Higher Berry Curvature in Quantum Systems

• Higher Berry curvature is a generalization of the standard Berry curvature, defined as a closed (d+2)-form on the parameter space of gapped quantum systems.
• It provides a flow interpretation that connects bulk topological invariants with boundary anomalies, ensuring gapless edge modes when the invariant is nonzero.
• Advanced formulations using wave-function techniques and tensor networks yield quantized invariants that correlate with second Chern numbers in lattice models.

Higher Berry curvature is a higher-form generalization of ordinary Berry curvature for parametrized gapped quantum systems. In the formulation introduced by Kapustin and Spodyneiko, a $d$-dimensional gapped many-body system depending smoothly on parameters carries a closed $(d+2)$-form $\Omega^{(d+2)}$ on parameter space; for $d=0$ this reduces to the usual Berry-curvature $2$-form, while for $d=1$ it becomes a $3$-form that measures a flow of ordinary Berry curvature (Kapustin et al., 2020, Wen et al., 2021). The phrase has also acquired broader usage in condensed-matter physics, where it can denote higher moments of ordinary Berry curvature or unusually large and singular Berry curvature in band problems, but these are distinct notions (Zhang et al., 2020, Wawrzik et al., 2023).

1. Definition and cohomological structure

For a finite-dimensional quantum system with unique gapped ground state, the usual Berry curvature can be written in resolvent form as

$$\Omega^{(2)} = i \oint \frac{dz}{2\pi i}\,\mathrm{Tr}\!\left(G\, dH\, G^2\, dH\right), \qquad G=(z-H)^{-1},$$

and satisfies $d\Omega^{(2)}=0$. Its integral over a closed $2$-cycle gives the first Chern class (Kapustin et al., 2020). Higher Berry curvature extends this structure to infinite many-body systems in spatial dimension $(d+2)$0, where naive global Berry geometry is obstructed by infinite volume and by the absence of a canonical bundle of ground states.

The central construction introduces local differential-form data $(d+2)$1 obeying descent relations

$(d+2)$2

and then contracts the top descendant with a coarse-cohomological $(d+2)$3-cocycle built from step-like functions $(d+2)$4. In the Kapustin–Spodyneiko formulation,

$(d+2)$5

The resulting $(d+2)$6-form is closed, $(d+2)$7, and its de Rham cohomology class is invariant under smooth deformations within a phase (Wen et al., 2021, Kapustin et al., 2020).

This establishes the basic hierarchy: ordinary Berry curvature is a $(d+2)$8-form in $(d+2)$9, higher Berry curvature is a $\Omega^{(d+2)}$0-form in $\Omega^{(d+2)}$1, and in general a $\Omega^{(d+2)}$2-dimensional parametrized system carries a $\Omega^{(d+2)}$3-form. The integrated invariant is often called the Kapustin–Spodyneiko, or KS, invariant; when parameter space $\Omega^{(d+2)}$4 is an oriented closed $\Omega^{(d+2)}$5-manifold, the KS number is

$\Omega^{(d+2)}$6

For short-range entangled families, integrals over spherical cycles are quantized, and for invertible systems the higher Berry class admits integral refinements (Kapustin et al., 2020, Artymowicz et al., 2023).

2. Flow interpretation and bulk–boundary correspondence

A defining physical interpretation of higher Berry curvature is as a flow law for lower-dimensional Berry data. In a semi-infinite geometry, the bulk $\Omega^{(d+2)}$7-form is related to a boundary $\Omega^{(d+2)}$8-form by

$\Omega^{(d+2)}$9

For $d=0$0, this becomes

$d=0$1

with $d=0$2 a boundary Berry curvature. In this sense, higher Berry curvature is a current of ordinary Berry curvature; more generally, in $d=0$3 spatial dimensions it is a flow of $d=0$4-dimensional higher Berry curvature to or from the boundary (Wen et al., 2021).

This flow picture yields a bulk–boundary correspondence. If the bulk KS invariant is nonzero, the boundary cannot be trivially gapped everywhere. In the solvable $d=0$5 model over $d=0$6, the boundary is a $d=0$7 effective system with a single gapless Weyl point, and the boundary Chern number jump matches the bulk KS number: $d=0$8 A related model over $d=0$9 pumps Chern number, with the KS number equal to the net Chern number transported in one cycle (Wen et al., 2021).

