Topological Hall Theorem

• Topological Hall Theorem is defined as the identification of Hall conductivity as a robust topological invariant, captured via band Chern numbers or Green-function winding numbers.
• It employs the Harper representation in a magnetic Brillouin zone to simplify complex star-product formulations into computationally tractable momentum-space integrals.
• Recent extensions address higher Hall conductivities and symmetry-preserving edge obstructions, while combinatorial variants generalize classical Hall-type theorems.

Searching arXiv for the specified paper and closely related usages of “Topological Hall Theorem.” The expression “Topological Hall Theorem” does not denote a single universally standardized theorem across the literature. In condensed-matter physics, the closest established usage is a theorem-like statement that the Hall conductivity of certain gapped two-dimensional systems is a topological invariant, expressible either through band Chern numbers or through Green-function winding numbers; in the presence of a constant magnetic field with rational flux, this invariant can be formulated in the magnetic Brillouin zone using the Harper representation (Suleymanov et al., 2021). In a different strand of recent work, the same label is used for wavefunction-based obstructions to symmetry-preserving gapped edges in $(2+1)$D topological orders via higher Hall conductivities and partial rotations (Kobayashi et al., 2024). Outside condensed matter, “topological Hall theorem” also refers to topological extensions of Hall’s transversal theorem in combinatorics, including geometric and reconfiguration variants (Holmsen et al., 2014). These usages are related by a common structural theme: Hall-type quantities are identified with robust invariants whose values are unchanged under admissible deformations.

1. Quantized Hall conductivity as a topological invariant

For a uniform, gapped two-dimensional electronic system, the dc Hall conductivity is quantized and equal to a topological invariant (Suleymanov et al., 2021). In the noninteracting, periodic case without magnetic field this is the TKNN/Chern number, while for interacting systems the same quantized value can be written as a Green-function topological invariant. The relevant physical setting is a uniform $2$D crystalline system with translational symmetry and a spectral (mobility) gap at the Fermi energy (Suleymanov et al., 2021).

In the homogeneous case without external magnetic field, a noninteracting, periodic, $2$D insulator with filled Bloch bands indexed by $n$ has Hall conductivity

$$\sigma_{xy} = \frac{e^2}{h} \sum_{\text{filled } n} C_n, \qquad C_n = \frac{1}{2\pi}\int_{\text{BZ}} d^2k\;\Omega_n(k),$$

with Berry curvature

$$\Omega_n(k) = i\sum_{m\neq n} \frac{ \langle u_n(k)|\partial_{k_x} H(k)|u_m(k)\rangle\, \langle u_m(k)|\partial_{k_y} H(k)|u_n(k)\rangle - (x\leftrightarrow y) } {\big(E_n(k)-E_m(k)\big)^2}.$$

Here $H(k)$ is the Bloch Hamiltonian, $E_n(k)$ its eigenvalues, and $|u_n(k)\rangle$ the periodic parts of Bloch eigenstates (Suleymanov et al., 2021).

For gapped systems, including those with interactions, one may instead write

$$N_3 = \frac{1}{24\pi^2} \int d\omega\, d^2k\; \epsilon^{\alpha\beta\gamma} \, \mathrm{Tr}\Big[ G\, \partial_{\alpha} G^{-1} \, G\, \partial_{\beta} G^{-1} \, G\, \partial_{\gamma} G^{-1} \Big], \quad \alpha,\beta,\gamma\in\{\omega,k_x,k_y\},$$

and, under the usual conditions,

$2$0

The integral equals the degree of the map from the $2$1-torus $2$2, parameterized by $2$3, to the manifold of invertible matrices $2$4, and hence is an integer (Suleymanov et al., 2021). Because $2$5, smooth deformations of $2$6 that keep it nonsingular do not change $2$7, which is the precise sense in which the Hall conductivity is topologically stable.

The assumptions are explicit: translational invariance, spectral or mobility gap, gauge invariance, and smoothness and nonsingularity of the full Green function throughout the integration domain (Suleymanov et al., 2021). For noninteracting systems, $2$8; for interacting systems, $2$9 is the full interacting Green function, assumed nonsingular in the gap (Suleymanov et al., 2021).

