Burkov–Balents Model in Topological Semimetals

• The Burkov–Balents model is a multilayer topological insulator framework defined by alternating topological and normal insulator layers, where surface Dirac states are coupled via intra-film (Δ_S) and inter-film (Δ_D) tunnelling.
• It distinguishes trivial and topological insulating regimes with a Dirac critical point at Δ_S = Δ_D, and incorporates Zeeman splitting (Δ_Z) to generate Weyl-semimetal and anomalous-quantum-Hall phases.
• The model’s extensions include detailed treatments of impurity effects, off-diagonal disorder, and ultra-thin-film variants, establishing it as a benchmark for understanding complex topological phase transitions.

The expression Burkov–Balents model is used most consistently for a multilayer topological-band model built from alternating topological-insulator and normal-insulator layers, in which the low-energy degrees of freedom are the top and bottom surface Dirac states of each topological-insulator film. In that usage, the model is parameterized by an intra-film hybridization $\Delta_S$, an inter-film tunnelling $\Delta_D$, and, in magnetic extensions, a Zeeman splitting $\Delta_Z$; it realizes trivial and topological insulating phases, a Dirac critical point, and, after time-reversal breaking, Weyl-semimetal and anomalous-quantum-Hall regimes (Alisultanov et al., 2024, Alisultanov et al., 28 Jul 2025). The label is nevertheless not fully univocal: the literature also contains a distinct Burkov–Balents spin-diffusion framework for disordered Rashba two-dimensional electron gases, and several Balents-associated kagome-spin-liquid papers explicitly state that they are not about a separate Burkov–Balents model but about the Balents–Fisher–Girvin construction (Szolnoki et al., 2017, Qi et al., 2015).

1. Terminology and scope

Within the cited literature, the multilayer topological-insulator/normal-insulator construction is the usage most directly tied to Weyl-semimetal physics, anomalous Hall response, and disorder in layered heterostructures (Alisultanov et al., 2024, Alisultanov et al., 28 Jul 2025). The resulting ambiguity is mainly terminological rather than physical: several distinct models contain “Balents” in the author list, but they describe different systems, different Hilbert spaces, and different observables.

Usage in the literature	Physical setting	Relation to the term
Burkov–Balents multilayer model	Alternating TI and NI layers; coupled surface Dirac states	Most direct match in Weyl-multilayer work
Balents–Fisher–Girvin model	Kagome XXZ/$\mathbb Z_2$ spin liquid	Explicitly not a separate Burkov–Balents model
Burkov–Balents spin-diffusion propagator	Disordered Rashba 2DEG	Distinct Burkov–Balents framework

The confusion with the Balents–Fisher–Girvin, or BFG, model is repeatedly addressed in the kagome literature. The vison-fractionalization analysis of kagome $\mathbb Z_2$ spin liquids states that it is not about a separate “Burkov–Balents model,” but about the BFG kagome spin liquid (Qi et al., 2015). The trapped-ion emulation paper and the finite-field kagome plateau study make the same point: their subject is the BFG model and its descendants, not the multilayer Burkov–Balents heterostructure (Nath et al., 2015, Plat et al., 2015).

2. Multilayer Hamiltonian and clean band topology

The clean multilayer Burkov–Balents Hamiltonian, in the form used for off-diagonal-disorder studies, is

$$\mathcal{H}_{\mathbf{k}}^{0} = v_{F}\tau^{z}\otimes \left(\hat{\mathbf z}\times\boldsymbol{\sigma}\right)\!\cdot\!\mathbf{k}_{\perp} + \tau^{x}\otimes\sigma_{0}\left(\Delta_{S}-\Delta_{D}\cos k_{z}d\right) + \tau^{y}\otimes\sigma_{0}\,\Delta_{D}\sin k_{z}d .$$

Here $\tau^{x,y,z}$ act in the top/bottom-surface pseudospin sector of a given topological-insulator film, $\sigma^{x,y,z}$ act in real spin space, $\mathbf{k}_\perp=(k_x,k_y)$ is the in-plane momentum, $d$ is the superlattice period, $\Delta_D$0 is the intra-film tunnelling amplitude, and $\Delta_D$1 is the inter-film tunnelling amplitude (Alisultanov et al., 28 Jul 2025). In equivalent layer-space formulations, the same model is written with local surface-Dirac terms, same-film hybridization $\Delta_D$2, and nearest-layer couplings through $\Delta_D$3 (Alisultanov et al., 2024).

