AdS-Schwarzschild,

$M/b$1

where the radial coordinate is interpreted as an RG scale, $M/b$2 being the UV and the horizon the IR (Landsteiner et al., 2015).

With background ansatz

$M/b$3

the equations of motion become

$M/b$4

$M/b$5

These equations already encode the basic competition: the axial gauge field suppresses scalar growth, while the scalar gives an effective mass to $M/b$6 and drives it toward zero in the interior. The backreacted zero-temperature model generalizes this setup by allowing anisotropic metric functions $M/b$7, $M/b$8, and a quartic scalar potential

$M/b$9

which makes the IR critical geometry and exact zero-temperature phases explicit (Landsteiner et al., 2015).

The consistent vector and axial currents are obtained by varying the renormalized on-shell action. In the standard conventions,

$z=2$0

$z=2$1

The distinction between consistent and covariant currents is nontrivial throughout the subject, because Chern-Simons contact terms contribute directly to Hall responses (Landsteiner et al., 2015, Landsteiner et al., 2015).

3. Infrared phases and quantum phase transitions

At zero temperature the backreacted model exhibits three classes of IR solutions: a topological Weyl semimetal phase, a topologically trivial phase, and a critical Lifshitz-like solution separating them. For the representative parameter choice

$z=2$2

the critical data are

$z=2$3

and the critical ratio is

$z=2$4

The critical geometry has anisotropic scaling

$z=2$5

with $z=2$6 (Landsteiner et al., 2015).

The two stable zero-temperature phases are distinguished by the infrared endpoint of the axial gauge field. In the topological phase,

$z=2$7

while in the trivial phase,

$z=2$8

and the scalar condenses to a nonzero IR value. This is naturally interpreted as restoration of time-reversal symmetry at the endpoint of the RG flow in the trivial phase: $z=2$9 explicitly breaks time reversal in the UV, but the flow can wash it out in the IR (Landsteiner et al., 2015).

In the original probe-limit finite-temperature analysis the transition appears as a crossover that sharpens as $\mathcal{L}=\bar\Psi\left(i\slashed\partial-e\slashed A-\gamma_z\gamma_5 b+M\right)\Psi,$0 is lowered. At low temperature, with $\mathcal{L}=\bar\Psi\left(i\slashed\partial-e\slashed A-\gamma_z\gamma_5 b+M\right)\Psi,$1, the anomalous Hall conductivity nearly vanishes around

$\mathcal{L}=\bar\Psi\left(i\slashed\partial-e\slashed A-\gamma_z\gamma_5 b+M\right)\Psi,$2

and later probe-limit quasinormal-mode work at $\mathcal{L}=\bar\Psi\left(i\slashed\partial-e\slashed A-\gamma_z\gamma_5 b+M\right)\Psi,$3 located the sharp crossover near

$\mathcal{L}=\bar\Psi\left(i\slashed\partial-e\slashed A-\gamma_z\gamma_5 b+M\right)\Psi,$4

These results are interpreted as remnants of the underlying $\mathcal{L}=\bar\Psi\left(i\slashed\partial-e\slashed A-\gamma_z\gamma_5 b+M\right)\Psi,$5 quantum critical point (Landsteiner et al., 2015, Rai et al., 2024).

The holographic RG-flow interpretation is one of the subject’s central organizing principles. The UV theory is specified by $\mathcal{L}=\bar\Psi\left(i\slashed\partial-e\slashed A-\gamma_z\gamma_5 b+M\right)\Psi,$6 and $\mathcal{L}=\bar\Psi\left(i\slashed\partial-e\slashed A-\gamma_z\gamma_5 b+M\right)\Psi,$7, but the low-energy phase is determined by the radial flow of $\mathcal{L}=\bar\Psi\left(i\slashed\partial-e\slashed A-\gamma_z\gamma_5 b+M\right)\Psi,$8 and $\mathcal{L}=\bar\Psi\left(i\slashed\partial-e\slashed A-\gamma_z\gamma_5 b+M\right)\Psi,$9. If $|b|>|M|$0 dominates, $|b|>|M|$1 remains nonzero in the IR and the system retains a nonzero Hall response; if $|b|>|M|$2 dominates, $|b|>|M|$3 drives $|b|>|M|$4 to zero and the IR becomes topologically trivial. This replaces the weak-coupling statement that topology is controlled by $|b|>|M|$5 with the strong-coupling statement that it is controlled by the IR bulk field $|b|>|M|$6 or $|b|>|M|$7 (Landsteiner et al., 2015, Landsteiner et al., 2015).