2. Magnetic Brillouin zone formulation in a constant field

A more involved situation takes place for the Hall effect in the presence of an external magnetic field. In a nonuniform or magnetic background, a Wigner–Moyal formulation gives

$2$0

with star product

$2$1

This formula is general but difficult to use numerically because of the noncommutative star product and explicit dependence on both $2$2 and $2$3 (Suleymanov et al., 2021).

The central result of "Hall conductivity as the topological invariant in magnetic Brillouin zone" (Suleymanov et al., 2021) is an alternative representation for a uniform $2$4D lattice in a constant magnetic field with rational flux per plaquette $2$5, equivalently $2$6 with coprime integers $2$7. Magnetic translation symmetry then allows one to work in the magnetic Brillouin zone (MBZ), whose area is reduced by $2$8. In the Harper representation, the nonuniformity due to the field is absorbed into an internal $2$9 matrix structure, while the functional dependence is on momentum $n$0 within the MBZ and Matsubara frequency $n$1 (Suleymanov et al., 2021).

Let

$n$2

Then the Hall conductivity becomes

$n$3

The factor $n$4 appears when one extends the MBZ integral to the full Brillouin zone and compensates the $n$5-fold replication of the MBZ (Suleymanov et al., 2021). This invariant is gauge invariant and replaces explicit spatial nonuniformity by an internal matrix structure.

In the simplest rectangular-lattice tight-binding model with Landau gauge and quantized flux $n$6, the Harper matrix is

$n$7

with $n$8, lattice spacing $n$9, hopping $$\sigma_{xy} = \frac{e^2}{h} \sum_{\text{filled } n} C_n, \qquad C_n = \frac{1}{2\pi}\int_{\text{BZ}} d^2k\;\Omega_n(k),$$0, and $$\sigma_{xy} = \frac{e^2}{h} \sum_{\text{filled } n} C_n, \qquad C_n = \frac{1}{2\pi}\int_{\text{BZ}} d^2k\;\Omega_n(k),$$1. The derivatives entering the invariant are

$$\sigma_{xy} = \frac{e^2}{h} \sum_{\text{filled } n} C_n, \qquad C_n = \frac{1}{2\pi}\int_{\text{BZ}} d^2k\;\Omega_n(k),$$2

This makes the MBZ formulation computationally tractable compared with the phase-space star-product expression (Suleymanov et al., 2021).

3. Degree of mapping, integrality, and robustness

The MBZ invariant has the same topological character as the zero-field Green-function invariant. The integral over $$\sigma_{xy} = \frac{e^2}{h} \sum_{\text{filled } n} C_n, \qquad C_n = \frac{1}{2\pi}\int_{\text{BZ}} d^2k\;\Omega_n(k),$$3 on the MBZ, or the full Brillouin zone after suitable extension, is the degree of the map

$$\sigma_{xy} = \frac{e^2}{h} \sum_{\text{filled } n} C_n, \qquad C_n = \frac{1}{2\pi}\int_{\text{BZ}} d^2k\;\Omega_n(k),$$4

restricted to the space of nonsingular Green functions (Suleymanov et al., 2021). The invariant counts how many times the image wraps around the nontrivial topology of $$\sigma_{xy} = \frac{e^2}{h} \sum_{\text{filled } n} C_n, \qquad C_n = \frac{1}{2\pi}\int_{\text{BZ}} d^2k\;\Omega_n(k),$$5 in three dimensions, again using $$\sigma_{xy} = \frac{e^2}{h} \sum_{\text{filled } n} C_n, \qquad C_n = \frac{1}{2\pi}\int_{\text{BZ}} d^2k\;\Omega_n(k),$$6.

The paper’s Appendix E shows explicitly that the Hall conductivity is unchanged under smooth variations of $$\sigma_{xy} = \frac{e^2}{h} \sum_{\text{filled } n} C_n, \qquad C_n = \frac{1}{2\pi}\int_{\text{BZ}} d^2k\;\Omega_n(k),$$7 and $$\sigma_{xy} = \frac{e^2}{h} \sum_{\text{filled } n} C_n, \qquad C_n = \frac{1}{2\pi}\int_{\text{BZ}} d^2k\;\Omega_n(k),$$8 that keep them nonsingular, using cyclicity of the trace and total-derivative arguments (Suleymanov et al., 2021). This topological stability is the mathematical basis for the insensitivity of the quantized response to weak perturbations that do not close the gap. A plausible implication is that the theorem-like content lies not only in quantization itself but in the deformation invariance of the Green-function expression.