In the clean, nonmagnetic problem, the topological distinction is set by the competition between $\Delta_D$4 and $\Delta_D$5. The multilayer is a trivial insulator for $\Delta_D$6, a topological insulator for $\Delta_D$7, and a Dirac critical point occurs at $\Delta_D$8 (Alisultanov et al., 2024, Alisultanov et al., 28 Jul 2025). The corresponding indicator is written as

$\Delta_D$9

so the sign of $\Delta_Z$0 separates the two insulating sectors (Alisultanov et al., 28 Jul 2025).

The location of the critical Dirac node is convention-dependent in the cited formulations. One momentum-space convention gives a Dirac point at $\Delta_Z$1 when $\Delta_Z$2 (Alisultanov et al., 2024), whereas the formulation with $\Delta_Z$3 above gives the direct gap closing at $\Delta_Z$4, $\Delta_Z$5 at the same parameter value (Alisultanov et al., 28 Jul 2025). This suggests a sign or gauge convention dependence in the $\Delta_Z$6 placement of the critical node rather than a disagreement about the phase boundary itself.

3. Weyl generation, magnetic extensions, and thin-film variants

The standard magnetic extension adds a Zeeman term,

$\Delta_Z$7

which breaks time-reversal symmetry and generates Weyl and anomalous-quantum-Hall regimes (Alisultanov et al., 28 Jul 2025). In the disorder-renormalized formulation, the Weyl-semimetal window is

$\Delta_Z$8

while

$\Delta_Z$9

corresponds to the normal-insulator regime and

$\mathbb Z_2$0

to the anomalous-quantum-Hall regime (Alisultanov et al., 2024). In the clean limit, the same inequalities organize the phase structure with $\mathbb Z_2$1 replaced by $\mathbb Z_2$2; this is a direct implication of the renormalized formulation.

A closely related but not identical implementation is the ultra-thin-film topological-insulator multilayer. There the basic unit is an ultra-thin TI film whose top and bottom surfaces already hybridize strongly, and the Hamiltonian contains a thin-film mass function

$\mathbb Z_2$3

together with Zeeman splitting $\mathbb Z_2$4 and, optionally, structure-inversion asymmetry $\mathbb Z_2$5 (Owerre, 2016). That model is described as “very similar” to Burkov–Balents, but with a different Hamiltonian arising from ultra-thin-film physics (Owerre, 2016).

In the ultra-thin-film variant, the $\mathbb Z_2$6 problem supports a Dirac-semimetal regime for $\mathbb Z_2$7, a transition at $\mathbb Z_2$8, and a $\mathbb Z_2$9D quantum spin Hall phase for $\mathbb Z_2$0, with

$\mathbb Z_2$1

as the parity criterion (Owerre, 2016). With Zeeman splitting but $\mathbb Z_2$2, the phase structure becomes ordinary insulator for $\mathbb Z_2$3, Weyl semimetal for $\mathbb Z_2$4, and $\mathbb Z_2$5D quantum anomalous Hall phase for $\mathbb Z_2$6, where

$\mathbb Z_2$7

When both $\mathbb Z_2$8 and $\mathbb Z_2$9, the model still supports a Weyl phase and, notably, the Weyl nodes remain at zero energy provided $$\mathcal{H}_{\mathbf{k}}^{0} = v_{F}\tau^{z}\otimes \left(\hat{\mathbf z}\times\boldsymbol{\sigma}\right)\!\cdot\!\mathbf{k}_{\perp} + \tau^{x}\otimes\sigma_{0}\left(\Delta_{S}-\Delta_{D}\cos k_{z}d\right) + \tau^{y}\otimes\sigma_{0}\,\Delta_{D}\sin k_{z}d .$$0 (Owerre, 2016).

4. Anomalous Hall conductivity and the Fermi-surface question

The Burkov–Balents framework became a focal point in the debate over how anomalous Hall conductivity should be interpreted in Weyl metals. A precise correction was given in the Comment by Vanderbilt, Souza, and Haldane, which addressed claims made about Weyl-node contributions to the intrinsic anomalous Hall conductivity in metallic ferromagnets (Vanderbilt et al., 2013). The central issue was not whether Weyl points contribute to anomalous Hall response—they do—but whether their presence invalidates Haldane’s statement that the non-quantized part of the intrinsic anomalous Hall conductivity is a Fermi-surface property (Vanderbilt et al., 2013).