The same logic extends to higher dimensions. A $2$0 model over $2$1 exhibits a boundary anomaly measured by a boundary $2$2-form $2$3, and if two $2$4-dimensional systems over the same closed oriented $2$5-manifold have different KS numbers, any interface between them must become gapless somewhere in parameter space or undergo a first-order transition; a gapped topologically ordered interface is ruled out (Wen et al., 2021).

A parallel field-theoretic description interprets stable gapless loci in coupling space as sources of higher Berry curvature. In effective actions with Wess–Zumino–Witten or generalized Thouless-pump terms, nonzero higher Berry number on a surrounding sphere forces the existence of diabolical points, which are explicitly compared to Weyl nodes, but in the space of couplings rather than momentum space (Hsin et al., 2020).

3. Wave-function and tensor-network formulations

A major development after the original construction was a wave-function-based formulation for infinite systems. In $2$6, the obstacle is that the total ordinary Berry curvature of the full state is extensive. The local approach decomposes the exterior derivative as $2$7, defines local $2$8-form contributions $2$9, and then assembles a $d=1$0-form flow across a cut. For a Schmidt decomposition

$d=1$1

the higher Berry curvature across the cut is

$d=1$2

This formula gives a direct interpretation: only the variation of the Schmidt weights contributes, and it couples to the Berry curvature of the Schmidt vectors on one side of the cut (Sommer et al., 2024).

The same work derives practical MPS expressions. In the left tangent-space gauge, the $d=1$3 higher Berry curvature becomes

$d=1$4

and for a translationally invariant uniform MPS,

$d=1$5

For essentially injective uniform MPS, this $d=1$6-form is the curvature of a gerbe, and its integral over a closed parameter manifold is quantized: $d=1$7 An exactly solvable dimerized spin-$d=1$8 chain over $d=1$9 realizes

$3$0

and iDMRG calculations on deformed models reproduce the invariant to within about $3$1 (Sommer et al., 2024).

Beyond one dimension, locally parameterized tensor-network states admit a systematic simplex-based construction. Starting from

$3$2

one defines

$3$3

where $3$4 is a simplex-exterior derivative and $3$5 is a suitable nontrivial cochain. The resulting $3$6-form characterizes a flow of $3$7-form higher Berry curvature and was illustrated explicitly in exactly solvable $3$8 lattice models with

$3$9

(Sommer et al., 2024).

4. Quantization, higher pumps, and second Chern numbers

Quantization was sharpened substantially for invertible phases. For smooth families of invertible $$\Omega^{(2)} = i \oint \frac{dz}{2\pi i}\,\mathrm{Tr}\!\left(G\, dH\, G^2\, dH\right), \qquad G=(z-H)^{-1},$$0 states, the higher Berry class $$\Omega^{(2)} = i \oint \frac{dz}{2\pi i}\,\mathrm{Tr}\!\left(G\, dH\, G^2\, dH\right), \qquad G=(z-H)^{-1},$$1 refines to an integral class: $$\Omega^{(2)} = i \oint \frac{dz}{2\pi i}\,\mathrm{Tr}\!\left(G\, dH\, G^2\, dH\right), \qquad G=(z-H)^{-1},$$2 In the same framework, for smooth families of invertible $$\Omega^{(2)} = i \oint \frac{dz}{2\pi i}\,\mathrm{Tr}\!\left(G\, dH\, G^2\, dH\right), \qquad G=(z-H)^{-1},$$3-invariant $$\Omega^{(2)} = i \oint \frac{dz}{2\pi i}\,\mathrm{Tr}\!\left(G\, dH\, G^2\, dH\right), \qquad G=(z-H)^{-1},$$4 states, the higher Thouless pump $$\Omega^{(2)} = i \oint \frac{dz}{2\pi i}\,\mathrm{Tr}\!\left(G\, dH\, G^2\, dH\right), \qquad G=(z-H)^{-1},$$5 satisfies

$$\Omega^{(2)} = i \oint \frac{dz}{2\pi i}\,\mathrm{Tr}\!\left(G\, dH\, G^2\, dH\right), \qquad G=(z-H)^{-1},$$6

The $$\Omega^{(2)} = i \oint \frac{dz}{2\pi i}\,\mathrm{Tr}\!\left(G\, dH\, G^2\, dH\right), \qquad G=(z-H)^{-1},$$7 Thouless pump can moreover be identified with an excess Berry curvature of flux insertion: $$\Omega^{(2)} = i \oint \frac{dz}{2\pi i}\,\mathrm{Tr}\!\left(G\, dH\, G^2\, dH\right), \qquad G=(z-H)^{-1},$$8 which makes the pump the Berry-curvature response of a fluxon (Artymowicz et al., 2023).