The quantization conditions are equally explicit: a spectral or mobility gap at the Fermi energy, magnetic translation symmetry with rational flux and a properly defined MBZ, and smoothness of $$\sigma_{xy} = \frac{e^2}{h} \sum_{\text{filled } n} C_n, \qquad C_n = \frac{1}{2\pi}\int_{\text{BZ}} d^2k\;\Omega_n(k),$$9 throughout the integration domain (Suleymanov et al., 2021). Under these conditions, $$\Omega_n(k) = i\sum_{m\neq n} \frac{ \langle u_n(k)|\partial_{k_x} H(k)|u_m(k)\rangle\, \langle u_m(k)|\partial_{k_y} H(k)|u_n(k)\rangle - (x\leftrightarrow y) } {\big(E_n(k)-E_m(k)\big)^2}.$$0, and for noninteracting systems it is an integer multiple of $$\Omega_n(k) = i\sum_{m\neq n} \frac{ \langle u_n(k)|\partial_{k_x} H(k)|u_m(k)\rangle\, \langle u_m(k)|\partial_{k_y} H(k)|u_n(k)\rangle - (x\leftrightarrow y) } {\big(E_n(k)-E_m(k)\big)^2}.$$1, leading to integer quantization of $$\Omega_n(k) = i\sum_{m\neq n} \frac{ \langle u_n(k)|\partial_{k_x} H(k)|u_m(k)\rangle\, \langle u_m(k)|\partial_{k_y} H(k)|u_n(k)\rangle - (x\leftrightarrow y) } {\big(E_n(k)-E_m(k)\big)^2}.$$2 (Suleymanov et al., 2021).

This formulation reproduces the standard Hofstadter–TKNN picture. In the presence of rational flux, the spectrum splits into $$\Omega_n(k) = i\sum_{m\neq n} \frac{ \langle u_n(k)|\partial_{k_x} H(k)|u_m(k)\rangle\, \langle u_m(k)|\partial_{k_y} H(k)|u_n(k)\rangle - (x\leftrightarrow y) } {\big(E_n(k)-E_m(k)\big)^2}.$$3 Harper subbands with well-defined Chern numbers over the MBZ, and

$$\Omega_n(k) = i\sum_{m\neq n} \frac{ \langle u_n(k)|\partial_{k_x} H(k)|u_m(k)\rangle\, \langle u_m(k)|\partial_{k_y} H(k)|u_n(k)\rangle - (x\leftrightarrow y) } {\big(E_n(k)-E_m(k)\big)^2}.$$4

The Green-function invariant $$\Omega_n(k) = i\sum_{m\neq n} \frac{ \langle u_n(k)|\partial_{k_x} H(k)|u_m(k)\rangle\, \langle u_m(k)|\partial_{k_y} H(k)|u_n(k)\rangle - (x\leftrightarrow y) } {\big(E_n(k)-E_m(k)\big)^2}.$$5 equals the sum of these Chern numbers times $$\Omega_n(k) = i\sum_{m\neq n} \frac{ \langle u_n(k)|\partial_{k_x} H(k)|u_m(k)\rangle\, \langle u_m(k)|\partial_{k_y} H(k)|u_n(k)\rangle - (x\leftrightarrow y) } {\big(E_n(k)-E_m(k)\big)^2}.$$6, thereby reproducing the TKNN quantization once the factor $$\Omega_n(k) = i\sum_{m\neq n} \frac{ \langle u_n(k)|\partial_{k_x} H(k)|u_m(k)\rangle\, \langle u_m(k)|\partial_{k_y} H(k)|u_n(k)\rangle - (x\leftrightarrow y) } {\big(E_n(k)-E_m(k)\big)^2}.$$7 is included in $$\Omega_n(k) = i\sum_{m\neq n} \frac{ \langle u_n(k)|\partial_{k_x} H(k)|u_m(k)\rangle\, \langle u_m(k)|\partial_{k_y} H(k)|u_n(k)\rangle - (x\leftrightarrow y) } {\big(E_n(k)-E_m(k)\big)^2}.$$8 (Suleymanov et al., 2021).