Later bottom-up work enlarged the phase diagram beyond the original continuous topological-to-trivial semimetal transition. A Stueckelberg Einstein-Maxwell-axial-scalar model was shown to realize a transition from a strongly interacting Weyl semimetal to an insulating phase with a hard gap in the real part of the diagonal optical conductivities, while the zero-frequency anomalous Hall conductivity remained nonzero. That insulating phase is therefore interpreted as a Chern insulator, and the transition is always first order, signified by a discontinuous anomalous Hall conductivity, in contrast to the continuous transition of the original model (Liu et al., 2018).

4. Transport diagnostics and anomaly structure

The anomalous Hall conductivity is the primary order parameter-like observable of the holographic Weyl semimetal. In the minimal model it is extracted from transverse vector perturbations

$|b|>|M|$8

and the Kubo formula

$|b|>|M|$9

After solving the helicity equations for $2b_{\mathrm{eff}}=2\sqrt{b^2-M^2},$0, one finds for the covariant current

$2b_{\mathrm{eff}}=2\sqrt{b^2-M^2},$1

while for the consistent current the anomalous Hall conductivity is

$2b_{\mathrm{eff}}=2\sqrt{b^2-M^2},$2

At zero temperature this becomes $2b_{\mathrm{eff}}=2\sqrt{b^2-M^2},$3, making the Hall response a pure IR observable controlled by the endpoint of the holographic RG flow (Landsteiner et al., 2015, Landsteiner et al., 2015).

This formula gives a direct strong-coupling analogue of the weak-coupling expression $2b_{\mathrm{eff}}=2\sqrt{b^2-M^2},$4 once the anomaly coefficient is matched by

$2b_{\mathrm{eff}}=2\sqrt{b^2-M^2},$5

The correspondence is then

$2b_{\mathrm{eff}}=2\sqrt{b^2-M^2},$6

A common misconception is that the topological phase requires a momentum-space node separation in the quasiparticle sense. In the holographic model, the defining low-energy quantity is instead the renormalized IR axial field that controls the Hall response (Landsteiner et al., 2015).

The diagonal conductivities are also determined by horizon or IR data. In the backreacted anisotropic background they are

$2b_{\mathrm{eff}}=2\sqrt{b^2-M^2},$7

at finite temperature, whereas in the probe-limit finite-temperature model the longitudinal conductivities satisfy

$2b_{\mathrm{eff}}=2\sqrt{b^2-M^2},$8

The latter was explicitly identified as a probe-limit artifact in the trivial gapped phase for large $2b_{\mathrm{eff}}=2\sqrt{b^2-M^2},$9, though the phase-transition behavior of the Hall response was expected to survive fully backreacted treatments (Landsteiner et al., 2015, Landsteiner et al., 2015).

The optical response provides a second major diagnostic. In a general class of bottom-up models, the zero-temperature AC conductivity is linear at low frequency in both semimetallic phases, but near the critical point an intermediate scaling regime appears. For the critical Lifshitz-like IR solution,

$|b|<|M|$0

with $|b|<|M|$1. The phase diagram can be reconstructed from the scaling of $|b|<|M|$2, yielding ultraviolet, Weyl-semimetal, trivial-semimetal, and quantum-critical regions in $|b|<|M|$3-space (Ammon et al., 2016).

The anomaly structure also governs less obvious Hall responses. In the study of axial Hall conductivity, the anomaly algebra predicts an axial Hall response equal to one third of the electric Hall response. Holographically, this relation is recovered only after including the nontrivial renormalization of the external axial gauge field,

$|b|<|M|$4

so that

$|b|<|M|$5

The paper’s point is that the anomaly coefficient is not renormalized, but the axial source that probes it is. This is one of the clearest places where holographic RG flow modifies the naive weak-coupling reading of anomaly constraints (Copetti et al., 2016).