The Comment accepts that a naive sliced-Brillouin-zone treatment can fail. In that treatment one writes

$$\mathcal{H}_{\mathbf{k}}^{0} = v_{F}\tau^{z}\otimes \left(\hat{\mathbf z}\times\boldsymbol{\sigma}\right)\!\cdot\!\mathbf{k}_{\perp} + \tau^{x}\otimes\sigma_{0}\left(\Delta_{S}-\Delta_{D}\cos k_{z}d\right) + \tau^{y}\otimes\sigma_{0}\,\Delta_{D}\sin k_{z}d .$$1

and then approximates the slice Hall response by Fermi-loop Berry phases,

$$\mathcal{H}_{\mathbf{k}}^{0} = v_{F}\tau^{z}\otimes \left(\hat{\mathbf z}\times\boldsymbol{\sigma}\right)\!\cdot\!\mathbf{k}_{\perp} + \tau^{x}\otimes\sigma_{0}\left(\Delta_{S}-\Delta_{D}\cos k_{z}d\right) + \tau^{y}\otimes\sigma_{0}\,\Delta_{D}\sin k_{z}d .$$2

Used naively, this can miss contributions from fully occupied bands, especially when Weyl crossings lie inside the occupied manifold (Vanderbilt et al., 2013).

The rebuttal is that this is not Haldane’s actual Fermi-surface formula. The defended expression is

$$\mathcal{H}_{\mathbf{k}}^{0} = v_{F}\tau^{z}\otimes \left(\hat{\mathbf z}\times\boldsymbol{\sigma}\right)\!\cdot\!\mathbf{k}_{\perp} + \tau^{x}\otimes\sigma_{0}\left(\Delta_{S}-\Delta_{D}\cos k_{z}d\right) + \tau^{y}\otimes\sigma_{0}\,\Delta_{D}\sin k_{z}d .$$3

so the anomalous Hall vector is written as a sum over Fermi sheets weighted by Berry-curvature flux (Vanderbilt et al., 2013). In this language, Weyl points are not omitted; they enter through the quantized Berry flux threading Fermi pockets,

$$\mathcal{H}_{\mathbf{k}}^{0} = v_{F}\tau^{z}\otimes \left(\hat{\mathbf z}\times\boldsymbol{\sigma}\right)\!\cdot\!\mathbf{k}_{\perp} + \tau^{x}\otimes\sigma_{0}\left(\Delta_{S}-\Delta_{D}\cos k_{z}d\right) + \tau^{y}\otimes\sigma_{0}\,\Delta_{D}\sin k_{z}d .$$4

The conceptual refinement is that the intrinsic anomalous Hall conductivity contains both a quantized or branch-dependent piece, tied to Chern numbers of filled bands or slices, and a continuously varying non-quantized metallic piece. The Comment maintains that the latter remains a Fermi-surface property even in Weyl metals (Vanderbilt et al., 2013).

For Burkov–Balents-type systems, this does not overturn the Weyl-node interpretation of anomalous Hall response. It instead reconciles two descriptions: Weyl-node separation continues to control the Hall response, but the non-quantized part can still be formulated as a Fermi-surface quantity when the Berry-flux structure of the Fermi sheets is treated correctly (Vanderbilt et al., 2013).

5. Impurities, off-diagonal disorder, and spectral stability

Local impurity scattering in lattice regularizations of Burkov–Balents-type Weyl models was analyzed using a $$\mathcal{H}_{\mathbf{k}}^{0} = v_{F}\tau^{z}\otimes \left(\hat{\mathbf z}\times\boldsymbol{\sigma}\right)\!\cdot\!\mathbf{k}_{\perp} + \tau^{x}\otimes\sigma_{0}\left(\Delta_{S}-\Delta_{D}\cos k_{z}d\right) + \tau^{y}\otimes\sigma_{0}\,\Delta_{D}\sin k_{z}d .$$5-matrix framework adapted from the Burkov–Hook–Balents continuum theory. The clean lattice Hamiltonian was written as