4. Computation, checks, and interacting extensions

The Harper-based method admits a direct computational workflow. One fixes the rational flux $$\Omega_n(k) = i\sum_{m\neq n} \frac{ \langle u_n(k)|\partial_{k_x} H(k)|u_m(k)\rangle\, \langle u_m(k)|\partial_{k_y} H(k)|u_n(k)\rangle - (x\leftrightarrow y) } {\big(E_n(k)-E_m(k)\big)^2}.$$9, constructs the Harper Hamiltonian $H(k)$0 on the MBZ, forms $H(k)$1, computes $H(k)$2, and evaluates the invariant numerically on a discretized $H(k)$3-grid (Suleymanov et al., 2021). At zero temperature, Matsubara sums become integrals over $H(k)$4 (Suleymanov et al., 2021).

Often, however, it is more efficient to compute band Chern numbers directly in the MBZ: $H(k)$5 with Berry curvature expressed through matrix elements of $H(k)$6 and $H(k)$7 (Suleymanov et al., 2021). The numerical notes are specific: gauge-covariant discretization schemes for Chern numbers, such as the link-variable method of Fukui–Hatsugai–Suzuki, should be used to avoid gauge singularities, and one should check convergence with respect to $H(k)$8-cutoff and grid refinements (Suleymanov et al., 2021).

Several consistency checks are emphasized. In a $H(k)$9D electron gas with uniform $E_n(k)$0, each filled Landau level contributes one to the Hall conductivity in units of $E_n(k)$1, matching the band-Chern-number picture (Suleymanov et al., 2021). For the Hofstadter model with flux $E_n(k)$2, the TKNN Diophantine equation

$E_n(k)$3

relates the subband index $E_n(k)$4 to its Chern number $E_n(k)$5. Numerical examples with $E_n(k)$6 and $E_n(k)$7 reproduce the expected plateaux $E_n(k)$8 when the chemical potential lies in the corresponding gaps, in agreement with the Diophantine solutions $E_n(k)$9 (Suleymanov et al., 2021).

The extension to interactions is more tentative. The Green-function invariant $|u_n(k)\rangle$0 is known to remain valid for interacting, gapped systems if the full Green function is nonsingular in the integration domain (Suleymanov et al., 2021). The paper hypothesizes that the Harper-representation formula with $|u_n(k)\rangle$1 taken as the full interacting Green function in the MBZ continues to hold, and proposes that it may be used for the topological description of fractional quantum Hall effect (Suleymanov et al., 2021). This is explicitly presented as a hypothesis rather than a proved theorem. The stated caveats are that Green-function zeros or poles may alter topology and that the degeneracy of ground states and topological order complicate direct single-particle formulations (Suleymanov et al., 2021).

5. Higher Hall conductivities and edge obstructions

A distinct recent use of the phrase “Topological Hall Theorem” appears in the study of $|u_n(k)\rangle$2D topological orders with $|u_n(k)\rangle$3 symmetry, where the central object is not the dc Hall conductivity in a band or Green-function setting but a hierarchy of higher Hall conductivity invariants extracted from a single wavefunction (Kobayashi et al., 2024). This usage concerns edge gappability rather than transport in a magnetic Brillouin zone.

For a fermionic topological order with $|u_n(k)\rangle$4 symmetry, the higher Hall conductivity invariants are

$|u_n(k)\rangle$5

defined for integers $|u_n(k)\rangle$6 coprime to the Frobenius–Schur exponent (Kobayashi et al., 2024). For $|u_n(k)\rangle$7,

$|u_n(k)\rangle$8

These invariants act as symmetry-protected obstructions to $|u_n(k)\rangle$9-preserving gapped edges beyond the conventional electric Hall conductivity $$N_3 = \frac{1}{24\pi^2} \int d\omega\, d^2k\; \epsilon^{\alpha\beta\gamma} \, \mathrm{Tr}\Big[ G\, \partial_{\alpha} G^{-1} \, G\, \partial_{\beta} G^{-1} \, G\, \partial_{\gamma} G^{-1} \Big], \quad \alpha,\beta,\gamma\in\{\omega,k_x,k_y\},$$0 and thermal Hall conductivity $$N_3 = \frac{1}{24\pi^2} \int d\omega\, d^2k\; \epsilon^{\alpha\beta\gamma} \, \mathrm{Tr}\Big[ G\, \partial_{\alpha} G^{-1} \, G\, \partial_{\beta} G^{-1} \, G\, \partial_{\gamma} G^{-1} \Big], \quad \alpha,\beta,\gamma\in\{\omega,k_x,k_y\},$$1 (Kobayashi et al., 2024).