A complementary discrete formulation based on translationally invariant MPS and iMPS evaluates higher Berry curvature on tetrahedra rather than loops, because the relevant transition data are only projectively defined. For an oriented triangle $$\Omega^{(2)} = i \oint \frac{dz}{2\pi i}\,\mathrm{Tr}\!\left(G\, dH\, G^2\, dH\right), \qquad G=(z-H)^{-1},$$9, one defines a discrete higher Berry phase $d\Omega^{(2)}=0$0; for a tetrahedron $d\Omega^{(2)}=0$1,

$d\Omega^{(2)}=0$2

and the quantized invariant is

$d\Omega^{(2)}=0$3

Numerical MPS calculations confirmed continuous higher Berry curvature during adiabatic evolution and quantization over closed $d\Omega^{(2)}=0$4-dimensional parameter spaces (Shiozaki et al., 2023, Heinsdorf et al., 24 Jun 2026).

A particularly explicit application rewrites the $d\Omega^{(2)}=0$5 lattice Dirac model as a family of translationally invariant infinite $d\Omega^{(2)}=0$6 chains over $d\Omega^{(2)}=0$7, computes the higher three-form Berry curvature from iMPS, and finds that the resulting Dixmier–Douady–Kapustin–Spodyneiko number is exactly congruent to the second Chern number of the original $d\Omega^{(2)}=0$8 Chern insulator. As a function of mass $d\Omega^{(2)}=0$9, the phase diagram is

$2$0

matching the analytic $2$1 result exactly (Heinsdorf et al., 24 Jun 2026). This construction also links higher Berry phase on $2$2 slices to the Chern–Simons axion angle $2$3, with excellent agreement in the $2$4 Dirac model, although the correspondence is not universal, as shown by the Hopf-insulator test (Heinsdorf et al., 24 Jun 2026).

5. Field-theoretic and boundary-conformal realizations

In effective field theory, slowly varying couplings are promoted to spacetime-dependent background fields $2$5. The topological response then takes Wess–Zumino–Witten form

$2$6

or, in the presence of symmetries, generalized Thouless-pump terms. Nonzero higher Berry number on a surrounding $2$7 forces a stable gapless locus in coupling space; boundary diabolical points and arcs are the boundary manifestation of the same topology (Hsin et al., 2020).

For free Dirac fermions with spacetime-dependent mass matrices, the higher Berry phase of the resulting parameterized invertible field theory is computed using the Atiyah–Patodi–Singer index theorem for superconnections. The local higher Berry curvature is

$2$8

with $2$9 the superconnection curvature. This produces gauged-WZW-type responses and, when the differential-form Berry curvature vanishes, the theory may still retain a nontrivial torsional Berry phase detectable by bordism and $(d+2)$00-invariants (Choi et al., 2022).

Boundary conformal field theory furnishes a distinct but closely related realization. On a boundary conformal manifold, the higher Berry connection is a $(d+2)$01-form

$(d+2)$02

constructed from the phase of the $(d+2)$03-point OPE coefficient of boundary-condition-changing operators. This geometry is additional to the Zamolodchikov metric: the metric comes from $(d+2)$04-point data of marginal operators, whereas the higher Berry connection comes from $(d+2)$05-point data of boundary-condition-changing operators. When the boundary conformal manifold is the position moduli space of a D-brane, $(d+2)$06 coincides with the NS-NS $(d+2)$07-field (Choi et al., 16 Jul 2025).

A related BCFT construction realizes higher Berry curvature as a Fock-space flow. For a family of Dirac-fermion conformal boundary conditions over $(d+2)$08, the total Berry curvature stored in occupied modes is a $(d+2)$09-form $(d+2)$10, and the higher Berry curvature is

$(d+2)$11

This is the BCFT analog of Berry-curvature flow in real space for $(d+2)$12 parametrized gapped systems (Wen, 16 Jul 2025).