$$\mathcal{H}_{\mathbf{k}}^{0} = v_{F}\tau^{z}\otimes \left(\hat{\mathbf z}\times\boldsymbol{\sigma}\right)\!\cdot\!\mathbf{k}_{\perp} + \tau^{x}\otimes\sigma_{0}\left(\Delta_{S}-\Delta_{D}\cos k_{z}d\right) + \tau^{y}\otimes\sigma_{0}\,\Delta_{D}\sin k_{z}d .$$6

with local impurity

$$\mathcal{H}_{\mathbf{k}}^{0} = v_{F}\tau^{z}\otimes \left(\hat{\mathbf z}\times\boldsymbol{\sigma}\right)\!\cdot\!\mathbf{k}_{\perp} + \tau^{x}\otimes\sigma_{0}\left(\Delta_{S}-\Delta_{D}\cos k_{z}d\right) + \tau^{y}\otimes\sigma_{0}\,\Delta_{D}\sin k_{z}d .$$7

and

$$\mathcal{H}_{\mathbf{k}}^{0} = v_{F}\tau^{z}\otimes \left(\hat{\mathbf z}\times\boldsymbol{\sigma}\right)\!\cdot\!\mathbf{k}_{\perp} + \tau^{x}\otimes\sigma_{0}\left(\Delta_{S}-\Delta_{D}\cos k_{z}d\right) + \tau^{y}\otimes\sigma_{0}\,\Delta_{D}\sin k_{z}d .$$8

In the time-reversal-breaking channel $$\mathcal{H}_{\mathbf{k}}^{0} = v_{F}\tau^{z}\otimes \left(\hat{\mathbf z}\times\boldsymbol{\sigma}\right)\!\cdot\!\mathbf{k}_{\perp} + \tau^{x}\otimes\sigma_{0}\left(\Delta_{S}-\Delta_{D}\cos k_{z}d\right) + \tau^{y}\otimes\sigma_{0}\,\Delta_{D}\sin k_{z}d .$$9, this lattice model is explicitly connected to the Burkov–Balents heterostructure proposal (Huang et al., 2012). The resulting classification is matrix-structural: scalar impurities $\tau^{x,y,z}$0 belong to the fully commuting class and can always induce resonances, whereas several non-scalar impurity channels possess stable energy windows in which no real impurity strength produces a resonance (Huang et al., 2012, Huang et al., 2013). The broader conclusion is that Weyl-node DOS suppression is not uniformly stable or unstable; its fate depends on the commutation algebra of $\tau^{x,y,z}$1 with the matrices appearing in the local Green function (Huang et al., 2013).

A distinct disorder problem is off-diagonal disorder in the multilayer tunnelling amplitudes. In the 2024 multilayer-topological-insulator study, nonmagnetic disorder inside each TI film renormalizes the effective intra-film coupling by first renormalizing the TI mass $\tau^{x,y,z}$2, then the penetration depth,

$\tau^{x,y,z}$3

and finally the effective Burkov–Balents parameter,

$\tau^{x,y,z}$4

This shifts the multilayer phase boundaries and can induce transitions between insulating, Weyl, and anomalous-quantum-Hall regimes (Alisultanov et al., 2024). The same work studies layer-to-layer fluctuations $\tau^{x,y,z}$5, treats them by locator-style disorder averaging, and finds that off-diagonal disorder inserts delocalized bulk states into the gap; the anomalous-quantum-Hall regime is thereby endangered, whereas the Weyl phase remains robust even under substantial off-diagonal disorder (Alisultanov et al., 2024).

The 2025 theory of off-diagonal disorder in multilayer topological insulators develops this point further. For a single Hermitian defect,

$\tau^{x,y,z}$6

the in-gap pole condition is

$\tau^{x,y,z}$7

The resulting defect bound state crosses zero energy in the trivial phase at $\tau^{x,y,z}$8, but in the topological phase it never crosses zero at finite defect strength (Alisultanov et al., 28 Jul 2025). The paper interprets this as a local marker of topology for off-diagonal disorder. It also analyzes Gaussian, Lorentzian, and uniform disorder, concluding that uniform disorder shortens the localization length slightly, while Gaussian and Lorentzian disorder enlarge it, and that Gaussian disorder can even delocalize the edges; chirality is maintained, but enhanced overlap between opposite edges pulls longitudinal conductance away from the quantized value (Alisultanov et al., 28 Jul 2025).