The extraction method uses the partial rotation unitary

$$N_3 = \frac{1}{24\pi^2} \int d\omega\, d^2k\; \epsilon^{\alpha\beta\gamma} \, \mathrm{Tr}\Big[ G\, \partial_{\alpha} G^{-1} \, G\, \partial_{\beta} G^{-1} \, G\, \partial_{\gamma} G^{-1} \Big], \quad \alpha,\beta,\gamma\in\{\omega,k_x,k_y\},$$2

and the expectation value

$$N_3 = \frac{1}{24\pi^2} \int d\omega\, d^2k\; \epsilon^{\alpha\beta\gamma} \, \mathrm{Tr}\Big[ G\, \partial_{\alpha} G^{-1} \, G\, \partial_{\beta} G^{-1} \, G\, \partial_{\gamma} G^{-1} \Big], \quad \alpha,\beta,\gamma\in\{\omega,k_x,k_y\},$$3

computed in a ground state $$N_3 = \frac{1}{24\pi^2} \int d\omega\, d^2k\; \epsilon^{\alpha\beta\gamma} \, \mathrm{Tr}\Big[ G\, \partial_{\alpha} G^{-1} \, G\, \partial_{\beta} G^{-1} \, G\, \partial_{\gamma} G^{-1} \Big], \quad \alpha,\beta,\gamma\in\{\omega,k_x,k_y\},$$4 on a cylinder (Kobayashi et al., 2024). The universal information resides in the phase of this expectation value. The paper states a theorem-like result: from $$N_3 = \frac{1}{24\pi^2} \int d\omega\, d^2k\; \epsilon^{\alpha\beta\gamma} \, \mathrm{Tr}\Big[ G\, \partial_{\alpha} G^{-1} \, G\, \partial_{\beta} G^{-1} \, G\, \partial_{\gamma} G^{-1} \Big], \quad \alpha,\beta,\gamma\in\{\omega,k_x,k_y\},$$5 and a set of partial rotation phases, one can extract $$N_3 = \frac{1}{24\pi^2} \int d\omega\, d^2k\; \epsilon^{\alpha\beta\gamma} \, \mathrm{Tr}\Big[ G\, \partial_{\alpha} G^{-1} \, G\, \partial_{\beta} G^{-1} \, G\, \partial_{\gamma} G^{-1} \Big], \quad \alpha,\beta,\gamma\in\{\omega,k_x,k_y\},$$6, $$N_3 = \frac{1}{24\pi^2} \int d\omega\, d^2k\; \epsilon^{\alpha\beta\gamma} \, \mathrm{Tr}\Big[ G\, \partial_{\alpha} G^{-1} \, G\, \partial_{\beta} G^{-1} \, G\, \partial_{\gamma} G^{-1} \Big], \quad \alpha,\beta,\gamma\in\{\omega,k_x,k_y\},$$7, and the higher central charges or higher Hall conductivities; in Abelian cases these data determine whether a symmetry-preserving gapped edge exists (Kobayashi et al., 2024).

This is called a “Topological Hall Theorem” in that work, but it addresses a different question from the MBZ theorem of Hall quantization. The shared feature is that Hall-type response data are encoded as topological invariants and recovered from global information insensitive to local deformations. A plausible implication is that the term has broadened from conductivity quantization proper to a wider family of Hall-related obstruction theorems in topological phases.

6. Relation to the Topological Hall Effect and other Hall-type theorems

In condensed-matter usage, “Topological Hall Theorem” can be confused with the topological Hall effect (THE), but the latter is a transport phenomenon rather than a theorem. THE arises when conduction electrons traverse noncoplanar spin textures and acquire a real-space Berry phase associated with scalar spin chirality

$$N_3 = \frac{1}{24\pi^2} \int d\omega\, d^2k\; \epsilon^{\alpha\beta\gamma} \, \mathrm{Tr}\Big[ G\, \partial_{\alpha} G^{-1} \, G\, \partial_{\beta} G^{-1} \, G\, \partial_{\gamma} G^{-1} \Big], \quad \alpha,\beta,\gamma\in\{\omega,k_x,k_y\},$$8