6. Relation to ordinary Berry geometry

Higher Berry curvature generalizes ordinary Berry curvature but is not interchangeable with it. In band theory, the ordinary Berry curvature is the momentum-space analog of magnetic field. In a single isolated Bloch band it appears as the antisymmetric part of the quantum geometric tensor, and its Brillouin-zone integral gives the Chern number (Varjas et al., 2021). The higher-form object $(d+2)$13, by contrast, lives on parameter space and encodes geometry of families of many-body states rather than local band geometry in the Brillouin zone.

This distinction is visible in lattice Chern-band engineering. Exactly constant ordinary Berry curvature is impossible in any topological two-band model, but can be achieved in lattice models with three or more bands per unit cell by deforming momentum space and using Moser’s theorem. Yet an “ideal flatband” cannot have constant Berry curvature, and the exact Landau-level density algebra cannot be realized in a tight-binding lattice system with finitely many degrees of freedom per unit cell (Varjas et al., 2021). These results concern ordinary momentum-space Berry curvature, not the higher-form many-body generalization.

Another conceptual refinement concerns the dependence of ordinary Berry curvature on eigenvectors rather than on explicit parameter dependence of the Hamiltonian. When the parameter derivative acts on states outside the domain of the Hamiltonian, an additional correction term $(d+2)$14 enters off-diagonal adiabatic formulas, and Berry curvature can be nonzero even when $(d+2)$15. The corrected expression is

$(d+2)$16

This is a statement about ordinary Berry curvature, but it sharpens the general geometric lesson that Berry data are properties of parameter-dependent eigenvectors, not of Hamiltonian coefficients alone (Konstantinou et al., 28 Jul 2025).

7. Distinct usages in transport and multipole literature

A recurrent source of confusion is terminological. In a separate transport literature, “higher Berry curvature” often refers not to a $(d+2)$17-form on parameter space, but to higher moments of ordinary Berry curvature near the Fermi energy. In this hierarchy, the Berry-curvature monopole controls the linear anomalous Hall effect, the Berry-curvature dipole controls the second-order nonlinear Hall effect, the quadrupole controls third-order nonlinear anomalous Hall response, and the hexapole controls fourth-order response (Zhang et al., 2020).

This multipole language was extended further in nonequilibrium transport. In time-reversal-symmetric, inversion-broken metals, the weak-field spontaneous Hall current is controlled by the Berry-curvature dipole, but when the field-induced momentum shift becomes comparable to the distance from the Fermi surface to Berry-curvature extrema, the system enters a fully nonequilibrium regime in which the Hall response “can no longer be attributed to the dipole of the Berry curvature”; all higher moments become comparable and the response becomes quasi-linear in the electric field (Sur et al., 2024). A related line of work shows that a symmetry-breaking dc electric field can induce a Berry-curvature dipole through the quantum metric, or Berry connection polarizability, in higher-wave-symmetric unconventional magnets, including altermagnets, thereby activating a second-harmonic nonlinear Hall signal even when equilibrium first- and second-order Hall responses vanish by symmetry (Korrapati et al., 23 Oct 2025).

Other recent studies engineer or exploit ordinary Berry-curvature multipoles directly. In multilayer graphene on a corrugated substrate, inhomogeneous interlayer sliding breaks three-fold rotational symmetry and generates a sizable Berry-curvature dipole, with the headline scaling

$(d+2)$18

so that a structural displacement of order $(d+2)$19 can produce a dipole of about $(d+2)$20 (Pan et al., 2024). In Weyl semimetals, surface Fermi arcs can carry a divergent surface Berry curvature with

$(d+2)$21

near the end of the arc, leading to a Berry-curvature dipole that grows linearly with slab thickness and enhancing nonlinear Hall, Magnus-Hall, and nonlinear chiral-anomaly responses (Wawrzik et al., 2023). Berry curvature also modifies thermoelectric transport by mixing anomalous Hall and anomalous Nernst channels into the Seebeck coefficient,

$(d+2)$22

with $(d+2)$23 when Berry curvature is ignored (Mizuta et al., 2014).

These developments are adjacent to, but distinct from, the Kapustin–Spodyneiko higher-form construction. The former concern ordinary Berry curvature in momentum space and its derivatives or singularities; the latter concerns closed $(d+2)$24-forms on parameter space of gapped many-body systems. The shared terminology reflects a common geometric origin, but the mathematical objects, physical interpretations, and topological invariants are different.