6. Distinct but related models, and recurrent misidentifications

Several models are regularly grouped with Burkov–Balents in informal usage even though the cited papers explicitly distinguish them. The clearest example is the Balents–Fisher–Girvin kagome model. The vison-fractionalization paper states that it is not about a separate “Burkov–Balents model,” but about the BFG $\tau^{x,y,z}$9 spin liquid on the kagome lattice, with anyons $\sigma^{x,y,z}$0, symmetry fractionalization classes in $\sigma^{x,y,z}$1, and a decisive role for the reduction of spin symmetry from $\sigma^{x,y,z}$2 to $\sigma^{x,y,z}$3 (Qi et al., 2015). The trapped-ion proposal and the finite-field plateau study likewise analyze BFG-type constrained kagome Hamiltonians with emergent $\sigma^{x,y,z}$4 gauge structure rather than any multilayer Weyl model (Nath et al., 2015, Plat et al., 2015).

A second distinct usage appears in spin transport. The 2017 study of spin relaxation with broken inversion symmetry and large spin-orbit coupling uses the spin-diffusion propagator by Burkov and Balents for a disordered Rashba two-dimensional electron gas,

$\sigma^{x,y,z}$5

with poles

$\sigma^{x,y,z}$6

That framework is a Burkov–Balents theory in the literal bibliographic sense, but it concerns D’yakonov–Perel’ spin relaxation, motional narrowing, and non-exponential spin dynamics rather than topological-insulator multilayers or Weyl nodes (Szolnoki et al., 2017).

The phrase Burkov–Balents model therefore requires contextual qualification. In Weyl-semimetal and multilayer-topological-insulator work, it denotes the layered surface-state Hamiltonian controlled by $\sigma^{x,y,z}$7, $\sigma^{x,y,z}$8, and often $\sigma^{x,y,z}$9. In other subfields, Burkov and Balents denotes different theoretical constructions, and several kagome-spin-liquid papers explicitly reject the identification altogether (Alisultanov et al., 2024, Qi et al., 2015).

7. Scientific role and continuing relevance

The enduring importance of the Burkov–Balents model lies in the fact that it unifies several themes within a single effective framework: band inversion in layered heterostructures, the splitting of Dirac criticality into Weyl nodes by symmetry breaking, anomalous Hall response in Weyl metals, and the reorganization of phases under both local impurities and random tunnelling (Vanderbilt et al., 2013, Huang et al., 2012, Alisultanov et al., 2024). Its layered construction also makes it unusually adaptable. The same logic survives in ultra-thin-film variants with structure-inversion asymmetry, in impurity-classification problems phrased in $\mathbf{k}_\perp=(k_x,k_y)$0-matrix algebra, and in off-diagonal-disorder theories where the tunnelling amplitudes themselves become random dynamical variables (Owerre, 2016, Huang et al., 2013, Alisultanov et al., 28 Jul 2025).

Equally significant is the model’s role as a reference point for what it is not. The kagome BFG model, the Chen–Balents $\mathbf{k}_\perp=(k_x,k_y)$1-$\mathbf{k}_\perp=(k_x,k_y)$2 spin-orbital model, and the Burkov–Balents Rashba spin-diffusion propagator all inhabit different theoretical domains, even though they share author names or certain formal motifs. The literature surveyed here repeatedly treats such distinctions as substantive rather than cosmetic, because the relevant symmetries, topological invariants, and low-energy observables differ from case to case (Qi et al., 2015, Szolnoki et al., 2017).

In that restricted and technically precise sense, the Burkov–Balents model is best understood not as a generic label for any Balents-associated topological-matter model, but as the multilayer coupled-surface-state framework whose control parameters $\mathbf{k}_\perp=(k_x,k_y)$3, $\mathbf{k}_\perp=(k_x,k_y)$4, and $\mathbf{k}_\perp=(k_x,k_y)$5 organize trivial, topological, Weyl, and anomalous-Hall phases, and whose modern extensions now include impurity classifications, off-diagonal disorder, local bound-state diagnostics, and thin-film generalizations (Alisultanov et al., 2024, Alisultanov et al., 28 Jul 2025, Owerre, 2016).