5. Hydrodynamics, odd viscosity, and bulk–boundary correspondence

The hydrodynamic and viscoelastic sectors reveal additional anomaly-controlled structure. In an axisymmetric $|b|<|M|$6-dimensional holographic Weyl semimetal there are two independent odd viscosities,

$|b|<|M|$7

with horizon formulas

$|b|<|M|$8

Both coefficients are nonzero only because of the mixed axial gravitational anomaly and become substantial only in the low-temperature quantum critical region, where both $|b|<|M|$9 and $$\mathbf J=\frac{e^2}{2\pi^2}\,\mathbf b_{\mathrm{eff}}\times \mathbf E.$$0 are simultaneously significant at the horizon. The same work also found anisotropic shear viscosities with

$$\mathbf J=\frac{e^2}{2\pi^2}\,\mathbf b_{\mathrm{eff}}\times \mathbf E.$$1

in the quantum critical regime, so the anisotropic longitudinal viscosity violates the KSS bound. Conductivities and viscosities are tied together by the exact horizon relation

$$\mathbf J=\frac{e^2}{2\pi^2}\,\mathbf b_{\mathrm{eff}}\times \mathbf E.$$2

which makes the anisotropic IR fixed point the common source of transport scaling (Landsteiner et al., 2016).

Quasinormal-mode analysis adds a dynamical criterion for the topological phase. In the probe-limit model, a longitudinal axial hydrodynamic mode exists only in the topological Weyl-semimetal phase. For $$\mathbf J=\frac{e^2}{2\pi^2}\,\mathbf b_{\mathrm{eff}}\times \mathbf E.$$3, where $$\mathbf J=\frac{e^2}{2\pi^2}\,\mathbf b_{\mathrm{eff}}\times \mathbf E.$$4, the axial longitudinal sector becomes diffusive,

$$\mathbf J=\frac{e^2}{2\pi^2}\,\mathbf b_{\mathrm{eff}}\times \mathbf E.$$5

For $$\mathbf J=\frac{e^2}{2\pi^2}\,\mathbf b_{\mathrm{eff}}\times \mathbf E.$$6 this remains, to very good approximation, a hydrodynamic mode; near the critical region it becomes pseudo-diffusive with a small purely imaginary gap; for $$\mathbf J=\frac{e^2}{2\pi^2}\,\mathbf b_{\mathrm{eff}}\times \mathbf E.$$7 it moves rapidly down the negative imaginary axis and ceases to be hydrodynamic. The result gives a dynamical counterpart to the Hall-response criterion: the topological phase is precisely the phase in which axial symmetry is effectively restored in the IR strongly enough to support a long-lived diffusive mode (Rai et al., 2024).

Bulk–boundary correspondence was studied directly by constructing inhomogeneous interfaces between phases with different topological response. In the presence of a vector chemical potential $$\mathbf J=\frac{e^2}{2\pi^2}\,\mathbf b_{\mathrm{eff}}\times \mathbf E.$$8, a current localized near the interface appears, and although the local profile $$\mathbf J=\frac{e^2}{2\pi^2}\,\mathbf b_{\mathrm{eff}}\times \mathbf E.$$9 depends on interface details, the integrated current is universal: $b$0 This result shows that the total current depends only on the difference of bulk topological data on the two sides, not on the microscopic shape of the interface, and therefore realizes bulk–boundary correspondence at strong coupling even without explicit Fermi-arc band structure (Grignani et al., 2016).

6. Top-down, finite-frequency, and finite-density realizations

A fully top-down holographic Weyl semimetal was constructed in the D3/D7 flavor system, where $b$1-dimensional $b$2 $b$3 SYM at large $b$4 is coupled to $b$5 $b$6 hypermultiplets. In that model a $b$7 subgroup of the R-symmetry acts on the hypermultiplet fermions as an axial symmetry, the hypermultiplet mass $b$8 comes from D3–D7 separation, and an external axial vector field $b$9 is implemented by a linearly varying mass phase $M$00. The model exhibits a first-order phase transition between a Weyl-semimetal phase at small $M$01 and an insulating phase at large $M$02. Remarkably, at $M$03 the anomalous Hall conductivity is independent of the hypermultiplet mass,

$M$04

throughout the topological phase, while in the trivial insulating phase it vanishes. At nonzero temperature the transition remains first order and the Hall conductivity acquires nontrivial dependence on mass and temperature (Fadafan et al., 2020).