or, in continuum form, with $$N_3 = \frac{1}{24\pi^2} \int d\omega\, d^2k\; \epsilon^{\alpha\beta\gamma} \, \mathrm{Tr}\Big[ G\, \partial_{\alpha} G^{-1} \, G\, \partial_{\beta} G^{-1} \, G\, \partial_{\gamma} G^{-1} \Big], \quad \alpha,\beta,\gamma\in\{\omega,k_x,k_y\},$$9 (Li et al., 16 Sep 2025, Liu et al., 2022). Recent work also introduces an interfacial topological Hall effect in Pt/h-LuFeO$2$00 bilayers, where an insulating magnet’s noncoplanar topology is imprinted into a heavy metal via magnetic proximity effect and read out electrically (Li et al., 16 Sep 2025). That literature explicitly notes that “Topological Hall Theorem” is a misnomer in this context and that the correct term is Topological Hall Effect (Li et al., 16 Sep 2025).

Other theorem-like statements also use related language. In solvable lattice Hamiltonians, a no-go theorem due to Kapustin and Fidkowski implies that a $2$01D gapped phase with nonzero $2$02 Hall conductivity cannot be realized by a local commuting-projector Hamiltonian with an on-site $2$03 symmetry (Han et al., 2022). This is sometimes framed as a theorem about topological Hall response, but it is structurally different from the MBZ formula: it is an obstruction theorem for microscopic realizations rather than an expression for $2$04 itself.

Outside condensed matter, “topological Hall theorem” refers to topological extensions of Hall’s transversal theorem. "A geometric Hall-type theorem" proves that for every integer $2$05 there exists a function $2$06 such that if

$2$07

for every non-empty $2$08, then the family $2$09 has a system of general-position representatives (Holmsen et al., 2014). The proof uses connectivity of the general-position complex and a colorful-simplex theorem, making it a literal topological Hall theorem in combinatorics (Holmsen et al., 2014). This is mathematically unrelated to the quantum Hall formulations, but the naming similarity reflects the same pattern: Hall-type existence conditions are lifted to topological invariants or connectivity criteria.

These multiple usages show that the phrase is context-dependent. In electronic transport, the most precise meaning is the theorem-like identification of Hall conductivity with a topological invariant, especially in the MBZ Green-function formulation for rational magnetic flux (Suleymanov et al., 2021). In topological order, it can denote wavefunction-extracted higher Hall obstructions (Kobayashi et al., 2024). In combinatorics, it denotes topological generalizations of Hall’s theorem (Holmsen et al., 2014).

7. Significance and scope

The significance of the MBZ formulation is that it converts a general but cumbersome Wigner–Moyal expression into an ordinary momentum-space integral over the magnetic Brillouin zone with matrix products and traces in an internal Harper space (Suleymanov et al., 2021). This yields a gauge-invariant and computationally tractable topological invariant that directly connects to Hofstadter subband Chern numbers and TKNN quantization (Suleymanov et al., 2021). It therefore unifies the zero-field Green-function viewpoint with the rational-flux magnetic-translation setting.

Its scope is nevertheless limited. The construction requires rational flux and magnetic translation symmetry; incommensurate fluxes or magnetic disorder obstruct MBZ construction (Suleymanov et al., 2021). Band touching within the MBZ or lack of a gap at the Fermi energy spoils quantization (Suleymanov et al., 2021). The interacting extension, including the suggested application to fractional quantum Hall states, remains a hypothesis rather than a theorem (Suleymanov et al., 2021).

Taken in the broadest scholarly sense, the “Topological Hall Theorem” is best understood as a family resemblance term. Its core condensed-matter content is that Hall response is governed by integer-valued topological data: occupied-band Chern numbers in periodic systems, Green-function winding numbers in interacting gapped phases, and an MBZ degree-of-mapping invariant in constant magnetic fields with rational flux (Suleymanov et al., 2021). Recent generalizations extend the Hall-theoretic viewpoint to higher edge obstructions derived from a single wavefunction (Kobayashi et al., 2024), while combinatorial analogues transplant Hall’s original theorem into topological and geometric settings (Holmsen et al., 2014). The unifying idea is not a single theorem statement but the robustness of Hall-type quantities under deformations that preserve the relevant gap, nonsingularity, or connectivity structure.