The existence of the phase transition is not limited to phenomenological bottom-up models. It was also shown to occur in a top-down model based on a consistent truncation of type IIB supergravity, strengthening the interpretation that holographic Weyl-semimetal physics is not merely an artifact of a specific potential choice (Copetti et al., 2016).

The dynamical transport content of the top-down flavor-brane model was developed further by computing full AC conductivities and the pole structure in complex frequency. Near the first-order Weyl-semimetal/insulator transition, the optical conductivities develop pronounced structures: $M$05 shows a peak, while $M$06 and $M$07 develop troughs. These features become strongest at low temperature and for masses just below the transition, and they are traced to quasinormal-mode poles that move close to the real axis near the critical embedding, producing relatively long-lived excitations (Furukawa et al., 2024).

Finite-density generalizations were achieved in a top-down flavor-brane construction extended by a worldvolume electric field. Two probe fermions of opposite chirality were introduced, and the resulting spectral function displayed at zero density and small $M$08 the hallmarks of a Weyl semimetal: four bands, two Weyl points, and linear dispersion near the nodes. At finite density, the spectral function at $M$09 revealed two distinct Fermi pockets, each centered around a Weyl point, which merged into a single connected Fermi surface as either $M$10 or $M$11 was increased. This is a Lifshitz transition of the Fermi-surface topology, and the work emphasized that it can be driven either by changing band shape through $M$12 or by shifting the bands relative to the Fermi level through $M$13 (Lu et al., 29 Sep 2025).

7. Deformations, driven states, and dislocations

Several deformations of the standard holographic Weyl semimetal have been studied. Adding a background magnetic field in the $M$14-direction shifts the critical coupling extracted from the peak of the horizon anisotropy

$M$15

At small field,

$M$16

while at large field,

$M$17

Thus a magnetic field stabilizes the topological phase against the scalar mass deformation, pushing the phase boundary to larger $M$18 (Bruni et al., 2023).

Breaking translation invariance has the opposite effect. With momentum relaxation introduced by linear axions, the critical ratio $M$19 decreases as $M$20 increases and, above a special threshold, goes to zero, implying that the Weyl-semimetal phase shrinks and finally disappears. The same qualitative behavior was reproduced in a massive-gravity implementation of momentum relaxation, where the critical value decreases with graviton mass and vanishes around

$M$21

at fixed $M$22. The agreement between axion and massive-gravity realizations was explicitly interpreted as a universal feature of momentum-relaxed holographic Weyl semimetals (Zhao, 2021, Zhao, 2021).

A distinct nonequilibrium extension realizes a holographic Floquet state in strongly coupled $M$23 supersymmetric massless QCD in a rotating electric field. There the Weyl-semimetal-like structure is generated dynamically by periodic driving rather than by a static axial background. The driven state is a nonequilibrium steady state stabilized by the large-$M$24 gluonic sector, and weak DC and AC probe analysis in the rotating background shows Hall currents as a linear response. The Hall response of Floquet Weyl semimetals therefore survives in the strong-coupling limit, and the system also exhibits frequency mixing characteristic of Floquet dynamics (Hashimoto et al., 2016).

Dislocations and torsion have motivated a more geometric line of work. One approach analyzes a dislocation in a three-dimensional Weyl semimetal and interprets the associated quantum singularity as a defect in momentum space that binds a topologically protected zero-energy mode; holographically, that singularity is mapped onto a domain wall in the bulk, with anomaly inflow and entanglement entropy used as organizing ideas (Tanaka, 2020). A later construction uses five-dimensional Chern-Simons gravitational theory in AdS with torsion to model holographic Weyl semimetals with dislocation defects at finite temperature. In that framework torsion is the holographic counterpart of crystalline dislocations, and the boundary chiral anomaly is proportional to the Nieh–Yan invariant (Juričić et al., 2024).

Taken together, these developments show that “holographic Weyl semimetal” denotes not a single model but a family of strongly coupled constructions unified by three recurrent structures: a competition between axial and mass deformations, anomaly-induced Hall response as the main topological diagnostic, and an infrared characterization of topology in terms of bulk fields rather than quasiparticle band geometry. Across bottom-up, top-down, finite-density, momentum-relaxed, Floquet-driven, and dislocation-based realizations, the subject has consistently recast Weyl-semimetal physics into a language of holographic RG flow, anomaly inflow, and strongly interacting transport (Landsteiner et al., 2015, Landsteiner et al., 2015).